ISSUE 2005 PROGRESS IN PHYSICS

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Chief Editor H. Arp Observational Cosmology: From High Redshift Galaxies to the Blue Dmitri Rabounski Pacific...... 3 [email protected] S. J. Crothers On the General Solution to Einstein’s Vacuum Field for Associate Editors the Point-Mass when λ = 0 and Its Consequences for Relativistic Cos- Prof. Florentin Smarandache mology...... 76 [email protected] Dr. Larissa Borissova H. Ratcliffe The First Crisis in Cosmology Conference. Mon¸cao,˜ Portugal, [email protected] June 23–25, 2005...... 19 Stephen J. Crothers [email protected] R. T. Cahill The Michelson and Morley 1887 Experiment and the Discovery of Absolute Motion...... 25 Department of Mathematics, University of New Mexico, 200 College Road, Gallup, R. T. Cahill Gravity Probe B Frame-Dragging Effect...... 30 NM 87301, USA R. Oros di Bartini Relations Between Physical Constants...... 34 S. J. Crothers Introducing Distance and Measurement in General Relativity: Copyright c Progress in Physics, 2005 Changes for the Standard Tests and the Cosmological Large-Scale...... 41 All rights reserved. Any part of Progress in J. Dunning-Davies A Re-Examination of Maxwell’s Electromagnetic Equa- Physics howsoever used in other publica- tions must include an appropriate citation tions...... 48 of this journal. R. T. Cahill Black Holes in Elliptical and Spiral Galaxies and in Globular Authors of articles published in Progress in Physics retain their rights to use their own Clusters...... 51 articles in any other publications and in any E. Gaastra Is the Biggest Paradigm Shift in the History of Science at Hand? . . 57 way they see fit. N. Kozyrev Sources of Stellar Energy and the Theory of the Internal Consti- This journal is powered by BaKoMa-TEX tution of Stars...... 61

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Observational Cosmology: From High Redshift Galaxies to the Blue Pacific

Halton Arp

Max-Planck-Institut fur¨ Astrophysik, Karl Schwarzschild-Str.1, Postfach 1317, D-85741 Garching, Germany E-mail: [email protected]

1 Birth of galaxies demonstrated in Arp and Russell, 2001 [1]. Individual cases of strong X-ray clusters are exemplified by elongations and Observed: Ejection of high redshift, low luminosity quasars connections as shown in the ejecting galaxy Arp 220, in Abell from active galaxy nuclei. 3667 and from NGC 720 (again, summarized in Arp, 2003 Shown by radio and X-ray pairs, alignments and lumin- [4]). Motion is confirmed by bow shocks and elongation is ous connecting filaments. Emergent velocities are much less interpreted as ablation trails. In short — if a quasar evolves than intrinsic redshift. Stripping of radio plasmas. Probabi- into a galaxy, a broken up quasar evolves into a group of lities of accidental association negligible. See Arp, 2003 [4] galaxies. for customarily supressed details. Observed: Evolution of quasars into normal companion ga- 2 Redshift is the key laxies. The large number of ejected objects enables a view of Observed: The whole quasar or galaxy is intrinsically red- empirical evolution from high surface brightness quasars shifted. through compact galaxies. From gaseous plasmoids to fo- Objects with the same path length to the observer have rmation of atoms and stars. From high redshift to low. much different redshifts and all parts of the object are shifted closely the same amount. Tired light is ruled out and also gravitational redshifting. The fundamental assumption: Are particle masses con- stant? The photon emitted in an orbital transition of an electron in an atom can only be redshifted if its mass is initially small. As time goes on the electron communicates with more and more matter within a sphere whose limit is expanding at velocity c. If the masses of electrons increase, emitted photons change from an initially high redshift to a lower redshift with time (see Narlikar and Arp, 1993 [6]) Fig. 1: Enhanced Hubble Space Telescope image showing ejection Predicted consequences: Quasars are born with high red- wake from the center of NGC 7319 (redshift z = 0.022) to within shift and evolve into galaxies of lower redshift. about 3.4 arcsec of the quasar (redshift z = 2.11) Near zero mass particles evolve from energy conditions Observed: Younger objects have higher intrinsic redshifts. in an active nucleus. (If particle masses have to be created In groups, star forming galaxies have systematically sometime, it seems easier to grow things from a low mass higher redshifts, e. g. spiral galaxies. Even companions in state rather than producing them instantaneously in a finished evolved groups like our own Andromeda Group or the nearby state.) M81 group still have small, residual redshift excesses relative DARK MATTER: The establishment gets it right, sort of. to their parent. In the Big Bang, gas blobs in the initial, hot universe Observed: X-ray and radio emission generally indicate have to condense into things we now see like quasars and early evolutionary stages and intrinsic redshift. galaxies. But we know hot gas blobs just go poof! Lots of Plasmoids ejected from an active nucleus can fragment dark matter (cold) had to be hypothesized to condense the or ablate during passage through galactic and intergalactic gas cloud. They are still looking for it. medium which results in the forming of groups and clusters But low mass particles must slow their velocities in order of proto galaxies. The most difficult result for astronomers to conserve momentum as their mass grows. Temperature is to accept is galaxy clusters which have intrinsic redshifts. internal velocity. Thus the plasmoid cools and condenses its Yet the association of clusters with lower redshift parents is increasing mass into a compact quasar. So maybe we

H. Arp. Observational Cosmology. From High Redshift Galaxies to the Blue Pacific 3 Volume 3 PROGRESS IN PHYSICS October, 2005

observed from the most reliable Cepheid distances is H0 = = 55 (Arp, 2002 [3]). What are the chances of obtaining the correct Hubble constant from an incorrect theory with no adjustable parameters? If this is correct there is negligible room for expansion of the universe. Observed: The current Hubble constant is too large. A large amount of observing time on the Hubble Space Telescope was devoted to observing Cepheid variables whose distances divided into their redshifts gave a definitive value of H0 = 72. That required the reintroduction of Einstein’s cosmological constant to adjust to the observations. But H0 = 72 was wrong because the higher redshift galaxies in the sample included younger (ScI) galaxies which had appreciable intrinsic redshifts. Independent distances to these galaxies by means of rotational luminosity distances (Tully-Fisher distances) also showed this class of galaxies had intrinsic redshifts which gave too high a Hubble constant (Russell, 2002 [8]) In fact well known clusters of galaxies gives H0’s in the 90’s (Russell, private communication) which clearly shows that Fig. 2: Schematic representation of quasars and companion galaxies neither do we have a correct distance scale or understanding found associated with central galaxies from 1966 to present. The of the nature of galaxy clusters. progression of characteristics is empirical but is also required by the variable mass theory of Narlikar and Arp, 1993 [6] DARK ENERGY: Expansion now claimed to be acceler- ation. have been observing dark matter ever since the discovery of As distance measures were extended to greater distances quasars! After all, what’s in a name? by using Supernovae as standard candles it was found that the distant Supernovae were somewhat too faint. This led Observed: Ambarzumian sees new galaxies. to a smaller H0 and hence an acceleration compared to In the late 1950’s when the prestigious Armenian astro- the supposed present day H0 = 72. Of course the younger nomer, Viktor Ambarzumian was president of the Interna- Supernovae could be intrinsically fainter and also we have tional Astronomical Union he said that just looking at pic- seen the accepted present day H0 is too large. Nevertheless tures convinced him that new galaxies were ejected out of astronomers have again added a huge amount of undetected old. Even now astronomers refuse to discuss it, saying that substance to the universe to make it agree with properties of big galaxies cannot come out of other big galaxies. But we a disproved set of assumptions. This is called the accordance have just seen that the changing redshift is the key that model but we could easily imagine another name for it. unlocks the growth of new galaxies with time. They are small when they come from the small nucleus. Ambarzumian’s superfluid just needed the nature of changing redshift. But 3 Physics — local and universal Oort and conventional astronomers preferred to condense hot gas out of a hot expanding universe. Instead of extrapolating our local phenomena out to the universe one might more profitably consider our local region Observed: The Hubble Relation. as a part of the physics of the universe. An article of faith in current cosmology is that the relation between faintness of galaxies and their redshift, the Hubble Note: Flat space, no curves, no expansion. Relation, means that the more distant a galaxy is the faster it The general solution of energy/momentum conservation is receding from us. With our galaxy redshifts a function of (relativistic field equations) which Narlikar made with age, however, the look back time to a distant galaxy shows it m = t2 gives a Euclidean, three dimensional, uncurved space. to us when it was younger and more intrinsically redshifted. The usual assumption that particle masses are constant in No Doppler recession needed! time only projects our local, snapshot view onto the rest of The latter non-expanding universe is even quantitative in the universe. that Narlikar’s general solution of the General Relativistic In any case it is not correct to solve the equations in a non- equations (m = t2) gives a Hubble constant directly in term general case. In that case the usual procedure of assigning of the age of our own galaxy. (H0 = 51 km/sec×Mpc for curvature and expansion properties to the mathematical term age of our galaxy = 13 billion years). The Hubble constant space (which has no physical attributes!) is only useful for

4 H. Arp. Observational Cosmology. From High Redshift Galaxies to the Blue Pacific October, 2005 PROGRESS IN PHYSICS Volume 3

excusing the violations with the observations caused by the ing sphere which is usual picture of the nucleus? Hence the inappropriate assumption of constant elementary masses. directional nature of the observed plasmoid ejections. Consequences: Relativity theory can furnish no gravity. Space (nothing) can not be a “rubber sheet”. Even if there 4 Planets and people could be a dimple — nothing would roll into it unless there was a previously existing pull of gravity. We need to find a In our own solar system we know the gas giant planets plausible cause for gravity other than invisible bands pulling increase in size as we go in toward the Sun through Neptune, things together. Uranus, Saturn and Jupiter. On the Earth’s side of Jupiter, however, we find the asteroid belt. It does not take anad- Required: Verysmall wave/particles pushing against bodies. vanced degree to come to the idea that the asteroids are the In 1747 the Genevoise philosopher-physicist George- remains of a broken up planet. But how? Did something Louis Le Sage postulated that pressure from the medium crash into it? What does it mean about our solar system? which filled space would push bodies together in accordance with the Newtonian Force =1/r2 law. Well before the cont- Mars: The Exploding Planet Hypothesis. inuing fruitless effort to unify Relativistic gravity and quan- We turn to a real expert on planets, Tom Van Flandern. tum gravity, Le Sage had solved the problem by doing away For years he has argued in convincing detail that Mars, with the need to warp space in order to account for gravity. originally bigger than Earth, had exploded visibly scarring the surface of its moon, the object we now call Mars. One Advantages: The Earth does not spiral into the Sun. detail should be especially convincing, namely that the pre- Relativistic gravity is assigned an instantaneous com- sent Mars, unable to hold an atmosphere, had long been ponent as well as a component that travels with the speed considered devoid of water, a completely arid desert. But of light, c. If gravity were limited to c, the Earth would recent up-close looks have revealed evidence for “water be rotating around the Sun where it was about 8 minutes dumps”, lots of water in the past which rapidly went away. ago. By calculating under the condition that no detectable Where else could this water have come from except the reduction in the size of the Earth’s orbit has been observed, original, close-by Mars as it exploded? Tom Van Flandern arrives at the minimum speed of gravity For me the most convincing progression is the increasing 10 of 2×10 c. We could call these extremely fast, extremely masses of the planets from the edge of the planetary system penetrating particles gravitons. toward Jupiter and then the decreasing masses from Jupiter A null observation saves causality. through Mercury. Except for the present Mars! But that The above reasoning essentially means that gravity can continuity would be preserved with an original Mars larger act as fast as it pleases, but not instantaneously because that than Earth and its moon larger than the Earth’s moon. would violate causality. This is reassuring since causality As for life on Mars, the Viking lander reported bacteria seems to be an accepted property of our universe (except for but the scientist said no. Then there was controversy about some early forms of quantum theory). organic forms in meteorites from Mars. But the most straight Black holes into white holes. forward statement that can be made is that features have now In its usual perverse way all the talk has been about black been observed that look “artificial” to some. Obviously no holes and all the observations have been about white holes. one is certain at this point but most scientists are trained to Forget for a moment that from the observer’s viewpoint it stop short of articulating the obvious. would take an infinity of time to form a black hole. The ob- Gravitons: Are planets part of the universe? servations show abundant material being ejected from stars, If a universal sea of very small, very high speed gravitons nebulae, galaxies, quasars. What collimates these out of a are responsible for gravity in galaxies and stars would not region in which everything is supposed to fall into? (Even these same gravitons be passing through the solar system ephemeral photons of light.) After 30 years of saying nothing and the planets in it? What would be the effect if a small comes out of black holes, Stephen Hawking now approaches percentage were, over time, absorbed in the cores of planets? the observations saying maybe a little leaks out. Speculation: What would we expect? Question: What happens when gravitons encounter a black Heating the core of a gas giant would cause the liquid/ hole? gaseous planet to expand in size. But if the core of a rocky If the density inside the concentration of matter is very planet would be too rigid to expand it would eventually high the steady flux of gravitons absorbed will eventually explode. Was the asteroid planet the first to go? Then the heat the core and eventually this energy must escape. After original Mars? And next the Earth? all it is only a local concentration of matter against the Geology: Let’s argue about the details. continuous push of the whole of intergalactic space. Is it Originally it was thought the Earth was flat. Then spher- reasonable to say it will escape through the path of least ical but with the continents anchored in rock. When Alfred resistance, for example through the flattened pole of a spinn- Wegener noted that continents fitted together like jigsaw

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puzzle and therefore had been pulled apart, it was violently References rejected because geologists said they were anchored in basalt- ic rock. Finally it was found that the Atlantic trench between 1. Arp H. and Russell D. G. A possible relationship between the Americas and Africa/Europe was opening up at a rate quasars and clusters of galaxies. Astrophysical Journal, 2001, of just about right for the Earth’s estimated age (Kokus, v. 549, 802–819 2002 [5]). So main stream geologists invented plate tecton- 2. Arp H. Pushing Gravity, ed. by M. R. Edwards, 2002, 1. ics where the continents skated blythly around on top of this 3. Arp H. Arguments for a Hubble constant near H0 = 55. Astro- anchoring rock! physical Journal, 2002, 571, 615–618. In 1958 the noted Geologist S. Warren Carey and in 1965 4. Arp H. Catalog of discordant redshift associations. Apeiron, K. M. Creer (in the old, usefully scientific Nature Magazine) Montreal, 2003. were among those who articulated the obvious, namely that 5. Kokus M. Pushing Gravity, ed. by M. R. Edwards, 2002, 285. the Earth is expanding. The controversy between plate tec- 6. Narlikar J. and Arp H. Flat spacetime cosmology — a unified tonics and expanding Earth has been acrid ever since. framework for extragalactic redshifts. Astrophysical Journal, (One recent conference proceedings by the latter adherents is 1993, 405, 51-56. “Why Expanding Earth?” (Scalera and Jacob, 2003 [7]). 7. Scalera G. and Jacob K.-H. (editors). Why expanding Earth? Let’s look around us. Nazionale di Geofisica e Vulcanologoia, Technisch Univ. The Earth is an obviously active place. volcanos, Earth Berlin, publ. INGV Roma, Italy, 2003. quakes, island building. People seem to agree the Atlantic 8. Russell D. G. Morphological type dependence in the Tully- is widening and the continents separating. But the Pacific is Fisher relationship. Astrophysical Journal, 2004, v. 607, violently contested with some satellite positioning claiming 241-246. no expansion. I remember hearing S. Warren Carey pains- 9. Van Flandern T. Dark matter, missing planets and new comets. takingly interpreting maps of the supposed subduction zone 2nd ed., North Atlantic Books, Berkely, 1999. where the Pacific plate was supposed to be diving under the 10. Van Flandern T. Pushing Gravity, ed. by M. R. Edwards, Andean land mass of Chile. He argued that there was no 2002, 93. debris scraped off the supposedly diving Pacific Plate. But in any case, where was the energy coming from to drive a huge Pacific plate under the massive Andean plate? My own suggestion about this is that the (plate) is stuck, not sliding under. Is it possible that the pressure from the Pacific Basin has been transmitted into the coastal ranges of the Americas where it is translated into mountain building? (Mountain building is a particularly contentious disagree- ment between static and expanding Earth proponents.) It is an impressive, almost thought provoking sight, to see hot lava welling up from under the southwest edge of the Big Island of Hawaii forming new land mass in front of our eyes. All through the Pacific there are underground vents, volcanos, mountain and island building. Is it possible this upwelling of mass in the central regions of the Pacific is putting pressure on the edge? Does it represent the emergence of material comparable to that along the Mid Atlantic ridge on the other side of the globe? The future: Life as an escape from danger. The galaxy is an evolving, intermittently violent environ- ment. The organic colonies that inhabit certain regions within it may or may not survive depending on how fast they recognize danger and how well they adapt, modify it or escape from it. Looking out over the beautiful blue Pacific one sees tropical paradises. On one mountain top, standing on barely cool lava, is the Earth’s biggest telescope. Looking out in the universe for answers. Can humankind collectively understand these answers? Can they collectively ensure their continued appreciation of the beauty of existence.

6 H. Arp. Observational Cosmology. From High Redshift Galaxies to the Blue Pacific October, 2005 PROGRESS IN PHYSICS Volume 3

On the General Solution to Einstein’s Vacuum Field for the Point-Mass when λ = 0 and Its Consequences for Relativistic Cosmology 6 Stephen J. Crothers Sydney, Australia E-mail: [email protected]

It is generally alleged that Einstein’s theory leads to a finite but unbounded universe. This allegation stems from an incorrect analysis of the metric for the point-mass when λ = 0. The standard analysis has incorrectly assumed that the variable r denotes a 6 radius in the gravitational field. Since r is in fact nothing more than a real-valued parameter for the actual radial quantities in the gravitational field, the standard interpretation is erroneous. Moreover, the true radial quantities lead inescapably to λ = 0 so that, cosmologically, Einstein’s theory predicts an infinite, static, empty universe.

1 Introduction these assumptions are still false, and further, that λ can only take the value of zero in Einstein’s theory. It has been shown [1, 2, 3] that the variable r which appears in the metric for the gravitational field is neither a radius 2 Definitions nor a coordinate in the gravitational field, and further [3], that it is merely a real-valued parameter in the pseudo- As is well-known, the basic spacetime of the General Theory Euclidean spacetime (Ms, gs) of Special Relativity, by which of Relativity is a metric space of the Riemannian geometry the Euclidean distance D = r r0 (Ms, gs) is mapped in- family, namely — the four-dimensional pseudo-Riemannian to the non-Euclidean distance| R− |( ∈M , g ), where (M , g ) p ∈ g g g g space with Minkowski signature. Such a space, like any denotes the pseudo-Riemannian spacetime of General Rela- Riemannian metric space, is strictly negative non-degenerate, tivity. Owing to their invalid assumptions about the variable i. e. the fundamental metric tensor gαβ of such a space has a r, the relativists claim that r = 3 defines a “horizon” for determinant which is strictly negative: g = det g < 0. λ || αβ|| the universe (e .g. [4]), by whichq the universe is supposed to Space metrics obtained from Einstein’s equations can have a finite volume. Thus, they have claimed a finite but be very different. This splits General Relativity’s spaces unbounded universe. This claim is demonstrably false. into numerous families. The two main families are derived The standard metric for the simple point-mass when from the fact that the energy-momentum tensor of matter λ = 0 is, Tαβ, contained in the Einstein equations, can (1) be linearly 6 proportional to the fundamental metric tensor gαβ or (2) have 2m λ ds2 = 1 r2 dt2 a more compound functional dependence. The first case is − r − 3 − much more attractive to scientists, because in this case one   (1) 1 can use gαβ, taken with a constant numerical coefficient, 2m λ 2 − 2 2 2 2 2 1 r dr r dθ + sin θdϕ . instead of the usual Tαβ, in the Einstein equations. Spaces − − r − 3 −   of the first family are known as Einstein spaces. The relativists simply look at (1) and make the following
From the purely geometrical perspective, an Einstein assumptions. space [5] is described by any metric obtained from (a) The variable r is a radial coordinate in the gravita- 1 tional field ; Rαβ gαβR = κTαβ λgαβ , − 2 − (b) r can go down to 0 ; where is a constant and , and therefore includes (c) A singularity in the gravitational field can occur only κ Tαβ gαβ all partially degenerate metrics.∝ Accordingly, such spaces where the Riemann tensor scalar curvature invariant become non-Einstein only when the determinant g of the (or Kretschmann scalar) f = R Rαβγδ is un- αβγδ metric becomes bounded .

The standard analysis has never proved these assum- g = det gαβ = 0 . ptions, but nonetheless simply takes them as given. I have || || demonstrated elsewhere [3] that when λ = 0, these assum- In terms of the required physical meaning of General ptions are false. I shall demonstrate herein that when λ = 0 Relativity I shall call a spacetime associated with a non- 6

S. J. Crothers. On the General Solution to Einstein’s Vacuum Field for the Point-Mass when λ = 0 7 6 Volume 3 PROGRESS IN PHYSICS October, 2005

degenerate metric, an Einstein universe, and the associated Transform (3) by setting, metric an Einstein metric. Cosmological models involving either λ = 0 or λ = 0, r∗ = C(D(r)) , (4) which do not result in a degenerate metric, I shall6 call rela- to carry (3) into, p tivistic cosmological models, which are necessarily Einstein universes, with associated Einstein metrics. 2 2 2 2 2 2 2 ds =A∗(r∗)dt B∗(r∗)dr∗ r∗ dθ + sin θdϕ . (5) Thus, any “partially” degenerate metric where g = 0 is − − 6 not an Einstein metric, and the associated space is not an For λ=0, one finds in the usual way that
the solution Einstein universe. Any cosmological model resulting in a to (5) is, 6 “partially” degenerate metric where g = 0 is neither a rela- 6 2 α λ 2 2 tivistic cosmological model nor an Einstein universe. ds = 1 r∗ dt − r∗ − 3 −   (6) 1 3 The general solution when λ = 0 α λ 2 − 2 2 2 2 2 6 1 r∗ dr∗ r∗ dθ + sin θdϕ . − − r − 3 −  ∗  The general solution for the simple point-mass [3] is,
α = const. 2 √C α √C C0 ds2 = n − dt2 n n dr2 Then by (4), √Cn − √Cn α 4Cn − (2)    −  2 α λ 2 C (dθ2 + sin2 θdϕ2) , ds = 1 C dt n − √ − 3 − −  C  2 1 02 n n n + α λ − C (7) Cn(r) = r r0 + α , n , 1 C dr2 | − | ∈ < − − √ − 3 4C −  C  α = 2m, r0  , ∈ < C dθ2 + sin2 θdϕ2 , where n and r0 are arbitrary and r is a real-valued parameter − in (M , g ).
s s C = C(D(r)),D = D(r) = r r0 , r0 , The most general static metric for the gravitational field | − | ∈ < [3] is, α = const ,

2 2 2 2 2 2 where r (Ms, gs) is a real-valued parameter and also ds =A(D)dt B(D)dr C(D) dθ + sin θdϕ , (3) r (M ,∈ g ) is an arbitrary constant which specifies the − − 0 ∈ s s
position of the point-mass in parameter space. D = r r , r , | − 0| 0 ∈ < When α = 0, (7) reduces to the empty de Sitter metric, which I write generally, in view of (7), as where analytic A, B, C > 0 r = r . ∀ 6 0 In relation to (3) I identify the coordinate radius D, the r- 1 2 λ 2 λ − 2 parameter, the radius of curvature R , and the proper radius ds = 1 F dt 1 F d√F c − 3 − − 3 − (8) (proper distance) Rp.     2 2 2 1. The coordinate radius is D = r r . F dθ + sin θdϕ , | − 0| − 2. The r-parameter is the variable r .
F = F (D(r)),D = D(r) = r r0 , r0 . 3. The radius of curvature is R = C(D(r)) . | − | ∈ < c 2 If F (D(r)) = r , r0 = 0, and r > r0, then the usual form 4. The proper radius is Rp = Bp(D(r))dr . of (8) is obtained, I remark that gives the mapping of the Euclid- Rp(D(r)) R p 1 λ λ − ean distance D = r r0 (Ms, gs) into the non-Euclidean ds2 = 1 r2 dt2 1 r2 dr2 distance R (M|, g−) [3].| ∈ Furthermore, the geometrical re- − 3 − − 3 − p g g     (9) lations between∈ the components of the metric tensor are invi- r2 dθ2 + sin2 θdϕ2 . olable and therefore hold for all metrics with the form of (3). − Thus, on the metric (2), The admissible forms for C(D(r)) and F (D
(r)) must now be generally ascertained. Rc = Cn(D(r)) , If C0 0, then B(D(r)) = 0 r, in violation of (3). ≡ ∀ p Therefore C0 = 0 r =r . 6 ∀ 6 0 √Cn Cn0 Now C(D(r)) must be such that when r , equa- Rp = dr . → ± ∞ s√Cn α 2√Cn tion (7) must reduce to (8) asymptotically. So, Z −

8 S. J. Crothers. On the General Solution to Einstein’s Vacuum Field for the Point-Mass when λ = 0 6 October, 2005 PROGRESS IN PHYSICS Volume 3

C(D(r)) as r , 1. and → ± ∞ F (D(r)) → 3α λα3 I have previously shown [3] that the condition for sin- β β3 = − ≈ 3 2λα2 − gularity on a metric describing the gravitational field of the − α 3 2λα2 3 2 point-mass is, ( − ) λ 3α λα (14b) 1 (3α λα3) 3 3 −2λα2 g (r ) = 0 . (10) − − − − 00 0  2  , − 3 2λα2 2λ  3α λα3  α 3α− λα3 3 3 −2λα2 Thus, by (7), it is required that,  − − −        α λ α λ etc., which satisfy the requirement that β = β(α, λ). 1 C(D(r )) = 1 β2 = 0 , (11) − − 3 0 − β − 3 3 C(D(r0)) Taking β1 = λ into (13) gives, p q having set C(D(r0)) = β. Thus, β is a scalar invariant for 3 α (7) that must contain the independent factors contributing to β β2 = + , (15a) p ≈ rλ α λ 2 the gravitational field, i .e. β = β(α, λ). Consequently it is 3 − required that when λ = 0, β = α = 2m to recover (2), when q 3 and α = 0, β = λ to recover (8), and when α = λ = 0, and 2 β = 0, C(Dq(r)) = r r0 to recover the flat spacetime of 3 α | − | β β3 = + Special Relativity. Also, when α = 0, C(D(r)) must reduce ≈ λ λ − r α 3 2 to F (D(r)). The value of β = β(λ) = F (D(r0)) in (8) is − also obtained from, q 2 p 1 α λ 3 + α 3 λ α√ λ 2 (15b) − 3 α − 3 − λ λ 2  √ λ +    g (r ) = 0 = 1 F (D(r )) = 1 β . α√ λ 2 q 00 0 − 3 0 − 3  3 −  , −    α 2λ 3 + α  Therefore,  2 3 λ α√ λ 2   − 3 −   √ 3 + α    3 λ λ q  α√ 2  β = . (12)   3 −   rλ   Thus, to render a solution to (7), C(D(r)) must at least etc., which satisfy the requirement that β = β(α, λ). satisfy the following conditions. However, according to (14a) and (14b), when λ = 0, 3 β = α = 2m, and when α = 0, β = λ . According to (15a), 1. C0(D(r)) = 0 r = r0 . 6 6 ∀ 6 q 3 2. As r , C(D(r)) 1 . (15b), when λ = 0, β = α = 2m, and when α = 0, β = . F (D(r)) 6 λ → ± ∞ → The required form for β, and therefore the required form 3. C(D(r )) = β2, β = β(α, λ) . q 0 for C(D(r)), cannot be constructed, i .e. it does not exist. 2 n n n 4. λ = 0 β = α = 2m and C = r r0 + α . There is no way C(D(r)) can be constructed to satisfy all ⇒ | − | the required conditions to render an admissible solution to 3
5. α = 0 β = λ and C(D(r)) = F (D(r)) . (7) in the form of (3). Therefore, the assumption that λ = 0 ⇒ 6 q 2 is incorrect, and so λ = 0. This can be confirmed in the 6. α = λ = 0 β = 0 and C(D(r)) = r r0 . ⇒ | − | following way. Both and must also be determined. α β(α, λ) The proper radius R (r) of (8) is given by, Since (11) is a cubic, it cannot be solved exactly for p β. However, I note that the two positive roots of (11) are d√F 3 λ 3 α λ 2 Rp(r) = = arcsin F (r) + K, approximately α and λ . Let P (β) = 1 β 3 β . Then λ λ 3 − − Z 1 F r r according to Newton’sq method, − 3 q where K is a constant. Now, the following condition must 1 α λ β2 P (βm) − βm − 3 m be satisfied, βm+1 = βm = βm . (13)  α 2λ  − P 0(βm) − β + β2 3 m as r r±,R 0 , m − → 0 p →   Taking β1 = α into (13) gives, and therefore,

3α λα3 3 λ β β = − , (14a) Rp(r ) = 0 = arcsin F (r ) + K, ≈ 2 3 2λα2 0 λ 3 0 − r r

S. J. Crothers. On the General Solution to Einstein’s Vacuum Field for the Point-Mass when λ = 0 9 6 Volume 3 PROGRESS IN PHYSICS October, 2005

and so, The “surface area” of the point is,

3 λ λ 12π R (r)= arcsin F (r) arcsin F (r ) . (16) A = . p 0 λ rλ " r 3 − r 3 # According to (8), De Sitter’s empty spherical universe has zero volume. Indeed, by (8) and (8b), 3 g00(r0) = 0 F (r0) = . r π 2π ⇒ λ 3 V = lim 0 dr sin θ dθ dϕ = 0 , But then, by (16), r → ±∞ λ Z Z0 Z0 r0 λ F (r) 1 , consequently, de Sitter’s empty spherical universe is indeed 3 ≡ r “empty”; and meaningless. It is not an Einstein universe. Rp(r) 0 . On (8) and (8b) the ratio, ≡ Indeed, by (16), 2π F (r) = r . λ λ Rp(r) ∞ ∀ F (r ) F (r) 1 , p 3 0 6 3 6 r r Therefore, the lone point which consitutes the empty de or Sitter “universe” (M , g ) is a quasiregular singularity and 3 3 ds ds 6 F (r) 6 , consequently cannot be extended. rλ rλ It is the unproven and invalid assumptions about the p and so variable r which have lead the relativists astray. They have 3 F (r) , (17) carried this error through all their work and consequently ≡ λ have completely lost sight of legitimate scientific theory, and producing all manner of nonsense along the way. Eddington Rp(r) 0 . (18) 2m αr2 ≡ [4], for instance, writes in relation to (1), γ = 1 r 3 for his equation (45.3), and said, − − Then F 0(D(r)) 0, and so there exists no function F (r) ≡ which renders a solution to (8) in the form of (3) when λ = 0 At a place where γ vanishes there is an impass- 6 and therefore there exists no function C(D(r)) which renders able barrier, since any change dr corresponds to a solution to (7) in the form of (3) when λ = 0. Consequently, an infinite distance ids surveyed by measuring 6 λ = 0. rods. The two positive roots of the cubic (45.3) Owing to their erroneous assumptions about the r-para- are approximately meter, the relativists have disregarded the requirement that 3 A, B, C > 0 in (3) must be met. If the required form (3) is r = 2m and r = α . relaxed, in which case the resulting metric is non-Einstein, The first root would representq
the boundary of and cannot therefore describe an Einstein universe, (8) can the particle — if a genuine particle could exist be written as, — and give it the appearance of impenetrability. 3 The second barrier is at a very great distance ds2 = dθ2 + sin2 θdϕ2 . (8b) − λ and may be described as the horizon of the world. This means that metric (8) (8b) maps
the whole of (M , g ) into the point R (D(r))≡ 0 of the de Sitter “space” Note that Eddington, despite these erroneous claims, did not s s p ≡ (Mds, gds). admit the sacred black hole. His arguments however, clearly Einstein, de Sitter, Eddington, Friedmann, and the mod- betray his assumption that r is a radius on (1). I also note that ern relativists all, have incorrectly assumed that r is a radial he has set the constant numerator of the middle term of his coordinate in (8), and consequently think of the “space” γ to 2m, as is usual, however, like all the modern relativists, associated with (8) as extended in the sense of having a he did not indicate how this identity is to be achieved. This volume greater than zero. This is incorrect. is just another assumption. As Abrams [6] has pointed out in The radius of curvature of the point Rp(D(r)) 0 is, regard to (1), one cannot appeal to far-field Keplerian orbits ≡ to fix the constant to 2m — but the issue is moot, since λ = 0. 3 There is no black hole associated with (1). The Lake- Rc(D(r)) . Roeder black hole is inconsistent with Einstein’s theory. ≡ rλ

10 S. J. Crothers. On the General Solution to Einstein’s Vacuum Field for the Point-Mass when λ = 0 6 October, 2005 PROGRESS IN PHYSICS Volume 3

4 The homogeneous static models 5 Einstein’s cylindrical cosmological model

It is routinely alleged by the relativists that the static homo- In this case, to reduce to Special Relativity, geneous cosmological models are exhausted by the line- elements of Einstein’s cylindrical model, de Sitter’s spherical ν = const = 0. model, and that of Special Relativity. This is not correct, as I shall now demonstrate that the only homogeneous universe Therefore, by (21), σ admitted by Einstein’s theory is that of his Special Theory e− 1 of Relativity, which is a static, infinite, pseudo-Euclidean, 8πP0 = 2 2 + λ , r∗ − r∗ empty world. and by (19), The cosmological models of Einstein and de Sitter are composed of a single world line and a single point respecti- 1 1 8πP0 = 2 2 + λ , vely, neither of which can be extended. Their line-elements B∗(r∗)r∗ − r∗ therefore cannot describe any Einstein universe. and by (4), If the Universe is considered as a continuous distribution 1 1 8πP0 = + λ , of matter of proper macroscopic density ρ00 and pressure BC − C P0 , the stress-energy tensor is, so 1 T 1 = T 2 = T 3 = P ,T 4 = ρ , = 1 λ 8πP0 C, 1 2 3 − 0 4 00 B − − μ
Tν = 0, μ = ν . C = C(D(r)),D(r) = r r ,B = B(D(r)) , 6 | − 0| Rewrite (5) by setting, r0 . ν ∈ < A∗(r∗) = e , ν = ν(r∗) , Consequently, Einstein’s line-element can be written as, σ B∗(r∗) = e , σ = σ(r∗) . (19) 1 2 ds2 = dt2 1 λ 8πP C − d√C Then (5) becomes, − − − 0 − 2 ν 2 σ 2 2 2 2 2 
2  2 2 ds = e dt e dr∗ r∗ dθ + sin θdϕ . (20) C dθ + sin dϕ = − − − (24)
02 It then follows in the usual way that, 1 C
= dt2 1 λ 8πP C − dr2 − − − 0 4C − σ νˉ 1 1 8πP = e− + + λ , (21) 0 2 2 
 2 r∗ r∗ − r∗ C dθ2 + sin dϕ2 ,   − σ σˉ 1 1
8πρ = e− + λ , (22) C = C(D(r)),D(r) = r r , r , 00 r − r 2 r 2 − | − 0| 0 ∈ <  ∗ ∗  ∗ where is arbitrary. dP ρ + P r0 0 = 00 0 νˉ , (23) It is now required to determine the admissible form of dr − 2 ∗ C(D(r)). where Clearly, if C0 0, then B = 0 r, in violation of (3). dν dσ ≡ ∀ νˉ = , σˉ = . Therefore, C0 = 0 r = r0. dr∗ dr∗ 6 ∀ 6 When P0 = λ = 0, (24) must reduce to Special Relativity, Since P0 is to be the same everywhere, (23) becomes, in which case, ρ + P 00 0 νˉ = 0 . P = λ = 0 C(D(r)) = r r 2 . 2 0 ⇒ | − 0| 1 Therefore, the following three possibilities arise, The metric (24) is singular when g11− (r0) = 0, i .e. when, dν 1. = 0 ; 1 λ 8πP C(r ) = 0 , dr∗ − − 0 0 2. ρ00 + P0 = 0 ;
1 dν C(r0) = . (25) 3. = 0 and ρ00 + P0 = 0 . ⇒ λ 8πP0 dr∗ − The 1st possibility yields Einstein’s so-called cylindrical Therefore, for C(D(r)) to render an admissible solution to model, the 2nd yields de Sitter’s so-called spherical model, (24) in the form of (3), it must at least satisfy the following and the 3rd yields Special Relativity. conditions:

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1. C0 = 0 r = r ; Clearly, if C0 0,B(D(r)) = 0 r, in violation of (3). 6 ∀ 6 0 ≡ ∀ 2 Therefore, C0 = 0 r = r . 2. P0 = λ = 0 C(D(r)) = r r0 ; 6 ∀ 6 0 ⇒1 | − | Since there is no matter present, it is required that, 3. C(r0) = λ 8πP . − 0 C(D(r)) Now the proper radius on (24) is, C(r ) = 0 and = 1 . 0 r r 2 | − 0| d√C This requires trivially that, Rp(r) = = 2 Z 1 λ 8πP0 C C(D(r)) = r r . − − | − 0| 1 q = arcsin (λ
8πP0 )C(r) + K, Therefore (27) becomes, λ 8πP0 − − q (r r )2 p ds2 = dt2 − 0 dr2 r r 2 dθ2 + sin2 dϕ2 = K = const. , − r r 2 −| − 0| | − 0| which must satisfy the condition, 2
= dt2 dr2 r r 2 dθ2 + sin dϕ2 , − − | − 0| + as r r±,R 0 . → 0 p → which is precisely the metric of Special Relativity,
according to the natural reduction on (2). Therefore, If the required form (3) is relaxed, in which case the 1 resulting metric is not an Einstein metric, Einstein’s cylindr- Rp(r0) = 0 = ical line-element is, λ 8πP × − 0 2 2 1 2 2 2 arcsinp (λ 8πP )C(r ) + K, ds = dt dθ + sin θdϕ . (28) × − 0 0 − λ 8πP0 q −
so This is a line-element which
cannot describe an Einstein universe. The Einstein space described by (28) consists of 1 Rp(r) = arcsin (λ 8πP0 )C(r) only one “world line”, through the point, λ 8πP − − − 0 h q (26) Rp(r) 0 . p ≡ arcsin (λ 8πP )C(r ) . − − 0 0 The spatial extent of (28) is a single point. The radius of q i curvature of this point space is, Now if follows from (26) that, 1 Rc(r) . (λ 8πP )C(r ) (λ 8πP )C(r) 1 , ≡ λ 8πP0 − 0 0 6 − 0 6 − q q For all r, the ratio 2πRc pis, so Rp 1 2π C(r0) 6 C(r) 6 , λ 8πP0 √λ 8πP − − 0 = . and therefore by (25),
Rp(r) ∞ Therefore R (r) 0 is a quasiregular singular point and 1 1 p ≡ 6 C(r) 6 . consequently cannot be extended. λ 8πP λ 8πP − 0 − 0 The “surface area” of this point space is, Thus,

4π 1 A = . C(r) , λ 8πP0 ≡ λ 8πP − − 0 The volume of the point space is, and so C0(r) 0 B(r) 0, in violation
of (3). Therefore ≡ ⇒ ≡ r π 2π there exists no C(D(r)) to satisfy (24) in the form of (3) 1 V = lim 0 dr sin θ dθ dϕ = 0 . when λ = 0,P0 = 0. Consequently, λ = P0 = 0, and (24) r → ±∞ λ 8πP0 6 6 − Z Z0 Z0 reduces to, r0
02 Equation (28) maps the whole of (Ms, gs) into a quasi- C 2 ds2 = dt2 dr2 C dθ2 + sin dϕ2 . (27) regular singular “world line”. − 4C − Einstein’s so-called “cylindrical universe” is meaning-
The form of C(D(r)) must still be determined. less. It does not contain a black hole.

12 S. J. Crothers. On the General Solution to Einstein’s Vacuum Field for the Point-Mass when λ = 0 6 October, 2005 PROGRESS IN PHYSICS Volume 3

6 De Sitter’s spherical cosmological model and so de Sitter’s line-element is,

In this case, λ + 8πρ ds2 = 1 00 C dt2 ρ + P = 0 . − 3 − 00 0   1 Adding (21) to (22) and setting to zero gives, λ + 8πρ − C 02 (32) 1 00 C dr2 − − 3 4C − σ σˉ νˉ   8π ρ + P = e− + = 0 , 00 0 r r 2 2 2  ∗ ∗  C dθ + sin θdϕ , or
−
νˉ = σˉ . C = C(D(r)),D(r) = r r , r , − | − 0| 0 ∈ < Therefore, where r0 is arbitrary. It remains now to determine the admissible form of ν(r∗) = σ(r∗) + ln K1 , (29) − C(D(r)) to render a solution to equation (32) in the form of

K1 = const. equation (3). If C0 0, then B(D(r)) = 0 r, in violation of (3). Since ρ00 is required to be a constant independent of ≡ ∀ Therefore C0 = 0 r = r . position, equation (22) can be immediately integrated to give, 6 ∀ 6 0 When λ = ρ00 = 0, (32) must reduce to that for Special Relativity. Therefore, σ λ + 8πρ00 2 K2 e− = 1 r∗ + , (30) − 3 r ∗ λ = ρ = 0 C(D(r)) = r r 2 . 00 ⇒ | − 0| K2 = const. Metric (32) is singular when g (r ) = 0, i .e. when According to (30), 00 0

λ + 8πρ00 λ + 8πρ00 2 K2 1 C(r ) = 0 , σ = ln 1 r∗ + , − 3 0 − − 3 r  ∗  3 and therefore, by (29), C(r0) = . (33) ⇒ λ + 8πρ00 λ + 8πρ00 2 K2 ν = ln 1 r∗ + K . Therefore, to render a solution to (32) in the form of (3), − 3 r 1  ∗   C(D(r)) must at least satisfy the following conditions: Substituting into (20) gives, 1. C0 = 0 r = r ; 6 ∀ 6 0 2. λ = ρ = 0 C(D(r)) = r r 2 ; λ + 8πρ K 00 ⇒ | − 0| 2 = 1 00 r 2 + 2 K dt2 3 ds ∗ 1 3. C(r0) = . − 3 r − λ+8πρ00  ∗   1 The proper radius on (32) is, λ + 8πρ00 2 K2 − 2 1 r∗ + dr∗ − − 3 r∗ − d√C   Rp(r) = = 2 2 2 2 λ+8πρ r∗ dθ + sin θdϕ , Z 1 00 C − − 3 r (34) which is, by (4),
  3 λ+8πρ = arcsin 00 C(r) +K, 2 λ + 8πρ00 K2 2 ds = 1 C + K dt sλ+8πρ00 s 3 − 3 √ 1 −    C   1 K = const , λ + 8πρ K − C 02 (31) 1 00 C + 2 dr2 − − 3 √ 4C − which must satisfy the condition,  C  2 2 2 + C dθ + sin θdϕ . as r r0±,Rp(r) 0 . − → →
Therefore, Now, when λ = ρ00 = 0, equation (31) must reduce to the metric for Special Relativity. Therefore, 3 λ + 8πρ00 Rp(r ) = 0 = arcsin C(r )+K, K = 1,K = 0 , 0 λ + 8πρ 3 0 1 2 r 00 s 

S. J. Crothers. On the General Solution to Einstein’s Vacuum Field for the Point-Mass when λ = 0 13 6 Volume 3 PROGRESS IN PHYSICS October, 2005

so (34) becomes, The volume of de Sitter’s “spherical universe” is,

r π 2π 3 λ+8πρ00 3 Rp(r)= arcsin C(r) V = lim 0 dr sin θ dθ dϕ = 0 . λ+8πρ 3 − r s 00 s λ + 8πρ00 → ±∞ h     Z Z0 Z0 (35) r0 λ + 8πρ00 arcsin C(r0) . For all values of r, the ratio, − s 3   i 2π 3 It then follows from (35) that, λ+8πρ 00 = . qRp(r) ∞ λ + 8πρ00 λ + 8πρ00 C(r0) 6 C(r) 6 1 , Therefore, R (r) 0 is a quasiregular singular point and s 3 s 3 p     consequently cannot be≡ extended. or According to (32), metric (37) maps the whole of 3 (Ms, gs) into a quasiregular singular point. C(r0) 6 C(r) 6 . λ + 8πρ00 Thus, de Sitter’s spherical universe is meaningless. It Then, by (33), does not contain a black hole. When ρ = 0 and λ = 0, de Sitter’s empty universe is 00 6 3 3 obtained from (37). I have already dealt with this case in 6 C(r) 6 . section 3. λ + 8πρ00 λ + 8πρ00

Therefore, C(r) is a constant function for all r, 7 The infinite static homogeneous universe of special relativity 3 C(r) , (36) ≡ λ + 8πρ00 In this case, by possibility 3 in section 4, and so, dν C0(r) 0 , νˉ = = 0, and ρ00 + P0 = 0 . ≡ dr∗ which implies that B(D(r)) 0, in violation of (3). Con- Therefore, sequently, there exists no function≡ C(D(r)) to render a ν = const = 0 by section 5 solution to (32) in the form of (3). Therefore, λ = ρ00 = 0, and (32) reduces to the metric of Special Relativity in the and same way as does (24). σˉ = νˉ by section 6 . − If the required form (3) is relaxed, in which case the Hence, also by section 6, resulting metric is not an Einstein metric, de Sitter’s line- element is, σ = ν = 0 . − 3 ds2 = dθ2 + sin2 θdϕ2 . (37) Therefore, (20) becomes, − λ + 8πρ00 2 2 2 2 2 2 2
ds = dt dr∗ r∗ dθ + sin θdϕ , This line-element cannot describe an Einstein universe. − − The Einstein space described by (37) consists of only one which becomes, by using (4),
point:

02 Rp(r) 0 . C ≡ ds2 = dt2 dr2 C dθ2 + sin2 θdϕ2 , The radius of curvature of this point is, − 4C −
C = C(D(r)),D(r) = r r0 , r0 , 3 | − | ∈ < Rc(r) , which, by the analyses in sections 5 and 6, becomes, ≡ sλ + 8πρ00 2 r r and the “surface area” of the point is, 2 2 − 0 2 2 2 2 2 ds = dt 2 dr r r0 dθ + sin θdϕ , − r r0
− | − | 12π | − | (38) A = .
λ + 8πρ r , 00 0 ∈ <

14 S. J. Crothers. On the General Solution to Einstein’s Vacuum Field for the Point-Mass when λ = 0 6 October, 2005 PROGRESS IN PHYSICS Volume 3

which is the flat, empty, and infinite spacetime of Special their false assumptions about the parameter r, the relativists Relativity, obtained from (2) by natural reduction. mistakenly think that C(D(r)) r2 in (32). Furthermore, ≡ When r0 = 0 and r >r0, (38) reduces to the usual form if the required form (3) is relaxed, thereby producing non- used by the relativists, Einstein metrics, de Sitter’s “spherical universe” is given by (37), and so, by (35), (36), and (40), ds2 = dt2 dr2 r2 dθ2 + sin2 θdϕ2 . − − λ + 8πρ C(D(r)) = r2 00 , The radius of curvature of (38) is,
≡ 3 D(r) = r r . and the transformations (39) and metric (40) are again utter | − 0| nonsense. The Lemaˆıtre-Robertson´ line-element is inevitably, The proper radius of (38) is, unmitigated claptrap. This can be proved generally as follows. The most general non-static line-element is

r r0 r | − | (r r ) ds2 = A(D, t)dt2 B(D, t)dD2 R (r) = d r r = − 0 dr = r r D. − − p | − 0| r r | − 0| ≡ (41) 0 2 2 2 Z0 rZ | − | C(D, t) dθ + sin θdϕ , 0 − The ratio, D = r r , r
| − 0| 0 ∈ <

2πD(r) 2π r r0 where analytic A, B, C > 0 r = r0 and t. = | − | = 2π r . Rewrite (41) by setting, ∀ 6 ∀ Rp(r) r r ∀ | − 0| ν Thus, only (38) can represent a static homogeneous uni- A(D, t) = e , ν = ν(G(D), t) , verse in Einstein’s theory, contrary to the claims of the B(D, t) = eσ, σ = σ(G(D), t) , modern relativists. However, since (38) contains no matter it cannot model the universe other than locally. C(D, t) = eμG2(D), μ = μ(G(D), t) , to get 8 Cosmological models of expansion ds2 = eν dt2 eσdG2 eμG2(D) dθ2 + sin2 θdϕ2 . (42) − − In view of the foregoing it is now evident that the models Now set,
proposed by the relativists purporting an expanding universe r∗ = G(D(r)) , (43) are also untenable in the framework of Einstein’s theory. The line-element obtained by the Abbe´ Lemaˆıtre´ and by to get Robertson, for instance, is inadmissible. Under the false 2 ν 2 σ 2 μ 2 2 2 2 ds = e dt e dr∗ e r∗ dθ + sin θdϕ , (44) assumption that r is a radius in de Sitter’s spherical universe, − − they proposed the following transformation of coordinates on ν = ν(r∗, t) , σ = σ(r∗, t) , μ = μ(r∗, t)
. the metric (32) (with ρ = 0 in the misleading form given 00 One then finds in the usual way that the solution in formula 9), 6 to (44) is, 2 r t 1 r g(t) W ˉ e rˉ= e− , t = t + W ln 1 2 , (39) 2 2 r2 2 − W ds = dt 2 1 W 2   − k 2 × − 1 + 4 r∗ (45) q 2 2 2 2 2 λ + 8πρ dr∗ + r∗
dθ + sin θdϕ , W 2 = 00 , × 3 where k is a constant.
 to get Then by (43) this becomes,

2 2 2tˉ 2 2 2 2 2 2 g(t) ds = dtˉ e W drˉ +r ˉ dθ +r ˉ sin θdϕ , e 2 − ds2 = dt2 dG2 + G2 dθ2 + sin θdϕ2 , − k 2 2 1
1 + 4 G or, by dropping the bar and setting k = W , 
 or,
ds2 = dt2 e2kt dr2 + r2dθ2 + r2 sin2 θdϕ2 . (40) − eg(t) ds2 = dt2 Now, as I have shown, (32) has no solution in
C(D(r)) − k 2 2 × 1 + 4 G (46) in the form (3), so transformations (39) and metric (40) are meaningless concoctions of mathematical symbols. Owing to G02dr2 +
G2 dθ2 + sin2 θdϕ2 , × h
i S. J. Crothers. On the General Solution to Einstein’s Vacuum Field for the Point-Mass when λ = 0 15 6 Volume 3 PROGRESS IN PHYSICS October, 2005

dG having assumed that G(D(r)) r, and erroneously take r G0 = , dr as a radius on the metric (50),≡ valid down to 0. Metric (50) is a meaningless concoction of mathematical symbols. G = G(D(r)) ,D(r) = r r0 , r0 . | − | ∈ < Nevertheless, the relativists transform this meaningless ex- The admissible form of G(D(r)) must now be determ- pression with a meaningless change of “coordinates” to ob- ined. tain the Robertson-Walker line-element, as follows. If G0 0, then B(D, t) = 0 r and t, in violation of Transform (46) by setting, ≡ ∀ ∀ (41). Therefore G0 = 0 r = r0. 6 ∀ 6 G(r) Metric (46) is singular when, Gˉ(ˉr) = . 1 + k G2 k 4 1 + G2(r ) = 0 , 4 0 This carries (46) into,

2 ˉ2 G(r ) = k < 0 . (47) 2 2 g(t) dG 2 2 2 2 ⇒ 0 √ ⇒ ds = dt e +Gˉ dθ + sin θdϕ . (51) k − 1 κGˉ2 − " − # The proper radius on (46) is,
This is easily seen to be
the familiar Robertson-Walker

1 g(t) dG line-element if, following the relativists, one incorrectly as- R (r, t) = e 2 = p k 2 sumes Gˉ rˉ, disregarding the fact that the admissible form 1 + 4 G Z of Gˉ must≡ be ascertained. In any event (51) is meaningless, 1 2 √k owing to the meaninglessness of (50), which I confirm as 2 g(t) = e arctan G(r) + K , follows. √k 2 ! Gˉ0 0 Bˉ = 0 rˉ, in violation of (41). Therefore ˉ ≡ ⇒ ∀ K = const , G0 = 0 rˉ=r ˉ0. 6 Equation∀ 6 (51) is singular when, which must satisfy the condition, 2 1 + ˉ ˉ as r r±,R 0 . 1 kG (ˉr0) = 0 G(ˉr0) = k > 0 . (52) → 0 p → − ⇒ √k ⇒ Therefore, The proper radius on (51) is,

1 2 √k 1 ˉ 2 g(t) g(t) dG Rp(r0, t) = e arctan G(r0) + K = 0 , Rˉp = e 2 √k 2 ! 1 kGˉ2 Z − and so 1 g(t) p1 = e 2 arcsin √kGˉ(ˉr) + K , √  k  1 g(t) 2 √k Rp(r, t) = e 2 arctan G(r) √k 2 − K = const. , h (48) √k which must satisfy the condition, arctan G(r ) . 0 + − 2 as rˉ rˉ±, Rˉ 0 , i → 0 p → Then by (47), so

1 2 √k 1 1 2 g(t) g(t) Rp(r, t)=e arctan G(r) arctan√ 1 , (49) Rˉp(ˉr , t) = 0 = e 2 arcsin √kGˉ(ˉr ) + K . √k 2 − − 0 √k 0 h i   k < 0 . Therefore,

Therefore, there exists no function G(D(r)) rendering a 1 g(t) 1 Rˉp(ˉr, t) = e 2 solution to (46) in the required form of (41). √k × The relativists however, owing to their invalid assum- (53) ptions about the parameter r, write equation (46) as, arcsin √kGˉ(ˉr) arcsin √kGˉ(ˉr ) . × − 0 h i eg(t) ds2 = dt2 dr2+r2 dθ2+ sin2 θdϕ2 , (50) Then − k 2 2 √kGˉ(ˉr ) √kGˉ(ˉr) 1 , 1+ 4 r 0 6 6 

16 S. J. Crothers. On the General Solution to Einstein’s Vacuum Field for the Point-Mass when λ = 0 6 October, 2005 PROGRESS IN PHYSICS Volume 3

or which changes with time. The proper radius is, ˉ ˉ 1 G(ˉr0) 6 G(ˉr) 6 . √k r Rp(r, t) = lim 0 dr 0 , Then by (52), r → ±∞ ≡ rZ 1 1 0 6 Gˉ(ˉr) 6 , √k √k and the volume of the point-space is so 1 r π 2π Gˉ(ˉr) . eg(t) ≡ √k V = lim 0 dr sin θ dθ 0 . r → ±∞ k ≡ Z Z0 Z0 Consequently, Gˉ0(ˉr) = 0 rˉ and t, in violation of r0 (41). Therefore, there exists no∀ function∀Gˉ(Dˉ(ˉr)) to render a solution to (51) in the required form of (41). Metric (55) consists of a single “world line” through the point R (r, t) 0. Furthermore, R (r, t) 0 is a quasi- If the conditions on (41) are relaxed in the fashion of p ≡ p ≡ the relativists, non-Einstein metrics with expanding radii of regular singular point-space since the ratio, curvature are obtained. Nonetheless the associated spaces 1 g(t) 2πe 2 have zero volume. Indeed, equation (40) becomes, . √kRp(r, t) ≡ ∞ λ + 8πρ ds2 = dt2 e2kt 00 dθ2 + sin2 θdϕ2 . (54) Therefore, Rp(r, t) 0 cannot be extended. − 3
≡
It immediately follows that the Friedmann models are This is not an Einstein universe. The radius of curvature all invalid, because the so-called Friedmann equation, with of (54) is, μν its associated equation of continuity, T;μ = 0, is based upon kt λ + 8πρ00 metric (51), which, as I have proven, has no solution in Rc(r, t) = e , r 3 G(r) in the required form of (41). Furthermore, metric (55) which expands or contracts with the sign of the constant k. cannot represent an Einstein universe and therefore has no Even so, the proper radius of the “space” of (54) is, cosmological meaning. Consequently, the Friedmann equa- tion is also nothing more than a meaningless concoction of r mathematical symbols, destitute of any physical significance Rp(r, t) = lim 0 dr 0 . whatsoever. Friedmann incorrectly assumed, just as the rela- r ≡ → ±∞ tivists have done all along, that the parameter r is a radius in rZ 0 the gravitational field. Owing to this erroneous assumption, The volume of this point-space is, his treatment of the metric for the gravitational field violates the inherent geometry of the metric and therefore violates r π 2π the geometrical form of the pseudo-Riemannian spacetime λ + 8πρ00 V = lim e2kt 0 dr sin θ dθ 0 . manifold. The same can be said of Einstein himself, who r → ±∞ 3 ≡
Z Z0 Z0 did not understand the geometry of his own creation, and by r0 making the same mistakes, failed to understand the impli- Metric (54) consists of a single “world line” through cations of his theory. the point Rp(r, t) 0. Furthermore, Rp(r, t) 0 is a quasi- Thus, the Friedmann models are all invalid, as is the ≡ ≡ regular singular point-space since the ratio, Einstein-de Sitter model, and all other general relativistic cosmological models purporting an expansion of the uni- 2πekt λ + 8πρ 00 . verse. Furthermore, there is no general relativistic substan- √p3Rp(r, t) ≡ ∞ tiation of the Big Bang hypothesis. Since the Big Bang hypo- thesis rests solely upon an invalid interpretation of General Therefore, R (r, t) 0 cannot be extended. p ≡ Relativity, it is abject nonsense. The standard interpretations Similarly, equation (51) becomes, of the Hubble-Humason relation and the cosmic microwave background are not consistent with Einstein’s theory. Ein- eg(t) ds2 = dt2 dθ2 + sin2 θdϕ2 , (55) stein’s theory cannot form the basis of a cosmology. − k
which is not an Einstein metric. The radius of curvature of 9 Singular points in Einstein’s universe (55) is, 1 g(t) e 2 It has been pointed out before [7, 8, 3] that singular points Rc(r, t) = , √k in Einstein’s universe are quasiregular. No curvature type

S. J. Crothers. On the General Solution to Einstein’s Vacuum Field for the Point-Mass when λ = 0 17 6 Volume 3 PROGRESS IN PHYSICS October, 2005

singularities arise in Einstein’s universe. The oddity of a point being associated with a non-zero radius of curvature is an inevitable consequence of Einstein’s geometry. There is nothing more pointlike in Einstein’s universe, and nothing more pointlike in the de Sitter point world or the Einstein cylindrical world line. A point as it is usually conceived of in Minkowski space does not exist in Einstein’s universe. The modern relativists have not understood this inescapable fact.

Acknowledgements

I would like to extend my thanks to Dr. D. Rabounski and Dr. L. Borissova for their kind advice as to the clarification of my definitions and my terminology, manifest as section 2 herein.

Dedication

I dedicate this paper to the memory of Dr. Leonard S. Abrams: (27 Nov. 1924 — 28 Dec. 2001).

References

1. Stavroulakis N. On a paper by J. Smoller and B. Temple. Annales de la Fondation Louis de Broglie, 2002, v. 27, 3 (see also in www.geocities.com/theometria/Stavroulakis-1.pdf). 2. Stavroulakis N. On the principles of general relativity and the SΘ(4)-invariant metrics. Proc. 3rd Panhellenic Congr. Geometry, Athens, 1997, 169 (see also in www.geocities.com/ theometria/Stavroulakis-2.pdf). 3. Crothers S. J. On the geometry of the general solution for the vacuum field of the point-mass. Progress in Physics, 2005, v. 2, 3–14. 4. Eddington A. S. The mathematical theory of relativity. Cambridge University Press, Cambridge, 2nd edition, 1960. 5. Petrov A. Z. Einstein spaces. Pergamon Press, London, 1969. 6. Abrams L. S. The total space-time of a point-mass when Λ = 0, and its consequences for the Lake-Roeder black hole. Physica6 A, v. 227, 1996, 131–140 (see also in arXiv: gr-qc/0102053). 7. Brillouin M. The singular points of Einstein’s Universe. Journ. Phys. Radium, 1923, v. 23, 43 (see also in arXiv: physics/ 0002009). 8. Abrams L. S. Black holes: the legacy of Hilbert’s error. Can. J. Phys., 1989, v. 67, 919 (see also in arXiv: gr-qc/0102055).

18 S. J. Crothers. On the General Solution to Einstein’s Vacuum Field for the Point-Mass when λ = 0 6 October, 2005 PROGRESS IN PHYSICS Volume 3

The First Crisis in Cosmology Conference Mon¸cao,˜ Portugal, June 23–25 2005

Hilton Ratcliffe Astronomical Society of Southern Africa E-mail: [email protected]

The author attended the first Crisis in Cosmology Conference of the recently associated Alternative Cosmology Group, and makes an informal report on the proceedings with some detail on selected presentations.

In May 2004, a group of about 30 concerned scientists His paper was on Modified Newtonian Dynamics (MOND), published an open letter to the global scientific community which I had eagerly anticipated and thoroughly appreciated. in New Scientist in which they protested the stranglehold of MOND is a very exciting development in observational ast- Big Bang theory on cosmological research and funding. The ronomy used to make Dark Matter redundant in the explan- letter was placed on the Internet∗ and rapidly attracted wide ation of cosmic gravitational effects like the anomalous rot- attention. It currently has about 300 signatories representing ational speeds of galaxies. Mordehai Milgrom of the Weiz- scientists and researchers of disparate backgrounds, and has mann Institute in Israel first noticed that mass discrepancies led to a loose association now known as the Alternative in stellar systems are detected only when the internal accel-

Cosmology Group†. This writer was one of the early signa- eration of gravity falls below the well-established value a0 = 8 2 tories to the letter, and holding the view that the Big Bang = 1.2×10− cm×s− . The standard Newtonian gravitational explanation of the Universe is scientifically untenable, pa- values fit perfectly above this threshold, and below a0 MOND tently illogical, and without any solid observational support posits a breakdown of Newton’s law. The dependence then whatsoever, became involved in the organisation of an intern- becomes linear with an asymptotic value of acceleration 1/2 ational forum where we could share ideas and plan our way a = (a0 g) , where g is the Newtonian value. Scarpa has forward. That idea became a reality with the staging of the called this the weak gravitational regime, and he and colle- First Crisis in Cosmology Conference (CCC-1) in the lovely, agues Marconi and Gilmozzi have applied it extensively to medieval walled village of Mon¸cao,˜ far northern Portugal, globular clusters with 100% success. What impressed me over 3 days in June of this year. most was that the clear empirical basis of MOND has been It was sponsored in part by the University of Minho in thoroughly tested, and is now in daily use by professional Braga, Portugal, and the Institute for Advanced Studies at astronomers at what is arguably the most sophisticated and Austin, Texas. Professor Jose´ Almeida of the Department advanced optical-infrared observatory in the world. In prac- of Physics at the University of Minho was instrumental tice, there is no need to invoke Dark Matter. Quote from in the organisation and ultimate success of an event that Riccardo: “Dark Matter is the craziest idea we’ve ever had is now to be held annually. The conference was arranged in astronomy. It can appear when you need it, it can do what in 3 sessions. On the first day, papers were presented on you like, be distributed in any way you like. It is the fairy observations that challenge the present model, the second day tale of astronomy”. dealt with conceptual difficulties in the standard model, and Big Bang theory depends critically on three first prin- we concluded with alternative cosmological world-views. ciples: that the Universe is holistically and systematically Since it is not practicable here to review all the papers expanding as per the Friedmann model; that General Relati- presented (some 34 in total, plus 6 posters), I’ll selectively vity correctly describes gravitation; and that Milne’s Cos- confine my comments to those that interested me particularly. mological Principle, which declares that the Universe at The American Institute of Physics will publish the proceed- some arbitrary “large scale” is isotropic and homogeneous, ings of the conference in their entirety in due course for those is true. The falsification of any one of these principles would interested in the detail. lead to the catastrophic failure of the theory. We saw at the First up was professional astronomer Dr. Riccardo Scarpa conference that all three can be successfully challenged on of the European Southern Observatory, Santiago, Chile. His the basis of empirical science. Retired electrical engineer job involves working with the magnificent Very Large Tele- Tom Andrews presented a novel approach to the validation scope array at Paranal, and I guess that makes him the envy (or rather, invalidation) of the expanding Universe model. It of just about every astronomer with blood in his veins! is well known that type 1A supernovae (SNe) show mea-

∗http://www.cosmologystatement.org/ surable anomalous dimming (with distance or remoteness †http://www.cosmology.info/ in time) in a flat expanding Universe model. Andrews used

H. Ratcliffe. The First Crisis in Cosmology Conference. Mon¸cao,˜ Portugal, June 23–25 2005 19 Volume 3 PROGRESS IN PHYSICS October, 2005

observational data from two independent sets of measure- of deuterium to hydrogen near the centre of the Milky Way ments of brightest cluster galaxies (defined as the brightest is 5 orders of magnitude higher than the Standard Model galaxy in a cluster). It was expected, since the light from the predicts, and measuring either for quasars produces deviation SNe and the bright galaxies traverses the same space to get to from predictions. Also problematic for BBN are barium and us, that the latter should also be anomalously dimmed. They beryllium, produced assumedly as secondary products of clearly are not. The orthodox explanation for SNe dimming supernovae by the process of spallation. However, observa- — that it is the result of the progressive expansion of space tions of metal-poor stars show greater abundance of Be than — is thereby refuted. He puts a further nail in the coffin possible by spallation. Van Flandern: “It should be evident to by citing Goldhaber’s study of SNe light curves, which did objective minds that nothing about the Universe interpreted not reveal the second predicted light-broadening effect due to with the Big Bang theory is necessarily right, not even the time dilation. Says Andrews: “The Hubble redshift of Fourier most basic idea in it that the Universe is expanding”. harmonic frequencies [for SNe] is shown to broaden the Problems in describing the geometry of the Universe light curve at the observer by (1 + z). Since this broadening were dealt with by several speakers, and we must here of spreads the total luminosity over a longer time period, the course drill down a bit to where the notion came from (in the apparent luminosity at the observer is decreased by the context of Big Bang theory). The theory originated in Father same factor. This accounts quantitatively for the dimming Georges Lemaˆıtre’s extensions to Friedmann’s solution of of SNe. On the other hand, no anomalous dimming occurs the Einstein General Relativity (GR) field equations, which for galaxies since the luminosity remains constant over time showed that the Universe described in GR could not be static periods much longer than the light travel time to the observer. as Einstein believed. From this starting point emerged some This effect is consistent with the non-expanding Universe irksome dilemmas regarding the fundamental nature of space model. The expanding model is logically falsified”. and the distribution of matter within it. It was here more than Professor Mike Disney of the School of Physics and anywhere that the rich diversity of opinion and approach Astronomy at Cardiff University calls a spade a spade. He within the Alternative Cosmology Group was demonstrated. has created an interesting benchmark for the evaluation of Professor Yurij Baryshev of the Institute of Astronomy at scientific models — he compares the number of free par- St. Petersburg State University quietly presented his argu- ameters in a theory with the number of independent mea- ment against the Cosmological Principle: large-scale struc- surements, and sets an arbitrary minimum of +3 for the ture is not possible in the Friedmann model, yet observation excess of measurements over free parameters to indicate shows it for as far as we can see. I had recently read Yurij’s that the theory is empirically viable. He ran through the book The Discovery of Cosmic Fractals, and knew that he exercise for the Big Bang model, and arrived at a figure of had studied the geometric fractals of Yale’s famous Professor 3 (17 free parameters against 14 measured). He therefore Benoit Mandelbrot, which in turn led to his extrapolation of argued− that the there is little statistical significance in the a fractal (inhomogeneous, anisotropic) non-expanding large- good fits claimed by Big Bang cosmologists since the surfeit scale universe. Baryshev discussed gravitation from the of free parameters can easily mould new data to fit a desired standpoint that the physics of gravity should be the focus of conclusion. Quote: “The study of some 60 cultures, going cosmological research. General Relativity and the Feynman back 12,000 years, shows that, like it or not, we will always field are different at all scales, although to date, all relativistic have a cosmology, and there have always been more free tests cannot distinguish between them. He pointed out that parameters than independent measurements. The best model if one reversed the flow and shrunk the radius, eventually is a compromise between parsimony (Occam’s razor) and the point would be reached where the energy density of the goodness-of-fit”. Universe would exceed the rest mass, and that is logically Disney has a case there, and it is amply illustrated when it impossible. He left us with this gem: Feynman to his wife comes to Big Bang Nucleosynthesis (which depends initially (upon returning from a conference) “Remind me not to attend on an arbitrarily set baryon/photon ratio), and the abundances any more gravity conferences!” of chemical elements. Dr. Tom van Flandern is another Conference co-ordinator Professor Jose´ Almeida present- straight talking, no frills man of science. He opened his ed a well-argued case for an interesting and unusual world- abstract with the words “The Big Bang has never achieved a view: a hyperspherical Universe of 4-D Euclidean space true prediction success where the theory was placed at risk (called 4-Dimensional Optics or 4DO) rather than the stand- of falsification before the results were known”. Ten years ard non-Euclidean Minkowski space. Dr. Franco Selleri of ago, Tom’s web site listed the Top Ten Problems with the the Universita` di Bari in Italy provided an equally interesting Big Bang, and today he has limited it to the Top Fifty. alternative — the certainty that the Universe in which we He pointed out the following contradictions in predicted live and breathe is a construction in simple 3-D Euclidean light element abundances: observed deuterium abundances space precludes the possibility of the Big Bang model. He don’t tie up with observed abundances of 4He and 7Li, and says: “No structure in three dimensional space, born from an attempts to explain this inconsistency have failed. The ratio explosion that occurred 10 to 20 billion years ago, could

20 H. Ratcliffe. The First Crisis in Cosmology Conference. Mon¸cao,˜ Portugal, June 23–25 2005 October, 2005 PROGRESS IN PHYSICS Volume 3

resemble the Universe we observe”. The key to Selleri’s at the conference as I had been keenly following his work theory is absolute simultaneity, obtained by using a term on the Wilkinson Microwave Anisotropy Probe (WMAP) e1 (the coefficient of x in the transformation of time) in data. Dr. Starkman has discovered some unexpected (for Big the Lorentz transformations, so that e1 = 0. Setting e1 = 0 Bangers) characteristics (he describes them as “bizarre”) in separates time and space, and a conception of reality is the data that have serious consequences for the Standard introduced in which no room is left for a fourth dimension. Model. Far from having the smooth, Gaussian distribution Both Big Bang and its progenitor General Relativity depend predicted by Big Bang, the microwave picture has distinct critically on 4-D Minkowski space, so the argument regressed anisotropies, and what’s more says Starkman, they are clearly even further to the viability of Relativity itself. And here is aligned with local astrophysical structures, particularly the where the big guns come in! ecliptic of the Solar System. Once the dipole harmonic is World-renowned mathematical physicist Professor Hu- stripped to remove the effect of the motion of the Solar seyin Yilmaz, formerly of the Institute for Advanced Studies System, the other harmonics, quadrupole, octopole, and so at Princeton University, and his hands-on experimentalist on reveal a distinct alignment with local objects, and show colleague Professor Carrol Alley of the University of Mary- also a preferred direction towards the Virgo supercluster. land, introduced us to the Yilmaz cosmology. Altogether 4 Conference chair, plasma physicist Eric Lerner concurred in papers were presented at CCC-1 on various aspects of Yilmaz his paper. He suggested that the microwave background is theory, and a fifth, by Dr. Hal Puthoff of the Institute for nothing more than a radio fog produced by plasma filaments, Advanced Studies at Austin, was brought to the conference which has reached a natural isotropic thermal equilibrium but not presented. It is no longer controversial to suggest of just under 3K. The radiation is simply starlight that has that GR has flaws, although I still feel awkward saying it been absorbed and re-radiated, and echoes the anisotropies out loud! Professor Yilmaz focussed on the fact that GR of the world around us. These findings correlate with the excludes gravitational stress-energy as a source of curvature. results of a number of other independent studies, including Consequently, stress-energy is merely a coordinate artefact that of Larson and Wandelt at the University of Illinois, in GR, whereas in the Yilmaz modification it is a true tensor. and also of former Cambridge enfant terrible and current Hal Puthoff described the GR term to me as a “pseudo- Imperial College theoretical physics prodigy, Professor Joao˜ tensor, which can appear or disappear depending on how Magueijo. Quote from Starkman: “This suggests that the you treat mass”. The crucial implication of this, in the words reported microwave background fluctuations on large ang- of Professor Alley, is that since “interactions are carried ular scales are not in fact cosmic, with important conseq- by the field stress energy, there are no interactive n-body uences”. Phew! solutions to the field equations of General Relativity”. In The final day saw us discussing viable alternative cos- plain language, GR is a single-body description of gravity! mologies, and here one inevitably leans towards personal The Yilmaz equations contain the correct terms, and they preferences. My own bias is unashamedly towards scientists have been applied with success to various vexing problems, who adopt the classical empirical method, and there is no for example the precession of Mercury’s perihelion, lunar better example of this than Swedish plasma physics pioneer laser ranging measurements, the flying of atomic clocks in and Nobel laureate Hannes Alfven. Consequently, I favoured aircraft, the relativistic behaviour of clocks in the GPS, and the paper on Plasma Cosmology presented by Eric Lerner, the predicted Sagnac effect in the one-way speed of light and as a direct result of that inclination find it very difficult on a rotating table. Anecdote from Professor Alley: at a here to be brief! Lerner summarised the basic premises: most lecture by Einstein in the 1920’s, Professor Sagnac was of the Universe is plasma, so the effect of electromagnetic in the audience. He questioned Einstein on the gedanken force on a cosmic scale is at least comparable to gravitation. experiment regarding contra-radiating light on a rotating Plasma cosmology assumes no origin in time for the Uni- plate. Einstein thought for a while and said, “That has got verse, and can therefore accommodate the conservation of nothing to do with relativity”. Sagnac loudly replied, “In energy/matter. Since we see evidence of evolution all around that case, Dr. Einstein, relativity has got nothing to do with us, we can assume evolution in the Universe, though not at reality!” the pace or on the scale of the Big Bang. Lastly, plasma The great observational “proof” of Big Bang theory is cosmology tries to explain as much of the Universe as pos- undoubtedly the grandly titled Cosmic Microwave Back- sible using known physics, and does not invoke assistance ground Radiation, stumbled upon by radio engineers Penzias from supernatural elements. Plasmas are scale invariant, so and Wilson in 1965, hijacked by Princeton cosmologist Jim we can safely infer large-scale plasma activity from what we Peebles, and demurely described by UC’s COBE data anal- see terrestrially. Gravity acts on filaments, which condense yser Dr. George Smoot as “like looking at the fingerprint of into “blobs” and disks form. As the body contracts, it gets rid God”. Well, it’s come back to haunt them! I was delighted of angular momentum which is conducted away by plasma. that despite some difficulties Glenn Starkman of Case West- Lerner’s colleague Anthony Peratt of Los Alamos Laboratory ern Reserve University was able to get his paper presented modelled plasma interaction on a computer and has arrived

H. Ratcliffe. The First Crisis in Cosmology Conference. Mon¸cao,˜ Portugal, June 23–25 2005 21 Volume 3 PROGRESS IN PHYSICS October, 2005

The First Crisis in Cosmology Conference, Mon¸cao,˜ Portugal, June 23–25 2005 Schedule of Presentations

Name Location Paper Title

Antonio Alfonso-Faus Madrid Polytech. Univ., Mass boom vs Big Bang [email protected] Spain Carrol Alley Univ. of Maryland, Going “beyond Einstein” with Yilmaz theory [email protected] USA Jose´ Almeida Universidade do Minho Geometric drive of Universal expansion [email protected] Thomas Andrews USA Falsification of the expanding Universe model [email protected] Yurij Baryshev St. Petersburg Univ., Conceptual problems of the standard cosmological model [email protected] Russia Yurij Baryshev St. Petersburg Univ., Physics of gravitational interaction [email protected] Russia Alain Blanchard Lab. d’Astrophys. The Big Bang picture: a wonderful success of modern science [email protected] Toulouse, France M. de Campos Univ. Federal de The Dyer-Roeder relation [email protected] Roraima, Brazil George Chapline Lawrence Livermore Tommy Gold revisited [email protected] National Lab., USA Mike Disney Univ. of Cardiff, The insignificance of current cosmology [email protected] Great Britain Anne M. Hofmeister and Washington Univ., USA Implications of thermodynamics on cosmologic models R. E. Criss [email protected] Michael Ibison Inst. for Adv. Studies, The Yilmaz cosmology [email protected] Austin, USA Michael Ibison Inst. for Adv. Studies, The steady-state cosmology [email protected] Austin, USA Michael Ivanov Belarus State Univ., Low-energy quantum gravity [email protected] Belarus Moncy John St. Thomas College, Decelerating past for the Universe? [email protected] India

Christian Joos and Josef Lutz Univ. of Gottingen;¨ [email protected]; Chemnitz Univ., Quantum redshift [email protected] Germany

Christian Joos and Josef Lutz Univ. of Gottingen;¨ [email protected]; Chemnitz Univ., Evolution of Universe in high-energy physics [email protected] Germany S. P. Leaning High redshift Supernovae data show no time dilation Eric Lerner Lawrenceville Plasma Is the Universe expanding? Some tests of physical geometry [email protected] Physics, USA

22 H. Ratcliffe. The First Crisis in Cosmology Conference. Mon¸cao,˜ Portugal, June 23–25 2005 October, 2005 PROGRESS IN PHYSICS Volume 3

The First Crisis in Cosmology Conference, Mon¸cao,˜ Portugal, June 23–25 2005 Schedule of Presentations (continue)

Eric Lerner Lawrenceville Plasma Overview of plasma cosmology [email protected] Physics, USA Sergey Levshakov Ioffe Phys. Tech. Inst., The cosmological variability of the fine-structure constant [email protected] St. Petersburg, Russia Martin Lopez-Corredoira´ Inst. de Astrofisica de Research on non-cosmological redshifts [email protected] Canarias, Spain Oliver Manuel University of Missouri, Isotopes tell Sun’s origin and operation [email protected] USA Jaques Moret-Bailly France Parametric light-matter interactions [email protected] Frank Potter and Howard Preston Univ. of California; Large-scale gravitational quantisation states [email protected] Preston Research, USA Eugene Savov Bulgarian Acad. Unique firework Universe and 3-D spiral code [email protected] of Sciences Riccardo Scarpa European Southern Modified Newtonian Dynamics: alternative to non-baryonic dark matter [email protected] Observatory, Chile Riccardo Scarpa, Gianni Marconi, and Roberto Gilmozzi European Southern Using globular clusters to test gravity [email protected]; [email protected]; Observatory, Chile [email protected] Donald Scott USA Real properties of magnetism and plasma [email protected] Franco Selleri Universita` di Bari, Italy Absolute simultaneity forbids Big Bang [email protected] Glenn Starkman Case Western Reserve Is the low-lambda microwave background cosmic? [email protected] Univ., USA Glenn Starkman Case Western Reserve Differentiating between modified gravity and dark energy [email protected] Univ., USA Tuomo Suntola Finland Spherically closed dynamic space [email protected]

Francesco Sylos Labini E. Fermi Centre, Italy Non-linear structures in gravitation and cosmology Y.P.Varshni Univ. of Ottawa, Common absorption lines in two quasars [email protected] Canada Y.P.Varshni, J. Talbot and Z. Ma Univ. of Ottawa; Chin. Peaks in emission lines in the spectra of quasars [email protected] Acad. of Sci. (China) Thomas van Flandern Meta Research, USA Top problems with Big Bang: the light elements [email protected] Mogens Wegener University of Aarhus, Kinematic cosmology [email protected] Denmark

Huseyin Yilmaz Princeton Univ., USA Beyond Einstein

H. Ratcliffe. The First Crisis in Cosmology Conference. Mon¸cao,˜ Portugal, June 23–25 2005 23 Volume 3 PROGRESS IN PHYSICS October, 2005

at a compelling simulation of the morphogenesis of galaxies. mentioned above. Data freely available from NASA’s SOHO Since plasma cosmology has no time constraints, the dev- and TRACE satellites graphically and unambiguously sup- elopment of large-scale structures — so problematic for Big port Manuel’s contentions (to the extent of images illustrating Bang — is accommodated. Lerner admits that there’s still a fixed surface formations revolving with a period of27.3 lot of work to be done, but with the prospect of more research days), and suggest that the standard Solar Model is grossly funding coming our way, he foresees the tidying up of the inaccurate. The implications, if Manuel’s ideas are validated, theory into a workable cosmological model. are exciting indeed. His words: “The question is, are neutron Dr. Alain Blanchard of the Laboratoire d’Astrophysique stars ‘dead’ nuclear matter, with tightly bound neutrons in Toulouse had come to CCC-1 explicitly to defend Big at minus 93 MeV relative to the free neutron, as widely Bang, and he did so admirably. My fears that the inclusion believed? Or are neutron stars the greatest known source of of a single speaker against the motion might amount to mere nuclear energy, with neutrons at plus 10 to 22 MeV relative tokenism were entirely unfounded. Despite the fact that many to free neutrons, as we conclude from the properties of the of us disagreed with much of what he said, he acquitted 2,850 known isotopes?” himself most competently and I would say ended up making The conference concluded with a stirring concert by a a number of good friends at the conference. Two quotes 3-piece baroque chamber music ensemble, and it gave me from Dr. Blanchard: “We are all scientists, and we all want cause to reflect that it appeared that only in our appreciation to progress. Where we differ is in our own prejudice.” “When of music did we find undiluted harmony. That the Big Bang you do an experiment, you can get a ‘yes’ or ‘no’ answer from theory will pass into history as an artefact of man’s obsession your equipment. When you work with astrophysical data, with dogma is a certainty; it will do so on its own merits, you are dealing with an altogether more complex situation, however, because it stands on feet of clay. For a viable infused with unknowns.” replacement theory to emerge solely from the efforts of No account of CCC-1 would be near complete without the Alternative Cosmology Group is unlikely unless the a summary of a paper that caught all of us by complete group can soon find cohesive direction, and put into practice surprise. Professor Oliver Manuel is not an astronomer. Nor the undertaking that we become completely interdisciplinary indeed is he a physicist. He is a nuclear chemist, chairman of in our approach. Nonetheless, that there is a crisis in the the Department of Chemistry at the University of Missouri, world of science is now confirmed. Papers presented at the and held in high enough esteem to be one of a handful of conference by some of the world’s leading scientists showed scientists entrusted with the job of analysing Moon rock beyond doubt that the weight of scientific evidence clearly brought back by the Apollo missions. His “telescope” is indicates that the dominant theory on the origin and destiny a mass spectrometer, and he uses it to identify and track of the Universe is deeply flawed. The implications of this isotopes in the terrestrial neighbourhood. His conclusions damning consensus are serious indeed, and will in time are astonishing, yet I can find no fault with his arguments. fundamentally affect not only the direction of many scientific The hard facts that emerge from Professor Manuel’s study disciplines, but also threaten to change the very way that we indicate that the chemical composition of the Sun beneath do science. the photosphere is predominantly iron! Manuel’s thesis has passed peer review in several mainstream journals, including Nature, Science, and the Journal of Nuclear Fusion. He derives a completely revolutionary Solar Model, one which spells big trouble for BBN. Subsequent investigation has shown that it is likely to represent a major paradigm shift in solar physics, and has implications also for the field of nuclear chemistry. He makes the following claims: 1. The chemical composition of the Sun is predominantly iron. 2. The energy of the Sun is not derived from nuclear fusion, but rather from neutron repulsion. 3. The Sun has a solid, electrically conducting ferrite surface beneath the photosphere, and rotates uniformly at all latitudes. 4. The solar system originated from a supernova about 5 billion years ago, and the Sun formed from the neutron star that remained. Manuel’s study contains much more than the sample points

24 H. Ratcliffe. The First Crisis in Cosmology Conference. Mon¸cao,˜ Portugal, June 23–25 2005 October, 2005 PROGRESS IN PHYSICS Volume 3

The Michelson and Morley 1887 Experiment and the Discovery of Absolute Motion

Reginald T. Cahill School of Chemistry, Physics and Earth Sciences, Flinders University, Adelaide 5001, Australia E-mail: [email protected]

Physics textbooks assert that in the famous interferometer 1887 experiment to detect absolute motion Michelson and Morley saw no rotation-induced fringe shifts — the signature of absolute motion; it was a null experiment. However this is incorrect. Their published data revealed to them the expected fringe shifts, but that data gave a speed of some 8 km/s using a Newtonian theory for the calibration of the interferometer, and so was rejected by them solely because it was less than the 30 km/s orbital speed of the Earth. A 2002 post relativistic-effects analysis for the operation of the device however gives a different calibration leading to a speed > 300 km/s. So this experiment detected both absolute motion and the breakdown of Newtonian physics. So far another six experiments have confirmed this first detection of absolute motion in1887.

1 Introduction fects; indeed the evidence is that absolute motion is the cause of these relativistic effects, a proposal that goes back The first detection of absolute motion, that is motion relative to Lorentz in the 19th century. Then of course one must to space itself, was actually by Michelson and Morley in use a relativistic theory for the operation of the Michelson 1887 [1]. However they totally bungled the reporting of their interferometer. What also follows from these experiments is own data, an achievement that Michelson managed again that the Einstein-Minkowski spacetime ontology is invalid- and again throughout his life-long search for experimental ated, and in particular that Einstein’s postulates regarding the evidence of absolute motion. invariant speed of light have always been in disagreement The Michelson interferometer was a brilliantly conceived with experiment from the beginning. This does not imply instrument for the detection of absolute motion, but only in that the use of a mathematical spacetime is not permitted; 2002 [2] was its principle of operation finally understood and in quantum field theory the mathematical spacetime encodes used to analyse, for the first time ever, the data from the 1887 absolute motion effects. An ongoing confusion in physics is experiment, despite the enormous impact of that experiment that absolute motion is incompatible with Lorentz symmetry, on the foundations of physics, particularly as they were laid when the evidence is that it is the cause of that dynamical down by Einstein. So great was Einstein’s influence that the symmetry. 1887 data was never re-analysed post-1905 using a proper relativistic-effects based theory for the interferometer. For 2 Michelson interferometer that reason modern-day vacuum Michelson interferometer experiments, as for example in [3], are badly conceived, The Michelson interferometer compares the change in the and their null results continue to cause much confusion: difference between travel times, when the device is rotated, only a Michelson interferometer in gas-mode can detect for two coherent beams of light that travel in orthogonal absolute motion, as we now see. So as better and better directions between mirrors; the changing time difference vacuum interferometers were developed over the last 70 being indicated by the shift of the interference fringes during years the rotation-induced fringe shift signature of absolute the rotation. This effect is caused by the absolute motion motion became smaller and smaller. But what went unnoticed of the device through 3-space with speed v, and that the until 2002 was that the gas in the interferometer was a key speed of light is relative to that 3-space, and not relative to component of this instrument when used as an “absolute the apparatus/observer. However to detect the speed of the motion detector”, and over time the experimental physicists apparatus through that 3-space gas must be present in the were using instruments with less and less sensitivity; and light paths for purely technical reasons. A theory is required in recent years they had finally perfected a totally dud in- to calibrate this device, and it turns out that the calibration strument. Reports from such experiments claim that absolute of gas-mode Michelson interferometers was only worked out motion is not observable, as Einstein had postulated, despite in 2002. The post relativistic-effects theory for this device is the fact that the apparatus is totally insensitive to absolute remarkably simple. The Fitzgerald-Lorentz contraction effect motion. It must be emphasised that absolute motion is not causes the arm AB parallel to the absolute velocity to be inconsistent with the various well-established relativistic ef- physically contracted to length

R. T. Cahill. The Michelson and Morley 1887 Experiment and the Discovery of Absolute Motion 25 Volume 3 PROGRESS IN PHYSICS October, 2005

v2 C C (1) L = L 1 2 . || r − c v The time tAB to travel AB is set by V tAB = L +vtAB, L || (a) (b) while for BA by V t = L vt , where V = c/n is the BA BA α speed of light, with n the refractive|| − index of the gas present (we ignore here the Fresnel drag effect for simplicity — an A BL A1 A2 B effect caused by the gas also being in absolute motion). For D D the total ABA travel time we then obtain Fig. 1: Schematic diagrams of the Michelson Interferometer, with beamsplitter/mirror at A and mirrors at B and C on arms from A, 2LV v2 with the arms of equal length L when at rest. D is the detector t = t + t = 1 . (2) ABA AB BA V 2 v2 − c2 screen. In (a) the interferometer is at rest in space. In (b) the − r interferometer is moving with speed v relative to space in the For travel in the AC direction we have, from the Pytha- direction indicated. Interference fringes are observed at D. If the goras theorem for the right-angled triangle in Fig. 1 that interferometer is rotated in the plane through 90◦, the roles of arms 2 2 2 (V tAC ) = L + (vtAC ) and that tCA = tAC . Then for the AC and AB are interchanged, and during the rotation shifts of total ACA travel time the fringes are seen in the case of absolute motion, but only if the apparatus operates in a gas. By measuring fringe shifts the speed v 2L may be determined. tACA = tAC + tCA = . (3) √V 2 v2 − Then fringe shifts from six (Michelson and Morley) or twenty Then the difference in travel time is (Miller) successive rotations are averaged, and the average (n2 1)L v2 v4 sidereal time noted, giving in the case of Michelson and Δt = − + O . (4) Morley the data in Fig. 2, or the Miller data like that in c c2 c4   Fig. 4. The form in (5) is then fitted to such data, by varying after expanding in powers of v/c (here the sign O means for the parameters vP and ψ. However Michelson and Morley “order”). This clearly shows that the interferometer can only implicitly assumed the Newtonian value k = 1, while Miller operate as a detector of absolute motion when not in vacuum used an indirect method to estimate the value of k, as he (n = 1), namely when the light passes through a gas, as in understood that the Newtonian theory was invalid, but had the early experiments (in transparent solids a more complex no other theory for the interferometer. Of course the Einstein phenomenon occurs and rotation-induced fringe shifts from postulates have that absolute motion has no meaning, and so absolute motion do not occur). A more general analysis effectively demands that k = 0. Using k = 1 gives only a [2, 9, 10], including Fresnel drag, gives nominal value for vP , being some 8 km/s for the Michelson and Morley experiment, and some 10 km/s from Miller; the Lv2 Δt = k2 P cos [2(θ ψ)] , (5) difference arising from the different latitude of Cleveland c3 − and Mt. Wilson. The relativistic theory for the calibration of where k2 n(n2 1), while neglect of the Fitzgerald- gas-mode interferometers was first used in 2002 [2]. Lorentz contraction≈ − effect gives k2 n3 1 for gases, ≈ ≈ which is essentially the Newtonian calibration that Michelson 3 Michelson-Morley data used. All the rotation-induced fringe shift data from the 1887 Michelson-Morley experiment, as tabulated in [1], is shown Fig.2 shows all the Michelson and Morley air-mode inter- in Fig. 2. The existence of this data continues to be denied ferometer fringe shift data, based upon a total of only 36 by the world of physics. rotations in July 1887, revealing the nominal speed of some The interferometers are operated with the arms horizont- 8 km/s when analysed using the prevailing but incorrect al, as shown by Miller’s interferometer in Fig. 3. Then in (5) Newtonian theory which has k = 1 in (5); and this value was θ is the azimuth of one arm (relative to the local meridian), known to Michelson and Morley. Including the Fitzgerald- while ψ is the azimuth of the absolute motion velocity Lorentz dynamical contraction effect as well as the effect of projected onto the plane of the interferometer, with projected the gas present as in (5) we find that nair = 1.00029 gives 2 component vP . Here the Fitzgerald-Lorentz contraction is a k = 0.00058 for air, which explains why the observed fringe real dynamical effect of absolute motion, unlike the Einstein shifts were so small. We then obtain the speeds shown in spacetime view that it is merely a spacetime perspective Fig. 2. In some cases the data does not have the expected form artefact, and whose magnitude depends on the choice of in (5); because the device was being operated at almost the observer. The instrument is operated by rotating at a rate of limit of sensitivity. The remaining fits give a speed in excess one rotation over several minutes, and observing the shift in of 300 km/s. The often-repeated statement that Michelson the fringe pattern through a telescope during the rotation. and Morley did not see any rotation-induced fringe shifts

26 R. T. Cahill. The Michelson and Morley 1887 Experiment and the Discovery of Absolute Motion October, 2005 PROGRESS IN PHYSICS Volume 3

7 hr July 8 13 hr July 8 0.04 181 km per sec 0.04 349 km per sec 0.02 0.02 0 0 -0.02 -0.02 -0.04 -0.04 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9

7 hr July 8 13 hr July 8 0.04 156 km per sec 0.04 313 km per sec 0.02 0.02 0 0 -0.02 -0.02 -0.04 -0.04 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 Fig. 3: Miller’s interferometer with an effective arm length of 7 hr July 9 13 hr July 09 L = 32 m achieved by multiple reflections. Used by Miller on 0.04 125 km per sec 0.04 275 km per sec Mt.Wilson to perform the 1925–1926 observations of absolute 0.02 0.02 motion. The steel arms weighed 1200 kilograms and floated in 0 0 a tank of 275 kilograms of Mercury. From Case Western Reserve -0.02 -0.02 University Archives. -0.04 -0.04 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 is completely wrong; all physicists should read their paper 7 hr July 9 13 hr July 09 [1] for a re-education, and indeed their paper has a table 0.04 343 km per sec 0.04 147 km per sec of the observed fringe shifts. To get the Michelson-Morley 0.02 0.02 Newtonian based value of some 8 km/s we must multiply 0 0 -0.02 -0.02 the above speeds by k = √0.00058 = 0.0241. They rejected -0.04 -0.04 their own data on the sole but spurious ground that the value 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 of 8 km/s was smaller than the speed of the Earth about the

7 hr July 11 13 hr July 12 Sun of 30 km/s. What their result really showed was that 0.04 355 km per sec 0.04 144 km per sec (i) absolute motion had been detected because fringe shifts 0.02 0.02 of the correct form, as in (5), had been detected, and (ii) 0 0 that the theory giving k2 = 1 was wrong, that Newtonian -0.02 -0.02 -0.04 -0.04 physics had failed. Michelson and Morley in 1887 should 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 have announced that the speed of light did depend of the direction of travel, that the speed was relative to an actual 7 hr July 11 13 hr July 12 0.04 296 km per sec 0.04 374 km per sec physical 3-space. However contrary to their own data they 0.02 0.02 concluded that absolute motion had not been detected. This 0 0 bungle has had enormous implications for fundamental the- -0.02 -0.02 ories of space and time over the last 100 years, and the -0.04 -0.04 resulting confusion is only now being finally corrected. 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9

4 Miller interferometer Fig. 2: Shows all the Michelson-Morley 1887 data after removal of the temperature induced linear fringe drifts. The data for each 360◦ It was Miller [4] who saw the flaw in the 1887 paper and full turn (the average of 6 individual turns) is divided into the 1st and realised that the theory for the Michelson interferometer must 2nd 180◦ parts and plotted one above the other. The dotted curve be wrong. To avoid using that theory Miller introduced the shows a best fit to the data using (5), while the full curves showthe scaling factor k, even though he had no theory for its value. expected forms using the Miller direction for v and the location and He then used the effect of the changing vector addition of times of the Michelson-Morley observations in Cleveland, Ohio in the Earth’s orbital velocity and the absolute galactic velocity July, 1887. While the amplitudes are in agreement in general with the Miller based predictions, the phase varies somewhat. Miller also of the solar system to determine the numerical value of k, saw a similar effect. This may be related to the Hick’s effect [4] because the orbital motion modulated the data, as shown in when, necessarily, the mirrors are not orthogonal, or may correspond Fig. 5. By making some 12,000 rotations of the interferometer to a genuine fluctuation in the direction of v associated with wave at Mt. Wilson in 1925/26 Miller determined the first estimate effects. We see that this data corresponds to a speed in excess of for k and for the absolute linear velocity of the solar system. 300 km/s, and not the 8 km/s reported in [1], which was based on Fig. 4 shows typical data from averaging the fringe shifts using Newtonian physics to calibrate the interferometer. from 20 rotations of the Miller interferometer, performed over a short period of time, and clearly shows the expected

R. T. Cahill. The Michelson and Morley 1887 Experiment and the Discovery of Absolute Motion 27 Volume 3 PROGRESS IN PHYSICS October, 2005

0.1 40 April 40 August 0.075 20 20 0.05 0 0 - 20 - 20 0.025 - 40 - 40 0 0 5 10 15 20 25 0 5 10 15 20 - 0.025 40 September 40 February - 0.05 20 20 - 0.075 0 0 - 20 - 20 - 40 - 40 0 50 100 150 200 250 300 350 arm azimuth 0 5 10 15 20 0 5 10 15 20

Fig. 4: Typical Miller rotation-induced fringe shifts from average Fig. 5: Miller azimuths ψ, measured from south and plotted aga- of 20 rotations, measured every 22.5◦, in fractions of a wavelength inst sidereal time in hrs, showing both data and best fit of theory hr Δλ/λ, vs azimuth θ (deg), measured clockwise from North, from giving v = 433 km/s in the direction (α = 5.2 , δ = 67◦), using − Cleveland Sept. 29, 1929 16:24 UT; 11:29 average sidereal time. n = 1.000226 appropriate for the altitude of Mt. Wilson. The This shows the quality of the fringe data that Miller obtained, and variation form month to month arises from the orbital motion of is considerably better than the comparable data by Michelson and the Earth about the Sun: in different months the vector sum of the Morley in Fig. 2. The curve is the best fit using the form in (5) galactic velocity of the solar system with the orbital velocity and but including a Hick’s [4] cos (θ β) component that is required sun in-flow velocity is different. As shown in Fig. 6 DeWitte using − when the mirrors are not orthogonal, and gives ψ = 158◦, or 22◦ a completely different experiment detected the same direction and measured from South, and a projected speed of vP = 351 km/s. This speed. value for v is different from that in Fig. 2 because of the difference in latitude of Cleveland and Mt. Wilson. This process was repeated by galactic gravitational in-flows∗. some 12,000 times over days and months throughout 1925/1926 Two old interferometer experiments, by Illingworth [5] giving, in part, the data in Fig. 5. and Joos [6], used helium, enabling the refractive index effect to be recently confirmed, because for helium, with n = form in (5) (only a linear drift caused by temperature effects =1.000036, we find that k2 = 0.00007. Until the refractive on the arm lengths has been removed — an effect also index effect was taken into account the data from the helium- removed by Michelson and Morley and also by Miller). In mode experiments appeared to be inconsistent with the data Fig. 4 the fringe shifts during rotation are given as fractions from the air-mode experiments; now they are seen to be of a wavelength, Δλ/λ = Δt/T , where Δt is given by (5) consistent. Ironically helium was introduced in place of air and T is the period of the light. Such rotation-induced fringe to reduce any possible unwanted effects of a gas, but we shifts clearly show that the speed of light is different in now understand the essential role of the gas. The data from different directions. The claim that Michelson interferome- an interferometer experiment by Jaseja et al [7], using two ters, operating in gas-mode, do not produce fringe shifts orthogonal masers with a He-Ne gas mixture, also indicates under rotation is clearly incorrect. But it is that claim that that they detected absolute motion, but were not aware of lead to the continuing belief, within physics, that absolute that as they used the incorrect Newtonian theory and so motion had never been detected, and that the speed of light considered the fringe shifts to be too small to be real, re- is invariant. The value of ψ from such rotations together miniscent of the same mistake by Michelson and Morley. lead to plots like those in Fig. 5, which show ψ from the The Michelson interferometer is a 2nd order device, as the 1925/1926 Miller [4] interferometer data for four different effect of absolute motion is proportional to (v/c)2, as in (5). months of the year, from which the RA = 5.2 hr is readily apparent. While the orbital motion of the Earth about the Sun 5 1st order experiments slightly affects the RA in each month, and Miller used this effect do determine the value of k, the new theory of gravity However much more sensitive 1st order experiments are required a reanalysis of the data [9, 11], revealing that the also possible. Ideally they simply measure the change in solar system has a large observed galactic velocity of some the one-way EM travel-time as the direction of propagation 420 30 km/s in the direction (RA = 5.2 hr, Dec = 67 deg). is changed. Fig. 6 shows the North-South orientated coaxial This± is different from the speed of 369 km/s in the− direction cable Radio Frequency (RF) travel time variations measured (RA = 11.20 hr, Dec = 7.22 deg) extracted from the Cosmic by DeWitte in Brussels in 1991 [9, 10, 11], which gives the Microwave Background− (CMB) anisotropy, and which de- same RA of absolute motion as found by Miller. That ex- scribes a motion relative to the distant universe, but not ∗See online papers http://www.mountainman.com.au/process_physics/ relative to the local 3-space. The Miller velocity is explained http://www.scieng.flinders.edu.au/cpes/people/cahill_r/processphysics.html

28 R. T. Cahill. The Michelson and Morley 1887 Experiment and the Discovery of Absolute Motion October, 2005 PROGRESS IN PHYSICS Volume 3

20 see footnote on page 28. In order to check Lorentz symmetry 15 we can use vacuum-mode resonant-cavity interferometers,

10 but using gas within the resonant-cavities would enable these devices to detect absolute motion with great precision. 5

ns 0 6 Conclusions - 5

- 10 So absolute motion was first detected in 1887, and again in at least another six experiments over the last 100 years. - 15 Had Michelson and Morley been as astute as their younger colleague Miller, and had been more careful in reporting their 0 10 20 30 40 50 60 70 Sidereal Time non-null data, the history of physics over the last 100 years would have totally different, and the spacetime ontology Fig. 6: Variations in twice the one-way travel time, in ns, for would never have been introduced. That ontology was only an RF signal to travel 1.5 km through a coaxial cable between mandated by the mistaken belief that absolute motion had Rue du Marais and Rue de la Paille, Brussels. An offset has been used such that the average is zero. The cable has a North-South not been detected. By the time Miller had sorted out that orientation, and the data is the difference of the travel times for bungle, the world of physics had adopted the spacetime NS and SN propagation. The sidereal time for maximum effect ontology as a model of reality because that model appeared to of 5 hr and 17 hr (indicated by vertical lines) agrees with the be confirmed by many relativistic phenomena, mainly from direction∼ found∼ by Miller. Plot shows data over 3 sidereal days and particle physics, although these phenomena could equally is plotted against sidereal time. DeWitte recorded such data from well have been understood using the Lorentzian interpreta- 178 days, and confirmed that the effect tracked sidereal time, and tion which involved no spacetime. We should now under- not solar time. Miller also confirmed this sidereal time tracking. stand that in quantum field theory a mathematical spacetime The fluctuations are evidence of turbulence in the flow. encodes absolute motion effects upon the elementary particle systems, but that there exists a physically observable foliation periment showed that RF waves travel at speeds determ- of that spacetime into a geometrical model of time and a ined by the orientation of the cable relative to the Miller separate geometrical model of 3-space. direction. That these very different experiments show the same speed and RA of absolute motion is one of the most References startling discoveries of the twentieth century. Torr and Kolen [8] using an East-West orientated nitrogen gas-filled coaxial 1. Michelson A. A. and Morley A. A. Philos. Mag., S. 5, 1887, cable also detected absolute motion. It should be noted that v. 24, No. 151, 449–463. analogous optical fibre experiments give null results for 2. Cahill R. T. and Kitto K. Michelson-Morley experiments the same reason, apparently, that transparent solids in a revisited and the cosmic background radiation preferred frame. Michelson interferometer also give null results, and so be- Apeiron, 2003, v. 10, No. 2, 104–117. have differently to coaxial cables. 3. Muller¨ H. et al. Modern Michelson-Morley experiment using Modern resonant-cavity interferometer experiments, for cryogenic optical resonators. Phys. Rev. Lett., 2003, v. 91(2), which the analysis leading to (5) is applicable, use vacuum 020401-1. with n = 1, and then k = 0, predicting no rotation-induced 4. Miller D. C. Rev. Mod. Phys., 1933, v. 5, 203–242. fringe shifts. In analysing the data from these experiments the 5. Illingworth K. K. Phys. Rev., 1927, v. 3, 692–696. consequent null effect is misinterpreted, as in [3], to imply 6. Joos G. Annalen der Physik, 1930, Bd. 7, 385. the absence of absolute motion. But it is absolute motion which causes the dynamical effects of length contractions, 7. Jaseja T. S. et al. Phys. Rev. A, 1964, v. 133, 1221. time dilations and other relativistic effects, in accord with 8. Torr D. G. and Kolen P. Precision Measurements and Fundam- Lorentzian interpretation of relativistic effects. The detection ental Constants, ed. by Taylor B. N. and Phillips W. D. Nat. of absolute motion is not incompatible with Lorentz sym- Bur. Stand. (U.S.), Spec. Pub., 1984, v. 617, 675. metry; the contrary belief was postulated by Einstein, and 9. Cahill R.T. Relativity, Gravitation, Cosmology, Nova Science has persisted for over 100 years, since 1905. So far the Pub., NY, 2004, 168–226. experimental evidence is that absolute motion and Lorentz 10. Cahill R. T. Absolute motion and gravitational effects. Apeiron, symmetry are real and valid phenomena; absolute motion is 2004, v. 11, No. 1, 53–111. motion presumably relative to some substructure to space, 11. Cahill R. T. Process Physics: from information theory to whereas Lorentz symmetry parameterises dynamical effects quantum space and matter. Nova Science Pub., NY, 2005. caused by the motion of systems through that substructure. There are novel wave phenomena that could also be studied;

R. T. Cahill. The Michelson and Morley 1887 Experiment and the Discovery of Absolute Motion 29 Volume 3 PROGRESS IN PHYSICS October, 2005

Novel Gravity Probe B Frame-Dragging Effect

Reginald T. Cahill School of Chemistry, Physics and Earth Sciences, Flinders University, Adelaide 5001, Australia E-mail: [email protected]

The Gravity Probe B (GP-B) satellite experiment will measure the precession of on- board gyroscopes to extraordinary accuracy. Such precessions are predicted by General Relativity (GR), and one component of this precession is the “frame-dragging” or Lense-Thirring effect, which is caused by the rotation of the Earth. A new theory of gravity, which passes the same extant tests of GR, predicts, however, a second and much larger “frame-dragging” precession. The magnitude and signature of this larger effect is given for comparison with the GP-B data.

1 Introduction celeration vector field g(r, t), and in differential form

The Gravity Probe B (GP-B) satellite experiment was launch- g = 4πGρ , (2) ∇∙ − ed in April 2004. It has the capacity to measure the precession of four on-board gyroscopes to unprecedented accuracy where ρ(r, t) is the matter density. However there is an [1, 2, 3, 4]. Such a precession is predicted by the Einstein alternative formulation [5] in terms of a vector “flow” field theory of gravity, General Relativity (GR), with two com- v(r, t) determined by ponents (i) a geodetic precession, and (ii) a “frame-dragging” ∂ precession known as the Lense-Thirring effect. The latter is ( v) + (v ) v = 4πGρ , (3) ∂t ∇∙ ∇∙ ∙∇ − particularly interesting effect induced by the rotation of the   Earth, and described in GR in terms of a “gravitomagnetic” with g now given by the Euler “fluid” acceleration field. According to GR this smaller effect will give apre- ∂v dv cession of 0.042 arcsec per year for the GP-B gyroscopes. g = + (v ) v = . (4) However a recently developed theory gives a different ac- ∂t ∙∇ dt count of gravity. While agreeing with GR for all the standard Trivially this g also satisfies (2). External to a spherical tests of GR this theory gives a dynamical account of the so- mass M of radius R a velocity field solution of (2) is called “dark matter” effect in spiral galaxies. It also success- fully predicts the masses of the black holes found in the 2GM globular clusters M15 and G1. Here we show that GR and the v(r) = ˆr , r > R , (5) −r r new theory make very different predictions for the “frame- dragging” effect, and so the GP-B experiment will be able which gives from (4) the usual inverse square law g field to decisively test both theories. While predicting the same GM earth-rotation induced precession, the new theory has an g(r) = ˆr , r > R . (6) − r2 additional much larger “frame-dragging” effect caused by the observed translational motion of the Earth. As well the However the flow equation (2) is not uniquely determined new theory explains the “frame-dragging” effect in terms of by Kepler’s laws because vorticity in a “substratum flow”. Herein the magnitude and ∂ signature of this new component of the gyroscope precession ( v) + (v )v + C(v) = 4πGρ , (7) is predicted for comparison with data from GP-B when it ∂t ∇∙ ∇∙ ∙∇ − becomes available. where   α C(v) = (trD)2 tr(D2) , (8) 8 − 2 Theories of gravity and   1 ∂v ∂v The Newtonian “inverse square law” for gravity, D = i + j , (9) ij 2 ∂x ∂x  j i  Gm m 1 2 (1) also has the same external solution (5), because C(v) = 0 F = 2 , r for the flow in (5). So the presence ofthe C (v) would was based on Kepler’s laws for the motion of the planets. not have manifested in the special case of planets in orbit Newton formulated gravity in terms of the gravitational ac- about the massive central sun. Here α is a dimensionless

30 R. T. Cahill. Novel Gravity Probe B Frame-Dragging Effect October, 2005 PROGRESS IN PHYSICS Volume 3

constant — a new gravitational constant, in addition to usual giving, as a generalisation of (4), the acceleration the Newtonian gravitational constant G. However inside a dv0 ∂v spherical mass we find [5] that C (v) = 0, and using the = + (v )v + ( v) vR 6 1 dt ∂t ∙∇ ∇ × × − Greenland borehole g anomaly data [4] we find that α− =   2 vR 1 d v (18) =139 5, which gives the fine structure constant α=e2~/c R . ± ≈ − v2 2 dt c2 1/137 to within experimental error. From (4) we can write 1 R   ≈ − c2 g = 4πGρ 4πGρ , (10) ∇∙ − − DM Formulating gravity in terms of a flow is probably un- where familiar, but General Relativity (GR) permits an analogous result for metrics of the Panleve-Gullstrand´ class [7], α ρ (r) = (trD)2 tr(D2) , (11) DM 1 2 32πG − 2 μ ν 2 (19) dτ = gμν dx dx = dt 2 dr v(r, t)dt . which introduces an effective “matter density” representing − c − the flow dynamics associated with the C (v) term. In [5] this The external-Schwarzschild metric belongs to this class dynamical effect is shown to be the “dark matter” effect. [8], and when expressed in the form of (19) the v field is The interpretation of the vector flow field v is that it is a identical to (5). Substituting (19) into the Einstein equations manifestation, at the classical level, of a quantum substratum 1 8πG to space; the flow is a rearrangement of that substratum, and Gμν Rμν R gμν = Tμν , (20) ≡ − 2 c2 not a flow through space. However (7) needs to be further generalised [5] to include vorticity, and also the effect of the gives motion of matter through this substratum via G = v v c2 v 00 i G ij j − G 0j j − i,j =1,2,3 j =1,2,3 vR r0(t), t = v0(t) v r0(t), t , (12) X X { } − { } 2 2 c vi i0 + c 00 , where v0(t) is the velocity of an object, at r0(t), relative to − G G i=1,2,3 the same frame of reference that defines the flow field; then X (21) 2 vR is the velocity of that matter relative to the substratum. Gi0 = ij vj + c i0 , The flow equation (7) is then generalised to, with d/dt = − G G j =1X,2,3 = ∂/∂t + v the Euler fluid or total derivative, ∙∇ G = , i, j = 1, 2, 3, ij G ij dDij δij 2 trD δij + tr(D ) + Dij trD + where the are given by dt 3 2 − 3 Gμν   1 δij α 2 2 + (trD)2 tr(D2) + (ΩD DΩ) = (13) 00 = (trD) tr(D ) , 3 8 − − ij G 2 − j    δ vi v  1 = 4πGρ ij + R R + ... , i, j = 1, 2, 3, = = ( v) , − 3 2c2 G i0 G0i −2 ∇ × ∇ × i   (22)   8πGρ d 1 ( v) = v , (14) = D δ trD + D δ trD trD ∇ × ∇ × c2 R G ij dt ij − ij ij − 2 ij −   1 ∂vi ∂vj 1 2
Ωij = = δ tr(D ) + (ΩD DΩ) , i, j = 1, 2, 3 2 ∂x − ∂x − 2 ij − ij  j i  (15) 1 1 and so GR also uses the Euler “fluid” derivative, and we =  ω =  ( v) , −2 ijk k −2 ijk ∇ × k obtain a set of equations analogous but not identical to (13)– and the vorticity vector field is ~ω = v. For zero vorticity (14). In vacuum, with Tμν = 0, we find that (22) demands that ∇× and vR c (13) reduces to (7). We obtain from (14) the Biot- (trD)2 tr(D2) = 0 . (23) Savart form for the vorticity − This simply corresponds to the fact that GR does not 2G ρ(r , t) 3 0 (16) permit the “dark matter” dynamical effect, namely that ~ω(r, t) = 2 d r0 3 vR(r0, t) (r r0) . c r r0 × − , according to (10). This happens because GR was Z | − | ρDM = 0 forced to agree with Newtonian gravity, in the appropriate The path r0(t) of an object through this flow is obtained by extremising the relativistic proper time limits, and that theory also has no such effect. The predictions from (13)–(14) and from (22) for the Gravity Probe B exper- v2 1/2 iment are different, and provide an opportunity to test both τ[r ] = dt 1 R (17) 0 − c2 gravity theories. Z  

R. T. Cahill. Novel Gravity Probe B Frame-Dragging Effect 31 Volume 3 PROGRESS IN PHYSICS October, 2005

S S

VE VE

v

Fig. 1: Shows the Earth (N is up) and vorticity vector field com- Fig. 2: Shows the Earth (N is up) and the much larger vorticity ponent ~ω induced by the rotation of the Earth, as in (24). The vector field component ~ω induced by the translation of the Earth, polar orbit of the GP-B satellite is shown, S is the gyroscope as in (27). The polar orbit of the GP-B satellite is shown, and S is starting spin orientation, directed towards the guide star IM Pegasi, the gyroscope starting spin orientation, directed towards the guide h h RA = 22 5302.2600, Dec = 16◦50028.200, and VE is the vernal star IM Pegasi, RA = 22 5302.2600, Dec = 16◦50028.200, VE is the h equinox. vernal equinox, and V is the direction RA = 5.2 , Dec = 67◦ of − the translational velocity vc. 3 “Frame-dragging” as a vorticity effect

Here we consider one difference between the two theories, induced “frame-dragging” or vorticity effect. Then vR(r) = = w r in (16), where w is the angular velocity of the Earth, namely that associated with the vorticity part of (18), leading × to the “frame-dragging” or Lense-Thirring effect. In GR giving G 3(r L)r r2 L the vorticity field is known as the “gravitomagnetic” field (24) ~ω (r) = 4 2 ∙ 5− , B = c ~ω. In both GR and the new theory the vorticity is c 2 r − given by (16) but with a key difference: in GR vR is only where L is the angular momentum of the Earth, and r is the the rotational velocity of the matter in the Earth, whereas in distance from the centre. This component of the vorticity field (13)–(14) vR is the vector sum of the rotational velocity and is shown in Fig. 1. Vorticity may be detected by observing the translational velocity of the Earth through the substratum. the precession of the GP-B gyroscopes. The vorticity term At least seven experiments have detected this translational in (18) leads to a torque on the angular momentum S of the velocity; some were gas-mode Michelson interferometers gyroscope, and others coaxial cable experiments [8, 9, 10], and the 3 translational velocity is now known to be approximately 430 ~τ = d r ρ(r) r ~ω (r) vR(r) , (25) h × × km/s in the direction RA = 5.2 , Dec = 67◦. This direction Z −   has been known since the Miller [11] gas-mode interfero- where ρ is its density, and where vR is used here to describe meter experiment, but the RA was more recently confirmed the rotation of the gyroscope. Then dS = ~τdt is the change in by the 1991 DeWitte coaxial cable experiment performed S over the time interval dt. In the above case vR(r) = s r, in the Brussels laboratories of Belgacom [9]. This flow is where s is the angular velocity of the gyroscope. This gives× related to galactic gravity flow effects [8, 9, 10], and sois 1 different to that of the velocity of the Earth with respect to the ~τ = ~ω S (26) Cosmic Microwave Background (CMB), which is 369 km/s 2 × h in the direction RA = 11.20 , Dec = 7.22◦. and so ~ω/2 is the instantaneous angular velocity of precession First consider the common but− much smaller rotation of the gyroscope. This corresponds to the well known fluid

32 R. T. Cahill. Novel Gravity Probe B Frame-Dragging Effect October, 2005 PROGRESS IN PHYSICS Volume 3

0.016 Here ΔS(t) is the integrated change in spin, and where 0.014 the approximation arises because the change in S(t0) on the 0.012 RHS of (28) is negligible. The plot in Fig. 3 shows this effect to be some 30 larger than the expected GP-B errors, 0.01 and so easily detectable,× if it exists as predicted herein. This 0.008 precession is about the instantaneous direction of the vorticity

arcsec 0.006 ~ω r(t) at the location of the satellite, and so is neither in 0.004 the plane, as for the geodetic precession, nor perpendicular to the
plane of the orbit, as for the earth-rotation induced 0.002 vorticity effect. Because the yearly orbital rotation of the Earth about 0 20 40 60 80 the Sun slightly effects [9] predictions for four months orbit minutes vC − throughout the year are shown in Fig. 3. Such yearly effects Fig. 3: Predicted variation of the precession angle ΔΘ = were first seen in the Miller [11] experiment. = ΔS (t) / S (0) , in arcsec, over one 97 minute GP-B orbit, | | | | from the vorticity induced by the translation of the Earth, as given References by (28). The orbit time begins at location S. Predictions are for the months of April, August, September and February, labeled by 1. Schiff L. I. Phys. Rev. Lett., 1960, v. 4, 215. increasing dash length. The “glitches” near 80 minutes are caused by the angle effects in (28). These changes arise from the effects of 2. Van Patten R. A. and Everitt C. W. F. Phys. Rev. Lett., 1976, the changing orbital velocity of the Earth about the Sun. The GP- v. 36, 629. B expected angle measurement accuracy is 0.0005 arcsec. Novel 3. Everitt C. W. F. et al. Near Zero: Festschrift for William M. gravitational waves will affect these plots. Fairbank, ed. by Everitt C. W. F., Freeman, S. Francisco, 1986. 4. Turneaure J. P., Everitt C. W. F., Parkinson B. W. et al. The Gravity Probe B relativity gyroscope experiment. Proc. of the result that the vorticity vector is twice the angular velocity Fourth Marcell Grossmann Meeting in General Relativity, ed. vector. For GP-B the direction of S has been chosen so that by R. Ruffini, Elsevier, Amsterdam, 1986. this precession is cumulative and, on averaging over an orbit, 6 5. Cahill R. T. “Dark matter” as a quantum foam in-flow effect. corresponds to some 7.7×10− arcsec per orbit, or 0.042 Trends In Dark Matter Research, ed. by Blain J. Val, Nova arcsec per year. GP-B has been superbly engineered so that Science Pub, NY, 2005; Cahill R. T. Gravitation, the “dark measurements to a precision of 0.0005 arcsec are possible. matter” effect and the fine structure constant. Apeiron, 2005, However for the unique translation-induced precession v. 12, No. 2, 144–177. if we use v v = 430 km/s in the direction RA = 5.2hr, R ≈ C 6. Ander M. E. et al. Phys. Rev. Lett., 1989, v. 62, 985. Dec = 67◦, namely ignoring the effects of the orbital motion − 7. Panleve´ P. Com. Rend. Acad. Sci., 1921, v. 173, 677; of the Earth, the observed flow past the Earth towards the Gullstrand A. Ark. Mat. Astron. Fys., 1922, v. 16, 1. Sun, and the flow into the Earth, and effects of the gravita- 8. Cahill R. T. Quantum foam, gravity and gravitational waves. tional waves, then (16) gives Relativity, Gravitation, Cosmology, ed. by Dvoeglazov V.V. 2GM v r and Espinoza Garrido A. A. Nova Science Pub., NY, 2004, ~ω (r) = C × . (27) 168–226. c2 r3 9. Cahill R. T. Absolute motion and gravitational effects. Apeiron, This much larger component of the vorticity field is 2004, v. 11, No. 1, 53–111. shown in Fig. 2. The maximum magnitude of the speed of this 10. Cahill R. T. Process Physics: from information theory to 2 6 precession component is ω/2 = gvC /c = 8×10− arcsec/s, quantum space and matter. Nova Science Pub., NY, 2005. where here g is the gravitational acceleration at the altitude of 11. Miller D. C. Rev. Mod. Phys., 1993, v. 5, 203–242. the satellite. This precession has a different signature: it is not cumulative, and is detectable by its variation over each single orbit, as its orbital average is zero, to first approximation. Fig. 3 shows ΔΘ = ΔS(t) / S(0) over one orbit, where, as in general, | | | |

t 1 ΔS(t) = dt0 ~ω r(t0) S(t0) 2 × ≈  Z0 
(28) t 1 dt0 ~ω r(t0) S(0) . ≈ 2 ×  Z0 

R. T. Cahill. Novel Gravity Probe B Frame-Dragging Effect 33 Volume 3 PROGRESS IN PHYSICS October, 2005

Relations Between Physical Constants

Roberto Oros di Bartini∗

This article discusses the main analytic relationship between physical constants, and applications thereof to cosmology. The mathematical bases herein are group theoretical methods and topological methods. From this it is argued that the Universe was born from an Inversion Explosion of the primordial particle (pre-particle) whose outer radius was that of the classical electron, and inner radius was that of the gravitational radius of the electron. All the mass was concentrated in the space between the radii, and was inverted outside the particle through the pre-particle’s surface (the inversion classical radius). This inversion process continues today, determining evolutionary changes in the fundamental physical constants.

As is well known, group theor- is the mapping which transfers images of A in accordance etical methods, and also topolog- with the pre-image of A. ical methods, can be effectively The specimen A, by definition, can be associated only employed in order to interpret with itself. For this reason it’s inner mapping can, accord- physical problems. We know of ing to Stoilow’s theorem, be represented as the superposition studies setting up the discrete in- of a topological mapping and subsequently by an analytic terior of space-time, and also rel- mapping. ationships between atomic quant- The population of images of A is a point-containing ities and cosmological quantities. system, whose elements are equivalent points; an n-dimen- However, no analytic relati- sional affine spread, containing (n + 1)-elements of the sys- onship between fundamental phy- tem, transforms into itself in linear manner sical quantities has been found. n+1 They are determined only by ex- x = a x . perimental means, because there i0 ik k k=1 is no theory that could give a the- Roberto di Bartini, 1920’s X oretical determination of them. (in Italian Air Force uniform) With all aik real numbers, the unitary transformation In this brief article we give the results of our own study, which, employing group theoretical methods and topological aik∗ alk = aki∗ akl , i, k = 1, 2, 3 . . . , n + 1 , methods, gives an analytic relationship between physical k k X X constants. is orthogonal, because det aik = 1. Hence, this transform- Let us consider a predicative unbounded and hence ation is rotational or, in other words,± an inversion twist. unique specimen A. Establishing an identity between this A projective space, containing a population of all images specimen A and itself of the object A, can be metrizable. The metric spread Rn 1 (coinciding completely with the projective spread) is closed, A A,A = 1 , ≡ A according to Hamel’s theorem. A coincidence group of points, drawing elements of the ∗Brief contents of this paper was presented by Prof. Bruno Pontecorvo to the Proceedings of the Academy of Sciences of the USSR (Doklady Acad. set of images of the object A, is a finite symmetric system, Sci. USSR), where it was published in 1965 [19]. Roberto di Bartini (1897– which can be considered as a topological spread mapped 1974), the author, was an Italian mathematician and aircraft engineer who, into the spherical space Rn. The surface of an (n + 1)- from 1923, worked in the USSR where he headed an aircraft project bureau. Because di Bartini attached great importance to this article, he signed it dimensional sphere, being equivalent to the volume of an with his full name, including his titular prefix and baronial name Oros — n-dimensional torus, is completely and everywhere densely from Orosti, the patrimony near Fiume (now Rijeka, located in Croatian filled by the n-dimensional excellent, closed and finite point- territory near the border), although he regularly signed papers as Roberto containing system of images of the object A. Bartini. The limited space in the Proceedings did not permit publication of n the whole article. For this reason Pontecorvo acquainted di Bartini with Prof. The dimension of the spread R , which consists only of Kyril Stanyukovich, who published this article in his bulletin, in Russian. the set of elements of the system, can be any integer n inside Pontecorvo and Stanyukovich regarded di Bartini’s paper highly. Decades the interval (1 N) to (N 1) where N is the number of later Stanyukovich suggested that it would be a good idea to publish di entities in the ensemble.− − Bartini’s article in English, because of the great importance of his idea of applying topological methods to cosmology and the results he obtained. We are going to consider sequences of stochastic transit- (Translated by D. Rabounski and S. J. Crothers.) — Editor’s remark. ions between different dimension spreads as stochastic vector

34 R. Oros di Bartini. Relations Between Physical Constants October, 2005 PROGRESS IN PHYSICS Volume 3

quantities, i. e. as fields. Then, given a distribution function of the real existence of the object A we are considering is a for frequencies of the stochastic transitions dependent on n, (3 + 3)-dimensional complex formation, which is the product we can find the most probable number of the dimension of of the 3-dimensional spatial-like and 3-dimensional time-like the ensemble in the following way. spreads (each of them has its own direction in the (3 + 3)- Let the differential function of distribution of frequencies dimensional complex formation). ν in the spectra of the transitions be given by

ϕ(ν) = νn exp[ πν2] . − If n 1, the mathematical expectation for the frequency of a transition from a state n is equal to

∞ n 2 n + 1 ν exp[ πν ]dν Γ 0 − 2 m(ν) = Z = .  n+1  ∞ 2 2 exp[ πν2]dν 2π − Z0 The statistical weight of the time duration for a given state is a quantity inversely proportional to the probability of this state to be changed. For this reason the most probable dimension of the ensemble is that number n under which the function m(ν) has its minimum. The inverse function of m(ν), is 1 Φ = = S = V , n m(ν) (n+1) T n One of the main concepts in dimension theory and combi- natorial topology is nerve. Using this term, we come to the where the function Φn is isomorphic to the function of the statement that any compact metric space of n dimensions surface’s value S(n+1) of a unit radius hypersphere located can be mapped homeomorphicly into a subset located in a in an (n+1)-dimensional space (this value is equal to the Euclidean space of (2n + 1) dimensions. And conversely, any volume of an n-dimensional hypertorus). This isomorphism compact metric space of (2n + 1) dimensions can be mapped is adequate for the ergodic concept, according to which the homeomorphicly way into a subset of n dimensions. There spatial and time spreads are equivalent aspects of a manifold. is a unique correspondence between the mapping 7 3 So, this isomorphism shows that realization of the object A and the mapping 3 7, which consists of the geometrical→ as a configuration (a form of its real existence) proceeds from realization of the abstract→ complex A. the objective probability of the existence of this form. The geometry of the aforementioned manifolds is determ- The positive branch of the function Φn is unimodal; ined by their own metrics, which, being set up inside them, for negative values of (n + 1) this function becomes sign- determines the quadratic interval alternating (see the figure). The formation takes its maximum length when n = 6, n ± 2 2 i k hence the most probable and most unprobable extremal dis- Δs = Φn gik Δx Δx , i, k = 1, 2, . . . , n , tributions of primary images of the object A are presented in Xik the 6-dimensional closed configuration: the existence of the which depends not only on the function gik of coordinates i total specimen A we are considering is 6-dimensional. and k, but also on the function of the number of independent Closure of this configuration is expressed by the finitude parameters Φn. of the volume of the states, and also the symmetry of distrib- The total length of a manifold is finite and constant, ution inside the volume. hence the sum of the lengths of all formations, realized in the Any even-dimensional space can be considered as the manifold, is a quantity invariant with respect to orthogonal product of two odd-dimensional spreads, which, having the transformations. Invariance of the total length of the form- same odd-dimension and the opposite directions, are emb- ation is expressed by the quadratic form edded within each other. Any spherical formation of n di- mensions is directed in spaces of (n + 1) and higher dim- 2 2 Niri = Nkrk , ensions. Any odd-dimensional projective space, if immersed in its own dimensions, becomes directed, while any even- where N is the number of entities, r is the radial equivalent dimensional projective space is one-sided. Thus the form of the formation. From here we see, the ratio of the radii is

R. Oros di Bartini. Relations Between Physical Constants 35 Volume 3 PROGRESS IN PHYSICS October, 2005

Rρ d2l = 1 , 4πq = SV˙ = 4πr2 . r2 dt2 where R is the largest radius; ρ is the smallest radius, realised Because the charge manifests in the spread Rn only as in the area of the transformation; r is the radius of spherical the strength of its field, and both parts of the equations are inversion of the formation (this is the calibre of the area). The equivalent, we can use the right side of the equation instead transformation areas are included in each other, the inversion of the left one. twist inside them is cascaded The field vector takes its ultimate value

Rr rρ l SV˙ = Re , Rρ = r, = ρe . c = = = 1 2π 2π r r t s4πri p Negative-dimensional configurations are inversion im- in the surface of the inversion sphere with the radius r. The 2 ages, corresponding to anti-states of the system. They have ultimate value of the field strength lt− takes a place in the 1 mirror symmetry if n = l(2m 1) and direct symmetry if same surface; ν = t− is the fundamental frequency of the n = 2(2m), where m = 1, 2, 3.− Odd-dimensional configurat- oscillator. The effective (half) product of the sphere surface ions have no anti-states. The volume of the anti-states is space and the oscillation acceleration equals the value of the pulsating charge, hence 1 V( n) = 4 − . 1 2 l 2 − V 4πq = 4πνr = 2πr c . n 2 i t i Equations of physics take a simple form if we use the LT In LT kinematic system of units the dimension of a kinematic system of units, whose units are two aspects l and charge (both gravitational and electric) is of the radius through which areas of the space n undergo t R 3 2 inversion: l is the element of the spatial-like spread of the dim m = dim e = L T − . subspace L, and t is the element of time-like spread of the In the kinematic system LT , exponents in structural subspace T . Introducing homogeneous coordinates permits formulae of dimensions of all physical quantities, including reduction of projective geometry theorems to algebraic equi- electromagnetic quantities, are integers. valents, and geometrical relations to kinematic relations. Denoting the fundamental ratio l/t as C, in the kinematic The kinematic equivalent of the formation corresponds system LT we obtain the generalized structural formula for the following model. physical quantities An elementary (3 + 3)-dimensional image of the object Σn γ n γ A can be considered as a wave or a rotating oscillator, D = c T − , which, in turn, becomes the sink and source, produced by where DΣn is the dimensional volume of a given physical the singularity of the transformation. There in the oscillator quantity, Σn is the sum of exponents in the formula of polarization of the background components occurs — the dimensions (see above), T is the radical of dimensions, n transformation L T or T L, depending on the direction → → and γ are integers. of the oscillator, which makes branching L and T spreads. Thus we calculate dimensions of physical quantities in The transmutation L T corresponds the shift of the field ↔ the kinematic LT system of units (see Table 1). vector at π/2 in its parallel transfer along closed arcs of radii Physical constants are expressed by some relations in n R and r in the affine coherence space R . the geometry of the ensemble, reduced to kinematic struc- The effective abundance of the pole is tures. The kinematic structures are aspects of the probability and configuration realization of the abstract complex A. The 1 1 e = Eds . most stable form of a kinematic state corresponds to the most 2 4π Zs probable form of the stochastic existence of the formation. A charge is an elementary oscillator, making a field The value of any physical constant can be obtained in the around itself and inside itself. There in the field a vector’s following way. The maximum value of the probability of the state we length depends only on the distance ri or 1/ri from the centre of the peculiarity. The inner field is the inversion map are considering is the same as the volume of a 6-dimensional torus, of the outer field; the mutual correspondence between the 3 16π 3 6 outer spatial-like and the inner time-like spreads leads to V6 = r = 33.0733588 r . torsion of the field. 15 The product of the space of the spherical surface and The extreme numerical values — the maximum of the the strength in the surface is independent of ri; this value positive branch and the minimum of the negative branches depends only on properties of the charge q of the function Φn are collected in Table 2.

36 R. Oros di Bartini. Relations Between Physical Constants October, 2005 PROGRESS IN PHYSICS Volume 3

Table 1

Quantity DΣn, taken under γ equal to:

Parameter Σn 5 4 3 2 1 0 1 2 − − 5 n 5 4 n 4 3 n 3 2 n 2 1 n 1 0 n 0 1 n+1 2 n+2 C T − C T − C T − C T − C T − C T − C− T C− T

3 5 Surface power L T − 2 4 Pressure L T − 1 3 Current density 2 L T − − Mass density, angular 0 2 L T − acceleration 1 1 Volume charge density L− T −

Electromagnetic field 2 3 L T − strength

Magnetic displacement, 1 2 1 L T − acceleration − 0 1 Frequency L T − 5 5 Power L T − 4 4 Force L T − 3 3 Current, loss mass L T − 2 2 Potential difference 0 L T − 1 1 Velocity L T − Dimensionless constants L0T 0 1 1 Conductivity L− T 2 2 Magnetic permittivity L− T 5 4 Force momentum, energy L T − 4 3 Motion quantity, impulse L T −

Mass, quantity of mag- 3 2 +1 L T − netism or electricity

Two-dimensional 2 1 L T − abundance

Length, capacity, self- 1 0 L T induction Period, duration L0T 1

Angular momentum, 5 3 L T − action 4 2 Magnetic momentum L T − 3 1 Loss volume +2 L T − Surface L2T 0 L1T 1 L0T 2 5 2 Moment of inertia L T − 4 1 L T − Volume of space +3 L3T 0 Volume of time L0T 3

R. Oros di Bartini. Relations Between Physical Constants 37 Volume 3 PROGRESS IN PHYSICS October, 2005

Table 2 Table 3 gives numerical values of physical constants, ob- tained analytically and experimentally. The appendix gives n + 1 +7.256946404 4.99128410 − experimental determinations in units of the CGS system (cm, gramme, sec), because they are conventional quantities, not Sn+1 +33.161194485 0.1209542108 − physical constants. The fact that the theoretically and experimentally obtain- The ratio between the ultimate values of the function ed values of physical constants coincide permits us to suppo- Sn+1 is se that all metric properties of the considered total and unique +S(n+1) specimen A can be identified as properties of our observed Eˉ = max = 274.163208 r12. S World, so the World is identical to the unique “particle” A. (n+1)min − In another paper it will be shown that a (3 + 3)-dimensional On the other hand, a finite length of a spherical layer structure of space-time can be proven in an experimental of Rn, homogeneously and everywhere densely filled by way, and also that this 6-dimensional model is free of logical doublets of the elementary formations A, is equivalent to a difficulties derived from the (3 + 1)-dimensional concept of vortical torus, concentric with the spherical layer. The mirror the space-time background∗. image of the layer is another concentric homogeneous double In the system of units we are using here the gravitational layer, which, in turn, is equivalent to a vortical torus coaxial constant is 1 l0 with the first one. Such formations were studied by Lewis κ = . and Larmore for the (3 + 1)-dimensional case. 4π t0   Conditions of stationary vortical motion are realized if If we convert its dimensions back to the CGS system, so 3 V rotV = gradϕ , 2vds = dΓ , that G = l , appropriate numerical values of the physic- × mt2 al quantities will be determined in another form (Column 5 in where ϕ is the potential of the circulation, Γ is the main h i Table 3). Reduced physical quantities are given in Column 8. kinematic invariant of the field. A vortical motion is stable Column 9 gives evolutionary changes of the physical quanti- only if the current lines coincide with the trajectory of the ties with time according to the theory, developed by Stanyu- vortex core. For a (3 + 1)-dimensional vortical torus we have kovich [17]†. Γ 4D 1 The gravitational “constant”, according to his theory, V = ln , x 2πD r − 4 increases proportionally to the space radius (and also the   world-time) and the number of elementary entities, according where r is the radius of the circulation, D is the torus to Dirac [18], increases proportional to the square of the diameter. space radius (and the square of world-time as well). There- 1 The velocity at the centre of the formation is 2 24 12 fore we obtain N = Tm B , hence B Tm . ' 40 ≈ 17 uπD Because Tm = t0ω0 10 , where t0 10 sec is the V = . ' c ' 23 1 2r space age of our Universe and ω0 = ρ = 10 sec− is the 10 The condition Vx = V , in the case we are considering, frequency of elementary interactions, we obtain B 10 3 = 1 ' 3 is true if n = 7 = 10 ×1000. 2 2 24 4D 2n + 1 In this case we obtain m e ~ Tm− B− , which ln = (2π + 0.25014803) = is in good agreement with the∼ evolution∼ ∼ concept∼ developed r 2n n by Stanyukovich. = 2π + 0.25014803 + = 7 , 2n + 1 D 1 Appendix = Eˉ = e7 = 274.15836 . r 4 Here is a determination of the quantity 1 cm in the CGS In the field of a vortical torus, with Bohr radius ofthe system of units. The analytic value of Rydberg constant is charge, r = 0.999 9028, the quantity π takes the numerical 1 6.9996968 Roberto di Bartini died before he prepared the second paper. He died value π∗ = 0.999 9514 π. So E = 4 e = 274.074996. ∗ In the LT kinematic system of units, and introducing the sitting at his desk, looking at papers with drawings of vortical tori and draft formulae. According to Professor Stanyukovich, Bartini was not in the habit relation B = V6E/π = 2885.3453, we express values of all of keeping many drafts, so unfortunately, we do not know anything about constants by prime relations between E and B the experimental statement that he planned to provide as the proof to his concept of the (3 + 3)-dimensional space-time background. — D. R. α β K = δE B , †Stanyukovich’s theory is given in Part II of his book [17]. Here T0m is the world-time moment when a particle (electron, nucleon, etc.) was born, where is equal to a quantized turn, and are integers. T is the world-time moment when we observe the particle. — D. R. δ e e α β m

38 R. Oros di Bartini. Relations Between Physical Constants October, 2005 PROGRESS IN PHYSICS Volume 3 1 2 1 2 3 1 2 2




1 2 11 12 1 12

3 m m
m m m m m m m m m m m m m 0 0 m 0 0 m m 0 2 m 0 0 m m T 0 m T m m 2 m T T m T 0 T T T 0 T m m 0 m T T 0 T T T T 0 0 T m T T T T time T 0 T T T T T T T T T ~ const const const const const const k Table on κ e e κe κμ κm n S m Dependence κγ √ √ √ e μ k m c TNM e ~ n κn E m κm κ M κM e κ n r κ κ κ S m TT RR C CGS H NN S κ 1 2 √ κ κ N κγ κ √ √ √ formula Structural 1 2 1 1 1 1 0 0 − 1 0 1 0 − − − 0 0 0 0 1 − sec − sec 0 sec 2 sec 0 1 sec sec sec sec sec 0 1 sec sec 0 sec sec sec 1 2 − sec 1 0 1 0 − sec 1 2 0 0 sec 0 1 0 1 0 gm 1 sec gm − sec gm 1 gm gm gm gm gm sec 0 0 gm gm 0 gm gm gm gm 0 0 2 3 2 1 1 gm 3 2 1 0 gm 1 0 0 5 2 gm 0 cm 3 gm cm 3 gm cm cm cm cm gm cm − 0 cm cm cm 0 cm cm cm 13 2 cm 24 cm 2 — 3 28 27 10 − 17 cm 55 10 28 56 17 − 21 cm cm 10 − 19 8 − 10 − − 10 10 − × 10 − 10 10 10 10 × 31 82 CGS-system 10 10 10 × × × 10 10 × − 10 × × × 10 10 > > > × × in × × 10 — cm > 29 57 19 — cm ∼ 10 10 10 1.83630 1.348 1.370374 9.1083 6.1781 Observed numerical values 6.670 1.67239 2.997930 2.817850 9.2734 6.62517 5.273058 4.80286 8 55 35 13 28 24 10 34 27 21 2 − 10 3 17 − − − 29 − − − 57 19 − 98 84 126 − − 19 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 × × × × × × × × × × × × × × × × × × × × × values CGS-system in 1.370375 6.670024 2.997930 1.836867 5.273048 1.346990 7.772329 2.817850 5.894831 9.108300 1.673074 4.802850 3.986064 1.966300 9.858261 4.426057 4.376299 9.155046 6.625152 9.273128 6.178094 1 2 2 2 2 2 3 3 2 1 0 0 0 0 0 0 0 0 1 0 0 − − − − − − − − − − t t t t t t t t t t t t t t t t t t t t t 0 0 0 0 1 1 1 1 0 0 0 1 3 3 3 3 0 5 5 4 0 l l l l l l l l l l l l l l l l l l l l l units 2 43 21 42 39 21 86 39 19 4 2 − 0 3 20 − − 0 42 − − − 43 42 − 86 84 126 − − − 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 × × × × × × × × × × × × × × × × × × × × × LT-system of Analytically obtained numerical 1.370375 7.986889 1.000000 1.836867 5.770146 2.753248 1.000000 2.091961 3.003491 5.517016 1.733058 1.314417 2.091961 7.273495 1.727694 4.376299 9.155046 2.586100 1.187469 5.806987 4.7802045 6 0 12 24 0 6 11 β 0 1 − 6 B − 12 − 12 0 6 0 − B 12 − 12 12 24 24 36 − 1 B B B B − − 0 B B B B B B 0 0 0 − B B B B B B B α B 0 0 0 0 0 1 B B E 0 0 0 0 0 0 0 0 E E E E 0 1 1 E E E E E E 1 1 1 0 E E E E E E E δE E 1 3 − 0 0 0 0 E E 1 1 2 1 4 2 3 − − 0 − π − 0 − 1 π π π π π 1 − π π π π π π = π π π π 2 2 0 0 0 π π π π 1 2 1 4 2 3 1 1 − 1 0 π 2 2 2 1 2 0 0 − − 2 2 2 2 2 2 2 2 2 1 − − 2 2 K 2 2 − − 2 2 2 2 2 6 3 11 ∗ 6 t r r r 2 12 B 2 R R 2 ρ c 4 2 12 E π 12 6 e /π T / R /ρ 2 π 2 πB πE πF 2 πB /πB l/t / 2 2 c B 2 4 πρc 2 2 πRc πρ πB πB 1 2 p 2 / 2 2 2 formula 2 2 c/ mcπE r/ 1 rc Structural

M/ 2

Er Notation b e k c c e r ρ r/ ~ κ n /α T AN R HNR Mc m ρ ν γ M μ 1 e/m B n/m 1.003662 1) = constant ratio velocity ge − constant ratio E frequency ( mass mass constant radius mass period density action E/ magneton = electron electron inversion F Gravitational Mass basic Electric radius of Space Number of actual ∗ Parameter Fundamental Charge basic Gravitational radius of Classical radius of Electron Nucleon Electron char Space Space Space Space entities Number of primary interactions Planck Bohr Compton Sommerfeld

R. Oros di Bartini. Relations Between Physical Constants 39 Volume 3 PROGRESS IN PHYSICS October, 2005

3 1 8 1 [R ] = (1/4πE )l− = 3.0922328×10− l− , the experime- 17. Stanyukovich K. P. Gravitational field and elementary particles. ntally∞ obtained value of the constant is (R )=109737.311 Nauka, Moscow, 1965. 1 ∞ ± 0.012cm− . Hence 1 cm is determined in the CGS system 18. Dirac P.A. M. Nature, 1957, v. 139, 323; Proc. Roy. Soc. A, ± 12 as (R )/[R ] = 3.5488041×10 l. 1938, v. 6, 199. ∞ ∞ Here is a determination of the quantity 1 sec in the CGS 19. Oros di Bartini R. Some relations between physical constants. system of units. The analytic value of the fundamental ve- Doklady Acad. Nauk USSR, 1965, v. 163, No. 4, 861–864. locity is [c] = l/t = 1, the experimentally obtained value of the velocity of light in vacuum is (c) = 2.997930 10 1 ± 0.0000080×10− cm×sec− . Hence 1 sec is determined in ± 23 the CGS system as (c)/l [c] = 1.0639066×10 t. Here is a determination of the quantity 1 gramme in the CGS system of units. The analytic value of the ratio e/mc is 6 20 1 [e/mc] = B = 5.7701460×10 l− t. This quantity, mea- 7 sured in experiments, is (e/mc)=1.758897 0.000032×10 1 ± 1 2 (cm×gm− )e . Hence 1 gramme is determined in the CGS 2 (e/mc) 15 3 2 system as = 3.297532510 10− l t− , so CGS’ l[e/mc]2 × 7 3 2 one gramme is 1 gm (CGS)=8.351217×10− cm sec− (CS).

References

1. Pauli W. Relativitatstheorie.¨ Encyclopaedie¨ der mathemati- schen Wissenschaften, Band V, Heft IV, Art. 19, 1921 (Pauli W. Theory of Relativity. Pergamon Press, 1958). 2. Eddington A. S. The mathematical theory of relativity, Cam- bridge University Press, Cambridge, 2nd edition, 1960. 3. Hurewicz W. and Wallman H. Dimension theory. Foreign Literature, Moscow, 1948. 4. Zeivert H. and Threphall W. Topology. GONTI, Moscow, 1938. 5. Chzgen Schen-Schen (Chern S. S.) Complex manifolds. Fore- ign Literature, Moscow, 1961 6. Pontriagine L. Foundations of combinatory topology. OGIZ, Moscow, 1947. 7. Busemann G. and Kelley P. Projective geometry. Foreign Literature, Moscow, 1957. 8. Mors M. Topological methods in the theory of functions. Foreign Literature, Moscow, 1951. 9. Hilbert D. und Cohn-Vossen S. Anschauliche Geometrie. Springer Verlag, Berlin, 1932 (Hilbert D. and Kon-Fossen S. Obvious geometry. GTTI, Moscow, 1951). 10. Vigner E. The theory of groups. Foreign Literature, Moscow, 1961. 11. Lamb G. Hydrodynamics. GTTI, Moscow, 1947. 12. Madelunge E. The mathematical apparatus in physics. PhysMathGiz, Moscow, 1960. 13. Bartlett M. Introduction into probability processes theory. Foreign Literature, Moscow, 1958. 14. McVittie G. The General Theory of Relativity and cosmology. Foreign Literature, Moscow, 1961. 15. Wheeler D. Gravitation, neutrino, and the Universe. Foreign Literature, Moscow, 1962. 16. Dicke R. Review of Modern Physics, 1957, v. 29, No. 3.

40 R. Oros di Bartini. Relations Between Physical Constants October, 2005 PROGRESS IN PHYSICS Volume 3

Introducing Distance and Measurement in General Relativity: Changes for the Standard Tests and the Cosmological Large-Scale

Stephen J. Crothers Sydney, Australia E-mail: [email protected]

Relativistic motion in the gravitational field of a massive body is governed bythe external metric of a spherically symmetric extended object. Consequently, any solution for the point-mass is inadequate for the treatment of such motions since it pertains to a fictitious object. I therefore develop herein the physics of the standard tests ofGeneral Relativity by means of the generalised solution for the field external to a sphere of incompressible homogeneous fluid.

1 π 1 Introduction arccos < χ χ < , 3 | a − 0| 2 The orthodox treatment of physics in the vicinity of a massive r r r r < , body is based upon the Hilbert [1] solution for the point- | a − 0| 6 | − 0| ∞ mass, a solution which is neither correct nor due to Schwarz- 2 where ρ0 is the constant density of the fluid, k is Gauss’ gra- schild [2], as the latter is almost universally claimed. vitational constant, the sign a denotes values at the surface In previous papers [3, 4] I derived the correct general of the sphere, χ χ parameterizes the radius of curvature solution for the point-mass and the point-charge in all their | − 0| of the interior of the sphere centred arbitrarily at χ0, r r0 standard configurations, and demonstrated that the Hilbert is the coordinate radius in the spacetime manifold of| Special− | solution is invalid. The general solution for the point-mass Relativity which is a parameter space for the gravitational is however, inadequate for any real physical situation since field external to the sphere centred arbitrarily at r0. the material point (and also the material point-charge) is a To eliminate the infinite number of coordinate systems fictitious object, and so quite meaningless. Therefore, Iavail admitted by (1), I rewrite the said metric in terms of the myself of the general solution for the external field of a only measurable distance in the gravitational field, i .e. the sphere of incompressible homogeneous fluid, obtained in a circumference G of a great circle, thus particular case by K. Schwarzschild [5] and generalised by 1 2 myself [6] to, 2 2πα 2 2πα − dG ds = 1 dt 1 2 2 − G − − G 4π − (√Cn α) √Cn C0     (2) ds2 = − dt2 n dr2 2 G 2 2 2 √Cn − (√Cn α) 4Cn − (1) dθ + sin θdϕ ,    −  − 4π2 2 2 2 Cn(dθ + sin θdϕ ) ,
− 3 2 α = sin3 χ χ , n n n a 0 C (r) = r r +  , sκρ0 | − | n − 0   3 3 3 α = sin χ χ , 2π sin χa χ0 G < , a 0 κρ | − | 6 ∞ sκρ0 − s 0

1 π 3 arccos < χa χ0 < . Rca = sin χa χ0 , 3 | − | 2 sκρ0 −

3 3  = sin3 χ χ 2 Distance and time κρ 2 a − 0 − s 0  1 According to (1), if t is constant, a three-dimensional mani- 9 1 3 cos χ χ χ χ sin 2 χ χ , fold results, having the line-element, − 4 a − 0 a − 0 − 2 a − 0   √C C 2 + ds2 = n n0 dr2 + C (dθ2 + sin2 θdϕ2) . (3) r0 , r , n , χ0 , χa , n ∈ < ∈ < ∈ < ∈ < ∈ < (√Cn α) 4Cn  − 

S. J. Crothers. Introducing Distance in General Relativity: Changes for the Standard Tests and the Cosmological Large-Scale 41 Volume 3 PROGRESS IN PHYSICS October, 2005

If α = 0, (1) reduces to the line-element of flat spacetime, which, by the use of (1) and (2), becomes

2 2 2 2 2 2 2 ds = dt dr r r0 (dθ + sin θdϕ ) , (4) G G − − | − | Rp(r) = Rp + α a 2π 2π − − 0 r r0 < , r 6 | − | ∞   since then ra r0. 3 3 ≡ sin χa χ sin χa χ α + The introduction of matter makes ra = r0, owing to the − κρ − 0 κρ − 0 − 6 sr 0 r 0  (8) extended nature of a real body, and introduces distortions from the Euclidean in time and distance. The value of α is G + G α effectively a measure of this distortion and therefore fixes 2π 2π + α ln − , the spacetime. 3 q q 3 sin χa χ + sin χa χ α κρ 0 κρ 0 When α = 0, the distance D = r r is the radius of a 0 − 0 − − | − 0| sphere centred at r . If r = 0 and r 0, then D r and is rq rq 0 0 > ≡ then both a radius and a coordinate, as is clear from (4). 3 2 α = sin3 χ χ . If r is constant in (3), then Cn(r) = Rc is constant, and a 0 sκρ0 − so (3) becomes,

2 2 2 2 2 According to (1), the proper time is related to the coord- ds = Rc (dθ + sin θdϕ ) , (5) inate time by, which describes a sphere of constant radius Rc embedded in Euclidean space. The infinitesimal tangential distances on α dτ = g00 dt = 1 dt . (9) (5) are simply, − C (r) q s n 2 2 2 p ds = Rc dθ + sin θdϕ . When α = 0, dτ = dt so that proper time and coordinate q When θ and ϕ are constant, (3) yields the proper radius, time are one and the same in flat spacetime. With the intro- duction of matter, proper time and coordinate time are no longer the same. It is evident from (9) that both τ and t are Cn(r) Cn0 (r) Rp = dr = finite and non-zero, since according to (1), s Cn(r) α 2 Cn(r) Z p − (6) 1 α α p p Cn(r) < 1 6 1 , = d Cn(r) , 9 − Cn(ra) − Cn(r) C (r) α Z s pn − p i .e. p p from which it clearlyp follows that the parameter r does not 1 2 α < cos χa χ0 6 1 , measure radial distances in the gravitational field. 9 | − | − C (r) Integrating (6) gives, n or 1 p Rp(r) = √Cn(√Cn α)+α ln √Cn + √Cn α +K, dt 6 dτ 6 dt , − − 3 q K = const p, p since In the far field, according to (9), which must satisfy the condition, Cn(r) dτ dt , + → ∞ ⇒ → r ra± Rp Rp , → ⇒ → a recovering flat pspacetime asymptotically. where ra is the parameter value at the surface of the body Therefore, if a body falls from rest from a point distant and Rpa the indeterminate proper radius of the sphere from from the gravitating mass, it will reach the surface of the outside the sphere. Therefore, mass in a finite coordinate time and a finite proper time. According to an external observer, time does not stop at the R (r) = R + C (r) C (r) α surface of the body, where dt = 3dτ, contrary to the orthodox p pa n n − − r analysis based upon the fictitious point-mass. p p 

Cn(ra) Cn(ra α + − r − (7) 3 Radar sounding p p  Cn(r) + Cn(r) α Consider an observer in the field of a massive body. Let the + α ln − , q q observer have coordinates, (r1, θ , ϕ ) . Let the coordinates pC (r ) + pC (r ) α 0 0 n a n a − of a small body located between the observer and the massive

qp qp

42 S. J. Crothers. Introducing Distance in General Relativity: Changes for the Standard Tests and the Cosmological Large-Scale October, 2005 PROGRESS IN PHYSICS Volume 3

body along a radial line be (r2, θ0, ϕ0) . Let the observer emit Then according to classical theory, the round trip time is a radar pulse towards the small body. Then by (1), Δˉτ = 2Rp , 1 − 02 α 2 α Cn (r) 2 so Δτ = Δˉτ . 1 dt = 1 dr = 6 α − Cn(r)! − Cn(r)! 4Cn(r) If is small for √Cn(r) 1 p p − α 2 C (r ) < C (r) < C (r ) , = 1 d Cn(r) , n 2 n n 1 − Cn(r)! p then p p p so p d C (r) Δτ 2 Cn(r1) Cn(r2) n α ≈ − − = 1 ,  pdt ± − Cn(r)! p p or α Cn(r1) Cn(r2) C (r ) p − + α ln n 1 , dr 2 Cn(r) α −   = 1 . p 2 Cn(rp1) spCn(r2)  dt ± Cn0 (r) − Cn(r)! p p p α Cn(r1) p Δˉτ 2 C (r ) C (r ) + ln . The coordinate time for the pulse to travel to the small ≈ n 1 − n 2 2  spCn(r2)  body and return to the observer is, p p Therefore, p √Cn(r2) √Cn(r1) √ √ d Cn d Cn Cn(r1) Δt = α + α = Δτ Δˉτ α ln − 1 1 − ≈ − Z − √Cn Z − √Cn spCn(r2) √Cn(r1) √Cn(r2)  p Cn(r1) Cn(r2) √Cn(r1) − = d√Cn   = 2 . − p Cn(pr1) 1 α i (10) Z − √Cn √Cn(r2) pG G G = α ln 1 1 − 2 = G − G The proper time lapse is, according to the observer, r 2 1 ! by formula (1), 3 3 G1 G1 G2 = sin χa χ0 ln − , √Cn(r1) G G sκρ0 − r 2 − 1 ! α α d√Cn Δτ = 1 dt = 2 1 α = − √Cn − √Cn 1 r r Z − √Cn √Cn(r2) G = G(r) = 2π Cn(D(r)) . Equation (10) gives the timep delay for a radar signal in α Cn(r1) α = 2 1 Cn(r1) Cn(r2)+α ln − . the gravitational field. − √C − n Cn(r2) α ! r p − p p p 4 Spectral shift The proper distance between the observer and the small body is, Let an emitter of light have coordinates (t , r , θ , ϕ ). √Cn(r1) E E E E √Cn Let a receiver have coordinates (tR, rR, θR, ϕR). Let u be Rp = d Cn an affine parameter along a null geodesic with the values u s√Cn α E Z − and u at emitter and receiver respectively. Then, √Cn(r2) p R 2 1 2 α dt α − d√C = C (r ) C (r ) α 1 = 1 n + n 1 n 1 − − − √C du − √C du r  n     n    p p  2 2 dθ 2 dϕ Cn(r2) Cn(r2) α + + Cn + Cn sin θ , − r − du du p p      so 1 Cn(r1) + Cn(r1) α 1 2 + α ln − . dt α − dxi dxj q q = 1 gˉ , pC (r ) + pC (r ) α du − √C ij du du n 2 n 2 − " n  #

qp qp

S. J. Crothers. Introducing Distance in General Relativity: Changes for the Standard Tests and the Cosmological Large-Scale 43 Volume 3 PROGRESS IN PHYSICS October, 2005

where gˉ = g . Then, 5 Advance of the perihelia ij − ij u 1 Consider the Lagrangian, R 1 2 α − dxi dxj tR tE = 1 gˉij du , 2 − " − √Cn du du # 1 α dt uZ   L = 1 E 2 − √C dτ − " n    # and so, for spatially fixed emitter and receiver, 1 2 1 α − d√Cn 1 (13) (1) (1) (2) (2) − 2 − √Cn dτ − tR tE = tR tE , "    # − − 2 2 1 dθ 2 dϕ and therefore, Cn + sin θ , − 2 " dτ dτ !# (2) (1) (2) (1)     ΔtR = tR tR = tE tE = ΔtE . (11) − − where τ is the proper time. Restricting motion, without Now by (1), the proper time is, π loss of generality, to the equatorial plane, θ = 2 , the Euler- Lagrange equations for (13) are, α ΔτE = 1 ΔtE , 1 2 2 − α − d √Cn α dt v Cn(rE ) 1 2 + u − √C dτ 2Cn dτ − u  n    (14) t q 2 2 2 and α − α d√Cn dϕ α 1 Cn =0, Δτ = 1 Δt . − − √Cn 2Cn dτ − dτ R R     p   v − C (r ) u n R α dt u 1 = const = K, (15) t q − √C dτ Then by (11),  n  dϕ 1 C = const = h , (16) 1 α 2 n dτ Δτ − √Cn(rR ) R = . (12) α and ds2 = g dxμdxν becomes, ΔτE 1  μν − √Cn(rE )   α dt 2 If z regular pulses of light are emitted, the emitted and 1 received frequencies are, −√Cn dτ −    (17) 1 2 2 z z α − d√Cn dϕ ν = , ν = , 1 Cn = 1 . E R − −√C dτ − dτ ΔτE ΔτR  n      so by (12), Rearrange (17) for,

1 ˙2 2 1 α 2 α t α d√Cn 1 1 1 Cn = . (18) ΔνR − √Cn(rE ) 2 2 = −√Cn ϕ˙ − −√Cn dϕ − ϕ˙ Δν 1 α  ≈      E C (r ) − √ n R Substituting (15) and (16) into (18) gives,   α 1 1 2 2 1 + , d√Cn Cn α K 2 + Cn 1 + 1 C = 0 . ≈ 2  −  dϕ h2 − √C − h2 n Cn(rR) Cn(rE )     n    q q Setting u = 1 reduces (18) to, whence, √Cn

du 2 α Δν νR νE α 1 1 + u2 = E + u + αu3 , (19) = − = dϕ h2 νE νE ≈ 2  −    Cn(rR) Cn(rE ) (K2 1) 1 1 q q  where E = − . The term αu3 represents the general- = πα = h2 relativistic perturbation of the Newtonian orbit. GR − GE   du Aphelion and perihelion occur when dϕ = 0, so by (19), 3 3 1 1 α = π sin χa χ0 . 3 2 κρ − G − G αu u + 2 u + E = 0 , (20) s 0  R E  − h

44 S. J. Crothers. Introducing Distance in General Relativity: Changes for the Standard Tests and the Cosmological Large-Scale October, 2005 PROGRESS IN PHYSICS Volume 3

Let u = u1 at aphelion and u = u2 at perihelion, so Let the radius of curvature of a great circle at closest u1 6 u 6 u2. One then finds in the usual way that the angle approach be Cn(rc). Now when there is no mass present, Δϕ between aphelion and subsequent perihelion is, (23) becomes p du 2 3α + u2 = F, Δϕ = 1 + u + u π . dϕ 4 1 2    
and has solution, Therefore, the angular advance ψ between successive perihelia is, u = u sin ϕ C (r ) = C (r) sin ϕ , c ⇒ n c n p p 3απ 3απ 1 1 and ψ = u1+u2 = + = 2 1 2 2 C (r ) C (r ) uc = = F. n 1 n 2 ! C (r )
(21) n c p p1 1 If = 3απ2 + , p G1 G2   Cn(r) α, Cn(r) > Cn(r ) ,  a where G and G are the measurable circumferences of 1 2 and p at closest approach,p thenp great circles at aphelion and at perihelion. Thus, to correctly u = uc > ua determine the value of ψ, the values of the said circum- du = 0 at u = u , ferences must be ascertained by direct measurement. Only dϕ c the circumferences are measurable in the gravitational field. 2 The radii of curvature and the proper radii must be calculated so F = uc (1 ucα), and (23) becomes, from the circumference values. − du 2 If the field is weak, as in the case of the Sun, one maytake + u2 = u2 (1 u α) + αu3 . (24) G 2πr, for r as an approximately “measurable” distance dϕ c − c   from≈ the gravitating sphere to a spacetime event. In such a Equation (24) must have a solution close to flat space- situation equation (21) becomes, time, so let 3απ 1 1 u = u sin ϕ + αw(ϕ) . ψ + . (22) c ≈ 2 r r  1 2  Putting this into (24) and working to first order in α, In the case of the Sun, α 3000 m, and for the planet gives Mercury, the usual value of ψ≈ 43 arcseconds per century dw ≈ 2 cos ϕ + 2w sin ϕ = u2 sin3 ϕ 1 , is obtained from (22). I emphasize however, that this value dϕ c −   is a Euclidean approximation for a weak field. In a strong or
field equation (22) is entirely inappropriate and equation d 1 (w sec ϕ) = u2 sec ϕ tan ϕ sin ϕ sec2 ϕ , (21) must be used. Unfortunately, this means that accurate dϕ 2 c − − solutions cannot be obtained since there is no obvious way
of obtaining the required circumferences in practise. This and so, aspect of Einstein’s theory seriously limits its utility. Since 1 w = u2 1 + cos2 ϕ sin ϕ + A cos ϕ , the relativists have not detected this limitation the issue has 2 c − not previously arisen in general. where A is an integration constant. If the
photon originates at 2 infinity in the direction ϕ = 0, then w(0) = 0, so A = uc , 6 Deflection of light and − 1 1 In the case of a photon, equation (17) becomes, u = u 1 αu sin ϕ + αu2 (1 cos ϕ)2 , (25) c − 2 c 2 c −   α dt 2 1 to first order in α. Putting u = 0 and ϕ = π + Δϕ into (25), −√C dτ −  n   then to first order in Δϕ, 1 2 2 α − d√Cn dϕ 2 1 Cn = 1 , 0 = ucΔϕ + 2αuc , − −√C dτ − dτ −  n      so the angle of deflection is, which leads to,

2 2α 2α 4πα du 2 3 Δϕ = 2αuc = = 1 = , + u = F + αu . (23) C (r ) n n Gc dϕ n c r r + n   c − 0 p  

S. J. Crothers. Introducing Distance in General Relativity: Changes for the Standard Tests and the Cosmological Large-Scale 45 Volume 3 PROGRESS IN PHYSICS October, 2005

Gc > Ga . not just as a radius in the gravitational field, but also asa measurable radius in the field. This is not correct. The only At a grazing trajectory to the surface of the body, measurable distance in the gravitational field is the aforesaid circumference of a great circle, from which the radius of Gc = Ga = 2π Cn(ra) , curvature Cn(r) and the proper radius Rp(r) must be p calculated, thus, 3 p C (r ) = sin χ χ , G n a κρ a − 0 C (r) = , s 0 n 2π p so then p Rp(r) = g11 dr . 3 3 − 2 κρ sin χa χ0 Z q 0 − 2 Δϕ = = 2 sin χa χ . (26) Only in the weak field, where the spacetime curvature q 3 − 0 κρ sin χa χ0 is very small, can Cn(r) be taken approximately as the 0 − Euclidean value r, thereby making R (r) C (r) r, q p p n as in flat spacetime. In a strong field this cannot≡ be done.≡ For the Sun [5], p Consequently, the problem arises as to how to accurately 1 measure the required great circumference? The correct de- sin χ χ , a − 0 ≈ 500 termination, for example, of the circumferences of great circles at aphelion and perihelion seem to be beyond practical so the deflection of light grazing the limb of the Sunis, determination. Any method adopted for determining the re- 2 quired circumference must be completely independent of any Δϕ 1.6500 . ≈ 5002 ≈ Euclidean quantity since, other than the great circumference itself, only non-Euclidean distances are valid in the gravita- Equation (26) is an interesting and quite surprising result, tional field, being determined by it. Therefore, anything short for sin χa χ0 gives the ratio of the “naturally measured” of physically measuring the great circumference will fail. fall velocity− of a free test particle falling from rest at infinity Consequently, General Relativity, whether right or wrong as down to the surface of the spherical body, to the speed of theories go, suffers from a serious practical limitation. light in vacuo. Thus, The value of the r-parameter is coordinate dependent the deflection of light grazing the limb ofa and is rightly determined from the coordinate independent spherical gravitating body is twice the square value of the circumference of the great circle associated of the ratio of the fall velocity of a free test with a spacetime event. One cannot obtain a circumference particle falling from rest at infinity down to the for the great circle of a given spacetime event, and hence surface, to the speed of light in vacuo, i .e ., the related radius of curvature and associated proper radius, from the specification of a coordinate radius, because the v 2 4GM Δϕ = 2 sin2 χ χ = 2 a = g , latter is not unique, being conditioned by arbitrary constants. a − 0 c c2R   ca The coordinate radius is therefore superfluous. It is for this

reason that I completely eliminated the coordinate radius where Rca is the radius of curvature of the body, Mg the active mass, and G is the gravitational constant. The quantity from the metric for the gravitational field, to describe the v is the escape velocity, metric in terms of the only quantity that is measurable in the a gravitational field — the great circumference (see also [6]). 2GMg The presence of the r-parameter has proved misleading to v = . a the relativists. Stavroulakis [8, 9, 10] has also completely s Rca eliminated the r-parameter from the equations, but does not make use of the great circumference. His approach is 7 Practical constraints and general comment formally correct, but rather less illuminating, because his resulting line element is in terms of the a quantity which is Owing to their invalid assumptions about the r-parameter [7], not measurable in the gravitational field. One cannot obtain the relativists have not recognised the practical limitations an explicit expression for the great circumference in terms associated with the application of General Relativity. It is of the proper radius. now clear that the fundamental element of distance in the As to the cosmological large-scale, I have proved else- gravitational field is the circumference of a great circle, where [11] that General Relativity adds nothing to Special centred at the heart of an extended spherical body and passing Relativity. Einstein’s field equations do not admit of solutions through a spacetime event external thereto. Heretofore the when the cosmological constant is not zero, and they do orthodox theorists have incorrectly taken the r-parameter, not admit of the expanding universe solutions alleged by

46 S. J. Crothers. Introducing Distance in General Relativity: Changes for the Standard Tests and the Cosmological Large-Scale October, 2005 PROGRESS IN PHYSICS Volume 3

the relativists. The lambda “solutions” and the expanding 5. Schwarzschild K. On the gravitational field of a sphere universe “solutions” are the result of such a muddleheaded- of incompressible fluid according to Einstein’s theory. ness that it is difficult to apprehend the kind of thoughtless- Sitzungsber. Preuss. Akad. Wiss., Phys. Math. Kl., 1916, ness that gave them birth. Since Special Relativity describes 424 (arXiv: physics/9912033, www.geocities.com/theometria/ an empty world (no gravity) it cannot form a basis for any Schwarzschild2.pdf). cosmology. This theoretical result is all the more interesting 6. Crothers S. J. On the vacuum field of a sphere of incom- owing to its agreement with observation. Arp [12], for in- pressible fluid. Progress in Physics, 2005, v. 2, 43–47. stance, has adduced considerable observational data which 7. Crothers S. J. On the geometry of the general solution for the is consistent on the large-scale with a flat, infinite, non- vacuum field of the point-mass. Progress in Physics, 2005, expanding Universe in Heraclitian flux. Bearing in mind v. 2, 3–14. that both Special Relativity and General Relativity cannot 8. Stavroulakis N. A statical smooth extension of Schwarz- yield a spacetime on the cosmological “large-scale”, there schild’s metric. Lettere al Nuovo Cimento, 1974, v. 11, 8 is currently no theoretical replacement for Newton’s cos- (www.geocities.com/theometria/Stavroulakis-3.pdf). mology, which accords with deep-space observations for a 9. Stavroulakis N. On the Principles of General Relativity and flat space, infinite in time and in extent. The all pervasive the SΘ(4)-invariant metrics. Proc. 3rd Panhellenic Congr. roleˆ given heretofore by the relativists to General Relativity, Geometry, Athens, 1997, 169 (www.geocities.com/theometria/ can be justified no longer. General Relativity is a theory of Stavroulakis-2.pdf). only local phenonomea, as is Special Relativity. 10. Stavroulakis N. On a paper by J. Smoller and B. Temple. Another serious shortcoming of General Relativity is its Annales de la Fondation Louis de Broglie, 2002, v. 27, 3 current inability to deal with the gravitational interaction of (www.geocities.com/theometria/Stavroulakis-1.pdf). two comparable masses. It is not even known if Einstein’s 11. Crothers S. J. On the general solution to Einstein’s vacuum field for the point-mass when λ = 0 and its implications for theory admits of configurations involving two or more mass- 6 es [13]. This shortcoming seems rather self evident, but app- relativistic cosmology, Progress in Physics, 2005, v. 3, 7–18. rently not so for the relativists, who routinely talk of black 12. Arp, H. Observational cosmology: from high redshift galaxies hole binary systems and colliding black holes (e .g. [14]), to the blue pacific, Progress in Physics, 2005, v. 3, 3–6. aside of the fact that no theory predicts the existence of black 13. McVittie G. C. Laplace’s alleged “black hole”. The Ob- holes to begin with, but to the contrary, precludes them. servatory, 1978, v. 98, 272 (www.geocities.com/theometria/ McVittie.pdf). 14. Misner C. W., Thorne K. S., Wheeler J. A. Gravitation. Acknowledgement W. H. Freeman and Company, New York, 1973. I am indebted to Dr. Dmitri Rabounski for drawing my attention to a serious error in a preliminary draft of this paper, and for additional helpful suggestions.

Dedication

I dedicate this paper to the memory of Dr. Leonard S. Abrams: (27 Nov. 1924 — 28 Dec. 2001).

References

1. Hilbert D. Nachr. Ges. Wiss. Gottingen, Math. Phys. Kl., 1917, 53 (arXiv: physics/0310104, www.geocities.com/theometria/ hilbert.pdf). 2. Schwarzschild K. On the gravitational field of a mass point according to Einstein’s theory. Sitzungsber. Preuss. Akad. Wiss., Phys. Math. Kl., 1916, 189 (arXiv: physics/9905030, www.geocities.com/theometria/schwarzschild.pdf). 3. Crothers S. J. On the general solution to Einstein’s vacuum field and it implications for relativistic degeneracy. Progress in Physics, 2005, v. 1, 68–73. 4. Crothers S. J. On the ramifications of the Schwarzschild spacetime metric. Progress in Physics, 2005, v. 1, 74–80.

S. J. Crothers. Introducing Distance in General Relativity: Changes for the Standard Tests and the Cosmological Large-Scale 47 Volume 3 PROGRESS IN PHYSICS October, 2005

A Re-Examination of Maxwell’s Electromagnetic Equations

Jeremy Dunning-Davies Department of Physics, University of Hull, Hull, England E-mail: [email protected]

It is pointed out that the usual derivation of the well-known Maxwell electromagnetic equations holds only for a medium at rest. A way in which the equations may be modified for the case when the mean flow of the medium is steady anduniformis proposed. The implication of this for the problem of the origin of planetary magnetic fields is discussed.

1 Introduction and the latter two to εμ ∂2 H Maxwell’s electromagnetic equations are surely among the 2H = . best known and most widely used sets of equations in phys- ∇ c2 ∂t2 ics. However, possibly because of this and since they have Therefore, in this special case, provided the medium is been used so successfully in so many areas for so many at rest, both E and H satisfy the well-known wave equation. years, they are, to some extent, taken for granted and used However, it has been shown [1] that, if the mean flow is stea- with little or no critical examination of their range of validity. dy and uniform, and, therefore, both homentropic and irro- This is particularly true of the two equations tational, the system of equations governing small-amplitude homentropic irrotational wave motion in such a flow reduces 1 ∂B E = to the equation ∇ × − c ∂t 1 D2ϕ 2ϕ = . and ∇ c2 Dt2 1 ∂D H = 4πj + . which is sometimes referred to as the convected, or progress- ∇ × c ∂t ive, wave equation. The question which remains is, for the Both these equations are used widely but, although the case of a medium not at rest, should Maxwell’s electromag- point is made quite clearly in most elementary, as well as netic equations be modified so as to reduce to this progressive more advanced, textbooks, it is often forgotten that these wave equation in the case of a non-conducting medium with equations apply only when the medium involved is assumed no charge present? to be at rest. This assumption is actually crucial in the derivation of these equations since it is because of it that 2 Generalisation of Maxwell’s equations it is allowable to take the operator d/dt inside the integral sign as a partial derivative and so finally derive each of In the derivation of the above equations. This leaves open the question of what μ ∂H happens if the medium is not at rest? E = ∇ × − c ∂t As is well known, for a non-conducting medium at rest, Maxwell’s electromagnetic equations, when no charge is it proves necessary to consider the integral present, reduce to μ d B dS μ ∂H − c dt ∙ E = 0 , E = , Z ∇ ∙ ∇ × − c ∂t and interchange the derivative and the integral. This operat- ε ∂E ion may be carried out only for a medium at rest. However, H = 0 , H = , ∇ ∙ ∇ × − c ∂t if the medium is moving, then the surface S in the integral will be moving also, and the mere change of S in the field where D = εE, B = μH and μ, ε are assumed constant in B will cause changes in the flux. Hence, following Abraham time. and Becker [2], a new kind of differentiation with respect to The first two equations are easily seen to lead to time is defined by the symbol B˙ as follows:

εμ ∂2 E d 2E = , B dS = B˙ dS . (a) 2 2 dt ∙ ∙ ∇ c ∂t Z Z

48 J. Dunning-Davies. A Re-Examination of Maxwell’s Electromagnetic Equations October, 2005 PROGRESS IN PHYSICS Volume 3

Here, B˙ is a vector, the flux of which across the moving or, noting that surface equals the rate of increase with time of the flux of (u B) = u ( B) B ( u) + (B ) u (u ) B , B across the same surface. In order to find B˙ , the exact ∇ × × ∇ ∙ − ∇ ∙ ∙ ∇ − ∙ ∇ details of the motion of the surface concerned must be ∂B known. Suppose this motion described by a vector , which B˙ = + (u ) B + B ( u) (B ) u . u ∂t ∙ ∇ ∇ ∙ − ∙ ∇ is assumed given for each element dS of the surface and is However, if the mean flow is steady and uniform and, the- the velocity of the element. refore, both homentropic and irrotational, the fluid velocity, Let S be the position of the surface S at time (t dt) 1 − u, will be constant and this latter equation will reduce to and S2 the position at some later time t. S2 may be obtained from S1 by giving each element of S1 a displacement udt ∂B DB . The surfaces and , together with the strip produced B˙ = + (u ) B = , S1 S2 ∂t ∙ ∇ Dt during the motion, bound a volume dt u dS. ∙ The rate of change with time of the flux of B across S that is, for such flow, B˙ becomes the well-known Euler R may be found from the difference between the flux across S2 derivative. It might be noted, though, that, for more general at time t and that across S1 at time (t dt); that is flows, the expression for B˙ is somewhat more complicated. − It follows that, if the mean flow is steady and uniform, d Bt dS2 Bt dt dS1 B dS = ∙ − − ∙ , the Maxwell equation, mentioned above, becomes dt ∙ dt Z R R μ DH μ ∂H where the subscript indicates the time at which the flux is E = = + (u ) H . ∇ × − c Dt − c ∂t ∙ ∇ measured.   The divergence theorem may be applied at time t to the Also, in this particular case, the remaining three Maxwell volume bounded by S1, S2 and the strip connecting them. equations will be Here the required normal to S2 will be the outward pointing normal and that to S1 the inward pointing normal. Also, a E = 0 , H = 0 , surface element of the side face will be given by ds udt. ∇ ∙ ∇ ∙ Then, the divergence theorem gives × ε DE ε ∂E H = = + (u ) E , ∇ × c Dt c ∂t ∙ ∇   Bt dS2 +dt B ds u Bt dS1 =dt ( B)u dS. with this form for the final equation following in a manner S ∙ ∙ × − S ∙ ∇∙ ∙ Z 2 I Z 1 Z similar to that adopted above when noting that, for a steady, Also uniform mean flow, ∂/∂t is replaced by D/Dt in the equation for E. ∂B ∇ × Bt dt dS1 = Bt dS1 dS1dt . These four modified Maxwell equations lead to both E − ∙ ∙ − ∂t Z Z Z and H satisfying the above mentioned progressive wave Hence, equation, as they surely must.

Bt dS2 Bt dt dS1 = dt B˙ dS1 + 3 The origin of planetary magnetic fields ∙ − − ∙ ∙ Z Z Z It is conceivable that use of these modified Maxwell electro- + ( B)u dS B ds u . ∇ ∙ ∙ 1 − ∙ × magnetic equations could provide new insight into the prob- Z I  lem of the origin of planetary magnetic fields. This is a Using Stokes’ theorem, the final term on the right-hand problem which has existed, without a really satisfactory side of this equation may be written explanation, for many years. It would seem reasonable to expect all such fields to arise from the same physical mechan- B ds u = u B ds = (u B) dS , ism, although the minute detail might vary from case to case. ∙ × × ∙ ∇ × × ∙ I I Z n o The mechanism generally favoured as providing the best and so finally explanation for the origin of these fields was the dynamo mechanism, although the main reason for its adoption was the d ∂B B dS = + u ( B) (u B) dS . failure of the alternatives to provide a consistent explanation. dt ∙ ∂t ∇ ∙ − ∇ × × ∙ Z Z   However, Cowling [3] showed that there is a limit to the de- gree of symmetry encountered in a steady dynamo mechan- Therefore, the B˙ , introduced in (a) above, is given by ism; this result, based on the traditional electromagnetic ∂B equations of Maxwell, shows that the steady maintenance B˙ = + u ( B) (u B) ∂t ∇ ∙ − ∇ × × of a poloidal field is simply not possible — the result isin

J. Dunning-Davies. A Re-Examination of Maxwell’s Electromagnetic Equations 49 Volume 3 PROGRESS IN PHYSICS October, 2005

reality an anti-dynamo theorem which raises difficulties in understanding the observed symmetry of the dipole field. Following Alfven´ [4], it might be noted that, in a stat- ionary state, there is no electromagnetic field along a neutral line because that would imply a non-vanishing E, and so a time varying B. The induced electric field v ∇×B vanishes on the neutral line since B does. Thus, there× can be no electromotive force along the neutral line, and therefore the current density in the stationary state vanishes, the conduct- ivity being infinite. On the other hand, B does not vanish on the neutral line. By Maxwell’s usual∇× equations, the non- vanishing B and the vanishing current density are in contradiction∇× and so the existence of a rotationally symmetric steady-state dynamo is disproved. However, this conclusion may not be drawn if the modified Maxwell equations, alluded to earlier, are used, since, even in the steady state where the partial derivatives with respect to time will all be zero, the equation for B will reduce to ∇ × 1 ∂E ε B = j + ε + εv E v E ∇ × μ ∂t ∙ ∇ → μ ∙ ∇   and there is no reason why this extra term on the right- hand side should be identically equal to zero. Also, the non- vanishing of E will not imply a time varying B since, once again, there∇ × is an extra term v B remaining to equate with the E. It follows that an− electromagnetic∙ ∇ field may exist along∇× the neutral line under these circumstances. Hence, no contradiction occurs; instead, a consistent system of differential equations remains to be solved.

References

1. Thornhill C. K. Proc. Roy. Soc. Lond., 1993, v. 442, 495. 2. Abraham M. , Becker, R. The Classical Theory of Electricity and Magnetism. Blackie and Son Ltd. , London, 1932. 3. Cowling T. G. Monthly Notices of the Royal Astronomical Society, 1934, v. 94, 39. 4. Alfven´ H., Falthammar¨ C. G. Cosmical Electrodynamics. Oxford at the Clarendon Press, 1963.

50 J. Dunning-Davies. A Re-Examination of Maxwell’s Electromagnetic Equations October, 2005 PROGRESS IN PHYSICS Volume 3

Black Holes in Elliptical and Spiral Galaxies and in Globular Clusters

Reginald T. Cahill School of Chemistry, Physics and Earth Sciences, Flinders University, Adelaide 5001, Australia E-mail: [email protected]

Supermassive black holes have been discovered at the centers of galaxies, and also in globular clusters. The data shows correlations between the black hole mass and the elliptical galaxy mass or globular cluster mass. It is shown that this correlation is accurately predicted by a theory of gravity which includes the new dynamics of self- interacting space. In spiral galaxies this dynamics is shown to explain the so-called “dark matter” rotation-curve anomaly, and also explains the Earth based bore-hole g anomaly data. Together these effects imply that the strength of the self-interaction dynamics is determined by the fine structure constant. This has major implications for fundamental physics and cosmology.

4 Introduction This form is mandated by Galilean covariance under change of observer. A minimalist non-relativistic modelling Our understanding of gravity is based on Newton’s modelling of the dynamics for this velocity field gives a direct account of Kepler’s phenomenological laws for the motion of the of the various phenomena noted above; basically the New- planets within the solar system. In this model Newton took tonian formulation of gravity missed a key dynamical effect the gravitational acceleration field to be the fundamental that did not manifest within the solar system. dynamical degree of freedom, and which is determined by In terms of the velocity field Newtonian gravity dynamics the matter distribution; essentially via the “universal inverse involves using . to construct a rank-0 tensor that can be square law”. However the observed linear correlation be- related to the matter∇ density ρ. The coefficient turns out to tween masses of black holes with the masses of the “host” be the Newtonian gravitational constant G. elliptical galaxies or globular clusters suggests that either the ∂v formation of these systems involves common evolutionary . + (v. )v = 4πGρ . (2) dynamical processes or that perhaps some new aspect to ∇ ∂t ∇ −   gravity is being revealed. Here it is shown that if rather This is clearly equivalent to the differential form of than an acceleration field a velocity field is assumed to Newtonian gravity, .g = 4πGρ. Outside of a spherical be fundamental to gravity, then we immediately find that mass M (2) has solution∇ − these black hole effects arise as a space self-interaction ∗ dynamical effect, and that the observed correlation is simply 2GM v(r) = ˆr , (3) that MBH /M = α/2 for spherical systems, where α is the −r r fine structure constant (α = e2/~c = 1/137.036), as shown in Fig. 1. This dynamics also manifests within the Earth, as for which (1) gives the usual inverse square law revealed by the bore hole g anomaly data, as in Fig. 2. It also GM g(r) = ˆr . (4) offers an explanation of the “dark matter” rotation-velocity − r2 effect, as illustrated in Fig. 3. This common explanation for The simplest non-Newtonian dynamics involves the two a range of seemingly unrelated effects has deep implications rank-0 tensors constructed at 2nd order from ∂v /∂x for fundamental physics and cosmology. i j ∂v α β . +(v. )v + (trD)2 + tr(D2)= 4πGρ, (5) 5 Modelling gravity ∇ ∂t ∇ 8 8 −   Let us phenomenologically investigate the consequences of 1 ∂vi ∂vj D = + , (6) using a velocity field v(r, t) to be the fundamental dynamical ij 2 ∂x ∂x  j i  degree of freedom to model gravity. The gravitational accel- and involves two arbitrary dimensionless constants. The ve- eration field is then defined by the Euler form locity in (3) is also a solution to (5) if β = α, and we then − v(r+v(r, t)Δt, t+Δt) v(r, t) define g(r, t) lim − = α 2 2 ≡ Δt 0 Δt C(v, t) = (trD) tr(D ) . (7) → ∂v (1) 8 − = + (v. )v.   We assume v = 0, then (v. )v = 1 (v2). ∂t ∇ ∗ ∇ × ∇ 2 ∇

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Hence the modelling of gravity by (5) and (1) now 7 Minimal black hole systems involves two gravitational constants G and α, with α being the strength of the self-interaction dynamics, but which was There are two classes of solutions when matter is present. not apparent in the solar system dynamics. We now show The simplest is when the black hole forms as a consequence that all the various phenomena discussed herein imply that of the velocity field generated by the matter, this generates α is the fine structure constant 1/137 up to experimental what can be termed an induced minimal black hole. This is in ≈ errors [1]. Hence non-relativistic gravity is a more complex the main applicable to systems such as planets, stars, globular phenomenon than currently understood. The new key feature clusters and elliptical galaxies. The second class of solutions is that (5) has a one-parameter μ class of vacuum correspond to non-minimal black hole systems; these arise (ρ = 0) “black hole” solutions in which the velocity field when the matter congregates around a pre-existing “vacuum” self-consistently maintains the singular form black hole. The minimal black holes are simpler to deal with, α/4 particularly when the matter system is spherically symmetric. v(r) = μr− ˆr . (8) − In this case the non-Newtonian gravitational effects are con- This class of solutions will be seen to account for the fined to within the system. A simple way to arrive atthis “black holes” observed in galaxies and globular cluster. As property is to solve (14) perturbatively. When the matter well this velocity field, from (1), gives rise to a non-“inverse density is confined to a sphere of radius R we find on square law” acceleration iterating (14) that the “dark matter” density is confined to that sphere, and that consequently g (r) has an inverse square αμ (1+α/4) g(r) = r− ˆr . (9) law behaviour outside of the sphere. Iterating (14) once we − 4 find inside radius R that This turns out to be the cause of the so-called “dark- matter” effect observed in spiral galaxies. For this reason we α R ρ (r) = sρ(s)ds + O(α2). (16) define DM 2r2 α Zr ρ (r) = (trD)2 tr(D2) , (10) DM 32πG −   and that the total “dark matter” so that (5) and (1) can be written as R 2 .g = 4πGρ 4πGρ , (11) MDM 4π r drρ (r) = ∇ − − DM ≡ DM Z0 which shows that we can think of the new self-interaction (17) 4πα R α dynamics as generating an effective “dark matter” density. = r2drρ(r) + O(α2) = M + O(α2) , 2 2 Z0 6 Spherical systems where M is the total amount of (actual) matter. Hence, to O(α), M /M = α/2 independently of the matter density It is sufficient here to consider time-independent and spheric- DM profile. This turns out to be a very useful property asknow- ally symmetric solutions of (5) for which v is radial. Then ledge of the density profile is then not required in order we have the integro-differential form for (5) to analyse observational data. Fig. 1 shows the value of

ρ(s) + ρDM v (s) MBH /M for, in particular, globular clusters M15 and G1 v2(r) = 2G d 3s , (12) r s and highly spherical “elliptical” galaxies M32, M87 and Z | − |
NGC 4374, showing that this ratio lies close to the “α/2- 2 α v vv0 line”, where α is the fine structure constant 1/137. How- ρ v(r) = + . (13) ≈ DM 8πG 2r2 r ever for the spiral galaxies their M /M values do not   DM 2 1 4
cluster close to the α/2-line. Hence it is suggested that these as r s = 4πδ (r s). This then gives ∇ | − | − − spherical systems manifest the minimal black hole dynamics r outlined above. However this dynamics is universal, so that 2 8πG 2 v (r) = s ds ρ(s) + ρDM v (s) + any spherical system must induce such a minimal black r Z0 hole mode, but for which outside of such a system only h
i (14) ∞ the Newtonian inverse square law would be apparent. So + 8πG sds ρ(s) + ρDM v (s) this mode must also apply to the Earth, which is certainly Zr h
i on doing the angle integrations. We can also write (5) as a a surprising prediction. However just such an effect has non-linear differential equation manifested in measurements of g in mine shafts and bore holes since the 1980’s. It will now be shown that data from vv0 2 these geophysical measurements give us a very accurate 2 +(v0) +vv00 = 4πGρ(r) 4πGρ v(r) . (15) r − − DM determination of the value of α in (5).
52 R. T. Cahill. Black Holes in Elliptical and Spiral Galaxies and in Globular Clusters October, 2005 PROGRESS IN PHYSICS Volume 3

Fig. 1: The data shows Log10[MBH /M] for the “black hole” or “dark matter” masses MBH for a variety of spherical matter systems with masses M, shown by solid circles, plotted against Log10[M/M0], where M0 is the solar mass, showing agreement with the “α/2-line” (Log [α/2] = 2.44) predicted by (17), and ranging over 15 orders of magnitude. The “black hole” effect is the same phenomenon as 10 − the “dark matter” effect. The data ranges from the Earth, as observed by the bore hole g anomaly, to globular cluster M15 [5, 6] and G1 [7], and then to spherical “elliptical” galaxies M32 (E2), NGC 4374 (E1) and M87 (E0). Best fit to the data from these star systems gives α = 1/134, while for the Earth data in Fig. 2 α = 1/139. A best fit to all the spherical systems in the plot gives α = 1/136. In these systems the “dark matter” or “black hole” spatial self-interaction effect is induced by the matter. For the spiral galaxies, shown by the filled boxes, where here M is the bulge mass, the black hole masses do not correlate with the “α/2-line”. This is because these systems form by matter in-falling to a primordial black hole, and so these systems are more contingent. For spiral galaxies this dynamical effect manifests most clearly via the non-Keplerian rotation-velocity curve, which decrease asymptotically very slowly, as shown in Fig. 3, as determined by the small value of α 1/137. The galaxy data is from Table 1 of [8, updated]. ≈

8 Bore hole g anomaly additional “dark matter mass” in (17). Inside the Earth we see that (18) gives a g (r) different from Newtonian gravity. To understand this bore hole anomaly we need to compute This has actually been observed in mine/borehole measure- the expression for g (r) just beneath and just above the ments of g (r) [2, 3, 4], though of course there had been no surface of the Earth. To lowest order in α the “dark-matter” explanation for the effect, and indeed the reality of the effect density in (16) is substituted into (14) finally gives via (1) was eventually doubted. The effect is that g decreases more the acceleration slowly with depth than predicted by Newtonian gravity. Here (1 + α ) GM the corresponding Newtonian form for g (r) is 2 , r > R , r2 G M  N , r > R ,  r r2  4πG  s2 ds ρ(s) + g (r) =  (19)  2 Newton  r g (r) =  r 0 (18)  4πGN  Z  s2 dsρ(s) , r < R ,  2παG r R r2 Z0 + 2 s0ds0ρ(s0) ds ,  r 0 s α  Z Z  with GN = (1 + )G. The gravity residual is defined as the  2  r < R . difference between the Newtonian g (r) and the measured   g (r), which we here identify with the g (r) from (18), This gives Newton’s “inverse square law” for r > R, but in which we see that the effective Newtonian gravitational Δg (r) g (r) g (r) . (20) α ≡ Newton − observed constant is GN = (1 + 2 )G, which is different to the fund- amental gravitational constant G in (2). This caused by the Then Δg (r) is found to be, to 1st order in R r, i. e. −

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mGal 1

km −1.5 −1 −0.5 0.5

−1

−2

−3

−4

−5

Fig. 2: The data shows the gravity residuals for the Greenland Ice Cap [4] measurements of the g (r) profile, defined as Δg (r) = 3 = g g , and measured in mGal (1 mGal = 10− Newton observed Fig. 3: Data shows the non-Keplerian rotation-speed curve v for 2 − ◦ cm/sec ), plotted against depth in km. Using (21) we obtain the spiral galaxy NGC 3198 in km/s plotted against radius in kpc/h. 1 α− = 139 5 from fitting the slope of the data, as shown. ± Lower curve is the rotation curve from the Newtonian theory for an exponential disk, which decreases asymptotically like 1/√r. The upper curve shows the asymptotic form from (24), with the decrease near the surface, determined by the small value of α. This asymptotic form is caused by the primordial black holes at the centres of spiral galaxies, and 0, r > R , which play a critical role in their formation. The spiral structure is Δg(r) = (21) caused by the rapid in-fall towards these primordial black holes.  2παG ρ(R)(R r) , r < R ,  − N − which is the form actually observed [4], as shown in Fig. 2. of the consequences of this time evolution: (i) because the Gravity residuals from a bore hole into the Greenland Ice acceleration field falls off much slower than the Newtonian Cap were determined down to a depth of 1.5km. The ice had inverse square law, as in (9), this in-fall would happen very a measured density of ρ = 930 kg/m3, and from (21), using rapidly, and (ii) the resultant in-flow would result in the 11 3 2 1 GN = 6.6742×10− m s− kg− , we obtain from a linear fit matter rotating much more rapidly than would be predicted 1 to the slope of the data points in Fig. 2 that α− = 139 5, by Newtonian gravity, (iii) so forming a quasar which, after ±1 which equals the value of the fine structure constant α− = the in-fall of some of the matter into the black hole has = 137.036 to within the errors, and for this reason we identify ceased, would (iv) result in a spiral galaxy exhibiting non- the constant α in (5) as being the fine structure constant. Then Keplerian rotation of stars and gas clouds, viz the so-called we arrive at the conclusion that there is indeed “black hole” “dark matter” effect. The study of this time evolution will be or “dark matter” dynamics within the Earth, and that from far from simple. Here we simply illustrate the effectiveness (17) we have again for the Earth that MBH /M = α/2, as is of the new theory of gravity in explaining this “dark matter” also shown in Fig. 1. or non-Keplerian rotation-velocity effect. This “minimal black hole” effect must also occur within We can determine the star orbital speeds for highly non- stars, although that could only be confirmed by indirect spherical galaxies in the asymptotic region by solving (15), observations. This effect results in g (r) becoming large at the for asymptotically where ρ 0 the velocity field will be center, unlike Newtonian gravity, which would affect nuclear approximately spherically symmetric≈ and radial; nearer in reaction rates. This effect may already have manifested in we would match such a solution to numerically determined the solar neutrino count problem [9, 10]. To study this will solutions of (5). Then (15) has an exact non-perturbative require including the new gravity dynamics into solar models. two-parameter (K and RS) analytic solution,

α 1/2 9 Spiral galaxies 1 1 RS 2 v (r) = K + ; (22) r RS r ! We now consider the situation in which matter in-falls around   an existing primordial black hole. Immediately we see some this velocity field then gives using (1) the non-Newtonian

54 R. T. Cahill. Black Holes in Elliptical and Spiral Galaxies and in Globular Clusters October, 2005 PROGRESS IN PHYSICS Volume 3

asymptotic acceleration the implicit question posed in this paper is that, given the physical existence of such a velocity field, are the Newtonian α K2 1 α R 2 and/or General Relativity formalisms the appropriate de- g (r) = + S , (23) 2 scriptions of the velocity field dynamics? The experimental 2 r 2rRS r !   evidence herein implies that a different dynamics is required applicable to the outer regions of spiral galaxies. to be developed, because when we generalise the velocity We then compute circular orbital speeds using v (r) = field modelling to include a spatial self-interaction dynamics, = rg(r) giving the predicted “universal rotation-speed◦ the experimental evidence is that the strength of this dynam- curve” ics is determined by the fine structure constant, α. This is an p α 1/2 extraordinary outcome, implying that gravity is determined K 1 α R 2 v (r) = + S . (24) by two fundamental constants, G and α. As α clearly is not ◦ 2 r 2R r S   ! in Newtonian gravity nor in General Relativity the various observational and experimental data herein is telling us that Because of the α dependent part this rotation-speed curve neither of these theories of gravity is complete. The modell- falls off extremely slowly with r, as is indeed observed for ing discussed here is non-relativistic, and essentially means spiral galaxies. This is illustrated in Fig. 3 for the spiral that Newtonian gravity was incomplete from the very beginn- galaxy NGC 3198. ing. This happened because the self-interaction dynamics did not manifest in the solar system planetary orbit motions, and 10 Interpretation and discussion so neither Kepler nor later Newton were aware of the intrinsic complexity of the phenomenon of gravity. General Relativity Section 2 outlines a model of space developed in [1, 11] was of course constructed to agree with Newtonian gravity in which space has a “substratum” structure which is in in the non-relativistic limit, and so missed out on this key differential motion. This means that the substratum in one non-relativistic self-interaction effect. region may have movement relative to another region. The Given both the experimental detection of the velocity substratum is not embedded in a deeper space; the substratum field, including in particular the recent discovery [11] ofan itself defines space, and requiring that, at some level ofdes- in-flow velocity component towards the Sun in the 1925/26 cription, it may be approximately described by a “classical” Miller interferometer data, and in agreement with the speed 3-vector velocity field v(r, t). Then the dynamics of space value from (3) for the Sun, together with the data from involves specifying dynamical equations for this vector field. various observations herein, all showing the presence of Here the coordinates r is not space itself, but a means of the α dependent effect, we should also discuss the physical labelling points in space. Of course in dealing with this interpretaion of the vacuum “black hole” solutions. These dynamics we are required to define v(r, t) relative to some are different in character from the so-called “black holes” set of observers, and then the dynamical equations must be of General Relativity: we use the same name only because such that the vector field transforms covariantly with respect these new “black holes” have an event horizon, but otherwise to changes of observers. As noted here Newtonian gravity they are completely different. In particular the mathematical itself may be written in terms of a vector field, as well existence of such vacuum “black holes” in General Relativity as in terms of the usual acceleration field g(r, t). General is doubtful. In the new theory of gravity these black holes Relativity also has a special class of metric known as the are exact mathematical solutions of the velocity equations Panleve-Gullstrand´ metrics in which the metrics are specified and correspond to self-sustaining in-flow singularities, that by a velocity field. Most significantly the major tests of is, where the in-flow speed becomes very large within the General Relativity involved the Schwarzschild metric, and classical description. This singularity would then require a this metric belongs to the Panleve-Gullstrand´ class. So in quantum description to resolve and explain what actually both cases these putatively successful models of gravity happens there. The in-flow does not involve any conserved involved, in fact, velocity fields, and so the spacetime metric measure, and there is no notion of this in-flow connecting to description was not essential. As well there are in total some wormholes etc. The in-flow is merely a self-destruction of seven experiments that have detected this velocity field [12], space, and in [11] it is suggested that space is in essence an so that it is more than a choice of dynamical degree of “information” system, in which case the destruction process freedom: indeed it is more fundamental in the sense that is easier to comprehend. As for the in-flow into the Earth, from it the acceleration field or metric may be mathematically which is completely analogous to the observed in-flow to- constructed. wards the Sun, the in-flow singularities or “black holes” are Hence the evidence, both experimental and theoretical, located at the centre of the Earth, but it is unclear whether is that space should be described by a velocity field. This there is one such singularity or multiple singularities. The implies that space is a complex dynamical system which is experimental existence of the Earth-centred in-flow singular- best thought of as some kind of “flow system”. However ity is indirect, as it is inferred solely by the anomalous var-

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iation of g with depth, and that this variation is determined compatible with these recent observations. These develop- by the value of α. In the case of the globular clusters and ments clearly have major implications for cosmology and elliptical galaxies, the in-flow singularities are observed by fundamental physics. The various experiments that detected means of the large accelerations of stars located near the the velocity field are discussed in [11, 12]. centres of such systems and so are more apparent, and as This research is supported by an Australian Research shown here in all case the effective mass of the in-flow Council Discovery Grant. singularity is α/2 times the total mass of these systems. It is important to note here that even if we disregard the References theoretical velocity field theory, we would still be left with the now well established α/2 observational effect. But then 1. Cahill R. T. Trends in Dark Matter Research, ed. by Blain J. this velocity field theory gives a simple explanation for this Val, Nova Science Pub, NY, 2005; Apeiron, 2005, v. 12, No. 2, data, although that in itself does not exclude other theories 155–177. offering a different explanation. It is hard to imagine however 2. Stacey F. D. et al. Phys. Rev. D, 1981, v. 23, 2683. how either Newtonian gravity or General Relativity could 3. Holding S. C., Stacey F. D. & Tuck G. J. Phys. Rev. D, 1986, offer such a simple explanation, seeing that neither involves v. 33, 3487. α, and involve only G. As well we see that the new theory 4. Ander M. E. et al. Phys. Rev. Lett., 1989, v. 62, 985. of gravity offers a very effective explanation for the rotation characteristics of spiral galaxies; the effect here being that 5. Gerssen J., van der Marel R. P., Gebhardt K., Guhathakurta P., the vacuum black hole(s) at the centres of such galaxies do Peterson R. & Pryor C. Astrophys. J., 2002, v. 124, 3270; Addendum 2003, v. 125, 376. not generate an acceleration field that falls off with distance according the inverse square law, but rather according to (23). 6. Murphy B. W., Cohn H. N., Lugger P.N., & Dull J. D. Bull. of Remarkably this is what the spiral galaxy data shows. This American Astron. Society, 1994, v. 26, No. 4, 1487. means that the so-called “dark matter” effect is not about 7. Gebhardt K., Rich R. M., & Ho L. C. Astrophys. J., 2002, a new and undetected form of matter. So the success of v. L41, 578. the new velocity field dynamics is that one theory explains 8. Kormendy J. & Richstone D. Astronomy and Astrophysics, a whole range of phenomena: this is the hallmark of any 1995, v. 33, 581. theory, namely economy of explanation. 9. Davies R. Phys. Rev. Lett., 1964, v. 12, 300. 10. Bahcall J. Phys. Rev. Lett., 1964, v. 12, 303. 11. Cahill R. T., Process Physics: From Information Theory to 11 Conclusion Quantum Space and Matter. Nova Science Pub., NY, 2005. 12. Cahill R. T. Progress in Physics, 2005, v. 3, 25–29. The observational and experimental data confirm that the massive black holes in globular clusters and galaxies are necessary phenomena within a theory for gravity which uses a velocity field as the fundamental degree of freedom. This involves two constants G and α and the data reveals that α is the fine structure constant. This suggests that the spatial self-interaction dynamics, which is missing in the Newtonian theory of gravity, may be a manifestation at the classical level of the quantum behaviour of space. It also emerges that the “black hole” effect and the “dark matter” effect are one phenomenon, namely the non-Newtonian acceleration caused by singular solutions. This effect must manifest in planets and stars, and the bore hole g anomaly confirms that for planets. For stars it follows that the structure codes should be modified to include the new spatial self-interaction dynamics, and to determine the effect upon neutrino count rates. The data shows that spherical systems with masses varying over 15 orders of magnitude exhibit the α-dependent dynamical effect. The non-Newtonian gravitational acceleration of pri- mordial black holes will cause rapid formation of quasars and stars, explaining why recent observations have revealed that these formed very early in the history of the universe. In this way the new theory of gravity makes the big bang theory

56 R. T. Cahill. Black Holes in Elliptical and Spiral Galaxies and in Globular Clusters October, 2005 PROGRESS IN PHYSICS Volume 3

Is the Biggest Paradigm Shift in the History of Science at Hand?

Eit Gaastra Groningen, The Netherlands E-mail: [email protected]

According to a growing number of scientists cosmology is at the end of an era. This era started 100 years ago with the publication of Albert Einstein’s special theory of relativity and came to its height in the 1920s when the theory of relativity was used to develop the big bang model. However, at this moment there is a crisis within cosmology. More and more scientists openly doubt the big bang. There are alternatives for the theory of relativity as well as for the big bang model, but so far most scientists are scared to pass over Einstein.

1 Introduction to this stretching as if it is caused by the recessional velocity of galaxies in the big bang universe. The big bang model rests on three pillars [1]. This trinity is The third pillar was discovered in 1965. In 1948 a group the cosmology of the twentieth century. of cosmologists calculated that in the case of a big bang The first pillar is the Theory of General Relativity. In certain radiation still had to be left over from a period shortly 1905 Einstein came with his Theory of Special Relativity after the big bang. In 1965 such radiation was measured. which describes the behaviour of light and in 1916 he pub- This radiation (of 3 Kelvin) is now known as the cosmic lished a theory about gravity, the Theory of General Relat- background radiation and since 1965 it is seen as the big ivity. In publications in 1922 and 1924 the Russian math- proof of the big bang model. ematician Alexander Friedmann used the formulae of the General Theory of Relativity to prove that the universe 2 Alternatives for the theory of relativity was dynamic: either it expanded or it shrunk. In 1927 it was the Belgian priest and astronomer-cosmologist Georges Einstein unfolded his special theory of relativity in an article Lemaˆıtre, using the cosmological equations of Friedmann, in 1905, in which he states that the velocity of light is always who suggested for the first time that the universe once could constant relative to an observer. But the apparent constancy have sprung from a point of very high-density, the primaeval of the velocity of light can be explained differently. atom. Another link in the realization of the big bang model Gravitons or other not yet detected particles may act as was the Dutch astronomer-cosmologist Willem de Sitter, the medium that is needed by light to propagate itself. This who suggested in 1917, together with Einstein, the de-Sitter- is somewhat comparable to air molecules that are needed universe, which was based on the formulae of the General as a medium by sound to propagate itself. A theory that Theory of Relativity. The de-Sitter-universe has no mass, calls a medium into existence to explain the propagation of but has the feature that mass particles that form in it will light is called an aether theory. Aether theories created a accelerate away from each other. furore in the nineteenth century, but fell into oblivion after The second pillar on which the big bang model rests is 1905, because of the rise of the theory of relativity. However, the stretching of light in an expanding universe. In the 1920s the last decennium the aether concept is making a come Edwin Hubble discovered that certain dots in the night sky back and is getting more and more advocates, among whom are not stars but galaxies instead. From 1924 on he measured is the Italian professor of physics Selleri [2]. (Also more the distances of the galaxies and in 1929 he announced advocates because despite the announcements by Michelson that the wavelength of light of galaxies is shifted towards and Morley about the “null result”, their famous interfer- a longer wavelength. The further away the galaxy the more ometer 1887 experiment actually may have detected both “stretched” the light. At the time this stretching of light was absolute motion and the breakdown of Newtonian phys- explained with the big bang model of Lemaˆıtre. The universe ics [3].) could have sprung from a point of very high-density mass and Albert Einstein’s theory of General Relativity of 1916 ever since the universe would expand as a balloon. Because describes the movement of light and matter with the curvature of the expansion of the universe space in the universe would of space-time more accurately than Isaac Newton’s universal stretch and in that case light would stretch along with space. law of gravitation from the seventeenth century. There are The stretching of light of faraway galaxies is still explained alternatives, both for the Theory of General Relativity and this way, although a lot of astronomers customarily to refer Newtonian gravity. The physics professors Assis [4] and

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Ghosh [5] look at inertia and gravity as forces that are which are dust and asteroids) of planets and stars may exist caused by all the matter in the universe. This is called the between the stars, between galaxies and between clusters extended Mach principle, after Ernst Mach who suggested in of galaxies. Such remnants will eventually reach the very the nineteenth century that the inertia of any body is caused cold temperature (3 Kelvin) of the universe and send out by its interaction with the rest of the universe. radiation that corresponds with that temperature. Other exam- There is also the so-called pushing gravity concept, a ples of alternatives that can explain an equilibrium temper- gravity model with gravitons going in and out of matter and ature are direct energy exchange between photons or indirect by doing so pushing objects towards each other (on a macro- energy exchange between photons via gravitons or other scale, for instance a teacup that falls to the ground or stars small particles. A growing number of scientists looks at the that are pushed towards each other; on a subatomic level cosmic background radiation as a result of the equilibrium things are different). Pushing gravity too is an alternative temperature of a universe infinite in space and time. for both the Theory of General Relativity and Newtonian In the sixteenth century Thomass Digges was the first gravity. The pushing gravity concept was first suggested by scientist to advance a universe filled with an infinite number Nicolas Fatio de Duillier in the seventeenth century [6]. of stars. In the last decennium more and more scientists have An aether theory, the extended Mach principle as well as taken the line of an infinite universe filled with an infinite pushing gravity, takes the line that smaller particles (like number of galaxies. (Also because, despite all beliefs to the gravitons) that we cannot yet detect do exist. The three contrary, General Relativity may not predict an expanding theories can stand alone, but can be combined as well. universe; the Friedmann models and the Einstein-de Sitter The pushing gravity concept for instance, can be used as model may be invalid [12].) an explanation for the extended Mach principle. In a bizarre way individual photons and individual atoms seem to interfere with themselves in the famous two-slit 4 Clusters of galaxies at large distances? experiment in Quantum Mechanics. An aether theory can explain the baffling interference in a very simple way [7,8]. If there was no big bang, and if we live in an infinite That is why, with an aether theory, Quantum Mechanics may universe, then distances of faraway galaxies are much larger also be unsettled. Next to that the intriguing black holes, than presently thought. A few years back big bang cos- sprung from the mathematics of the theory of relativity, mologists concluded that the big bang ought to have taken may vanish by embracing the pushing gravity concept. place 13.7 billion years ago. Therefore within the big bang (Besides, black holes may not be predicted by General Rela- model objects are always less than 13.7 years old. Big bang tivity [9, 10].) astronomers observe certain galaxies with enormous shifts of the wavelength of light and therefore think these objects sent out their light very long ago, for instance 13 billion 3 Alternatives for the big bang years. With the tired light model in an infinite universe objects with such large shifts of the wavelength of light Fritz Zwicky suggested in 1929 that photons may lose energy will be at distances of more than 70 billion light-years. The while travelling through space, but so far his idea has always galaxies, which big bang astronomers now think they observe been overshadowed by the big bang explanation with stretch- at these large distances, may therefore be clusters of galaxies ing space. Zwicky’s explanation is known as the tired light in reality. concept and it is used by alternative thinking scientists as In the 1920s Edwin Hubble inaugurated a new era by part of a model that looks at the universe as infinite in finding that certain dots in the night sky are not stars,but time and space. In a tired light theory photons lose energy galaxies instead. Only then did scientists realize that certain by interaction with gravitons or other small particles. The objects are at much larger distances than accepted at the tired light model can be combined with an aether theory, the time. Within the years to come new telescopes will deliver extended Mach principle and pushing gravity. sharper images of faraway objects which are now addressed Next to alternatives for the theory of relativity and the as galaxies. The big bang model already has difficulty ex- stretching of light, scientists have found alternatives for the plaining galaxies in the very early universe, because in the third pillar of current conventional cosmology, the cosmic big bang formed, loose matter, needs time to aggregate into background radiation discovered in 1965. That a cosmic stars and galaxies. If it turns out that not only galaxies but background radiation can originate as a result of the equi- also big clusters of galaxies exist in the very early universe librium temperature of the universe was already suggested the big bang model will probably go down. In that case by many scientists in the half century preceding 1948, the there will be a lot of change within cosmology, and also the year in which cosmologists predicted the cosmic background theory of relativity will then be highly questioned. With the radiation of the big bang universe [11]. In a space and time festivities of 100 years of relativity we may have come close infinite universe many old cooled down remnants (amongst to the end of a scientific era.

58 E. Gaastra. Is the Biggest Paradigm Shift in the History of Science at Hand? October, 2005 PROGRESS IN PHYSICS Volume 3

5 Knowledge and power At this moment career-fear is the big obstacle when it comes to progress in physics, cosmology and astronomy. If the big bang model goes down then of course the first question is: What will replace it? If the here named alternat- ives break through then also another question rises: Why did 6 Are time and space properties of our reason? the alternatives need so much time to break through? Isaac Newton (1642–1726) thought that there was something A good theory needing a lot of time to break through has like “absolute space” and “absolute time” and two centuries happened before. In the third century BC the Greek philo- later Albert Einstein (1879–1955) melted these two together sopher and scientist Aristarchus published a book in which in the “space-time” concept. Newton and Einstein argued he proposed that the Earth rotates daily and revolves annually that space and time do exist physically, and ever since con- about the Sun. Eighteen hundred years later Copernicus was ventional scientists think that way too. However, it has been aware of the proposition by Aristarchus. Aristarchus and argued for centuries by scientists and philosophers (often sci- Copernicus were the heroes of the Copernican Revolution entists and philosophers at the same time) that space and time that followed after the publication of Copernicus’ book Revo- are not physically existing entities. Examples of such alt- lutions of the Celestial Spheres in 1543 [1]. The power of ernative thinkers are the Frenchman Rene Descartes (1596– the Sun-centred model was its simplicity compared to the 1650), the Dutchman Christiaan Huygens (1629–1695), the epicycles of the Earth-centred model. German Gottfried Leibniz (1646–1716), the Irishman George It took a long time, after the publication of Copernicus’ Berkeley (1685–1753), the East-Prussian Immanuel Kant greatest work, before the Earth-centred model was left en (1724–1804) and the already mentioned Austrian, Ernst Mach masse for the Sun-centred model. One of the reasons for this (1838–1916). was that, for a long time, the Earth-centred model described Our current natural sciences have their origin in Newton’s the movement of planets more accurately than the Sun- laws and formulae. Many physicists, cosmologists and astro- centred model of Copernicus. Formulae of wrong models nomers dismiss philosophy because they think it is misty. stay dominant when alternatives are not sufficiently develop- They feel safe with the basics and mathematics of the current ed. The gravity formulae of the theory of relativity and the conventional standard theories. Still, though mathematics is law of universal gravitation by Newton don’t explain how needed to do good predictions, sooner or later the whole gravity works, but they can be used to calculate with. The bastion falls apart if mathematics is based upon wrong prin- pushing gravity model explains, in a very simple way, how ciples. Thinking about basic principles needs philosophy. gravity works, but when it comes to formulae the concept is, Centuries ago it was the generalists, with philosophy and as was the model of Copernicus four centuries ago, still in its all the natural sciences in their package, who advocated that infancy. The same applies for aether theories, the extended space and time were properties of our reason in the first place Mach principle, the tired light model and the equilibrium and not properties of the world. The theory of relativity has temperature of the universe as an explanation for the cosmic time as the fourth dimension. If time does not exist then background radiation. The power of the aforementioned alt- the theory of relativity can be dismissed, and also the string ernatives is that they form, in a very simple way, a coherent theory, which has run wild with the mathematics of the theory whole within an infinite universe model. of relativity and works with eleven dimensions. Another reason for the late definitive capitulation of the Processes in an atomic clock slow down when the clock Sun-centred model was that the new model endangered the moves fast, and often this is seen as evidence for the existence position of authority held by the Catholic Church. Four cen- of time. But in the case of an aether, processes in fast moving turies ago scientific knowledge was dictated by the Catholic atomic clocks slow down because more aether slows down Church. Those who wanted to make a career as a scientist, or the processes in the clock. Our brains use time to compare just wanted to stay alive as a human, were forced to canonize the movement of mass with the movement of other mass. For the Earth-centred model. instance the rotation of our Earth (24 hours or one day) and Right now established science institutes dictate know- the orbit of our Earth around the Sun (365 days or one year). ledge when it comes to the fields of physics, cosmology That is all; it does not mean that time really exists. If time and astronomy. Physics professors Assis (Brazil) and Ghosh does not exist physically then the whole scientific bastion as (India) independently developed the same alternative for we have known it since Newton and, especially, as we have the theory of relativity. Both have published their work, known it the last 100 years, falls apart. but within the established science institutes they don’t find an audience. Professor of physics, the late Paul Marmet (Canada), attached questions to the fundamental laws of 7 Revolution by computer? nature (like the theory of relativity) and had to leave the sci- ence institute where he did his research. Right now students One can draw a parallel between what is happening now and learn to canonize the big bang and the theory of relativity. what happened four centuries ago. Before Copernicus en-

E. Gaastra. Is the Biggest Paradigm Shift in the History of Science at Hand? 59 Volume 3 PROGRESS IN PHYSICS October, 2005

tered the scene, the Catholic Church had passed on more dissidents argued at the first ever crisis in cosmology confer- or less definitely settled knowledge for more than thousand ence in Mon¸cao,˜ Portugal [13] that the big bang theory fails years. However, where knowledge did not change much with to explain certain observations. The biggest revolution in the respect to its contents, a strong development took place with history of science may be at hand. respect to the passing on and propagation of the knowledge. In the early Middle Ages convents arose, in the twelfth References century came the cathedral-schools and around 1200 the first universities were founded. In the course of centuries these 1. Harrison E. R. Cosmology: the science of the universe. universities gained an ever more independent position with Cambridge University Press, Cambridge, 2000. respect to the church, which finally made the church lose its 2. Selleri F. Lezioni di relativita’ da Einstein all’ etere di Lorentz. position of authority with respect to science. Progedit, Bari, 2003. Next to that in the late Middle Ages the church lost its 3. Cahill R. T. The Michelson and Morley 1887 experiment and monopoly with respect to knowledge, faster, because of the the discovery of absolute motion. Progress in Physics, 2005, invention of the art of printing. From that moment on more v. 3, 25–29. people could master knowledge themselves and could have 4. Assis A. K. T. Relational Mechanics. Apeiron, Montreal, 1999. their own thoughts about it and propagate those thoughts by 5. Ghosh A. Origin of Inertia. Apeiron, Montreal, 2000. printing and distributing their own books. 6. Van Lunteren F. Pushing Gravity, ed. by M. R. Edwards, The third development, at the end of the Middle Ages, 2002, 41. that would help the Copernican Revolution, was the invention of the telescope, which brought new possibilities for astro- 7. Edwards M. R. Pushing Gravity, ed. by M. R. Edwards, 2002, 137. nomy. A few decennia ago the computer was developed. It 8. Buonomano V. Pushing Gravity, ed. by M. R. Edwards, brought the internet, which split itself from science and 2002, 303. obtained its own independent position. The internet brings 9. Crothers S. J. On the general solution to Einstein’s vacuum knowledge to a lot of people all over the world. Now people field and its implications for relativistic degeneracy. Progress can publish their ideas with respect to physics, cosmology in Physics, 2005, v. 1, 68–73. and astronomy, independently of the universities and estab- 10. Crothers S. J. On the ramifications of the Schwarzschild space- lished periodicals. The universities lose more and more their time metric. Progress in Physics, 2005, v. 1, 74–80. monopoly as guardians of science, and the same goes for 11. Assis A. K. T. and Neves M. C. D. History of the 2.7 K the periodicals that serve as their extension piece. Before the temperature prior to Penzias and Wilson. Apeiron, 1995, v. 2, internet alternative thinking scientists were unknown isolated 79–84. islands who could not publish their ideas and did not know 12. Crothers S. J. On the general solution of Einstein’s vacuum field for the point-mass when λ = 0 and its implications for of each other’s existence. Now there are web pages which 6 form a vibrating net of interacting alternative models, a net relativistic cosmology. Progress in Physics, 2005, v. 3, 7–18. that grows every day. Next to that it is thanks to the computer 13. Ratcliffe H. The first crisis in cosmology conference. Progress that very strong telescopes have been put into use these last in Physics, 2005, v. 3, 19–24. decennia, and that ever stronger and better telescopes are on their way. Perhaps the science historians of the future will conclude that it was the computer that brought the Second Copernican Revolution.

8 Conclusions

Established conventional physicists and cosmologists behave as the church at the time of Galileo. Not by threatening with the death penalty, but simply by sniffing at alternative ideas. This will change as soon as the concerning noses smell funding money instead of career-fear. In our current society money and careers are the central issues where it comes to our necessities of life. Like four centuries ago the worries about the necessities of life are the driving forces behind the impasse. Still, just as at the time of Copernicus and Galileo: under the surface of the current standard theories the revolution may be going on at full speed. In June 2005

60 E. Gaastra. Is the Biggest Paradigm Shift in the History of Science at Hand? October, 2005 PROGRESS IN PHYSICS Volume 3

Sources of Stellar Energy and the Theory of the Internal Constitution of Stars

Nikolai Kozyrev∗

This is a presentation of research into the inductive solution to the problem on the internal constitution of stars. The solution is given in terms of the analytic study of regularities in observational astrophysics. Conditions under which matter exists in stars are not the subject of a priori suppositions, they are the objects of research. In the first part of this research we consider two main correlations derived from observations: “mass-luminosity” and “period — average density of Cepheids”. Results we have obtained from the analysis of the correlations are different to the standard theoretical reasoning about the internal constitution of stars. The main results are: (1) in any stars, including even super-giants, the radiant pressure plays no essential part — it is negligible in comparison to the gaseous pressure; (2) inner regions of stars are filled mainly by hydrogen (the average molecular weight is close to 1/2); (3) absorption of light is derived from Thomson dispersion in free electrons; (4) stars have an internal constitution close to polytropic structures of the class 3/2. The results obtained, taken altogether, permit calculation of the physical conditions in the internal constitution of stars, proceeding from their observational characteristics L, M, and R. For instance, the temperature obtained for the centre of the Sun is about 6 million degrees. This is not enough for nuclear reactions. In the second part, the Russell-Hertzsprung diagram, transformed according to physical conditions inside stars shows: the energy output inside stars is a simple function of the physical conditions. Instead of the transection line given by the heat output surface and the heat radiation surface, stars fill an area in the plane of density and temperature. The surfaces coincide, being proof of the fact that there is only one condition — the radiation condition. Hence stars generate their energy not in any reactions. Stars are machines, directly generating radiations. The observed diagram of the heat radiation, the relation “mass-luminosity-radius”, cannot be explained by standard physical laws. Stars exist in just those conditions where classical laws are broken, and a special mechanism for the generation of energy becomes possible. Those conditions are determined by the main direction on the diagram and the main point located in the direction. Physical coordinates of the main point have been found using observational data. The constants (physical coordinates) should be included in the theory of the internal constitution of stars which pretend to adequately account for observational data. There in detail manifests the inconsistency of the explanations of stellar energy as given by nuclear reactions, and also calculations as to the percentage of hydrogen and helium in stars. Also considered are peculiarities of some sequences in the Russell-Hertzsprung diagram, which are interesting from the theoretical viewpoint.

∗Editor’s remark: This is the doctoral thesis of Nikolai Aleksandrovich Crimean branch of the Observatory). In 1936 he was imprisoned for 10 Kozyrev (1908–1983), the famous astronomer and experimental physicist years without judicial interdiction, by the communist regime in the USSR. — one of the founders of astrophysics in the 1930’s, the discoverer of lunar Set free in 1946, he completed the draft of this doctoral thesis and published volcanism (1958), and the atmosphere of Mercury (1963) (see the article it in Russian in the local bulletin of the Crimean branch of the Observatory Kozyrev in the Encyclopaedia Britannica). Besides his studies in astronomy, (Proc. Crimean Astron. Obs., 1948, v. 2, and 1951, v. 6). Throughout the Kozyrev contributed many original experimental and theoretical works in subsequent years he continued to expand upon his thesis. Although this physics, where he introduced the “causal or asymmetrical mechanics” which research was started in the 1940’s, it remains relevant today, because the takes the physical properties of time into account. See his articles reporting basis here is observational data on stars of regular classes. This data has on his many years of experimental research into the physical properties of not changed substantially during the intervening decades. (Translated from time, Time in Science and Philosophy (Prague, 1971) and On the Evolu- the final Russian text by D. Rabounski and S. J. Crothers.) tion of Double Stars, Comptes rendus (Bruxelles, 1967). Throughout his scientific career Kozyrev worked at the Pulkovo Astronomical Observatory near St. Petersburg (except for the years 1946–1957 when he worked at the

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The author dedicates this paper to the blessed memory of Introduction Prof. Aristarch A. Belopolski Energy, radiated by the Sun and stars into space, is maintained by Contents special sources which should keep stars radiating light during Introduction ...... 62 at least a few billion years. The energy sources should be depen- PART I dent upon the physical conditions Chapter 1 Deducing the Main Equations of Equilibrium of matter inside stars. It follows in Stars from this fact that stars are stable 1.1 Equation of mechanical equilibrium ...... 63 space bodies. During the last de- 1.2 Equation of heat equilibrium ...... 64 cade, nuclear physics discover- 1.3 The main system of the equations. Transformation of the variables ...... 65 ed thermonuclear reactions that could be the energy source satis- Chapter 2 Analysis of the Main Equations and the Rela- tion “Mass-Luminosity” fying the above requirements. The reactions between protons 2.1 Observed characteristics of stars...... 66 Prof. Nikolai Kozyrev, 1970’s 2.2 Stars of polytropic structure ...... 67 and numerous light nuclei, which 2.3 Solution to the simplest system of the equations ...... 68 result in transformations of hydrogen into helium, can be 2.4 Physical conditions at the centre of stars ...... 71 initiated under temperatures close to the possible temperature 2.5 The “mass-luminosity” relation...... 72 of the inner regions of stars — about 20 million degrees. 2.6 The radiant pressure inside stars ...... 74 Comparing different thermonuclear reactions, Bethe con- 2.7 Comparing the obtained results to results obtained by other cluded that the energy of the Sun and other stars of the main researchers ...... 75 sequence is generated in cyclic reactions where the main 2.8 The roleˆ of convection inside stars...... 76 part is played by nitrogen and carbon nuclei, which capture Chapter 3 The Internal Constitution of Stars, Obtained protons and then produce helium nuclei [1]. This theory, from the Analysis of the Relation “Period — Average developed by Bethe and widely regarded in recent years, Density of Cepheids” and Other Observational Data has had no direct astrophysical verification until now. Stars 3.1 The main equation of pulsation...... 78 produce various amounts of energy, e. g. stars of the giants 3.2 Calculation of the mean values in the pulsation equation by sequence have temperatures much lower than that which is the perturbation method ...... 79 necessary for thermonuclear reactions, and the presence of 3.3 Comparing the theoretical results to observational data ...... 80 bulk convection in upper shells of stars, supernova explos- 3.4 Additional data about the internal constitution of stars...... 81 ions, peculiar ultra-violet spectra lead to the conclusion that 3.5 Conclusions about the internal constitution of stars...... 82 energy is generated even in the upper shells of stars and, PART II sometimes, it is explosive. It is quite natural to inquire as to a general reason for all the phenomena. Therefore we should Chapter 1 The Russell-Hertzsprung Diagram and the Ori- gin of Stellar Energy be more accurate in our attempts to apply the nuclear reaction 1.1 An explanation of the Russell-Hertzsprung diagram by theory to stars. It is possible to say (without exaggeration), the theory of the internal constitution of stars ...... 83 that during the last century, beginning with Helmholtz’s 1.2 Transforming the Russell-Hertzsprung diagram to the physical contraction hypothesis, every substantial discovery in physics characteristics, specific to the central regions of stars . . . . . 85 led to new attempts to explain stellar energy. Moreover, 1.3 The arc of nuclear reactions...... 86 after every attempt it was claimed that this problem was 1.4 Distribution of stars on the physical conditions diagram ...... 87 finally solved, despite the fact that there was no verification 1.5 Inconsistency of the explanation of stellar energy by Bethe’s in astrophysical data. It is probable that there is an energy thermonuclear reactions ...... 90 generation mechanism of a particular kind, unknown in an 1.6 The “mass-luminosity” relation in connection with the Russell- Earthly laboratory. At the same time, this circumstance can- Hertzsprung diagram ...... 92 not be related to a hypothesis that some exclusive conditions 1.7 Calculation of the main constants of the stellar energy state. . .93 occur inside stars. Conditions inside many stars (e. g. the Chapter 2 Properties of Some Sequences in the Russell- infrared satellite of ε Aurigae) are close to those that can Hertzsprung Diagram be realized in the laboratory. The reason that such an energy 2.1 The sequence of giants ...... 95 generation mechanism remained elusive in experiments is 2.2 The main sequence ...... 96 due to peculiarities in the experiment statement and, possibly, 2.3 White dwarfs ...... 97 in the necessity for large-scale considerations in the experi- ment. Considering physical theories, it is possible that their

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inconsistency in the stellar energy problem arises for the chanism generating stellar energy). The formulae obtained reason that the main principles of interaction between matter are completely unexpected from the viewpoint of theoretical and radiant energy need to be developed further. physics. At the same time they are so typical that we have Much of the phenomena and empirical correlations dis- in them a possibility of studying the physical process which covered by observational astrophysics are linked to the prob- generates stellar energy. lem of the origin of stellar energy, hence the observational This gives us an inductive method for determining a sol- data have no satisfying theoretical interpretation. First, it is ution to the problem of the origin of stellar energy. Follow- related to behaviour of a star as a whole, i. e. to problems ing this method we use some standard physical laws in associated with the theory of the internal constitution of subsequent steps of this research, laws which may be violated stars. Today’s theories of the internal constitution of stars by phenomenology . However this circumstance cannot in- are built upon a priori assumptions about the behaviour of validate this purely astrophysical method. It only leads to matter and energy in stars. One tests the truth or falsity the successive approximations so characteristic of the phe- of the theories by comparing the results of the theoretical nomenological method. Consequently, the results we have analysis to observational data. This is one way to build obtained in Part I can be considered as the first order of various models of stars, which is very popular nowadays. approximation. But such an approach cannot be very productive, because The problem of the internal constitution of stars has been the laws of Nature are sometimes so unexpected that many very much complicated by many previous theoretical studies. such trials, in order to guess them, cannot establish the correct Therefore, it is necessary to consider this problem from the solution. Because empirical correlations, characterizing a star outset with the utmost clarity. Observations show that a star, as a whole, are surely obtained from observations, we have in its regular duration, is in a balanced or quasi-balanced therein a possibility of changing the whole statement of state. Hence matter inside stars should satisfy conditions of the problem, formulating it in another way — considering mechanical equilibrium and heat equilibrium. From this we the world of stars as a giant laboratory, where matter and obtain two main equations, by which we give a mathematical radiant energy can be in enormously different scales of states, formulation of our problem. Considering the simplest case, and proceeding from our analysis of observed empirical we neglect the rotation of a star and suppose it spherically correlations obtained in the stellar laboratory, having made symmetric. no arbitrary assumptions, we can find conditions governing the behaviour of matter and energy in stars as some un- PARTI known terms in the correlations, formulated as mathematical equations. Such a problem can seems hopelessly intractable, Chapter 1 owing to so many unknown terms. Naturally, we do not Deducing the Main Equations of Equilibrium in Stars know: (1) the phase state of matter — Boltzmann gas, Fermi gas, or something else; (2) the manner of energy transfer — 1.1 Equation of mechanical equilibrium radiation or convection — possible under some mechanism of energy generation; (3) the roleˆ of the radiant pressure inside Let us denote by P the total pressure, i. e. the sum of the stars, and other factors linked to the radiant pressure, namely gaseous pressure p and the radiant energy pressure B, taken — (4) the value of the absorption coefficient; (5) chemical at a distance r from the centre of a star. The mechanical composition of stars, i. e. the average numerical value of the equilibrium condition requires that the change of P in a unit molecular weight inside stars, and finally, (6) the mechanism of distance along the star’s radius must be kept in equilibrium generating stellar energy. To our good fortune is the fact by the weight of a unit of the gas volume that the main correlation of observational astrophysics, that dP between mass and luminosity of stars, although giving no = gρ , (1.1) answer as to the origin of stellar energy, gives data about the dr − other unknowns. Therefore, employing the relation “period where ρ is the gas density, g is the gravity force acceleration. — average density of Cepheids”, we make more precise our If ϕ is the gravitational potential conclusions about the internal constitution of stars. As a g = grad ϕ , (1.2) result there is a possibility, even without knowledge of the − origin of stellar energy, to calculate the physical conditions and the potential satisfies Poisson equation inside stars by proceeding from their observable charac- 2ϕ = 4πGρ , teristics: luminosity L, mass M, and radius R. On this ∇ − 8 basis we can interpret another correlation of observational where G = 6.67×10− is the gravitational constant. For astrophysics, the Russell-Hertzsprung diagram — the cor- spherical symmetry, relation between temperature and luminosity of stars, which 1 dr2 grad ϕ 2ϕ = div grad ϕ = . (1.4) depends almost exclusively on the last unknown (the me- ∇ r2 dr

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Comparing the equalities, we obtain the equation of me- First we determine FR. Radiations, being transferred chanical equilibrium for a star through a layer of thickness ds, change their intensity I through the layer of thickness ds, according to Kirchhoff’s 1 d r2dP = 4πG , (1.5) law ρr2 dr ρdr − dI   = κρ I E , (1.12) ds − − where where κ is the absorption coefficient
per unit mass, E is the P = p + B. (1.6) radiant productivity of an absolute black body (calculated Radiations are almost isotropic inside stars. For this per unit of solid angle ω). In polar coordinates this equation reason B equals one third of the radiant energy density. is ∂I sin θ ∂I As we show in the next paragraph, we can put the radiant cos θ = κρ I E , (1.12a) energy density, determined by the Stephan-Boltzmann law, ∂r − r ∂θ − − in a precise form. Therefore, where θ is the angle between the direction
of the normal to the layer (the direction along the radius r) and the radiation 1 B = αT 4, (1.7) direction (the direction of the intensity I). The flow FR and 3 the radiant pressure B are connected to the radiation intensity 15 by the relations where α = 7.59×10− is Stephan’s constant, T is the absolute temperature. The pressure P depends, generally 2 speaking, upon the matter density and the temperature. This FR = I cos θdω , Bc = I cos θ dω , (1.13) correlation is given by the matter phase state. If the gas is Z Z ideal, it is where c is the velocity of light, while the integration is taken T over all solid angles. We denote p = nkT = < ρ . (1.8) μ Here n is the number of particles in a unit volume of the Idω = J. (1.14) 16 7 gas, k=1.372×10− is Boltzmann’s constant, =8.313×10 Z is Clapeyron’s constant, μ is the average molecular< weight. Multiplying (1.12a) by cos θ and taking the integral over For example, in a regular Fermi gas the pressure depends all solid angles dω, we have only on the density dB 1 c (J 3Bc) = κρF . 5/3 5/3 12 R p = Kρ ,K = μe KH ,KH = 9.89×10 , (1.9) dr − r − − In order to obtain F we next apply Eddington’s approx- where μe is the number of the molecular weight units for R each free electron. imation We see that the pressure distribution inside a star can 3Bc = J = 4πE , (1.15) be obtained from (1.5) only if we know the temperature thereby taking FR to within high order terms. Then distribution. The latter is determined by the heat equilibrium condition. c dB F = . (1.16) R −κρ dr 1.2 Equation of heat equilibrium Let us consider the convective energy flow Fc . Everyday we see huge convection currents in the surface of the Sun (it Let us denote by ε the quantity of energy produced per second is possible this convection is forced by sudden production of by a unit mass of stellar matter. The quantity ε is dependent energy). To make the convective energy flow substantial, upon the physical conditions of the matter in a star, so ε is a Fc convection currents of matter should be rapid and cause function of the radius r of a star. To study ε is the main task of this research. The heat equilibrium condition (known also transfer of energy over long distances in a star. Such condit- as the energy balance condition) can be written as follows ions can be in regions of unstable convection of matter, where free convection can be initiated. Schwarzschild’s pioneering div F = ερ , (1.10) research [2], and subsequent works by other astrophysicists (Unsold,¨ Cowling, Bierman and others) showed that although where F is the total flow of energy, being the sum ofthe a star is in the state of stable mechanical and heat equi- radiant energy flow FR, the energy flow Fc dragged by librium as a whole, free convection can start in regions where convection currents, and the heat conductivity flow FT (1) stellar energy sources rapidly increase their power, or (2) the ionization energy is of the same order as the heat F = FR + Fc + FT . (1.11) energy of the gas.

64 N. Kozyrev. Sources of Stellar Energy and the Theory of the Internal Constitution of Stars October, 2005 PROGRESS IN PHYSICS Volume 3

We assume convection currents flowing along the radius Using formulae (1.10), (1.16), (1.17), (1.21), we obtain of star. We denote by Q the total energy per unit of convection the heat equilibrium equation current mass. Hence, Q is the sum of the inner energy of the 1 1 r2db 1 1 dp ε gas, the heat function, the potential and kinetic energies. r2Au = . (1.22) ρr2 dr κρdr − cρr2 dr dr − c We regularly assume that a convection current retains its     own energy along its path, i. e. it changes adiabatically, We finally note that, because ε is tiny value in comparison and dissipation of its energy occurs only when the current to the radiation per mass unit, even tiny changes in the state stops. Then the energy flow transferred by the convection, of matter should break the equalities. Therefore even for large according to Schmidt [3], is regions in stars the heat equilibrium condition (1.10) can be locally broken. The same can be said about the equation for dQ F = Aρ ,A =v ˉλˉ . (1.17) the convective energy flow, because huge convections in stars c − dr can be statistically interpreted in only large surfaces like that of a whole star. Therefore the equations we have obtained ˉ The quantity A is the convection coefficient, λ is the can be supposed as the average along the whole radius of a average length travelled by the convection current, vˉ is the star, and taken over a long time. Then the equations are true. average velocity of the current. If the radiant pressure is The aforementioned limitations do not matter in our negligible in comparison to the gaseous pressure, in an ideal analysis because we are interested in understanding the be- gas (according to the 1st law of thermodynamics) we have haviour of a star as a whole.

1 dQ dT d ρ = cv + p , (1.18) 1.3 The main system of the equations. Transformation dr dr dr of the variables or, in another form, In order to focus our attention on the main task of this dQ dT 1 dp research, we begin by considering the equations obtained = c , (1.18) dr p dr − ρ dr for equilibrium in the simplest case: (1) in the mechanical equilibrium equation we assume the radiant pressure B neg- where cv is the heat capacity of the gas under constant ligible in comparison to the gaseous pressure p, while (2) in volume, cpis the heat capacity under constant pressure the heat equilibrium equation we assume the convection term negligible. Then we obtain the main system of the equations in the form cp = cv + < . 1 d r2dp μ = 4πG , ρr2 dr ρdr − Denoting   (I) cp 1 d r2dB ε = Γ , = . cv ρr2 dr κρdr − c   we have The radiant pressure depends only on the gas temperature Γ c = < . (1.20) T , according to formula (1.7). The absorption coefficient κ p Γ 1 μ − (taken per unit mass) depends p and B. This correlation is After an obvious transformation we arrive at the formulae unknown. Also unknown is the energy ε produced by a unit mass of gas. Let us suppose the functions known. Then in dQ 1 dp Γ pdB order to solve the system we need to have the state equation = u , u = 1 , (1.21) dr −ρ dr − 4(Γ 1) Bdp of matter, connecting ρ, p, and B. In this case only two − functions remain unknown: for instance p and B, whose (for a monatomic gas Γ = 5/3). dependence on the radius r is fully determined by equations The heat conductivity flow has a formula analogous to (I). These functions should satisfy the following boundary (1.17). Because particles move in any direction in a gas, in conditions. In the surface of a star the total energy flow is the formula for A we have one third of the average velocity ( ). According formula (1.13), F0 = FR0 Fc = FT = 0 of particles instead of vˉ. In this case dQ/dr is equal to only 1 3 the first term of equation (1.18), and so dQ/dr has the same- FR0 = J0 = cB0 , order numerical value that it has in the energy convective 2 2 so, taking formula (1.16) into account, we obtain the con- flow Fc . Therefore, taking A from Fc (1.17) into account, dition in the surface of a star we see that Fc is much more that FT . In only very rare exceptions, like a degenerate gas, can the heat conductivity 2 dB under p = 0 we have B = , (1.23) flow FT be essential for energy transfer. −3 κρdr

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From equations (I) we see that the finite solution con- Here we can write the main system of the equations dition under r = 0 is the same as in terms of the new variables (Ia), taking convection into dp dB account. Because of (1.22), we obtain under r = 0 we have = 0 , = 0 . (1.24) dr dr 1 d x2dp 1 = 1 , The boundary conditions are absolutely necessary, they 2 ρ1 x dx ρ1 dx − are true at the centre of any real star. The theory of the inner   (II) 2 constitution of stars by Milne [4], built on solutions which do 1 d x dB κc ρc 1 dp 1 x2Au 1 = λε . not satisfy these boundary conditions, does not mean that the ρ x2 dx κ ρ dx − cγ ρ x2 dx − 1 1  1 1  c 1   boundary conditions are absolutely violated by the theory. In layers located far from the centre the boundary solutions can For an ideal gas, equation (1.21) leads to a very simple be realized, if derivatives of physical characteristics of matter formula for u are not continuous functions of the radius, but have breaks. Γ p dB u = 1 1 1 . (1.30) Hence, Milne’s theory permits a break a priori in the state − 4(Γ 1) B dp equation of matter, so the theory permits stellar matter to exist − 1 1 in at least two different states. Following this hypothetical Owing to (1.5) and (1.6) it follows at last that the main approach as to the properties of stellar matter, we can deduce system of the equations, taking the radiant pressure into conclusions about high temerpatures and pressures in stars. account in the absence of convection, takes the form Avoiding the view that “peculiar” conditions exist in stars, 1 d x2d(p + γ B ) we obtain a natural way of starting our research into the 1 c 1 = 1 , ρ x2 dx ρ dx − problem by considering the phase state equations of matter. 1  1  Hence we carry out very important transformations of the (III) 1 d x2dB variables in the system (I). Instead of r and other variables we 1 = λε . ρ x2 dx κ ρ dx − 1 introduce dimensionless quantities bearing the same physical 1  1 1  conditions. We denote by index c the values of the functions in the centre of a star (r = 0). Instead of r we introduce a dimensionless quantity x according to the formula Chapter 2 4πG Analysis of the Main Equations and x = ar , a = ρc , (1.25) the Relation “Mass-Luminosity” s pc and we introduce functions 2.1 Observed characteristics of stars ρ p B ρ1 = , p1 = ,B1 = ,... (1.26) Astronomical observations give the following quantities ρc pc Bc characterizing star: radius R, mass M, and luminosity L (the Then, as it is easy to check, the system (I) transforms to total energy radiated by a star per second). We are going to the form 2 consider correlations between the quantities and parameters 1 d x dp1 of the main system of the star equilibrium equations. As 2 = 1 , ρ1 x dx ρ1 dx − a result, the main system of the equations considered under   (Ia) 2 any phase state of stellar matter includes only two parameters 1 d x dB1 = λε , characterizing matter and radiation inside a star: Bc and pc. ρ x2 dx κ ρ dx − 1 1  1 1  Because of formula (1.25), we obtain where ε κ B λ = c c , γ = c . (1.27) 1 pc c R = x0 , (2.1) 4πGc γc pc ρ 4πG c r Numerical values of all functions in the system (Ia) are between 0 and 1. Then the conditions at in the centre of a where x0 is the value of x at the surface of a star, where star (x = 0) take the form p1 = B1 = 0. With this formula, and introducing a state eq- uation of matter, we can easily obtain the correlation R = dp1 dB1 = f (Bc, pc). It should be noted that in the general case the p1 = 1 , = 0 ,B1 = 1 , = 0 . (1.28) dx dx value of x0 in formula (2.1) is dependent on Bc and pc. At the same time, because the equation system consists of In the surface of a star (x = x0), instead of (1.23), we can use the simple conditions functions variable between 0 and 1, the value of x0 should be of the same order (i. e. close to 1). Therefore the first

B1 = 0 , p1 = 0 . (1.29) multiplier in (2.1) plays the main role.ˆ

66 N. Kozyrev. Sources of Stellar Energy and the Theory of the Internal Constitution of Stars October, 2005 PROGRESS IN PHYSICS Volume 3

Because of Because all the functions included in the main system of R equations can be expressed through B and p , we can find M = 4π ρr2dr , 1 1 0 the functions from the system of the differential equations Z with respect to two parameters and . Boundary condit- we have Bc pc 3/2 ions (1.28) are enough to find the solutions at the centre of pc M = Mx , (2.2) 3/2 2 0 a star. Hence, boundary conditions at the surface of a star G √4π ρc (1.29) are true under only some relations between Bc and pc. where Therefore all quantities characterizing a star are functions of x0 2 only one of two parameters, for instance Bc: R = f (Bc), Mx0 = ρ1 x dx . 1 Z0 M = f2 (Bc), L = f3 (Bc). This circumstance, with the same At last, the total luminosity of star is chemical composition of stars, gives the relations: (1) “mass- luminosity” L = ϕ1 (M) and (2) the Russell-Hertzsprung dia- R 2 gram L = ϕ2 (R). L = 4π ερr dr , From the above we see that the equilibrium of stars 0 Z has this necessary consequence: correlations between M, L, and we obtain and R. Thus the correlations discovered by observational astrophysics can be predicted by the theory of the inner L L x0 = ε x0 ,L = ε ρ x2dx . (2.3) constitution of stars. M c M x0 c 1 x0 Z0 2.2 Stars of polytropic structure Values of the quantities Mx0 and Lx0 should change a little under changes of pc and Bc, remaining close to 1. If x0, Solutions to the main system of the equations give functions Mx0 , and Lx0 are the same for numerous stars, such stars are homological, so the stars actually have the same structure. p1 (x) and B1 (x). Hence, solving the system we can as well As it is easy to see, the average density ρˉ of star is obtain B1 (p1). If we set up a phase state, we can as well connected to by the formula obtain the function p1 (ρ1). ρc Γ Let us assume p1 (ρ1) as p1 (ρ1 ), where Γ is a constant.

3Mx0 Such a structure for a star is known as polytropic. Having ρˉ = ρc 3 . (2.4) stars of polytropic structure, we can easily find all the func- x0 tions of x. Therefore, in order to obtain a representation of We find a formula for the total potential energy Ω of star the solutions in the first instance, we are going to consider thus stars of polytropic structure. Emden’s pioneering research on R Mr the internal constitution of stars was done in this way. Ω = G dMr . − 0 r The aforementioned polytropic correlation can be used Z instead of the heat equilibrium equation, so only the first Multiplying the term under the integral by R, and divid- equation remains in the system. We introduce a new variable ing by M 2, we obtain T1 which, in an ideal gas, equals the reduced temperature 2 GM p Ω = Ωx (2.5) 1 Γ 1 − R 0 = ρ1 − = T1 , (2.7) ρ1 and also x0 or, in another form, x0 Ωx = x ρ Mx dx . 0 M 2 1 n 1 n+1 x0 0 ρ = T , n = , p = T , (2.7a) Z 1 1 Γ 1 1 1 Under low radiant pressure, taking the equation of me- − chanical equilibrium into account, the system (I) gives so that we obtain n x0 x0 x0 dp1 = (n + 1) T1 dT . 3 2 x ρ1Mx dx = x dp1 = 3 x p1dx , (2.5a) 0 − 0 0 Substituting the formulae into the first equation of the Z Z Z main system (I), we obtain from which we obtain 1 1 2 dT1 n x0 E T10 = x1 = T1 , (2.8) 2 x2 dx dx − 3x0 p1 x dx 1 1  1  0   Ωx0 = Z . (2.6) where a new variable x is introduced instead of x x0 2 1 2 ρ1x dx x = √n + 1 x1 . (2.9)  Z0 

N. Kozyrev. Sources of Stellar Energy and the Theory of the Internal Constitution of Stars 67 Volume 3 PROGRESS IN PHYSICS October, 2005

Emden’s equation (2.8) can be integrated very easily if Table 1 n = 0 or n = 1. Naturally, under n = 0 (a star of constant 2 x0 density) we obtain n x0 Mx0 Ωx0 3Mx0 2 x1 p = T = 1 , (2.10) 0 2.45 4.90 1.0 3/5 1 1 − 6 1 4.52 9.04 3.4 3/4 so the remaining characteristics can be calculated just as 3/2 5.81 11.1 5.9 6/7 easily. Under n = 1 the substitution n = T1 x1 reduces the differential equation (2.8) to the simple form n00 = n. 2 7.65 12.7 11.4 1 − Hence, under n = 1, we have 2.5 10.2 14.4 24.1 6/5

2 3 sin x1 sin x1 3 13.8 16.1 54.4 /2 T1 = , p1 = . (2.11) 2 12 x1 x1 3.25 17.0 17.5 88.2 /7 With other polytropic indices n, we obtain solutions which are in series. All odd derivatives of the operator E Formula (2.5a) leads to another relation between the integrals. As a result we obtain should become zero under x1 = 0. For even derivatives, we have 6 x0 M 2 2i + 3 2 x0 (2i) (2i+2) 1 p1 x dx = , E0 T10 = T1 (0) . (2.12) − n + 1 −x (n + 1) 2i + 1   Z0 0 Now, differentiating  equation (2.8), we obtain derivatives and, substituting this into (2.6), we obtain Ritter’s formula in different orders of the function T under x = 0, so we 1 1 3 obtain the coefficients of the series expansion. As a result we Ω = . (2.15) x0 5 n obtain the series − This formula, in addition to other conclusions, leads to x2 n n(8n 5) T = 1 1 + x4 − x6 + the fact that a star can have a finite radius only if n < 5. 1 − 3! 5! 1 − 3 7! 1 × (2.13) 2 n(122n 183n + 70) 8 2.3 Solution to the simplest system of the equations + − x1 + ... 9×9! Using (2.13), we move far away from the special point To begin, we consider the system (Ia), which is true in the absence of convection and if the radiant pressure is low. The x1 = 0. Subsequent solutions can be obtained by numerical integration. As a result we construct a table containing char- absorption coefficient κ, the quantity of produced energy ε, acteristics of stellar structures under different n (see Table 1). and the phase state equation of matter, can be represented as products of different power functions , , . Then the The case of n = 3/2 corresponds to an adiabatic change of p B ρ functions , , and the phase state equation, the state of monatomic ideal gas (Γ = 5/3) and also a regular κ1 = κ/κc ε1 = ε/εc Fermi gas (1.9). If n = 3, we get a relativistic Fermi gas or an are dependent only on p1, B1 , ρ1; they have no parameters pc, Bc, ρc. In this case the coefficient λ remains the sole ideal gas under B1 = p1 (the latter is known as Eddington’s solution). parameter of the system. In this simplest case we study the In polytropic structures we can calculate exact values of system (Ia) under further limitations: we assume an ideal gas and κ independent of physical conditions. Thus, we have the Ωx0 . Naturally, the integral of the numerator of (2.6) can be transformed to correlations x0 x0 x0 dT κ=const: κ =1, p =B1/4ρ , ε =f (p ,B ) , (2.16) p x2dx = T dM = M 1 dx . 1 1 1 1 1 1 1 1 1 x − x dx Z0 Z0 Z0 1 d x2dp 1 = 1 , Emden’s equation leads to 2 ρ1 x dx ρ1 dx −   (2.17) 2 dT1 1 d x2dB Mx = (n + 1) x , (2.14) 1 = λε , − dx ρ x2 dx ρ dx − 1 1  1  so we obtain where ε κ B λ = c c γ = c . (2.18) x x c 0 1 0 M 2 4πGc γc pc p x2dx = x dx = 1 n + 1 x2 Taking integrals on the both parts of (2.17), we obtain Z0 Z0 2 Mx0 2 Mx0 2 Mx x dB1 x dp1 = + dM . = λLx , = Mx , (2.19) −x (n + 1) n + 1 x x ρ dx − ρ dx − 0 Z0 1 1

68 N. Kozyrev. Sources of Stellar Energy and the Theory of the Internal Constitution of Stars October, 2005 PROGRESS IN PHYSICS Volume 3

where we have introduced the notation The surface condition B1 = 0, being applied to this form- x x ula under p1 = 0, gives an equation determining λ. This 2 2 Lx = ε1 ρ1 x dx , Mx = ρ1 x dx . (2.20) method gives a numerical value of λ which can be refined by Z0 Z0 numerical integration of the system (2.17). This integration Integrating (2.19) using boundary conditions, we obtain can be done step-by-step.

x0 The centre of a star, i. e. the point where x = 0, is the l ρ1 λ = x , l = Mx 2 dx , singular point of the differential equations (2.17). We can 0 ρ 0 x L 1 dx Z move far away from the singular point using series and x x2 Z0 then (as soon as their convergence becomes poor) we apply numerical integration. We re-write the system (2.7) as follows hence x0 ρ M 1 dx x 2 1/4 0 x B1 dp1 1/4 λ = Z . (2.21) E = p B− , x0 ρ p dx − 1 1 L 1 dx  1  x x2 (2.23) Z0 1/4 B1 dB1 1/4 From formulae (2.21) and (2.20) we conclude that the E = λε p B− . p dx − 1 1 1 more concentrated are the sources of stellar energy, the  1  greater is λ. If the source’s productivity ε increases towards Formula (2.12) gives the centre of a star, λ > 1. If ε = const along the radius, ε1 = 1 and hence . If stellar energy is generated mostly in the (2i) 2i + 3 (2i+1) λ = 1 E [u] = [u] . (2.24) surface layers of a star, λ < 1. Equations (2.19) lead to 0 2i + 1 0

dB1 λLx Then, differentiating formula (2.23) step-by-step using = . (2.22) (2.24), we obtain different order derivatives of the functions dp1 Mx p1 (x) and B1 (x) under x = 0 that yields the possibility of Because of the boundary conditions p1 = 0, B1 = 0 and expanding the functions into Laurent series. Here are the first p1 = 1, B1 = 1, the derivative dB1 /dp1 always takes the few terms of the expansions average value 1. Owing to 1 x2 2 x4 dB dB λL p = 1 + 4 λ ... 1 = λ , 1 = x0 , 1 − 3 2! 15 − 4! − dp dp Mx  1 x=0  1 x=x0 0   λ x2 we come to the following conclusions: if energy sources B = 1 + (2.25) 1 − 3 2! are located at the centre of a star, λLx0 /Mx0 < 1; if energy sources are located on the surface, λL /M > 1. If energy 2λ 3 ∂ε ∂ε x4 x0 x0 + (4 λ) + 1 + λ 1 ... sources are homogeneously distributed inside a star, 15 − 2 ∂p ∂B 4! −   1 1 0  λLx0 /Mx0 = 1 and B1 = p1, so we have polytropic class 3, considered in the previous paragraph. This particular solution In order to carry out numerical integration we use form- is the basis of Eddington’s theory of the internal constitution ulae which can be easily obtained from the system (2.23), namely of stars. If n > 3, (dB1 /dp1)x0 so we have Lx0 . Therefore we conclude that polytropic→ ∞ classes n > 3→char- ∞ 2 1/2 p10 B10 2 acterize stars where energy sources concentrate near the p00 = p B− + p0 , 1 1 1 1 x surface. Polytropic classes n < 3 correspond to stars where − p1 − 4B1 −   (2.23a) energy sources concentrate at the centre. Therefore the data 2 1/2 p10 B10 2 of Table 1 characterize the most probable structures of stars. B00 = λε p B− + B0 . 1 − 1 1 1 1 p − 4B − x It should be noted that if n < 3, formulae (2.7) and (2.7a)  1 1  lead to , and hence . So polytropic (dB1 /dp1)x0 = 0 Lx0 = 0 In this system, we introduce the reduced temperature T1 structures of stars where energy sources concentrate at the 1/4 instead of B1 , and a new variable u1 = p1 instead of p1 centre can exist only if there is an energy drainage in the 5 upper layer of a star. u1 u10 T10 2 u00 = + u0 , Differentiating formula (2.22) step-by-step and using the 1 −4T 2 1 u − T − x 1  1 1   system (2.17) gives derivatives of B (p ) under p = 1 and, (2.23b) 1 1 1 8 hence, expansion of into a Taylor series. The first λε1 u1 u10 T10 2 B1 (p1) T 00 = + T 0 4 . 1 − 4T 5 1 u − T − x terms of the expansion take the form 1   1 1  

3 ∂ε1 ∂ε1 2 This substitution gives a great advantage, because of small B1 = 1+λ(p1 1)+ λ +λ (p1 1) + ... − 10 ∂p ∂B − slow changes of the functions T1 and u1.  1 1 1

N. Kozyrev. Sources of Stellar Energy and the Theory of the Internal Constitution of Stars 69 Volume 3 PROGRESS IN PHYSICS October, 2005

A numerical solution can be obtained close to the surface Table 2 layer, but not in the surface itself, because the equations λLx0 (2.23) can be integrated in the upper layers without problems. ε λ x0 Mx Lx 1 0 0 3 Mx0 Naturally, assuming Mx = Mx0 = const and Lx = Lx0 = const in formula (2.19), we obtain 3 = 1 1 13.8 16.1 16.1 3.8×10− 3 dp M dB λL B 1.76 10 12.4 2.01 1.8 10− 1 = x0 dx , 1 = x0 dx , 1 × 2 2 3 ρ − x ρ − x B p 2.32 9 11.5 1.57 2.2 10− 1 1 (2.26) 1 1 × λL B = x0 p . 1 M 1 x0 The variability of κ can be determined by a function of The ideal gas equation and the last relation of (2.26) the general form α permit us to write down p1 κ1 = . Bβ dp1 1/4 dp1 3/4 1 = B1 = B1− dB1 . ρ1 p1 At first we consider the simplest case where energy Integrating the first equation of (2.26), we obtain sources are homogeneously distributed inside a star. In this case ε1 = 1, Lx = Mx, and equation (2.22a) can be integrated x0 x 4T1 = Mx0 − , (2.27) x0 x 1 + β B1+β = λ p1+α . which gives a linear law for the temperature increase within 1 1 + α 1 the uppermost layers of a star. To obtain λ by step-by-step integration, we need to have Proceeding from the conditions at the centre of any star a criterion by which the resulting value is true. It is easy to (B1 = p1 = 1), we obtain see from (2.26) that such a criterion can be a constant value 1 + α λ for the quotient B1/p1 starting from x located far away from λ = ,B1 = p1 . the centre of a star. Solutions are dependent on changes of λ, 1 + β therefore an exact numerical value of this parameter should Hence the star has polytropic structure of class be found. Performing the numerical integration, values of the functions near the surface of a star are not well determined. 4 Therefore, in order to calculate and in would be n = 1 . Lx0 Mx0 λ − better to use their integral formulae (2.20). If energy sources increase their productivity towards the centre of a star, we Looking from the physical viewpoint, the most probable effects are: decrease in the absorption coefficient of a star obtain an exact value for Lx0 even in a very rough solution for the system. The calculation of x is not as good, but it with depth, and also α 1. Because 0 > − can be obtained for fixed M and x far away from the centre x0 α β α β − − through formula (2.27) 1+β 1+α κ1 = p1 = B1 , x x0 = . (2.27a) 4T1 κ1 decreases with increase of p1 and B1 only if α < β. Then 1 M x − x0 it is evident that λ < 1 and n > 3. Hence, variability of κ Using the above method, exact solutions to the system are results in an increase of polytropic class. According to the obtained. Table 2 contains the characteristics of the solutions theory of photoelectric absorption, in comparison to the characteristics of Eddington’s model∗. The last column contains a characteristic that is very ρ1 κ1 = 3.5 . important for the “mass-luminosity” relation (as we will see T1 later). Let us determine what changes are expected in the char- In this case α = 1, β = 1.125 and hence n = 3.25. Table 1 acteristics of the internal constitution of stars if the absorption gives respective numerical values of the characteristics x0 and . Other calculated characteristics are 0.94 and coefficient κ is variable. If κ is dependent on the physical Mx0 λ = 3 2 3 3.06 10− . All the numerical val- conditions, equation (2.22) takes the form λLx0 /Mx0 = λ/Mx0 = × ues are close to those calculated in Table 2. This is expected, dB κ λL 1 = 1 x . (2.22a) if variability of κ leads to the same order effect for the other dp1 Mx classes of energy source distribution inside stars. Looking at Table 1 and Table 2 we see that the charact- ∗In his model ε = 1, so the energy sources productivity is ε = const 1 3 3 along the radius (see the first row in the table). — Editor’s remark. eristics x M 10 and λL /M 2 10− have tiny 0 ' x0 ' x0 x0 ' ×

70 N. Kozyrev. Sources of Stellar Energy and the Theory of the Internal Constitution of Stars October, 2005 PROGRESS IN PHYSICS Volume 3

changes under different suppositions about the internal con- ion of the gaseous pressure at the centre of a star stitution of stars (the internal distribution of energy sources) . ∗ 2 2 4 M There are three main cases: (1) sources of stellar energy, G M x0 M pc = . (2.29) homogeneously distributed inside a star, (2) energy sources 2 2 R 4 4π R Mx   0 R
are so strongly concentrated at the centre of star that their productivity is proportional to the 8th order of the temp- 33
10 Because M =1.985×10 and R =6.95×10 , we obtain erature, (3) polytropic structures where energy sources are 2 concentrated at the surface — there is a drainage in the surface 4 M 14 x0 M layer of a star. It should be noted that we did not consider pc = 8.9×10 . (2.30) 2 R 4 Mx other possible cases of distributed energy sources in a star, 0 R
such as production of energy in only an “energetically active” Thus the pressure at the centre of the Sun
should be about layer at a middle distance from the centre. In such distributed 1016 dynes/cm2 (ten billion atmospheres). It should be noted, energy sources, as it is easy to see from the second equation as we see from the deductive method, the formulae for ρ of the main system, there should be an isothermal core inside c and p are applicable to any phase state of matter. a star, and such a star is close to polytropic structures higher c Let us assume stars consisting of an ideal gas. Then than class 3. In this case, instead of the former ε , we can 1 taking the ratio of (2.30) to (2.28) and using the ideal gas build ε/ε = ε , 0 ε 1, which will be subsumed into max 1 6 1 6 equation (1.8), we obtain the temperature at the centre of a λ. However in such a case ε , and hence all characteristics 1 star obtained as solutions to the system, is dependent on pc and M 7 x0 M B , and the possibility to solve the system everywhere inside c Tc = 2.29×10 μ R . (2.31) a star sets up as well correlations between the parameters. Mx0 R

At last we reach the very natural conclusion that energy is Hence, the temperature at the centre of the Sun should generated inside a star only under specific relations between be about 10 million degrees. As another example, consider B and p in that quantity which is required by the com- the infrared satellite of ε Aurigae. For this star we have patibility of the equilibrium equations. In order to continue M = 24.6M , log L/L = 4.46, R = 2,140R [5]. Calcul- this research and draw conclusions, it is very important to ating the central density and temperature by formulae (2.30) 3 note the fact that the characteristic λLx /M is actually the
5 5 0 x0 and (2.31), we obtain Tc 2×10 and pc 2×10 : thus the same for any stellar structure (see the last column in Table 2). temperature is about two' hundred thousand' degrees and the This characteristic remains almost constant under even exotic pressure about one atmosphere. Because the star is finely distributions of energy sources in stars (exotic sources of located in the “mass-luminosity” diagram (Fig. 1) and the stellar energy), because of a parallel increase/decrease of its Russell-Hertzsprung diagram, we have reason to conclude: numerator and denominator. Following a line of successive the star has the internal constitution regular for all stars. approximations, we have a right to accept the tables as This conclusion can be the leading arrow pointing to the the first order approximation which can be compared to supposition that heat energy is generated in stars under phys- observational data. All the above conclusions show that it ical conditions close to those which can be produced in an is not necessary to solve the main system of the equilibrium Earthly laboratory. equations (2.17) for more detailed cases of the aforement- Let us prove that only inside white dwarfs (the stars ioned structures of stars. Therefore we did not prove the of the very small radii — about one hundredth of R ), the uniqueness of the parameter λ. degenerate Fermi gas equation (1.9) can be valid. Naturally, if gas at the centre of a star satisfies the Fermi equation, 13 5/3 5/3 2.4 Physical conditions at the centre of stars we obtain pc = 1×10 ρc μ−e . Formulae (2.28) and (2.30) show that this condition is true only if

The average density of the Sun is ρˉ = 1.411. Using this 1/3

R 3 x0 Mx0 5/3 numerical value in (2.4), we obtain a formula determining = 3.16 10− μ− . (2.32) R × M 1/3 e the central density of stars M

3 M This formula remains true independently of the state of x0 M

ρc = 0.470 3 . (2.28) matter in other parts of the star. The last circumstance can M R 1/3 x0 affect only the numerical value of the factor . At the R x0 Mx0 same time, table 1 shows that we can assume the numerical
Taking this into account, formula (2.1) permits calculat- value approximately equal to 10.† Formula (2.32) shows that

∗It should be noted that the tables characterize the structure of stars †For stars of absolutely different structure, including such boundary only if the radiant pressure is low. In the opposite case all the characteristics instances as the naturally impossible case of equally dense stars, and the x0, Mx0 , and others are dependent on γc. cases where energy sources are located at the surface. — Editor’s remark.

N. Kozyrev. Sources of Stellar Energy and the Theory of the Internal Constitution of Stars 71 Volume 3 PROGRESS IN PHYSICS October, 2005

numerical values for radii as Chandrasekhar’s table (his well- known relation between the radius and mass of star). The exact numerical value of the ultimate mass calculated by him coincides with our 5.7M . In Chandrasekhar’s formula,

as well as in our formula (2.32), radius is correlated opposite to mass. Today we surely know masses and radii of only three white dwarfs: the white dwarfs do not confirm the opposite correlation mass-radius. So, save for the radius of Sirius’ satellite coinciding with our formula (2.32), we have no direct astrophysical confirmation about degeneration of gas inside white dwarfs. Considering stars built on an ideal gas, we deduce a formula determining the mass of a star dependent on internal physical conditions. We can use formulae (2.30) and (2.31) or formula (2.2) directly. Applying the Boyle-Mariotte equation (1.8) to formula (2.2), and taking the Stephan-Boltzmann law Fig. 1: The “mass-luminosity” relation. Here points are visual (1.7) into account, we obtain binaries, circles are spectral-binaries and eclipse variable stars, γ1/2 2 crosses are stars in Giades, squares are white dwarfs, the crossed c 2.251 1033 M = C 2 Mx0 ,C = < = × . (2.33) circle is the satellite of ε Aurigae. μ 3/2 4 G 3 πα

q 33 for regular degeneration of gas, stars (under M = M ) should Introducing the mass of the Sun M = 1.985×10 into have approximately the same radius R 2 109, i. e. about the equation, we obtain ' × 20,000 km (R = 0.03R ). Such dimensions are attributed 1/2 γc to white dwarfs. For example, the satellite of Sirius has M = 1.134M Mx . (2.34) μ2 0 M = 0.94M and R = 0.035R [6]. If the density is more than the above mentioned (if the radius is less than R = As we will see below, the “mass-luminosity” correlation = 0.03R ) and the mass of the star increases, formula (2.32) shows γc is close to 1 for blue super-giants. Hence formula shows that regular degeneration can become relativistic deg- (2.34) gives the observed numerical values for masses of eneration stars. The fact that we obtain true orders for numerical values of the masses of stars, proceeding only from numerical 4/3 4/3 15 − p = Kρ ,K = KH μe ,KH = 1.23×10 . values of the fundamental constants G, , α, is excellent confirmation of the theory. < We apply these formulae to the centre of a star, and take equations (2.28) and (2.30) into account. As a result we 2.5 The “mass-luminosity” relation see that the radius drops out of the formulae, so relativistic degeneration can be realized in a star solely in terms of the In deducing the “mass-luminosity” correlation, we assume: mass (1) the radiant pressure is negligible in comparison to the M gaseous pressure everywhere inside a star; (2) stars consist = 0.356M , (μ = 1). (2.32a) M x0 e of an ideal gas; (3) ε and κ can be approximated by functions like pαBβ. Then the main system of the equilibrium equa-

Because of Table 1, we see: n = 0 only if Mx0 = 16.1. tions takes the form Hence, the lower boundary of the mass of a non-degenerated 1 d x2dp gaseous star is 5.7M . In order to study degenerated gaseous 1 = 1 , 2 stars in detail, we should use the phase state equation that ρ1 x dx ρ1 dx −   (2.35) includes the regular state, the boundary state between the 1 d x2dB regular and degenerated states, and the degenerated state. 1 = λε , ρ x2 dx κ ρ dx − 1 Applying formulae (2.28) and (2.30) to the above ratio, 1  1 1  we obtain a correlation between the radius and the mass where of a star, which is unbounded for small radii. It should be εc κc Bc λ = , γc = . noted that introduction of a mass-radius correlation is the 4πGc γc pc essence of Chandrasekhar’s theory of white dwarfs [7]. On Solving the system, as we know, is possible under a the other hand, having observable sizes of white dwarfs, numerical value of λ close to 1. Hence a star can be in equi- equation (2.32) taken under 1/3 10 gives the same librium only if the energy generated inside it is determined x0 Mx0 =

72 N. Kozyrev. Sources of Stellar Energy and the Theory of the Internal Constitution of Stars October, 2005 PROGRESS IN PHYSICS Volume 3

by the formula stars, according to today’s data. The diagram has been built λ4πGc on masses of stars taken from Kuiper’s data base [8], and the εc = γc . (2.36) κc monograph by Russell and Moore [9]. We excluded Trumpler If a star produces another quantity of energy, it will stars [10] from the Kuiper data, because their masses were contract or expand until its new shape results in production measured uncertainly. Naturally, Trumpler calculated masses of energy exactly by formula (2.36). Because γc determines of such stars, located in stellar clusters, with the supposition the mass of a star (2.34) and εc determines the luminosity of that the K-term (the term for radiant velocities with respect a star, the “mass-luminosity” correlation should be contained to the whole cluster) is fully explained by Einstein’s red shift. in formula (2.36). In other words, the “mass-luminosity” For this reason the calculated masses of Trumpler stars can correlation is the condition of equilibrium of stars. be much more than their real masses. Instead of Trumpler Because of (2.3), stars, in order to fill the spaces of extremely bulky stars in the diagram, we used extremely bulky eclipse variable stars L Mx0 (VV Cephei, V 381 Scorpii) and data for Plasckett’s spectral- εc = . M Lx0 variable star BD +6◦ 1309. As we see in Fig. 1, our obtained correlation L M 3 is Substituting this equation into (2.36), we obtain in good accord with the observational data in all spectra∼ of

4πGc λLx0 observed masses (having a small deviation inside 1.5m). The L = Mγc . dashed line L M 10/3 is only a little different from our line. κc Mx0 Parenago [11],∼ Kuiper [8], Russell [9], and others accept this The quantity γc can be removed with the mass of a star L M 10/3 line as the best representation of observational by (2.33) data.∼ Some researchers found the exponent of mass more than our’s. For instance, Braize [12] obtained L M 3.58. 4πG44πα λL L = μ4 x0 M 3. (2.37) Even if such maximal deviation from our exponential∼ index 3κ 4 M 3 c <  x0  3 is real, the theoretical result is excellent for most stars. 33 The coefficient of proportionality in our formula (2.38) is The luminosity of the Sun is L = 3.78×10 . Proceeding from formula (2.37), we obtain very susceptible to μ. For this reason, coincidence of our theoretical correlation and observational data is evidence that L μ4 λL M 3 the chemical composition of stars is the same on the average. 1.04 104 x0 = × 3 . (2.38) The same should be said about the absorption coefficient κ: L κc Mx M  0   because physical conditions inside stars can be very different The formula (2.38) gives a very simple correlation: the even under the same luminosity (for example, red giants and luminosity of a star is proportional to the third order of blue stars located in the main direction), it is an unavoidable its mass. In deducing this formula, we accepted that ε is conclusion that the absorption coefficient of stellar matter is determined by a function ε pαBα, so ε depends on p independent of pressure and temperature. The conclusions ∼ 1 1 and B1 . It is evident that rejection of this assumption cannot justify our assumption in §1.3, when we solved the main substantially change the obtained correlation (2.38). Natur- system of equilibrium equations. ally, under arbitrary ε, the quantity ε1 depends on pc and Bc. The fact that white dwarfs lie off the main sequence Thus the multiplier 3 in formula (2.38) will have can be considered as a confirmation of degenerate gas inside λLx0 /Mx0 different numerical values for different stellar structures. At them. Because a large increase of the absorption coefficient in the same time Table 2 shows that this multiplier is approxim- white dwarfs in comparison to regular stars is not very plau- ately the same for absolutely different structures, including sible, another explanation can be given only if the structural boundary structures which are exotic. Therefore the “mass- multiplier λLx0 /Mx0 in white dwarfs is 100 times more luminosity” correlation gives no information about sources than in other stars. The location of white∼ dwarfs in the of stellar energy — the correlation is imperceptible to their Russell-Hertzsprung diagram can give a key to this problem. properties. However, other assumptions are very important. At last, proceeding from observational data, we calculate 4 As we see from the deductive path to formula (2.33), the the coefficient μ /κc in our theoretical formula (2.38). The correlation between mass and luminosity can be deduced line L M 3, which is the best representation of observ- only if the pressure depends on temperature, so our formula ational∼ data, lies a little above the point where the Sun is (2.38) can be obtained only if the gas is ideal. It is also located. For this reason, under M = M , we should have important to make the absorption coefficient κ constant for L = 1.8L in our formula (2.38). According to table 2, we all stars. The roleˆ of the radiant pressure will be considered assume 3 2 10 3. Then we obtain λLx0 /Mx0 = × − in the next paragraph. μ4 And so forth we are going to compare formula (2.38) to = 0.08 . (2.39) observational data. Fig. 1 shows masses and luminosities of κc

N. Kozyrev. Sources of Stellar Energy and the Theory of the Internal Constitution of Stars 73 Volume 3 PROGRESS IN PHYSICS October, 2005

2.6 The radiant pressure inside stars Eddington [13] and others showed an inclination of the “mass-luminosity” line to this direction in the region of bulky In the above we neglected the radiant pressure in comparison stars (the upper right corner of the diagram). But further more to the gaseous one in the equation of mechanical equilibrium exact data, as it was especially shown by Russell [9] and of a star. Now we consider the main system of the equation Baize [12], do not show the inclination for even extremely (III), which takes the radiant pressure into account. If the bulky stars (see our Fig. 2). Therefore we can conclude that absorption coefficient κ is constant (κ1 = 1), this system there are no internal structures of stars for γc > 1; the ultimate takes the form case of possible masses of stars is the case where γc = 1. Having no suppositions about the origin of energy sources 1 d x2dp 1 in stars , it is very difficult to give an explanation of this 2 = (1 λγc ε1) , † ρ1 x dx ρ1 dx − − fact proceeding from only the equilibrium of stars. The very   (2.40) 1 d x2dB exotic internal constitution of stars under γc > 1 suggests 1 = λε . that if such stars really exist in nature, they are very rare ρ x2 dx ρ dx − 1 1  1  exceptions. After calculations analogous to those carried out in de- In order to ascertain what influence γc has on the structure ducing formula (2.21), we obtain of a star, we consider the simplest (abstract) case where energy sources are distributed homogeneously throughout a x0 ρ 1 star (ε1 = 1). In this case, as we know, Mx 2 dx 0 x λ (1 + γc) = Z . (2.41) 1 x0 ρ λ = , (2.43) L 1 dx 1 + γ x x2 c Z0 and the system (2.40) takes the form The ratio of integrals in this formula depends on the distribution of energy sources inside a star, i. e. on the struct- 2 1 d x dp1 1 ure of a star. This ratio maintains a numerical value close to 1 = , ρ x2 dx ρ dx −1 + γ under any conditions. Thus λ(1 + γ ) 1. If energy sources 1  1  c c ∼ (2.44) are distributed homogeneously throughout the volume of a 1 d x2dB 1 1 = . star, we have ε1 = 1, Lx = Mx and hence the exact equality 2 ρ1 x dx ρ1 dx −1 + γc λ(1 + γc) = 1. If energy sources are concentrated at the   centre of a star, λ(1 + γc) > 1. In this case, if the radiant Introducing a new variable xγc=0 instead of x pressure takes high values (γc > 1), the internal constitution of a star becomes very interesting, because in this case x = 1 + γc xγc=0 , (2.45) λγ > 1 and the right side term in the first equation of c p (2.40) is positive at the centre of a star, our formula (2.41) we obtain the main system in the same form as that in the absence of the radiant pressure. So, in this case the main leads to p100 > 0, and hence at the centre of such a star the gaseous pressure and the density have a minimum, while characteristics of the internal constitution of star are their maximum is located at a distance from the centre . ∗ x = x 1 + γ 1/2, From this we conclude that extremely bulky stars having 0 0(γc=0) c high γ can be in equilibrium only if λ(1 + γ ) 1, or, in 3/2 c c M = M 1 +
γ , other words, if the next condition is true ∼ x0 x0(γc=0) c (2.46)

3
/2 λγc=0 4πGc Lx0 = Lx0(γc=0) 1 + γc , λ = . εc . (2.42) 1 + γc ∼ κ
Characteristics indexed by γ = 0 are attributed to the Thus, starting from an extremely bulky stellar mass c structures of stars where γ 1; their numerical values can wherein γ > 1, the quantity of energy generated by a unit c c be taken from our Table 2. Because Table 2 shows very of the mass should be constant for all such extremely bulky small changes in M for very different structures of stars, stars. The luminosity of such stars, following formulae (2.3) x0 formulae (2.46) should as well give an approximate picture and (2.2), should be directly proportional to their mass: for other structures of stars. Under high γ , the mass of a star L M. This correlation is given by the straight line drawn c (2.34) becomes in∼ the upper right corner of Fig. 1. Original data due to 1/2 γ 3/2 ∗This amazing conclusion about the internal constitution of a star is true c M 1.134M 1 + γc Mx (γ =0) . (2.47) under only high values of the radiant pressure. In regular stars the radiant ' μ2 0 c pressure is so low that we neglect it in comparison to the gaseous pressure
(see previous paragraphs). — Editor’s remark. †That is the corner-stone of Kozyrev’s research. — Editor’s remark.

74 N. Kozyrev. Sources of Stellar Energy and the Theory of the Internal Constitution of Stars October, 2005 PROGRESS IN PHYSICS Volume 3

Astronomical observations show that maximum masses quite exact with respect to the real one. If κ = κT, as a result of stars reach 120M — see Fig. 1, showing an inclination of (2.39) we have μ = 0.43. Because μ cannot be less than ∼ of the “mass-luminosity” correlation near log (M/M ) = 2. 1/2, the obtained ultimate value of κ = 0.8 is twice κ = 0.40. T Assuming this mass in (2.47), and assuming γc = 1 and This fact can be explained by the circumstance that, in this

Mx0 = 10 for it, we obtain the average molecular weight case of extremely bulky masses, the structural coefficient in μ = 0.51. formula (2.38) should be twice as small. It is evident that Then in such stars, by formula (2.39), we obtain κ = 0.8. we can accept μ = 1/2 to within 0.05. If all heavy nuclei are On the other hand, because the “mass-luminosity” correlation ionized, their average molecular weight is 2. If we assume has a tendency to the line L M for extremely bulky masses the average molecular weight in a star to be 0.55 instead of ∼ 4 1 χ (see Fig. 1), we obtain the ultimate value εˉ= 5×10 . For /2, the percentage of ionized atoms of hydrogen H becomes homogeneously distributed energy sources, formula (2.42) 1 1 leads to κ = 0.5. If they are concentrated at the centre, 2χ + (1 χ ) = , χ 90% . H 2 − H 0.55 H ' εc > εˉ= εc (Lx0 /Mx0 ). Even in this case formula (2.42) leads to εc > εˉ. There is some compensation, so the calculated Thus the maximum admissible composition of heavy numerical value of the absorption coefficient κ is true. An nuclei inside stars, permitted by the “mass-luminosity” cor- exact formula for εˉ can be easily obtained as relation, is only a few percent. Under μ = 1/2 the mass of x0 a star, where γc = 1, is obtained as 130M . This value is ρ1 Mx 2 dx indicated by the vertical line in Fig. 1. L 4πGc Lx x γc =ε ˉ = 0 Z0 . (2.48) At last we calculate the radiant pressure at the centre x0 M κ Mx0 ρ 1 + γc 3 L 1 dx of the Sun. Formula (2.34) leads to γc 10− . In this x x2 ' Z0 case the radiant pressure term in the equation of mechanical So, having considered the “mass-luminosity” correlation, equilibrium can be neglected. we draw the following important conclusions: 1. All stars (except possibly for white dwarfs) are built 2.7 Comparing the obtained results to results obtained on an ideal gas; by other researchers 2. In their inner regions, where stellar energy is generated, To deduce the “mass-luminosity” correlation by the explana- all stars have the same chemical composition, μ = tion according to the regular theory of the internal constitut- 1 = const = /2, so they are built on a mix of protons ion of stars, becomes very complicated because the theore- and electrons without substantial percentage of other ticians take a priori the absorption coefficient as dependent nuclei; on the physical conditions. They supposed the absorption 3. The absorption coefficient per unit of mass κ is inde- of light inside stars due to free-connected transitions of pendent of the physical conditions inside stars, it is a electrons (absorption outside spectral series) or transitions little less than 1. of electrons from one hyperbolic orbit to another in the field Thomson dispersion of light in free electrons has the of positive charged nuclei. The theory of such absorption was same properties. Naturally, the Thomson dispersion coeffi- first developed by Kramers, and subsequently by Gaunt, and cient per electron is especially, by Chandrasekhar [14]. According to Chandra- sekhar, the absorption coefficient depends on physical cond- 8π e2 2 itions as 6.66 10 25 ρ σ0 = 2 = × − , (2.49) 25 2 3 m c κCh = 3.9×10 1 χ , (2.51)  e  T 3.5 − H where e and m are the charge and the mass of the electron. χ2
e where H is the percentage of hydrogen, the numerical factor In the mix of protons and electrons we obtain is obtained for Russell’s composition of elements. In order

25 to clarify the possible roleˆ of such absorption in the “mass- σ 6.66 10− 0 × luminosity” correlation, we assume (for simplicity) κT = = 24 = 0.40 . (2.50) mH 1.66×10− κ κ = 0 . (2.52) The fact that our calculated approximate value of κ is Ch γ close to κT = 0.40 shows that the interaction between light and matter inside stars is determined mainly by the Thomson In this case, having small γc, formulae (2.38) and (2.33) process — acceleration of free electrons by the electric field show L M 5. This exponent is large, so we cannot neglect ∼ of light waves. γc in comparison to 1. If γc is large, the formulae show Because μ stays in the “mass-luminosity” correlation L M 3/2. Thus, in order to coordinate theory and observa- (2.38) in fourth degree, the obtained theoretical value of μ is tions,∼ we are forced to consider “middle” numerical values of

N. Kozyrev. Sources of Stellar Energy and the Theory of the Internal Constitution of Stars 75 Volume 3 PROGRESS IN PHYSICS October, 2005

γc and reject the linear correlation between log L and log M. see from (2.56), the theoretical value of κ0 decreases slightly. Formulae (2.47) and (2.48) show On the other hand, the previous paragraph showed that the value of κ0, obtained from observations, decreases much 2 γc 1 + γc 1 β more. As a result, the theoretical and observational values of M 2 ,M 2 − , 4 4 4 κ can be matched (which is in accordance with Stromgren’s¨ ∼ μ
∼ μ β (2.53) 2 conclusion). All theoretical studies by Stromgren’s¨ followers, γ 3/2 L M c ,L M 3/2 1 β μ . who argued for evolutionary changes of relative amounts of ∼ 1 + γ ∼ − c hydrogen in stars, were born from the above hypothesis.
Here are formulae where γc has been replaced with the The hypothesis became very popular, because it provided constant β, one regularly uses in the theory of the internal an explanation of stellar energy by means of thermonuclear constitution of stars reactions, as suggested by Bethe. It is evident that the above theories are very strained. On p 1 β = c = . (2.54) the other hand, the simplicity of our theory and the general p0 1 + γc way it was obtained are evidence of its truth. It should be Thus the “mass-luminosity” correlation, described by the noted that our two main conclusions two formulae (2.53), becomes very complicated. The form- 2 (1) μ = 1/2 , χ = 1 ; (2) κ = κ , ulae are in approximate agreement with Eddington’s form- H T ulae [15] and others. The exact formula for (2.51) introduces obtained independently of each other, are physically con- the central temperature Tc into them. Under large γc, as we χ2 nected. Naturally, if H = 1, Chandrasekhar’s formula (2.51) see from formulae (2.46), the formula for Tc (2.31) includes becomes inapplicable. Kramers absorption (free-connected the multiplier β transitions) becomes a few orders less; it scarcely reaches M the Thomson process. At the same time, our main result is 7 x0 M that γc < 1 for all stars, and this led to all the results of our Tc = 2.29×10 μ β R . (2.55) Mx0 γ =0 theory. Therefore this result is so important that we mean  c R to verify it by other astrophysical data. We will do it in the Then, through Tc, the radius and the reduced temperature next chapter, analysing the correlation “period — average of a star can be introduced into the “mass-luminosity” cor- density of Cepheids”. In addition, according to our theory, relation. This is the way to obtain the well-known Eddington the central regions of stars, where stellar energy is generated, temperature correction. consist almost entirely of hydrogen. This conclusion, despite In order to coordinate the considered case of “middle” γc, its seemingly paradoxical nature, must be considered as an we should accept γc = 1 starting from masses M 10M . empirically established fact. We will see further that study of ' So, for the Sun we obtain γc = 0.08. As we see from form- the problem of the origin of stellar energy will reconcile this ula (2.47), it is possible if μ 2. Then formula (2.39), using result with spectroscopic data about the presence of heavy the numerical value '3 3.8 10 3 given by Eddin- λLx0 /Mx0 = × − elements in the surface layers of stars. gton’s model, gives κc = 170 and κ0 = 14. The theoretical value of κ can be obtained by comparing (2.52) and (2.51); 0 2.8 The roleˆ of convection inside stars it is αμ κ = 3.9 1025 1 χ2 . (2.56) 0 × H In §1.3 we gave the equations of equilibrium of stars (II), 3 Tc − < which take convective transfer of energy into account. As-
7 According to (2.55)p we obtain Tc = 4×10 . Then, by suming the convection coefficient const, the second eq- A = (2.56), we have κ0 = 0.4. So, according to Eddington’s uation of the system (the heat equilibrium equation) can be model, the theoretically obtained value of the absorption written as coefficient κ0 = 14 is 30 times less than the κ0 = 0.4 re- 2 quired, consistent with the observational data∗. This diver- 1 d x dB1 κc ρcA 1 2 dp1 2 2 x u = λε1 , (2.57) gence is the well-known “difficulty” associated with Edding- ρ x dx ρ dx − c γc ρ x dx − 1  1  1   ton’s theory, already noted by Eddington himself. According to Stromgren¨ [16], this difficulty can be removed if we accept where Γ p dB the hypothesis that stars change their chemical composition u = 1 1 1 . (2.58) − 4(Γ 1) B dp with luminosity. Supposing the maximum hydrogen content, − 1 1 μ can vary within the boundaries 1/2 6 μ 6 2. Then, as we The convection term in (2.57) plays a substantial roleˆ only if ∗As it was shown in the previous paragraph, Kozyrev’s theory κc ρcA c γc gives κ0 = 0.5–0.8 for stars having different internal constitutions, which > 1 , A > . (2.59) corresponds well to observations. — Editor’s remark. c γc κc ρc

76 N. Kozyrev. Sources of Stellar Energy and the Theory of the Internal Constitution of Stars October, 2005 PROGRESS IN PHYSICS Volume 3

Table 3

3 x0 λLx0 κ x Mx λLx x Mx 1 1 0 0 0 3 3Mx0 Mx0 3 const 2.4913 3.570 3.018 8.9 11.46 20.5 1.97×10− 3 κCh 1.88 1.25 1.25 11.2 12.4 37.0 0.65×10−

Hence the convection coefficient for the Sun should sat- Table 2 shows that even when ε1 = B1 , any star should 7 isfy A >5×10 . In super-giants, convection would be sub- contain a convective core. If ε1 depends only on temperature 16 m stantial only under A >10 . The convection coefficient A, and can be approximated by function ε1 = T , the calc- as we see from formula (1.17), equals the product of the ulations show that λ reaches its critical value of 1.6 when convective current velocity vˉ and the average length λˉ tra- m = 3.5. Thus a star has a convective core if m > 3.5. The velled by the current. Thus convection can influence energy radius x1 of the convective core is determined by equality transfer inside super-giants if convection currents are about between the temperature gradients (see above). Writing the size of the star (which seems improbable). At the same (2.62) as an equality, we obtain time, if a convection instability occurs in a star, the average length of travel of the current becomes the size of the whole B1 λLx = 1.6Mx = 1.6Mx ρx . (2.63) convection zone. Then the coefficient A increases so much 1 1 p 1 1  1 x1 that it can reach values satisfying (2.59). If A is much more than the right side of (2.59), taking into account the fact that It is evident that the size of the convective core increases all terms of the equilibrium equation (2.57) are about 1, the if the energy sources become more concentrated at the centre term in square brackets is close to 0. Then, if A is large, of a star. In the case of a strong concentration, all energy sources become concentrated inside the convective core. 4(Γ 1) − Γ Then inside the region of radiant equilibrium we have u = 0 , hence B1 = p1 , (2.60) λLx = λLx1 = const. Because the border of the convective which is the equation of adiabatic changes of state. For a core is determined by equality of the physical characteristics’ monatomic gas, Γ = 5/3 (n = 3/2) and hence gradients in regions of radiant equilibrium and convective equilibrium, not only are p1 and T1 continuous inside such 8/5 B1 = p1 . (2.61) stars but so are their derivatives. Therefore such a structure for a star can be finely calculated by solving the main system

Because, according our conclusions, stars are built up of the equilibrium equations under ε1 = 0 and boundary 5 almost entirely of hydrogen, Γ can be different from /3 in conditions: (1) under some values of x =x1, quantities p1, B1 only the upper layers of stars, which is insufficient in our and their derivatives should have numerical values satisfying consideration of a star as a whole. Therefore zones of free the solution to Emden’s equation under n = 3/2; (2) under convection can appear because of an exotic distribution of some value x =x0 we should have p1 = B1 = 0. The four energy sources. boundary conditions fully determine the solution. We can

Free convection can also start in another case, as soon find x1 by step-by-step calculations as done in §2.3 for λ. as the temperature gradient of radiant equilibrium exceeds The formulated problem, known as the problem of the the temperature gradient of convective equilibrium. This is internal constitution of a star having a point-source of energy Schwarzschild’s condition, and it can be written as and low radiant pressure, was first set up by Cowling [17]. In his calculations the absorption coefficient was taken as d log B d log B 1 > 1 , variable according to Chandrasekhar’s formula (2.51): d log p1 Rad d log p1 Con . However in §2.6 we showed that inside     κ = κCh κ = κT all stars. Only in the surface layers of a star should κ which, taking (2.22) and (2.61) into account, leads to increase to κT. But, because of very slow changes of physical conditions along the radius of a star, κ remains κ in the λLx B T > 1.6 1 . (2.62) greater part of the volume of a star. Therefore it is very M p x 1 interesting to calculate the internal structure of a star under From this formula we see that free convection is im- given values of κ = const. We did this, differing thereby possible in the surface layers of stars. In central regions we from Cowling’s model, so that there are two alternatives: obtain the next condition for free convection our model (κ = const) and Cowling’s model (κT). All the calculations were carried out by numerical integration of

λ > 1.6. (2.23b) assuming there that ε1 = 0.

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Table 4 ical characteristics of matter inside stars if their structural characteristics are known. In order to be sure of the calc- x T1 p1 ρ1 ulations, besides our general theoretical considerations, it 0.00 1.000 1.000 1.000 would be very important to obtain the structural character- istics proceeding from observational data, related at least to 0.50 0.983 0.958 0.975 some classes of stars. 1.00 0.935 0.845 0.904 Properties of the internal structure of a star should mani- 1.50 0.856 0.677 0.791 fest in its dynamical properties. Therefore we expect that 2.00 0.762 0.507 0.665 the observed properties of variable stars would permit us to learn of their structures. For instance, the pulsation period 2.50 0.652 0.346 0.530 of Cepheids should be dependent on both their physical 3.00 0.544 0.211 0.388 characteristics and the distribution of the characteristics in- 3.50 0.451 0.117 0.259 side the stars. Theoretical deduction of this correlation can 1 be done very strictly. Therefore we have a basis for this 4.00 0.370 0.598 10− 0.161 × deduction in all its details. 1 1 4.50 0.328 0.284 10− 0.936 10− × × Radiation of energy by an oscillating star must result in 1 1 5.00 0.245 0.125×10− 0.510×10− a dispersion of mechanical energy of its oscillations. It is 2 1 most probable that the oscillation energy of variable stars is 5.50 0.195 0.52×10− 0.266×10− 2 1 generated and supported by energy sources connected to the 6.00 0.154 0.20 10− 0.129 10− × × oscillation and radiation processes. In other words, such stars 3 2 6.50 0.118 0.67×10− 0.57×10− are self-inducing oscillating systems. Observable arcs of the 3 2 7.00 0.087 0.20×10− 0.23×10− oscillating luminosity and speed reveal a nonlinear nature for 4 3 the oscillations, which is specific to self-inducing oscillating 7.50 0.060 0.49 10− 0.82 10− × × systems. The key point of a self-inducing oscillating system 5 3 8.00 0.036 0.64 10− 0.18 10− × × is a harmonic frequency equal to the natural frequency of 6 4 8.50 0.015 0.19×10− 0.79×10− the whole oscillating system. Therefore, making no attempt 8.90 0.000 0.000 0.000 to understand the nature of the oscillations, we can deduce the oscillation period as the natural period of weak linear Table 3 gives the main characteristics of the “convective” oscillations. model of a star under κ = const and κ = κCh. The κCh are taken from Cowling’s calculations. Values of λL were x0 3.1 The main equation of pulsation found by formula (2.62). In this model, distribution of energy sources inside the convective core does not matter. For this Typical Cepheids have masses less than 10 solar masses. For reason, the quantities λ and L are inseparable. If we would x0 instance, δ Cephei has M 9M . In this case equation (2.34) like to calculate them separately, we should set up the dis- leads to γ < 0.1, so Cepheids' should satisfy L M 3, i. e. the tribution function for them inside the convective core. c “mass-luminosity” relation. Therefore the radiant∼ pressure We see that the main characteristics of the structure of a plays no roleˆ in such stars, so considering their internal star, the quantities x , M , and λL , are only a little dif- 0 x0 x0 constitutions we should take into account only the gaseous ferent from those calculated in Table 2. The main difference pressure. In solving this problem we will consider linear between structures of stars under the two values const κ = oscillations, neglecting higher order terms. This problem and is that under our const the convective core is κ = κCh κ = becomes much simpler because temperature changes in such larger, so such stars are close to polytropic structures of class a star satisfy adiabatic oscillations in almost its whole volume, 3 2, and there we obtain a lower concentration of matter at the / except only for the surface layer. Naturally, in order to centre: 20.5 . Table 4 gives the full list of calculations ρc = ρˉ obtain the ratio between observed temperature variations for our convective model ( const). κ = and adiabatic temperature variations close to 1, the average change of energy inside 1 gram in one second should be Chapter 3 about εˉ, i. e. 102. This is 108 per half period. On the other ∼ The Internal Constitution of Stars, Obtained from the hand, the heat energy of a unit of mass should be about Ω/M (according to the virial theorem), that is 1015 ergs Analysis of the Relation “Period — Average Density of ∼ Cepheids” and Other Observational Data by formula (2.5). Thus during the pulsation the relative change of the energy is only 107, so pulsations of stars are In the previous chapter we deduced numerous theoretical adiabatic, with high precision. We assume that the pulsation correlations, which give a possibility of calculating the phys- of a star can be determined by a simple standing wave with

78 N. Kozyrev. Sources of Stellar Energy and the Theory of the Internal Constitution of Stars October, 2005 PROGRESS IN PHYSICS Volume 3

a frequency n/2π where n2 2 ∂ V 2 λ = 4 . (3.8) V (r, t)=V (r) sin nt, a= = n V (r) sin nt, (3.1) 4πGρc Γ ∂t2 − − 3 where V (r) is the relative amplitude of the pulsation So the problem of finding the pulsation
period has been reduced to a search for those numerical values of λ by which δr V (r) = . the differential equation (3.7) has a solution satisfying the r “natural” boundary conditions By making the above assumptions, Eddington had solved dV x0 the problem of pulsation of a star. x4 p = 0. (3.9) 1 dx Linking the coordinate r to the same particle inside a 0 star, we have the continuity equation as follows Formula (3.8) gives the correlation “period — average 2 Mr = const , r ρdr = const . (3.2) density of Cepheids” and, hence, the general correlation “period — average density of a star”. It is evident that λ Using the condition of adiabatic changes δp = Γ δρ and p ρ depends on the internal structure of a star. Its expected taking variation from the second equality, we obtain numerical value should be about 1. For a homogeneously 3 δp dV dense star, x /(3Mx) = 1 everywhere inside it. In this case = Γ 3V + r . (3.3) p − dr the differential equation (3.7) has the solution: V = const,   λ = 1. This solution determines the main oscillation of such It is evident that the equations of motion a star. In order to find the main oscillations of differently structured stars, we proceed from the solution by applying dp GMr = (g + a) , g = the method of perturbations. ρdr − r2 give, neglecting higher order terms, 3.2 Calculation of the mean values in the pulsation dδp dp equation by the perturbation method = aρ + 4V . dr − dr Substituting formula (3.3) into this equation, we obtain We write the pulsation equation in general form Eddington’s equation of pulsation py 0 0 + q y 1 λρ = 0 . (3.10) − d2V 1 dV r dp + 4 + +

dr2 r dr p dr If we know a solution to this equation under another   (3.4) function ρ = ρ V 1 dp n2 r 0 + (3Γ 4) = 0 . rΓ p dr − − g py0 0 + q y 1 λ ρ = 0 , (3.11)   0 0 − 0 0 We introduce a dimensionless variable x instead of r (we

used this variable in our studies of the internal constitution hence we know the function y0 and the parameter λ0. After of stars). As it is easy to see multiplying (3.10) by y0 and (3.11) by y, we subtract one from the other. Then we integrate the result, taking the limits g ρˉr Mx 0 and . So, we obtain = 4πG = 4πGρ . (3.5) x0 r 3 c x3 x0 Substituting (3.5) into formula (3.4), we transform the qyy [λ ρ λρ] dx = 0 , 0 0 0 − pulsation equation to the form Z0 2 hence d V 1 dV x dp1 + 4 + x0 dr2 x dr p dr −  1  qyy0 ρ0 dx (3.6) 0 V 1 dp n2 x3 λ = λ0 Z . (3.12) 1 (4 3Γ) + = 0 . x0 − xΓ p1 dr − 4πGρc 3Mx qyy0 ρdx   0 We transform this equation to self-conjugated form. Mul- Z tiplying it by x4 p , we obtain If the oscillations are small, equation (3.10) is the same 1 as (3.11) with only an infinitely small correction 3 d 4 dV 3 dp1 (4 3Γ) x x p1 V x − 1 λ =0, (3.7) dx dx − dx Γ − 3M ρ = ρ0 + δρ , y = y0 + δy , λ = λ0 + δλ .    x 

N. Kozyrev. Sources of Stellar Energy and the Theory of the Internal Constitution of Stars 79 Volume 3 PROGRESS IN PHYSICS October, 2005

Then the exact formula for δλ ρˉ. This result was obtained by Ledoux [18] by a completely

x0 different method. It is interesting that our equations (3.16) qyy0 δρ dx and (3.18) coincide with Ledoux’s formulae. 0 δλ = λ0 Z We next calculate λ for stars of polytropic structures. In − x0 such cases Ix0 is qyy0 ρdx 0 Z x0 I = x2 M 6(n + 1) T x2 dx , (3.20) can be replaced by x0 0 x0 − 1 Z0 x0 2 where n is the polytropic exponent. Thus qy0 δρ dx 0 x δλ = λ0 Z , 0 − x0 T x2 dx qy2 ρdx 1 5 n 1 0 = − 1 6 (n + 1) 0 . (3.21) 0 ˉ  Z 2  Z λ 3 − Mx0 x0   and thus we have     x0 2 Calculations of the numerical values of λˉ for cases of qy0 ρ0 dx 0 different polytropic exponents are given in Table 5. λ = λ0 Z . (3.13) x0 Under large λˉ, much different from 1, 2 Table 5 qy0 ρdx the calculations for Table 5 are less Z0 precise. Therefore, in order to check n λˉ In our case y0 = 1 and λ0 = 1. Comparing formulae the calculated results, it is interesting (3.10) and (3.7), using (3.13), we obtain 0 1.00 to compare the results for n = 3 to those obtained by Eddington via his x0 1 1.91 exact solution of his adiabatic oscil- 3 xρ1 Mx dx 0 3/2 2.52 lation equation for his stellar model. λ = λ0 Z . (3.14) x0 4 2 3.85 For the stars we consider, he obtained, x ρ1 dx n2 3 0 2.5 7.00 = (3 4/Γ). Hence, com- Z πGρc Γ 10 − We re-write this equation, according to (2.5a), as follows 3 13.1 paring his result to our formula (3.8), 9 ρc we obtain λ = 9/40 and λˉ = = x0 40 ρˉ 2 9 p1 x dx = 12.23. This is in good agreement with our result λˉ = 13.1 0 λ = λ0 Z . (3.15) given in Table 5. x0 4 ρ1 x dx Z0 3.3 Comparing the theoretical results to observational If we introduce the average density ρˉ into formula (3.8) data instead of the central one ρc, then according to (2.4), We represent the “period — average density” correlation in 2 the next form ˉ n λ = 4 , (3.16) 4πGρˉ Γ P ρˉ0 = c1 , (3.22) − 3 q 3
where P is the period (days), ρˉ is the average density ρc x 0 λˉ = λ = 0 λ . (3.17) expressed in the multiples of the average density of the Sun ρˉ 3Mx0 2π Using formulae (2.6) and (3.17) we re-write (3.15) as n = , ρˉ = 1.411ρˉ . 86,400 P 0 x2 Ω M λˉ = 0 x0 x0 , (3.18) I Employing formula (3.16), it is easy to obtain a correla- x0 ˉ tion between the coefficients λ and c1 where Ix0 is the dimensionless moment of inertia 4 2 ˉ − x0 λ Γ = 0.447 10c1 . (3.23) 4 − 3 Ix = ρ x dx . (3.19)   0 1
Z0 By analysis of the “mass-luminosity” relation we have Formulae (3.16) and (3.18) determine the oscillation per- previously shown that the radiant pressure is much less than iod of a star, P = 2π/n, independently of its average density the gaseous pressure in a star. Therefore, because the inner

80 N. Kozyrev. Sources of Stellar Energy and the Theory of the Internal Constitution of Stars October, 2005 PROGRESS IN PHYSICS Volume 3

regions of a star are primarily composed of hydrogen, the Equation (3.23) shows that when λˉ = 12.23 and the ob- heat energy there is much more than the energy of ionization. servable c1 = 0.075 we have Γeff = 1.40. At the same time Γeff 5 So we have all grounds to assume Γ = /3, the ratio of the should undergo changes independently of γc, i. e. depending heat capacities for a monatomic gas. Hence upon the roleˆ of the radiant pressure. For a monatomic gas, Eddington [21] and others obtained this correlation as 2 ˉ − λ = 1.34 10c1 . (3.24) 4 1 1 + 4γc
Γeff = . (3.26) In order to express c1 in terms of the observed char- − 3 3 (1 + γc)(1 + 8γc) acteristics of a star, we replace ρˉ0 in formula (3.22) with the reduced temperature and the luminosity, via the “mass- Under Γeff = 1.40 we obtain γc = 1.5. We accept this luminosity” formula. The “mass-luminosity” correlation has numerical value in accordance with the average period of d a general form L M α for any star. We denote by Tˉ the Cepheids, P = 10 . Then, by the “mass-luminosity” relation, ∼ M = 12M . It is possible to think that this result is in good reduced temperature of a star (with respect to the temperature ˆ of the Sun), and by Mb its reduced stellar magnitude. Then, agreement with the conventional viewpoint on the role of the by formula (3.22), we obtain radiant pressure inside stars (see §2.7). However, because λc depends on the mass of a star, other periods give different 1 ˉ Γeff (by formula 3.26) and hence other numerical values 0.30 Mb M +log P +3 log T = log c1 . (3.25) − 5α − of c . Using formulae (3.26) and (3,23), we can calculate   1
c1 for variable stars having longer pulsations, with periods From this formula we see that, in order to find c1, it is d d 20 < P < 30 . Instead of the average value log c1 = 1.12 unnecessary to know the exact value of α (if, of course, α − found by Becker for the stars, there should be log c1 = 1.00. has a large numerical value). Eddington’s formula for the Despite the small change, observations show no such increase− “mass-luminosity” relation, taken for huge masses, gives of c1 [20]. Therefore, our conclusion about the negligible α 2 (compare with 2.53). Therefore, Eddington’s value ∼ roleˆ of the radiant pressure in stars, even inside super-giants, of c1 = 0.100 is overstated. Applying another correlation, finds a new verification. This result verifies as well our results 10/3, Parenago [19] obtained 0.071. Becker [20] L M c1 = μ = 1/2 and κ = κ , obtained in chapter 3. carried∼ out a precise analysis of observational data using T Kuiper’s empirical “mass-luminosity” arc. He obtained the 3.4 Additional data about the internal constitution of average value of c = 0.076 for Cepheids. Formula (2.4) 1 stars gives λˉ = 2.7 or λˉ = 2.3, so that Table 5 leads us to conclude 3 that Cepheids have structures close to the polytropic class /2, Some indications of the internal structure of stars can be like all other stars. Hence Cepheids have a low concentration obtained from analysis of the elliptic effect in the luminosity of matter at the centre: ρc = 6ρˉ. arcs of eclipse variable stars. Observations of such binaries This result is in qualitative agreement with the “natural gives the ratio of diameters at the equator of a star, which viewpoint” that sources of stellar energy increase their prod- becomes elliptic because of the flow-deforming effect in uctivity towards the centre of a star. However (as we saw such binary systems. For synchronous rotations of the whole in §2.8) the model for a point-source of energy and for system and each star in it, the compressed polar diameter of a constant absorption coefficient, giving stars of minimal each star should be different (in the first order approximation) average densities, leads to a strong concentration at the from the average equatorial one with a multiplier dependent centre, ρc/ρˉ= 20.5. Thus λˉ for such a model should be more x0 on their masses. Thus, proceeding from the observed com- 2 than an observable one. Really, having p1 x dx = 6.06 pression we can calculate the meridian compression . Ac- 0 cording to Clairaut’s theory  is proportional to ϕ, the ratio and I = 140.0 calculated by Table 4, formulaeZ (3.15) and x0 of the centrifugal force at the equator to the force of gravity (3.17) give λˉ = 8.0 for models with the ultimate concentra- tion of energy sources. So, such stars are of the polytropic ω2  = αϕ , ϕ = , class n = 2.5. If the absorption coefficient is variable 3πGρˉ (Cowling’s model), calculations give even more: λˉ = 8.4. Eddington and others, in their theoretical studies of the where α is a constant dependent on the structure of the star. pulsation period within the framework of Eddington’s model, This constant was calculated for stars of polytropic structures explain the deviation between the theoretical and observed by numerous researchers: Russell, Chandrasekhar and others. values of λˉ by an effect of the radiant pressure. Studies of If n = 0 (homogeneous star), α = 1.25. If n = 1, we have 2 pulsations under γc close to 1 show that the obtained formula α = 15/(2π ) = 0.755. If n = 5 (the ultimate concentration, for the period under low γc is true even if Γ is the reduced Roche’s model), α = 0.50. We see that the constant α is ratio of the heat capacities (which is, depending on the roleˆ sensitive to changes in the structure of a star. Therefore of the radiant pressure, 4/3 6 Γ 6 5/3). determination of the numerical values of n in this way

N. Kozyrev. Sources of Stellar Energy and the Theory of the Internal Constitution of Stars 81 Volume 3 PROGRESS IN PHYSICS October, 2005

requires extremely precise observations. The values of n Table 6 with Table 6, we come to the same so obtained are very uncertain, despite the simplicity of conclusion that we have obtained by the theory. Shapley first concluded that stars are almost n k completely different methods: that homogeneous. This was verified by Luiten [22] who found stars have polytropic structures of the average value α = 0.57 for a large number of stars like β 0 1.00 class n = 3/2. Lyrae, and α = 0.71 for stars like Algol. His results corres- 1 0.65 For the convective model of a star pond to the polytropic structures n = 3/2 and n = 1 respect- (calculated in §2.8) we obtain k=0.26. 3/2 0.52 ively. This is much less than required. The The observed motion of the line of apsides in numerous 2 0.40 same convective model with a vari- eclipse binaries can be explained, in numerous cases, by their 2.5 0.28 able absorption coefficient (Cowling’s elliptic form. Because matter is more strongly concentrated in 3 0.20 model) gives even less: k = 0.19. a binary system than in regular stars, the binary components The agreement of our value n = 3/2 interact like two point-masses, so there should be no motion with other data, obtained by very dif- of the line of apsides. Therefore the velocity of the line of ferent methods, verifies Blackett’s law. It is possible his apsides should be proportional (in the first order approxi- formula (3.27) should be written without β, but with the mation) to α 1/2, where α is sensitive to changes in the denominator 2πc. structure of a− star (as we showed above). Many theoretical studies on this theme give contradictory formulae for the velocity, depending on hypotheses about the properties of 3.5 Conclusions about the internal constitution of stars rotation in the pair. Russell, in his initial studies of this problem, supposed the rotating components solid bodies. The most certain conclusions about the structure of stars This theory, being applied to the system Y Cygni by Russell are derived from the theory of pulsation of Cepheids. We and Dugan [23], gave α 1/2 = 0.034, which is the polytropic have concluded that Cepheids have structures close to the − 3 structure 1/2 < n <2. Other researchers, having made other polytropic one of class n = /2, for which ρc = 6ρ ˉ. This suppositions, obtained larger n: n 3. It is probable that conclusion is verified by other data, whereas each ofthem we can be most sure only that, because' we observe motion could be doubtful when being considered in isolation. At of the line of apsides in binaries, the stars have no strong the same time all the data, characterizing stars of different concentration of matter at the centre. classes, lead to the same result. It is probable that stars are Blackett supposed a law according to which the ratio really close to being homogeneous, having a low concentr- between the magnetic momentum PH and angular moment- ation of matter at the centre like the bulky planets, Jupiter um U is constant for all rotating space bodies. If this law and Saturn. Such a distribution of matter, as we saw in the is correct, we could have a possibility of determining the ultimate case of the convective model, cannot be explained structures of stars in an independent way. We denote by k by a strong concentration of an energy source at the centre, the ratio between the moments of the inertia of an arbitrary or by a special kind of absorption coefficient. The real reason structured star rotating with the angular velocity ω and of the is that the radiant pressure B is included in the mechanical same star if it would be homogeneous throughout. Then equilibrium equation through the gaseous pressure in the 1 exponent /4. Therefore the structural characteristics Mx0 2 5 I 2 x0 and X0, determined by the function ρ1, have small changes U = kωMR , k = 2 , 5 3 x Mx0 even in very different models. Hence, in order to obtain the observable low concentration of matter at the centre of stars, where is the dimensionless moment of inertia. Using Ix0 we can search for the reason only in the heat equilibrium Blanchett’s formula [24] equation. The polytropic model n = 3/2 differs from other

1/2 polytropic models by a smaller value of x0. In order to PH G = β (3.27) make x0 smaller, the gaseous pressure should decrease more U 2c strongly in the upper layers of a star. Such a rapid decrease in (β is a dimensionless multiplier, equal to about 1), and having the pressure is possible only if the surface layers are heavy. 3 the magnetic magnitude at the pole H = 2PH /R , we can In other words, in the case of the strong increase of the calculate k. For the Earth (k = 0.88), we obtain β = 0.3. molecular weight in the surface layers of a star. Such an Supposing k = 0.16 for stars, Blackett has found: β = 1.14 explanation is in complete agreement with our conclusion for the Sun and β = 1.16 for 78 Virginis (its magnetic field about the high concentration of hydrogen in the internal has been measured by Babcock). regions of stars. If the average molecular weight changes If Blackett’s law (3.27) is valid throughout the Universe from μ = 1/2 at the centre to μ = 2 at the surface of a star, and β = 0.3 for all space bodies, not just for the Earth, then such a change of the molecular weight can be sufficient. k = 0.60 should be accepted for stars. Comparing k = 0.60 What is the goal of introducing the variable μ? Let us

82 N. Kozyrev. Sources of Stellar Energy and the Theory of the Internal Constitution of Stars October, 2005 PROGRESS IN PHYSICS Volume 3

assume that μ depends on the temperature as substantial. Therefore on the average we have μ < 2 inside a star, so the problem about homogeneity of the molecular 1 weight of stars is not completely solved with the above. μ1 = s , (3.28) T1 We saw that the dimensionless mass Mx0 is almost the where s is a positive determined exponent. Increase of the same in completely different models of stars. For polytropic 3 molecular weight at the surface should result in an increase structures of the classes n = /2 and n = 2, convective mo- dels, and models described in Table 2, we obtained approxi- of the absorption coefficient κ (transition from κ = κT to mately the same numerical values of Mx0 . Therefore we can κ = κCh). At the same time, under energy sources concentr- surely accept Mx0 = 11. What about x0? According to obser- ated at the centre, the quantity κ1Lx/Mx can remain almost ved structures of stars, we accept x0 = 6. Hence ρˉc = 6.5ρˉ. In the same. If κ1Lx/Mx = const = 1, equation (2.22a) leads to order to obtain κ = κ1 from the observed “mass-luminosity” 3 3 4 relation, we should have λL /M = 1.0 10− . Thus we p1 = B1 = T1 , λ = 1 . (3.29) x0 x0 ×

obtain λLx0 = 1.5. As a result, using these numerical values Instead of T1 we introduce the characteristics in formulae (2.28), (2.30), and (2.31), we have a way of cal- culating the physical conditions at the centre of any star. We 1+s T1 1+s s 1 u = = T = μ− , (3.30) now make this calculation for the Sun. Assuming μc = /2, 1 1 1 μ1 we obtain which keeps the ideal gas equation in the regular form 15 2 ρc = 9.2, pc = 9.5×10 dynes/cm , p = u ρ . According to (3.29), we have 1 1 1 3 12 2 γc = 0.4×10− ,Bc = 3.8×10 dynes/cm , (3.34) 4 1+s p1 = u1 . (3.31) 6 Tc = 6.3×10 degrees. where we should equate the exponent 4/(1 + s) to n + 1 Of the data the most soundly calculated is γc , because according to formula (2.7a). it is dependent only on Mx0 . Thus a low temperature at Thus we have the centre of the Sun, about 6 million degrees, is obtained

n 3 s because of low numerical values of μc and x0. Having such ρ = u , n = − , (3.32) 1 1 1 + s low temperatures, it is scarcely possible to explain the origin of stellar energy by thermonuclear reactions. so the function u1 is determined by Emden’s equation of The results indicate possible ways to continue our research class n. Hence, in order to obtain the structure n = 3/2, there into the internal constitution of stars. They open a way for a should be s = 3/5 — the very low increase of the molecular physical interpretation of the Russell-Hertzsprung diagram, weight: for instance, under such s the molecular weight μ which is directly linked to the origin of stellar energy. increases 4 times at the distance x1 where

8 5 1 3 1 3 μ = = 0.025,T = = 0.10. (3.33) 1 4 1 4 P A R T II     Chapter 1 At x > x1 the molecular weight remains unchanged, the equilibrium of a star is determined by the regular system of The Russell-Hertzsprung Diagram and the Origin the equilibrium equations. However at the numerical values of Stellar Energy (3.33) almost the whole mass of a star is accounted for (see Table 4, for instance), so we obtain small corrections to 1.1 An explanation of the Russell-Hertzsprung diagram the polytropic structure n = 3/2. Naturally, tables of Emden’s by the theory of the internal constitution of stars 3 function taken under n = /2 show that x1 = 5.6 and Mx1 = = 11.0. Applying formula (2.27a), we obtain x0 = 7.0 instead The Russell-Hertzsprung diagram connects the luminosity L of x0 = 6, as expected for such a polytropic structure. These of a star to its spectral class or, in other words, the reduced calculations show that the observed structures of stars∗ verify temperature Teff . The theory of the internal constitution of our result about the high content of hydrogen in the internal stars uses the radius R of a star instead of the effective regions of a star, obtained from the “mass-luminosity” rel- temperature Teff . It follows from the Stephan-Boltzmann law ation. At the same time, it should be taken into account that 1 the hydrogen content in the surface layers of stars is also L = 4πR2σ T 4 , σ = αc , eff 4 ∗The fact that the molecular weight is variable does not change the formulas, determining the pulsation period of Cepheids. The variability of where c is the velocity of light, α is the radiant energy μ can include a goal only if the whole structure of star has been changed. density constant. Thus, the Russell-Hertzsprung diagram is

N. Kozyrev. Sources of Stellar Energy and the Theory of the Internal Constitution of Stars 83 Volume 3 PROGRESS IN PHYSICS October, 2005

the same for the correlation L(R) or M (R), if we use the If we know how κ depends on B and ρ, formula (1.4a) “mass-luminosity” relation. Due to the existence of numerous leads to equation (1.4). So formulae (1.1), (1.2), and (1.4a) sequences in the Russell-Hertzsprung diagram (the main permit calculation of the average numerical values of the sequence, the sequences of giants, dwarfs, etc.) the cor- density, the pressure, and the temperature for any star. Exact relations L(R) and M (R) are not sufficiently clear. In this numerical values of the physical parameters at a given point paragraph we show that for most stars the correlations L(R) inside a star (at the centre, for instance) can be obtained, and M (R) are directly connected to the mechanism gener- if we multiply the formulae by dimensionless “structural” ating stellar energy. The essence of the correlation L(R) coefficients. We studied the structural coefficients in detail in becomes clear, as soon as we replace the observable charact- Part I of this research. We studied them by both mathematical eristics of stars (the masses M, the luminosities L, and the methods (solving the system of the dimensionless differential radii R) with the parameters which determine the physical equations and mechanical equilibrium and heat equilibrium conditions inside stars. The method of such calculations and of a star) and empirical methods (the analysis of observable the precision of the obtained results were discussed in detail properties of stars). in Part I of this research. Values of ρ, p, and T , calculated by formulae (1.1), First we calculate the average density of a star (1.2), and (1.4), should be connected by the equation of the phase state of matter. Hence, we obtain the first theoretical 3M ρ = . (1.1) correlation 4πR3 F1 (L, M, R) = 0 , (1.7) Then, having the mechanical equilibrium of a star, we calculate the average pressure within. This internal pressure which almost does not depend on the kind of energy gener- is in equilibrium with the weight of the column whose ation in stars. aperture is one square centimeter and whose length is the For instance, a star built on an ideal gas has radius of the star. The pressure is p = gρR. Because of T g = GM/R2, p = < ρ . 3G M 2 μ p = 4 . (1.2) 4π R Dividing (1.2) by (1.1), we obtain What can be said about the temperature of a star? It should be naturally calculated by the energy flow of excess G M αG4 M 4 T μ ,B 4 μ 4 , (1.8) radiation FR ' R ' 3 R L < < F = , (1.3) R 4πR2 B γ = M 2 μ4. (1.9) because the flow is connected to the gradient of the temper- p ' atures. If we know what mechanism transfers energy inside a Comparing (1.8) to formula (1.4a), obtained for the en- star, we can calculate the temperature T by formula ergy transfer by radiation, we obtain the correlation (1.7) in (1.1) or (1.2) clear form T = f (L, M, R) . (1.4) μ4 L M 3 . (1.7a) For instance, if energy is dragged by radiations, according ' κ to §1.2, we have Another instance — a star built on a degenerate gas c dB F = , (1.5) R κρ dr 5 − p ρ 3 , where κ is the absorption coefficient per unit mass, B is the ' radiant pressure then formulae (1.1) and (1.2) lead to 1 B = αT 4. (1.6) 3 RM 1/3 = const , (7.b) We often use the radiant pressure B instead of the temp- erature. By formula (1.3) we can write so in this case we just obtain the correlation like (1.7), where there is no L. κF B R ρR , Formula (1.7a), which is true for an ideal gas, can include ' c R only through κ. Therefore this formula is actually the which, by using (1.1) and (1.3), gives “mass-luminosity” relation, which is in good agreement with observational data L M 3, if μ4/κ = const = 0.08. The calc- 3LM ulations are valid under∼ the low radiant pressure . As B κ . (1.4a) γ < 1 ' (4π)2 cR4 we see from formula (1.9), inside extremely bulky stars the

84 N. Kozyrev. Sources of Stellar Energy and the Theory of the Internal Constitution of Stars October, 2005 PROGRESS IN PHYSICS Volume 3

value of γ can be more than 1. In such cases formula (1.2) We denote by a bar all the quantities expressed in terms will determine the radiant pressure of their numerical values in the Sun. Assuming, according to our conclusion in Part I, that stars have the same structure, M 2 B , we can, by formulae (1.1), (1.2), and (1.10), strictly calculate ' R4 the central characteristics of stars not the gaseous one. Comparing to formula (1.4a), we have Mˉ 2 Mˉ Lˉ pˉ = , ρˉ = , εˉ = . (1.12) c ˉ 4 c ˉ 3 c ˉ M R R M L . (1.7b) Even for very different structures of stars, it is impossible ' κ to obtain distorted results by the formulae. As we saw in Astronomical observations show that super-giants do not the previous paragraph, we can calculate the temperature have the huge variations of M which are predicted by this (or, which is equivalent, the radiant pressure) in two ways, formula. Therefore, in Part I, we came to the conclusion that either way being connected to suppositions. First, the radiant γ 1 for stars of regular masses M 100M , so formula 6 6 pressure can be obtained through the flow of energy, i. e. (1.9) gives for them: μ = 1/2. Hence, κ = 0.8, which is ap- through ε by formula (1.4a). The exact formula of that proximately equal to Thomson’s absorption coefficient. This relation, by equations (1.27) in §1.3 (Part I), is is very interesting, for we have obtained that the radiant εc κc pressure places a barrier to the existence of extremely large Bc = pc , (1.13) 4πGc λ masses for stars, although there is no such barrier in the where λ is the structural parameter of the main system of the theory based on the equilibrium equations of stars. dimensionless equations of equilibrium: its numerical value Until now, we hardly used the heat equilibrium equation, is about 1. Second, for an ideal gas, the radiant pressure can which requires that the energy produced inside a star should be calculated directly from formulae (1.12) be equal to its radiation into space. According to the heat equilibrium equation, the average productivity of energy by Bˉ pˉ 4 Mˉ 4 c = c = . (1.14) one gram of stellar matter can be calculated by the formula μˉ4 ρˉ Rˉ 4  c L Formulae (1.13) and (1.14) must lead to the same result. ε = . (1.10) M This requirement leads to the “mass-luminosity” relation. Our conclusion that all stars (except for while dwarfs) are On the other hand, if the productivity of energy is de- built on an ideal gas is so well grounded that it is fair to use termined by some other reactions, ε would be a function of ρ formula (1.14) in order to calculate the temperature or the and T . This function would also be dependent on the kinetics radiant pressure in stars. Naturally, Eddington [21] showed: of the supposed reaction. Thus formulae (1.10), (1.1), (1.4), under temperatures of about a few million degrees, because and the equation of the reaction demand the existence of the of the ionization of matter, the atoms of even heavy elements second correlation take up so little space (about one millionth of their normal F (L, M, R) = 0 , (1.11) sizes) that van der Waals’ corrections are negligible if the 2 density is even much more than 1. However, because of which is fully determined by the mechanism that generates plasma, there could be substantial electrostatic interactions energy in the reaction. For an ideal gas, R disappears from between particles, making the pressure negative, and the gas the first correlation F1 = 0 (1.7). For this reason formula approaches properties of a super-ideal one. The approximate (1.11) transforms into the relation L(R) or M (R), which theory of such phenomena in strong electrolytes has been become directly dependent on the kind of energy sources developed by Debye and Huckell.¨ Eddington and Rosseland in stars. For a degenerate gas we obtain another picture: as applied the theory to a gas inside stars. They came to the we saw above, in this case M (R) is independent of energy conclusion that the electric pressure cannot substantially sources, and then M and L are connected by equation (1.11). change the internal constitution of stars. Giving no details of that theory, we can show directly that the electric pressure is 1.2 Transforming the Russell-Hertzsprung diagram to negligible in stars built on hydrogen. We compare the kinetic the physical characteristics specific to the central energy of particles to the energy of Coulomb interaction regions of stars z2 e2 kT > . r Our task is to find those processes which generate energy in As soon as the formula becomes true in a gas, the gas stars. In order to solve this problem, we must know physical becomes ideal. Cubing the equation we obtain conditions inside stars. In other words, we should proceed from the observed characteristics , , to physical para- (kT )3 (kT )4 L M R = > z6 e6, meters. n p

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where n is the number of particles in a unit volume. Because where Tm is temperature expressed in millions of degrees. the radiant pressure is given by the formula For instance, for the proton-proton reaction, the constants a and A take the values π2 (kT )4 B = 3 , (1.6a) 3 45 (~c) a = 33.8,A = 4×10 . (1.17) a gas becomes ideal as soon as the ratio between the radiant In order to find the arc of the relation between ρc and Bc, pressure and the gaseous pressure becomes on which stars should be located if nuclear reactions are the sources of their energy, we eliminate εc from formula (1.16) π2 z6 e2 3 γ > . by formula (1.13) 45 ~c   2 τ λ4πGcB = Aκ p ρ τ e− c . (1.18) Because of formula (1.9), this ratio is determined by the c c c c c mass of a star. Because γ = 1 under Mˉ = 100, we obtain As the exponent indicates (see formula (1.16)), ε is very 1 γ 2 M/ˉ 100. So, for an ideal gas, we obtain the condition sensitive to temperature. Therefore, inside such stars, a core ≈ of free convection should exist, as was shown in detail in 3/2 100π e2 Part I, §2.8. We showed there that cannot be calculated 100M > M > z3 M . (1.15) λ √45 c separately for stars within which there is a convective core: ~  the equilibrium equations determine only λL , where L which is dependent only on the mass of a star. x0 x0 is the dimensionless luminosity For hydrogen or singly ionized elements, we have z = 1. Hence, for hydrogen contents of stars, the electric pressure x0 L = ε ρ x2dx . (1.19) can play a substantial roleˆ only in stars with masses less than x0 1 1 0 0.01–0.02 of the mass of the Sun. Z It is amazing that of all possible states of matter in stars In this formula x0 is the dimensionless radius (see Part I). there are realized those states which are the most simple from The subscript 1 on ε and ρ means that the quantities are taken the theoretical point of view. in terms of their numerical values at the centre of a star. In the Now, if we know Mˉ and Rˉ for a star, assuming the case under consideration (stars inside which thermonuclear same molecular weight μˉ = 1 for all stars (by our previous reactions occur). conclusions), we can calculate its central characteristics ρˉc x0 2 2 (τ1 τc) 2 ˉ − − and Tc by formulae (1.12) and (1.14). The range, within Lx0 = ρ1 τ1 e x dx . which the calculated physical parameters are located, is so Z0 8 6 2 ˉ 2 3 4 large (10− < ρˉc <10 , 10− < Tc <10 , 10− < εˉc <10 ), Because this integral includes the convective core (where that we use logarithmic scales. We use the abscissa for 3/2 ρ1 = T1 ), logρ ˉc, while the ordinate is used for log Bˉc (or equivalently, x0 1/3 4 log Tˉ ). If an energy generation law like ε = f (ρ ,T ) ex- τc T − 1 c c c c L (τ ) = T 1/3 x2 e− 1 − dx . (1.20) ists in Nature, the points plotted along the -coordinate x0 c 1 logε ˉc z 0 axis will build a surface. On the other hand, the equilibrium Z
condition requires formula (1.13), so the equilibrium states of The integral Lx0 (τc) can be easily taken by numerical stars should be possible only at the transection of the above methods, if we use Emden’s solution T1 (x) for stars of the 3 2 surfaces∗. Hence, stars should be located in the plane (logρ ˉc, polytropic structure / . The calculations show that numerical log Bˉc) along a line which is actually the relation M (R) values of the integral taken under very different τc are very transformed to the physical characteristics inside stars. There little different from 1. For instance, in the diagram, we draw the numerical values of logε ˉ in c L (33.8) = 0.67,L (7.3) = 1.15. order to picture the whole volume. x0 x0 For the proton-proton reactions formula (1.17), the first 1.3 The arc of nuclear reactions value of Lx0 is 1 million degrees at the centre of a star, the second value is one hundred million degrees. Assuming The equation for the generation of energy by thermonuclear Lx0 1 (according to our conclusions in Part I), Table 3 gives reactions is λ ≈3 in stars where the absorption coefficient is constant. ≈ 2 τ a In Part I of this research we found the average molecular ε = A ρτ ε− , τ = , (1.16) 1/3 1 Tm weight /2 for all stars. We also found that all stars have structures very close to the polytropic structure of the class ∗The energy generation surface, drawn from the energy generation 3 2 law εc = f (ρc,Tc), and the energy drainage surface, drawn from for- / . Under these conditions, the central temperature of the Sun 6 mula (1.13). should be 6×10 degrees. Therefore Bethe’s carbon-nitrogen

86 N. Kozyrev. Sources of Stellar Energy and the Theory of the Internal Constitution of Stars October, 2005 PROGRESS IN PHYSICS Volume 3

cycle is improbable as the source of stellar energy. As an log Bˉc, logε ˉc. We calculated the characteristics for every star example, we consider proton-proton reactions. Because of on our list. The results are given in Fig. 2∗. There the abscissa the numerical values obtained for the constants a and A takes the logarithm of the matter density, logρ ˉc, while the (1.17), formula (1.18) gives ordinate takes the logarithm of the radiant energy density, log Bˉc, where both values are taken at the centre of a star†. 1 κc log ρc = 0.217τc 5.5 log τc + 5.26 log . (1.21) Each star is plotted as a point in the numerical value of logε ˉc − − 2 μ — the energy productivity per second from one gramme of matter at the centre of a star with respect to the energy Taking κc/μ constant in this formula, we see that ρc has the very slanting minimum (independent of the temperature) productivity per second at the centre of the Sun. In order to 6 make exploration of the diagram easier, we have drawn the at τc = 11 that is Tc = 30×10 degrees. In a hydrogen star where the absorption coefficient is Thomson, the last term net values of the fixed masses and radii. Bold lines at the left side and the right side are the boundaries of that area where of (1.21) is zero and the minimal value of ρc is 100. Hence, stars undergoing proton-proton reactions internally should be the ideal gas law is true (stars land in exactly this area). The left bold line is the boundary of the ultimately large located along the line ρc 100 in the diagram for (ρc, Bc). It appears that stars of the≈ main sequence satisfy the require- radiant pressure (γ = 1). The bold line in the lower part of ment (in a rude approximation). Therefore, it also appears the diagram is the boundary of the ultimately large electric that the energy produced by thermonuclear reactions could pressure, drawn for hydrogen by formula (1.15). This line explain the luminosity of most of stars. But this is only an leads to the right side bold lines, which are the boundaries illusion. This illusion disappears completely as soon as we of the degeneration of gas calculated for hydrogen (the first line) and heavy elements (the second line). construct the diagram for ( logρ ˉc, log Bˉc) using the data of observational astronomy. We built the right boundary lines in the following way. We denote by ne the number of free electrons inside one cubic centimetre, and μ the molecular weight per electron. 1.4 Distribution of stars on the physical conditions dia- e Then gram ρ = μe mH ne , Currently we know all three parameters (the mass, the bolo- so Sommerfeld’s condition of degeneration metric absolute stellar magnitude, and the spectral class) for 3 approximately two hundred stars. In our research we should ne ~ 1 use only independent measurements of the quantities. For 3/2 > 1 (1.23) 2 (2πme kT ) this reason, we cannot use the stellar magnitudes obtained by the spectroscopic parallax method, because the basis of can be re-written as this method is the “mass-luminosity” relation. 8 3/2 For stars of the main sequence we used the observational ρ > 10− μe T . (1.24) data collection published in 1948 by Lohmann [26], who generalized data by Parenago and Kuiper. For eclipse variable For the variables p and ρ, we obtain the degeneration stars we used data collections mainly by Martynov [27], boundary equation‡ Gaposchkin [28], and others. Finally, we took particularly 5/3 5/3 interesting data about super-giants from collections by Pare- p = kμe ρ , nago [29], Kuiper [7], and Struve [30]. Some important data 5/3 (1.25) ρ 5/3 5/3 about the masses of sub-dwarfs were given to the writer pˉ = k ρˉ μ , p e by Prof. Parenago in person, and I’m very grateful to him for his help, and critical discussion of the whole research. 5/3 Consequently, we used the complete data of about 150 stars. which coincides with the Fermi gas state equation p=Kρ = 5/3 5/3 The stellar magnitudes were obtained by the above ment- = KH μe ρ (formula 1.9 in Part I), if ioned astronomers by the trigonometric parallaxes method K K = 9.89 1012. and the empirically obtained bolometric corrections (Petit, ≈ H ×

Nickolson, Kuiper). In order to go from the spectral class ∗Of course not all the stars are shown in the diagram, because that to the effective temperature, we used Kuiper’s temperature would produce a very dense concentration of points. At the same time, scale. Then we calculated the radius of a star by the formula the plotted points show real concentrations of stars in its different parts. — Editor’s remark. 5 log Rˉ = 4.62 m 10 log Tˉ , (1.22) †The bar means that both values are expressed in multiples of the − b − eff corresponding values at the centre of the Sun. — Editor’s remark. ‡The degeneration boundary equation is represented here in two forms: where mb is the bolometric stellar magnitude of the star. expressed in absolute values of p and in multiples of the pressure in Then, by formulae (1.12) and (1.14), we calculated logρ ˉc, the Sun. — Editor’s remark.

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At the centre of the Sun, as obtained in Part I of this of stars we see two opposite tendencies of the isoergs to be research (see formula 3.34), curved. We have a large dataset here, so the isoergs were drawn very accurately. The twists are in exact agreement 15 ρc = 9.2, pc = 9.5×10 , with the breaks, discovered by Lohmann [26], in the “mass- 3 12 luminosity” relation for stars of the main sequence. It is γc = 0.4×10− ,Bc = 3.8×10 , (1.26) wonderful that this tendency, intensifying at the bottom, 6 gives the anomalously large luminosities for sub-giants (the Tc = 6.3×10 , satellites of Algol) — the circumstance, considered by Struve then we obtain [30]. For instance, the luminosity of the satellite of XZ Sagittarii, according to Struve, is ten thousand times more 2 5/3 5/3 pˉ = 4×10− ρˉ μe . than that calculated by the regular “mass-luminosity” rel- ation. There we obtain also the anomalously large luminosity, The right side boundaries drawn in the diagram are con- discovered by Parenago [29], for sub-dwarfs of small masses. structed for μ = 1 and μ = 2. At the same time these are e e The increase of the opposite tendency at the top verifies the lines along which stars built on a degenerate gas (the lines of low luminosity of extremely hot stars, an increase which Chandrasekhar’s “mass-radius” relation) should be located. leads to Trumpler stars. It is very doubtful that masses of In this case the ordinate axis has the meaning log (ˉp/ρˉ)4 that Trumpler stars measured through their Einstein red shift are becomes the logarithm of the radiant energy density log Bˉ valid. For this reason, the diagram contains only Trumpler for ideal gases only. In this sense we have drawn white stars of “intermediate” masses. Looking at the region of sub- dwarfs and Jupiter on the diagram. Under low pressure, near giants and sub-dwarfs (of large masses and of small ones) the boundary of strong electric interactions, the degeneration we see that ε is almost constant there, and independent of the lines bend. Then the lines become constant density lines, masses of the stars. Only by considering altogether the stars because of the lowering of the ionization level and the located in the diagram we can arrive at the result obtained in appearance of normal atoms. The lines were constructed Part I of this research: L M 3. according to Kothari’s “pressure-ionization” theory [31]. So, the first conclusion∼ that can be drawn from our Here we see a wonderful consequence of Kothari’s theory: consideration of the diagram is: deviations from the “mass- the maximum radius which can be attained by a cold body luminosity” relation are real, they cannot be related to sys- is about the radius of Jupiter. tematic errors in the observational data. The possibility of Finally, this diagram contains the arc along which should drawing the exact lines of constant εˉ itself is wonderful: it be located stars whose energy is generated by proton-proton c shows that ε is a simple function of ρ and B. Hence, the reactions. The arc is built by formula (1.21), where we used luminosity L is a simple function of M and R. Some doubts the central characteristics of the Sun (1.26) obtained in Part I. can arise from the region located below and a little left of The values logε ˉ plotted for every star builds the system c the central region of the diagram, where the isoergs do not of isoergs — the lines of the same productivity of energy. coincide with L for sub-dwarfs of spectral class F–G and L The lines were drawn through the interval of ten changes of of normal dwarfs of class M. It is most probable that the εˉ . If a “mass-luminosity” relation for stars does not contain c inconsistency is only a visual effect, derived from errors in their radii, εˉ should be a function of only the masses of c experimental measurements of the masses and radii of the stars. Hence, the isoergs should be parallel to the constant sub-dwarfs. mass lines. In general, we can suppose the “mass-luminosity” As a whole our diagram shows the plane image of the relation as the function surface ε (ρ, B). We obtained much more than expected: L M α, (1.27) we should obtain only one section of the surface, but we ∼ obtained the whole surface, beautifully seen in the central then the interval between the neighbouring isoergs should region of the diagram. Actually, we see no tendency for stars decrease with increasing α according to the picture drawn to be distributed along a sequence ε = const. Thus, of the two in the upper left part of the diagram. We see that the real equations determining ε, there remains only one: the energy picture does not correspond to formula (1.27) absolutely. productivity in stars is determined by the energy drainage Only for giants, and the central region of the main sequence (radiation) only. This conclusion is very important. Thus the (at the centre of the diagram) do the isoergs trace a path ap- mechanism that generates energy in stars is not of any kid of proximately parallel to the constant mass lines at the interval reactions, but is like the generation of energy in the process α = 3.8. In all other regions of the diagram the isoergs εˉc of its drainage. The crude example is the energy production are wonderfully curved, especially in the regions of super- when a star, radiating energy into space, is cooling down: giants (the lower left part of the diagram) and hot sub- the star compresses, so the energy of its gravitational field dwarfs (the upper right part). As we will soon see, the becomes free, cooling the star (the well-known Helmholtz- curvilinearity can be explained. In the central concentration Kelvin mechanism). Naturally, in a cooling down (compress-

88 N. Kozyrev. Sources of Stellar Energy and the Theory of the Internal Constitution of Stars October, 2005 PROGRESS IN PHYSICS Volume 3

Fig. 2: The diagram of physical conditions inside stars (the stellar energy diagram): the productivity of stellar energy sources independence of the physical conditions in the central regions of stars. The abscissa is the logarithm of the density of matter, the ordinate is the logarithm of the radiant energy density (both are taken at the centre of stars in multiples of the corresponding values at the centre of the Sun). The small diagram at the upper left depicts the intervals between the neighbouring isoergs.

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ing) star the quantity of energy generated is determined by diagram support a stellar energy mechanism like reactions. the speed of this process. At the same time the speed is In the real situation the equation of the main direction (1.28) regulated by the heat drainage. Of course, the Helmholtz- contradicts the kinetics of any reaction. Naturally, equation Kelvin mechanism is only a crude example, because of the (1.28) can be derived from the condition of energy drainage inapplicable short period of the cooling (a few million years). (1.13) only if At the same time the mechanism that really generates energy 1 in stars should also be self-regulating by the radiation. In ε , under ρ T 4, (1.29) contrast to reaction, such a mechanism should be called a ∼ T ∼ machine. i. e. only if the energy productivity increases with decrease It should be noted that despite many classes of stars in in temperature and hence the density. The directions of all the diagram, the filling of the diagram has some limitations. the isoergs in the diagram, and also the numerical values First there is the main direction along which stars are ε = 103–104 in giants and super-giants under the low temp- concentrated under a huge range of physical conditions — eratures inside them (about a hundred thousands degrees) from the sequence of giants, then the central concentration in cannot be explained by nuclear reactions. It is evident there- our diagram (the so-called main sequence of the Hertzsprung- fore, that the possibility for nuclear reactions is just limited Russel diagram), to sub-dwarfs of class A and white dwarfs. by the main sequence of the Russell-Hertzsprung diagram In order to amplify the importance of this direction, we (the central concentration of stars in our diagram). indicated the main location of normal giants by a hatched The proton-proton reaction arc is outside the main seq- strip. The main direction wonderfully traces an angle of uence of stars. If we move the arc to the left, into the region exactly 45◦. Hence, all stars are concentrated along the line, of the main sequence stars, we should change the constant determined by the equation∗ A in the reaction equation (1.16) or change the physical characteristics at the centre of the Sun (1.26) as we found B ρμ4. (1.28) ∼ in Part I. Equation (1.18) shows that the shift of the proton- Because stars built on a degenerate gas satisfy this dir- proton reaction arc along the density axis is proportional to ection, a more accurate formula is the square of the change of the reaction constant A. Hence, in order to build the proton-proton reaction arc through the main 5/5 p ρ (1.28a) concentration of stars we should take at least A = 105–106 ∼ 3 Second, there is in the main direction (1.28) a special instead of the well-known value A = 4×10 . This seems very improbable, for then we should ignore the central charact- point — the centre of the main sequence†, around which stars are distributed at greater distances, and in especially large eristics of the Sun that we have obtained, and hence all numbers. conclusions in Part I of this research which are in fine Thus, there must exist two fundamental constants which agreement with observational data. Only in a such case could determine the generation of energy in stars: we arrive at a temperature of about 20 million degrees at the centre of the Sun; enough for proton-proton reactions and 1. The coefficient of proportionality of equation (1.28); also Bethe’s carbon-nitrogen cycle. 2. One of the coordinates of the “main point”, because All theoretical studies to date on the internal constitution its second coordinate is determined by the eq. (1.28). of stars follow this approach. The sole reason adduced as The above mentioned symmetry of the surface ε (ρ, B) is proof of the high concentration of matter in stars, is the slow connected to the same two constants. motion of the lines of apsides in compact binaries. However Concluding the general description of the diagram, we the collection published by Luyten, Struve, and Morgan [32] note: this diagram can also give a practical profit in calculat- shows no relation between the velocity of such motion and ions of the mass of a star by its luminosity and the spectral the ratio of the star radius to the orbit semi-axis. At the same class. Naturally, having the radius calculated, we follow the time, such a relation would be necessary if the motion of line R = const to that point where logε ˉ+ log Mˉ gives the the lines of apsides in a binary system is connected to the observed value of log Lˉ. deformations of the stars. Therefore we completely agree with the conclusion of those astronomers, that no theory 1.5 Inconsistency of the explanation of stellar energy by correctly explains the observed motions of apsides. Even if Bethe’s thermonuclear reactions we accept that the arc of nuclear reactions could intersect the central concentration of stars in our diagram (the stars of It is seemingly possible that the existence of the uncovered the main sequence in the Russell-Hertzsprung diagram), we main direction along which stars are concentrated in our should explain why the stars are distributed not along this arc, but fill some region around it. One could explain this See formula (1.14). — Editor’s remark. ∗ circumstance by a “dispersion” of the parameters included in †The main sequence in the sense of the Russell-Hertzsprung diagram, is here the central concentration of stars. — Editor’s remark. the main equations. For instance, one relates this dispersion

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to possible differences in the chemical composition of stars, clusters are different according to their hydrogen percentage, their structure etc. Here we consider the probability of such one can perceive an evolutionary meaning — the proof of the explanations. nuclear transformations of elements in stars. The idea that stars can have different chemical com- Stromgren’s¨ research prepared the ground for checking positions had been introduced into the theory in 1932 by the whole nuclear hypothesis of stellar energy: substituting Stromgren¨ [16], before Bethe’s hypothesis about nuclear the obtained correlation X (ρ, B) into the reaction equation sources for stellar energy. He used only the heat drainage (II), we must come to the well-known relation (I). The condition (1.13), which leads to the “mass-luminosity” relat- nuclear reaction equation (1.16), where X is included through ion (1.7a) for ideal gases. In chapter 2 of Part I we showed in A, had not passed that examination. Therefore they intro- detail that the theoretical relation (1.7a) is in good agreement duced the second parameter Y into the theory — the percent- (to within the accuracy of Stromgren’s¨ data) with the observ- age of helium. As a result, every function f1 and f2 can ed correlation for hydrogen stars (where we have Thomson’s be separately equated to the function ε(ρ, B) known from absorption coefficient, which is independent of physical con- observations. Making the calculations for many stars, it is ditions). Introducing some a priori suppositions (see §2.7, possible to obtain two surfaces: X (ρ, B) and Y (ρ, B). How- Part I), Eddington, Stromgren¨ and other researchers followed ever, both surfaces are not a consequence of the equilibrium another path; they attempted to explain non-transparency of conditions of stars. It remains unknown as to why such stellar matter by high content of heavy elements, which build surfaces exist, i. e. why the observed ε is a simple function the so-called Russell mix. At the same time the absorption of ρ and B? It is very difficult to explain this result by theory gives such a correlation κ(ρ, B) for this mix which, evolutionary transformations of X and Y , if the transform- being substituted into formula (1.7a), leads to incompatibility ation of elements procedes in only one direction. Of course, with observational data. Stromgren¨ showed that such a “dif- taking a very small part of the plane (ρ, B), the evolution ficulty” can be removed if we suppose different percentages of elements can explain changes of X and Y . For instance, of heavy elements in stars, which substantially changes the calculations made by Masevich [34] gave a monotone de- resulting absorption coefficient κ. Light element percentages crease of hydrogen for numerous stars located between the X can be considered as the hydrogen percentage. Comparing spectral classes B and G. To the contrary, from the class the theoretical formula to the observable “mass-luminosity” G to the class M, the hydrogen percentage increases again relation gives the function X (ρ, B) or X (M,R). Looking (see the work of Lohmann work [26] cited above). As a at the Stromgren¨ surface from the physical viewpoint we result we should be forced to think that stars evolve in two can interpret it as follows. As we know, the heat drainage different ways. In such a case the result that the chemical equation imposes a condition on the energy generation in composition of stars is completely determined by the physical stars. This is condition (1.13), according to which κ and conditions inside them can only be real if there is a balanced μ depend on the chemical composition of a star. Let us transformation of elements. Then the mechanism that gen- suppose that the chemical composition is determined by one erates energy in stars becomes the Helmholtz-Kelvin mech- parameter X. Then anism, not reactions. Nuclear transformations of elements only become an auxiliary circumstance which changes the ε = f1 (ρ, B, X) . (I) thermal capacity of the gas. At the same time, the balanced For processes like a reaction, the energy productivity ε transformation of elements is excluded from consideration, is dependent on the same variables by the equation of this because it is possible only if the temperature becomes tens reaction of billions of degrees, which is absolutely absent in stars. ε = f (ρ, B, X) . (II) All the above considerations show that the surfaces 2 X (ρ, B) and Y (ρ, B) obtained by the aforementioned res- So we obtain the condition f1 = f2 , which will be true earchers are only a result of the trimming of formulae (I) only if a specific relation X (ρ, B) is true in the star. The and (II) to the observed relation ε(ρ, B). Following this parameter X undergoes changes within the narrow range approach, we cannot arrive at a solution to the stellar energy 0 6 X 6 1, so stars should fill a region in the plane (ρ, B). problem and the problem of the evolution of stars. This con- Some details of the Russell-Hertzsprung diagram can be clusion is related not only to nuclear reactions; it also shows obtained as a result of an additional condition, imposed the impossibility of any sources of energy whose productivity on X (ρ, B): Stromgren¨ showed that arcs of X = const can is not regulated by the heat drainage condition. Naturally, be aligned with the distribution of stars in the Russell- the coincidence of the surfaces (I) and (II) manifests their Hertzsprung diagram. Kuiper’s research [33] is especially identity. In a real situation the second condition is not present∗. interesting in this relation. He discovered that stars collected in open clusters are located along one of Stromgren’s¨ arcs ∗For reactions, the energy productivity increases with the increase of the density. In the heat drainage condition we see the opposite: equation X = const and that the numerical values of X are different for (1.13). Therefore the surfaces (I) and (II), located over the plane (ρ, B), different clusters. Looking at this result, showing that stellar should be oppositely inclined — their transection should be very sharp.

N. Kozyrev. Sources of Stellar Energy and the Theory of the Internal Constitution of Stars 91 Volume 3 PROGRESS IN PHYSICS October, 2005

So we get back to our conclusion of the previous paragraph: By formulae (1.30) and (1.31), we obtain there are special physical conditions, the main direction κ (1.28) and the main point in the plane (ρ, B), about which c = 8. (1.32) μ4 stars generate exactly as much energy as they radiate into 1 space. In other words, stars are machines which generate If μ = /2, we obtain κc = 0.5. This result implies that radiant energy. The heat drainage is the power regulation the non-transparency of giants is derived from Thomson’s mechanism in the machines. dispersion of light in free electrons (κT = 0.40), as it should be in a pure hydrogen star. 1.6 The “mass-luminosity” relation in connection with The main peculiarity of the “mass-luminosity” relation the Russell-Hertzsprung diagram is the systematic curvilinearity of the isoergs in the plane (ρ, B). Let us show that this curvilinearity cannot be ex- The luminosity of stars built on an ideal gas, radiant transfer plained by the changes of the coefficient in formula (1.30). of energy and low radiant pressure, is determined by formula First we consider the multiplier containing the molecular (1.7a). This formula is given in its exact form by (2.38) in weight and the absorption coefficient. Part I. We re-write formula (2.38) as The curvilinearity of the isoergs shows that for the same mass the diagram contains anomalous low luminosity stars at Lˉ μ4 λL the top and anomalous bright stars at the bottom. Hence, the εˉ = = 1.04 104 x0 Mˉ 2, (1.30) ˉ × 3 left part of (1.32) should increase under higher temperatures, M κc Mx  0  and should decrease with lower temperatures. Looking from where Mx0 is the dimensionless mass of a star, κc is the the viewpoint of today’s physics, such changes of the absorpt- absorption coefficient at its centre. It has already been shown ion coefficient are impossible. Moreover, for the ultimate that the structural multiplier of this formula has approxim- inclinations of the isoergs, we obtain absolutely impossible ately the same numerical value numerical values of the coefficient (1.32). For instance, in the case of super-giants, the lower temperature stars, this λL x0 2 10 3 coefficient is 100 times less than that in giants. Even ifwe 3 × − (1.31) Mx0 ' imagine a star built on heavy elements, we obtain that κ is about 1. In hot super-giants (the direction of Trumpler for all physically reasonable models of stars. The true “mass- stars) the coefficient (1.32) becomes 200. Because of high luminosity” relation is shown in Fig. 2 by the system of temperatures in such stars, the absorption coefficient cannot isoergs εˉ= L/ˉ Mˉ = const. If we do not take the radius of a be so large. star into account, we obtain the correlation shown in Fig. 1, In order to explain the curvilinearity by the structural Part I. There L is approximately proportional to the cube of multiplier (1.31), we should propose that it be anomalously M, although we saw a dispersion of points near this direction large in stars of high luminosity (sub-giants) and anomal- 3 L M . As we mentioned before, in Part I, the comparison ously small in stars like Trumpler stars. We note that the of∼ this result to formula (1.30) indicates that: (1) the radiant dimensionless mass Mx0 included in (1.31) cannot be sub- pressure plays no substantial roleˆ in stars, (2) stars are built stantially changed, as shown in Part I. So the structural on hydrogen. multiplier (1.31) can be changed by only λLx0 . Employing Now we know that the dispersion of points near the the main system of the dimensionless equations of equilibr- average direction L M 3 is not stochastic. So we could ium of stars, we easily obtain the equation compare the exact correlation∼ to the formula (1.30), and also check our previous conclusions. dB λLx 1 = , (1.33) Our first conclusion about the negligible roleˆ of the rad- dp1 Mx iant pressure is confirmed absolutely, because of the mech- anical equilibrium of giants. Naturally, comparing formula which is equation (2.22) of Part I, where B1 and p1 are (1.7b) to (1.7a), we see that the greater the roleˆ of the the radiant pressure and the gaseous pressure expressed in radiant pressure, the less ε is dependent on M, so the interval multiples of their values at the centre of a star. Here the between the neighbouring isotherms should increase for large absorption coefficient κ is assumed constant from the centre masses. Such a tendency is completely absent for bulky to the surface, i. e. κ1 = 1. Applying this equation to the stars (see the stellar energy diagram, Fig. 2). This result, surface layers of a star, we deduce that the structural coef- in combination with formula (1.9) (its exact form is formula ficient is λL B 2.47, Part I), leads to the conclusion that giants are built x0 = 1 . (1.34) M mainly on hydrogen (the molecular weight 1/2). Thus we x0 p1 calculate the absorption coefficient for giants. We see inthe We denote the numerical values of the functions at the diagram that red giants of masses 20M have logε ˉ= 3. boundary between the surface layer and the “internal” layers ≈

92 N. Kozyrev. Sources of Stellar Energy and the Theory of the Internal Constitution of Stars October, 2005 PROGRESS IN PHYSICS Volume 3

of a star by the subscript 0. We consider two ultimate cases considered as the incompatibility of both the values of B. of the temperature gradients within the “internal” layers: So we denote by B∗ the radiant pressure calculated by the 1. The “internal” zone of a star is isothermal: ideal gas equation. For the radiant transport of energy in a star, formulae (1.4a) and (1.13) lead to

λLx0 1 = , (1.34a) Bˉ∗ Mx0 p1 =ε ˉpˉ . (1.35) 0 κˉ 2. The “internal” zone of a star is convective ( 8/5): B1 = p1 By this formula we can calculate Bˉ∗/κˉ for every star of the stellar energy diagram (Fig. 2). As a result we can λLx 0 = p . (1.34b) find the correlation of the quantity Bˉ /κˉ to pˉ and ρˉ. Fig. 3 M 10 ∗ x0 shows the stellar energy diagram transformed in this fashion. In the first theoretical case, spreading the isothermal zone Here the abscissa is logρ ˉ, while the ordinate is logp ˉ. In to almost the surface of a star, we can make the structural order to make the diagram readable, we have not plotted all coefficient as large as we please. This case is attributed stars. We have plotted only the Sun and a few giants. At the to sub-giants and anomalous bright stars in general. The same time we drawn the lines of constant Bˉ∗/κˉ through ten second theoretical case can explain stars of anomalously low intervals. The lines show the surface log Bˉ∗/κˉ (logρ, ˉ logp ˉ). luminosity. Following this way, i. e. spreading the convective For the constant absorption coefficient κ, the lines show the zone inside stars, Tuominen [35] attempted to explain the low system of isotherms. If B∗ = B, there should be a system luminosity of Trumpler stars. of parallel straight lines, inclined at 45◦ to the logp ˉ axis The isothermy can appear if energy is generated mainly and following through the interval 0.25. As we see, the real in the upper layers of a star. The spreading of the convective picture is different in principle. There is in it a wonderful zone outside the Schwarzschild boundary can occur if energy symmetry of the surface log Bˉ∗/κˉ. Here the origin of the is generated in moved masses of stellar gas, i. e. under forced coordinates coincides with the central point of symmetry of convection. A real explanation by physics should connect the the isoergs. At the same time it is the main point mentioned above peculiarities of the energy generation to the physical in relation in the stellar energy diagram. The coordinates of conditions inside stars or their general characteristics L, M, the point with respect to the Sun are R. Before attempting to study the theoretical possibility of logρ ˉ0 = 0.58, logp ˉ0 = 0.53, such relations, it is necessary to determine them first from − − (1.36) observational data. Dividing εˉ by Mˉ 2 for every star, we ˉ ˉ log B0 = +0.22, log B0∗ = +0.50. obtain the relation of the structural coefficient of formula (1.30) for ρ and B. But, at the same time, the determination Using the data, we deduce that the main point is attributed of this relation in this way is somewhat unclear. There are no to a star of the Russell-Hertzsprung main sequence, which clear sequences or laws, so we do not show it here. Generally has spectral class A4. Rotating the whole diagram around the speaking, a reason should be simpler than its consequences. main point by 180◦, we obtain almost the same diagram, only Therefore, it is most probable that the structural coefficient the logarithms of the isotherms change their signs. Hence, if is not the reason. It is most probable that the reason for the B∗ p ρ incompatibility of the observed “mass-luminosity” relation κ B = f , , 0∗ p ρ with formula (1.30) is that equation (1.30) itself is built κ  0 0  incorrectly. This implies that the main equations of equi- we have librium of stars are also built incorrectly. This conclusion is p ρ p0 ρ0 in accordance with our conclusion in the previous paragraph: f , f , = 1. (1.37) p0 ρ0 p ρ energy is generated in stars like in machines — their workings     are incompatible with the standard principles of today’s The relation (1.37) is valid in the central region of the mechanics and thermodynamics. diagram. An exception is white dwarfs, in which B∗/κ is 100 times less than that required by formula (1.37), i. e. 100 times less than that required for the correspondence to giants 1.7 Calculation of the main constants of the stellar en- after the 180◦ rotation of the diagram. It is probable that this ergy state circumstance is connected to the fact that white dwarfs are located close to the boundary of degenerate gas. The theoretical “mass-luminosity” relation (1.30) is obtained Besides the isotherms, we have drawn the main direction as a result of comparing the radiant energy B calculated by along which stars are distributed. Now the equation of the the excess energy flow (formula 1.4a or 1.13) to the same direction (1.28) can be written in the more precise form B calculated by the phase state equation of matter (through p and ρ by formula 1.14). Therefore the incompatibility of Bˉ log = +0.80. (1.38) the theoretical correlation (1.30) to observational data can be ρˉ

N. Kozyrev. Sources of Stellar Energy and the Theory of the Internal Constitution of Stars 93 Volume 3 PROGRESS IN PHYSICS October, 2005

Fig. 3: Isotherms of stellar matter. The coordinate axes are the logarithms of the matter density and the gaseous pressure. Dashed lines show isotherms of an ideal gas.

Because of the very large range of the physical states in is about the ionization energy of the hydrogen atom. Fig. 3 the diagram, the main direction is drawn very precisely (to shows that, besides the main direction, the axis ρ = ρ0 is within 5%). It should be noted that, despite their peculiarities, also important. Its equation can be formulated through the white dwarfs satisfy the main direction like all regular stars. average distance between particles in a star A theory of the internal constitution of stars, which could 1/3 explain observational data (the relation 1.37, for instance), r = 0.55 (ne)− should be built on equations containing the coordinates of the as follows main point. This circumstance is very interesting: it shows 2 8 e that there is an absolute system of “physical coordinates”, r = 0.51×10− = rH = , (1.43) 2χ where physical quantities of absolutely different dimensions 0 can be combined. Such combinations can lead to a com- where rH is the radius of the hydrogen atom, e is the charge of pletely unexpected source of stellar energy. Therefore it is the electron. As a result we obtain the very simple correlation very important to calculate the absolute numerical values of between the constants of the lines (1.42) and (1.43), which the constants (1.36). Assuming in (1.36) a mostly hydrogen bears a substantial physical meaning. content for stars μ = 1/2, and using the above calculated In the previous paragraph we showed that the peculiarit- physical characteristics at the centre of the Sun (1.26), we ies of the “mass-luminosity” relation∗ cannot be explained by obtain changes of the absorption coefficient κ. Therefore the lines 15 12 B∗/κ = const should bear the properties of the isotherms. ρ0 = 2.4, p0 = 2.8×10 ,B0 = 6.3×10 . (1.39) The isotherms drawn in Fig. 3 are like the isotherms of the We calculate B0∗ by formula (1.13). Introducing the aver- van der Waals gas. The meaning of this analogy is that there age productivity of energy ε is a boundary near which the isotherms become distorted, at which the regular laws of thermodynamics are violated. εκc pc Mx0 Bc∗ = , (1.40) The asymptotes of the boundary line (the boundary between 4πGc λLx0 two different phases in the theory of van der Waals) are axes (1.42) and (1.43). The distortion of the isotherms increases assuming κc equal to Thomson’s absorption coefficient, ε =

with approach to the axis ρ = ρ0 or r = rH. That region is = 1.9, Mx0 = 11, and the structural multiplier according to 12 filled by stars of the Russell-Hertzsprung main sequence. The (1.31). We then obtain for the Sun, Bc∗ = 1.1×10 instead 12 wonderful difference from van der Waals’ formula is the fact of Bc = 3.8×10 . Hence, that there are two systems of the distortions, equation (1.37), 12 B∗ = 4.1 10 B . (1.41) which become smoothed with the distance from the axis 0 × ≈ 0 ρ = ρ0 (for both small densities and large densities). We introduce the average number of electrons in one Stars can radiate energy for a long time only under cubic centimetre ne instead of the density of matter: ρ = 1.66 conditions close to the boundaries (1.42) and (1.43). This 24 ×10− ne. Then the equation of the main direction becomes most probably happens because the mechanism generating energy in stars works only if the standard laws of classical 3B 11 physics are broken. = 1.4×10− = 8.7 eV, (1.42) ne The results are completely unexpected from the view- point of contemporary theoretical physics. The results show which is close to the hydrogen ionization potential, i. e. χ0 = 13.5 eV. Thus the average radiant energy per particle in = ∗The dispersion of showing-stars points around the theoretically stars (calculated by the ideal gas formula) is constant and calculated direction “mass-luminosity”. — Editor’s remark.

94 N. Kozyrev. Sources of Stellar Energy and the Theory of the Internal Constitution of Stars October, 2005 PROGRESS IN PHYSICS Volume 3

that in stars the classical laws of mechanics and thermody- namics are broken much earlier than predicted by Einstein’s theory of relativity, and it occurs under entirely different cir- cumstances. The main direction constants (1.42) and (1.43) show that the source of stellar energy is not Einstein’s con- version of mass and energy (his mass-energy equivalence principle), but by a completely different combination of physical quantities. Here we limit ourselves only to conclusions which follow from the observational data. A generalization of the results and subsequent theoretical consequences will be dealt with in the third part of this research. In the next chapter we only Fig. 4: Finding the exponent index α in the L M α relation for consider some specific details of the Russell-Hertzsprung Cepheids. ∼ diagram, not previously discussed. one attempts to explain the luminosity of giants by nuclear reactions, one attributes to them an exotic internal constit- Chapter 2 ution (the large shell which covers a normal star). Therefore, Properties of Some Sequences in the Russell- the simple structure of giants we have obtained gives an Hertzsprung Diagram additional argument for the inconsistency of the nuclear sources of stellar energy. At the same time, because of their 2.1 The sequence of giants simple structure, giants and super-giants are quite wonderful. For instance, for a giant like the satellite of ε Aurigae we 4 The stellar energy diagram (see Fig. 2) shows that the “mass- obtain its central density at 10− of the density of air, and luminosity” relation has the most simple form for stars of the the pressure at about 1 atmosphere. Therefore, it is quite Russell-Hertzsprung main sequence possible that in moving forward along the main direction we can encounter nebulae satisfying the condition (1.42). Such L M α, α = 3.8. (2.1) ∼ nebulae can generate their own energy, just like stars. Because of the physical conditions in giants, obtained Cepheids, denoted by crosses in the diagram, also satisfy above, the huge amounts of energy radiating from them can- the relation (2.1). Using the pulsation equation P ρ = c we √ 1 not be explained by nuclear reactions. Even if this were true, obtained (see formula 3.25 of Part I) their life-span would be very short. For reactions, the upper 1 limit of the life-span of a star (the full transformation of its 0.30 m 4.62 + log P +3 log Tˉ = log c , (2.2) − 5α b − eff 1 mass into radiant energy) can be obtained as the ratio of εˉ to   2
c . So, by formula (1.40), we obtain ˉ where Teff is the reduced temperature of a star, expressed in multiples of the reduced temperature of the Sun, mb is the t0 Mx0 absolute stellar magnitude, P is the pulsation period (days). t = , (2.3) 4γc λLx0 We plot stars in a diagram where the abscissa is m 4.62,   b − where while the ordinate is log P + log Tˉeff . As a result we should obtain a straight line, which gives both the constant c (see κT c 1 t = = 6 1016 sec = 2 109 years (2.4) §3.3 of Part I) and the angular coefficient 0.30 1/5α. Fig. 4 0 πG × × shows this diagram, built using the collected data− of Becker ˉ and γc = Bc/pc is the ratio of the radiant pressure to the [36], who directly calculated Teff and mb by the radiant velocities arc (independently of the distances). As a result the gaseous one. As obtained, the structural multiplier here is average straight line satisfying all the stars has the angular about 4. Therefore t0 coefficient 0.25 and c = 0.075. Hence, α = 4, which is in t = . (2.5) 1 γ fine accordance with the expected result (α = 3.8). Such a c coincidence makes Melnikov’s conclusion unreasonable: that In giants γ 1, so we obtain that t is almost the same as c ≈ Cepheids have the same masses (α = ), as shown by the t0. At the same time, as we know, the percentage of energy dashed line in Fig. 4. ∞ which could be set free in nuclear reactions is no more than In §1.5 of Part I we showed that the “mass-luminosity” 0.008. Hence, the maximum life-span of a giant is about 7 relation for giants is explained by the fact that the structural 1.6×10 years, which is absolutely inapplicable. This gives coefficient 3 has the same value 2 10 3 (1.31) additional support for our conclusion that the mechanism of λLx0 /Mx0 × − for all stars. In order to obviate difficulties' which appear if stellar energy is not like reactions.

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It is very interesting that the constant (2.4) has a numer- In the stellar energy diagram, the Russell-Hertzsprung ical value similar to the time constant in Hubble’s relation main sequence is the ring of radius c filled by stars. The (the red shift of nebulae). It is probable that the exact form boundary equation of this region is of the Hubble equation should be log2 Bˉ + log2 ρˉ = c2. (2.11) t/t0 ν = ν0 e− , (2.6) We transform this equation to the variables Mˉ and Rˉ by where ν is the observed frequency of a line in a nebula formulae (1.12) and (1.14). We obtain spectrum when it is located at t light years from us, ν0 is 17 log2 Mˉ 38 log Mˉ log Rˉ + 25 log2 Rˉ = c2. (2.12) its normal frequency. According to the General Theory of − Relativity the theoretical correlation between the constant t0 and the average density ρˉ of matter in the visible part of the As we have found, for stars located in this central region Universe (the Russell-Hertzsprung main sequence), the exponent of 1 the “mass-luminosity” relation is about 4. Therefore, using t , (2.7) 0 ' √πGρˉ formulae which, independently of its theoretical origin, is also the very log Mˉ = 0.1mb , 5 log Rˉ = mb x , interesting empirical correlation. Because of (2.4) and (2.7), − − − we re-write equation (2.6) as follows we transform (2.12) to the form

κTρxˉ 2 2 2 ν = ν0 e− , (2.8) mb + 2×1.51mb x + 2.44x = c1 . (2.13) where x = ct is the path of a photon. Formula (2.8) is like the The left side of this equation is almost a perfect square, formula of absorption, and so may give additional support to hence we have the equation of a very eccentric ellipse, with the explanation of the nebula red shift by unusual processes an angular coefficient close to 1.51. The exact solution can which occur in photons during their journey towards us. It is be found by transforming (2.13) to the main axes using the possible that in this formula ρˉ is the average density of the secular equation. As a result we obtain intergalactic gas. a = 8.9, α = 1.58 , (2.14) b − 2.2 The main sequence where a and b are the main axis and the secondary axis The contemporary data of observational astronomy has suf- of the ellipse respectively, α is the angle of inclination of ficiently filled the Russell-Hertzsprug diagram, i. e. the “lum- its main axis to the abscissa’s axis. Because of the large inosity — spectral class” plane. As a result we see that there eccentricity, there is in the Russell-Hertzsprung diagram the ˉ are no strong arcs L(Teff ) and L(R), but regions filled illusion that stars are concentrated along the line a, the by stars. In the previous chapter we showed that such a main axis of the ellipse. The calculated angular coefficient dispersion of points implies that the energy productivity in α = 1.58 (2.14) is in close agreement with the empirically stars is regulated exclusively by the energy drainage (the determined− α = 1.62 (2.9). radiation). So the mechanism generating stellar energy is not Thus the Russell-Hertzsprung− main sequence has no like any reactions. It is possible that only the main sequence physical meaning: it is the result of the scale stretching used of the Russell-Hertzsprung diagram can be considered a line in observational astrophysics. In contrast, the reality of the along which stars are located. According to Parenago [38], scale used in our stellar energy diagram (Fig. 2) is confirmed this direction is by the homogeneous distribution of the isoergs. As obtained in Part I of this research, from the viewpoint ˉ mb = m 1.62x, x = 10 log Teff . (2.9) − of the internal constitution of stars, stars located at the op- posite ends of the main sequence (the spectral classes O An analogous relation had been found by Kuiper [8] as the and M) differ from each other no more than stars of the M (R) relation same spectral class, but of different luminosity. Therefore the log Rˉ = 0.7 log M.ˉ (2.10) “evolution of a star along the main sequence” is a senseless Using formulae (1.12) and (1.14), we could transform term. formula (2.10) to a correlation B(ρ). At the same time, The results show that the term “sequence” was applied looking at the stellar energy diagram (Fig. 2), we see that the very unfortunately to groups of stars in the Russell- stars of the Russell-Hertzsprung “main sequence” have no Hertzsprung diagram. It is quite reasonable to change this B(ρ) correlation, but fill instead a ring at the centre ofthe terminology, using the term “region” instead of “sequence”: diagram. This incompatibility should be considered in detail. the region of giants, the main region, etc.

96 N. Kozyrev. Sources of Stellar Energy and the Theory of the Internal Constitution of Stars October, 2005 PROGRESS IN PHYSICS Volume 3

2.3 White dwarfs The radiant flow can be written as 4 cαT 3 dT There is very little observational data related to white dwarfs. F = , (2.21) R 3 κρ dr Only for the satellite of Sirius and for o2 Eridani do we know − values of all three quantities L, M, and R. For Sirius’ satellite hence we obtain F zT 1/2 αce4 π3/2 m1/2 2.6zT 1/2 R = e = . (2.22) 2 7/2 ˉ ˉ FT κρ k 18√2 κρ M = 0.95, R = 0.030, ε = 1.1×10− ,   (2.15) Using (2.15) it is easily seen that even if κ 1, F < F ρ = 104, ρ = 3 105, p = 1 1022. R T c × c × in the internal regions of white dwarfs. We can' apply the formulae obtained to the case of the conductive transport For an ideal gas and an average molecular weight μ = 1/2, 8 of energy, if we eliminate κ with the effective absorption we obtain Tc = 2×10 degrees. The calculations show that coefficient κ∗ white dwarfs generate energy hundreds of times smaller 2.6zT 1/2 than regular stars. Looking at the isoergs in Fig. 2 and κ∗ = . (2.23) ρ the isotherms in Fig. 3, we see that the deviation of white dwarfs from the “mass-luminosity” relation is of a special Thus, if white dwarfs are built on an ideal gas whose kind; not the same as that for regular stars. At the same time state is about the degeneration boundary, their luminosity white dwarfs satisfy the main direction in the stellar energy should be more than that calculated by the “mass-luminosity” diagram: they lie in the line following giants. Therefore it formula (the heat equilibrium condition). would be natural to start our brief research into the internal We consider the regular explanation for white dwarfs, constitution of white dwarfs by proceeding from the general according to which they are stars built on a fully degenerate supposition that they are hot stars whose gas is at the bound- gas. For the full degeneration, we use Chandrasekhar’s ˉ ary of degeneration “mass-radius” formula (see formula 2.32, Part I). With M = 1 we obtain 3/2 8 ρ = AT ,A = 10− μe . (2.16) Rˉ = 0.042 (μe = 1), Rˉ = 0.013 (μe = 2). We now show that, because of high density of matter in The observable radius (2.15) cannot be twice as small, so white dwarfs, the radiant transport of energy FR is less than we should take Sirius’ satellite as being composed of at least the transport of energy by the electron conductivity FT 50% hydrogen. From here we come to a serious difficulty: because of the high density of white dwarfs, even for a 1 dT few million degrees internally, they should produce much F = vˉe λˉ cˉv ne , R −3 dr more energy than they can radiate.We now show that such temperatures are necessary for white dwarfs. where λˉ is the mean free path of electrons moved at the Applying the main equations of equilibrium to the surface average velocity vˉ , cˉ is the average heat capacity per e v layer of a star, we obtain particle. Also B Lκ εκ 1 n 3 = = , (2.24) λ = , n = e , σ = πr2, c = k , (2.18) p 4πGcM 4πGc n σ i z v 2 i where κ is the absorption coefficient in the surface layer. At where ni is the number of ions deviating the electrons, σ is the boundary of degeneration we can transform the left side the ion section determined by the 90◦ deviation condition by (2.16) 3εκ A2 2 ρ0 = < 2 ze 4πGc μα me ve = , (2.19) r so that μ2 i. e. the condition to move along a hyperbola. ρ = 125 εκ e , 0 μ Substituting (2.19) and (2.18) into formula (2.17) and   (2.25) μ eliminating vˉ by the formula T 3/2 = 1.25 1010 εκ e . 0 × μ 5/2   12 2kT vˉ5 = , We see from formula (2.22) that even in the surface layer √π me the quantity F can be greater than F . Substituting κ   T R ∗ (2.23) into (2.25), we obtain we obtain 24 2k7 T 5 1/2 dT z 2/5 F = . (2.20) T = 2.5 107 ε2/5 . (2.26) T −ze4 π3 m dr 0 × μ  e   

N. Kozyrev. Sources of Stellar Energy and the Theory of the Internal Constitution of Stars 97 Volume 3 PROGRESS IN PHYSICS October, 2005

2 For ε = 10− , μ = 1, and z = 1, we obtain 3. Schmidt W. Der Massenaustausch in freier Luft und verwandte Erscheinungen. Hamburg, 1925 (Probleme der Kosmischen 6 T0 = 4×10 , ρ0 = 80, κ0∗ = 65, Physik, Bd. 7). 4. Milne E. A. The analysis of stellar structure. Monthly Notices thus for such conditions, κ > κ . ∗ of the Royal Astronomical Society, 1930, v. 91, No. 1, 4–55. We know that in the surface layer the temperature is linked to the depth h as follows 5. Kuiper G. P. Note on Hall’s measures of ε Aurigae. Astrophys. Journal, 1938, v. 87, No. 2, 213–215. gμ T = h . (2.27) 6. Parenago P P. Physical characteristics of sub-dwarfs. Astron. 4 Journal — USSR, 1946, v. 23, No. 1, 37. < 7 7. Chandrasekhar S. The highly collapsed configurations of a In the surface of Sirius’ satellite we have g = 3×10 . Hence h = 3 107. Therefore the surface layer is about 2% stellar mass. Second paper. Monthly Notices of the Royal 0 × Astronomical Society, 1935, v. 88, No. 4, 472–507. of the radius of the white dwarf, so we can take the radius at the observed radius of the white dwarf. 8. Kuiper G. P. The empirical mass-luminosity relation. Astro- It should be isothermal in the degenerated core, because phys. Journal, 1938, v. 88, No. 4, 472–507. the absorption coefficient rapidly decreases with increasing 9. Russell H. N., Moore Ch. E. The masses of the stars with a density. For a degenerate gas we can transform formula (2.23) general catalogue of dynamical parallaxes. Astrophys. Mono- in a simple way, if we suppose the heat capacity proportional graphs, Chicago, 1946. 10. Trumpler R.J. Observational evidence of a relativity red shift in to the temperature. Then, in the formula for FR (2.20), the 3/2 class O stars. Publications of the Astron. Society of the Pacific, temperature remains in the first power, while T0 should be eliminated with the density by (2.16). As a result we obtain 1935, v. 47, No. 279, 254. 11. Parenago P.P. The mass-luminosity relation. Astron. Journal FR ρT and also ∼ — USSR, 1937, v. 14, No. 1, 46. 2 8 T 12. Baize P. Les masses des etoiles´ doubles visuelles et la relation κ∗ 2.6×10− zμe . (2.28) 1 ' ρ empirique masse-luminosite.´ Astronomique — Paris, 1947,   t. 13, fasc. 2, 123–152. Even for 4 106 degrees throughout a white dwarf, the × 13. Eddington A. S. The internal constitution of the stars. Cam- average productivity of energy calculated by the proton- bridge, 1926, 153. proton reaction formula (1.16) is ε = 102 erg/sec, which is much more than that observed. In order to remove the contra- 14. Chandrasekhar S. An introduction to the study of stellar structure. Astrophys. Monographs, Chicago, 1939, 412. diction, we must propose a very low percentage of hydrogen, 15. Eddington A. S. The internal constitution of the stars. Cam- which contradicts the calculation above,∗ which gives hyd- rogen as at least 50% of its contents. So the large observed bridge, 1926, 135. value of the radius of Sirius’ satellite remains unexplained. 16. Stromgren¨ B. The opacity of stellar matter and the hydrogen So we should return to our initial point of view, according content of the stars. Zeitschrift fur¨ Astrophysik, 1932, Bd. 4, to which white dwarfs are hot stars at the boundary of H. 2, 118–152; On the interpretation of the Hertzsprung- Russell-Diagram. Zeitschrift fur Astrophysik, 1933, Bd. 7, H. 3, degeneration, but built on heavy elements. The low lumi- ¨ 222–248. nosity of such stars is probably derived from the presence of endothermic phenomena inside them. That is, besides energy 17. Cowling T. G. The stability of gaseous stars. Second paper. Monthly Notices of the Royal Astronomical Society, 1935, v. 96, generating processes, there are also processes where ε is No. 1, 57. negative. This consideration shows again that the luminosity of stars is unexplained within the framework of today’s 18. Ledoux P. On the radial pulsation of gaseous stars. Astrophys. thermodynamics. Journal, 1945, v. 102, No. 2, 143–153. 19. Parenago P.P. Stellar astronomy. Moscow, 1938, 200. 20. Becker W. Spektralphotometrische Untersuchungen an δ References Cephei-Sternen X. Zeitschrift fur¨ Astrophysik, 1940, Bd. 19, H. 4/5, 297. 1. Bethe H. A. Energy production in stars. Physical Review, 1939, 21. Eddington A. S. The internal constitution of the stars. Cam- v. 55, No. 5, 434–456. bridge, 1926, 191. 2. Schwarzschild K. Uber¨ das Gleichgewicht der Sonnenatmo- 22. Luyten W. J. On the ellipticity of close binaries. Monthly sphare.¨ Nachrichten von der Koniglichen¨ Gesellschaft der Notices of the Royal Astronomical Society, 1938, v. 98, No. 6, Wissenschaften zu Gottingen.¨ Mathematisch — physicalische 459–466. Klasse. 1906, H. 6, 41–53. 23. Russell H. N., Dugan R. S. Apsidal motion in Y Cygni and

∗By Chandrasekhar’s formula for a fully degenerate gas. — Editor’s other stars. Monthly Notices of the Royal Astronomical Society, remark. 1930, v. 91, No. 2, 212–215.

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24. Blackett P.M. S. The magnetic field of massive rotating bodies. Nature, 1947, v. 159, No. 4046, 658–666. 25. Eddington A. S. The internal constitution of the stars. Cam- bridge, 1926, 163. 26. Lohmann W. Die innere Struktur der Masse-Leuchtkraft- funktion und die chemische Zusammensetzung der Sterne der Hauptreihe. Zeitschrift fur¨ Astrophysik, 1948, Bd. 25, H. 1/2, 104. 27. Martynov D. Ya. Eclipse variable stars. Moscow, 1939. 28. Gaposchkin S. Die Bedeckungsveranderlichen.¨ Veroffentli-¨ chungen der Universitatssternwarte¨ zu Berlin-Babelsberg, 1932, Bd. 9, H. 5, 1–141. 29. Parenago P.P. Physical characteristics of sub-dwarfs. Astron. Journal — USSR, 1946, v. 23, No. 1, 31–39. 30. Struve O. The masses and mass-ratios of close binary systems. Annales d’Astrophys., 1948, t. 11, Nom. 2, 117–123. 31. Kothari D. S. The theory of pressure-ionization and its applications. Proceedings of the Royal Society of London, Ser. A, 1938, v. 165, No. A923, 486–500. 32. Luyten W. J., Struve O., Morgan W. W. Reobservation of the orbits of ten spectroscopic binaries with a discussion of apsidal motions. Publications of Yerkes Observatory, Univ. of Chicago, 1939, v. 7, part. 4, 251–300. 33. Kuiper G. P. On the hydrogen contents of clusters. Astrophys. Journal, 1937, v. 86, No. 2, 176–197. 34. Masevich A. G. Stellar evolution where is corpuscular radiat- ions, considered from the viewpoint of the internal constitution of stars. Astron. Journal — USSR, 1949, v. 26, No. 4, 207–218. 35. Tuominen J. Uber¨ die inneren Aufbau der TRUMPLERschen¨ Sterne. Zeitschrift fur¨ Astrophysik, 1943, Bd. 22, H. 2, 90–110. 36. Becker W. Spektralphotometrische Untersuchungen an δ Cephei-Sternen X. Zeitschrift fur¨ Astrophysik, 1940, Bd. 19, H. 4/5, 297; Spektralphotometrische Untersuchungen an δ Cephei-Sternen. Zeitschrift fur¨ Astrophysik, 1941, Bd. 20, H. 3, 229. 37. Melnikov O. A. On the stability of masses of ling-periodical Cepheids. Proceedings of Pulkovo Astron. Observatory, 1948, v. 17, No. 2 (137), 47–62. 38. Parenago P.P. The generalized relation mass-luminosity. Astron. Journal — USSR, 1939, v. 16, No. 6, 13.

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