Mach's Principle and a Variable Speed of Light

Home , Variable speed of light

arXiv:gr-qc/0511038v4 6 Oct 2007 h rprinlt fieta n rvttn as h equivalenc the mass, gravitating and c inertial of proportionality the a hsc,Sc ecie sanwpooa how proposal new a as describes 5 variable Sec. physics, a cal by described relativity the be c alternatively how general can repeats of (GR) 4 tests sec. experimental (1986), similar respective Will Using theory. precursor as scalar-tensor a arguments so-called paper, the (1957) Dicke’s of has in effect outlined of electromagnetic approach been an of similar as perspective very gravity A historical describing given. brief are theories a and VSL time and on considerations scales general length 3 speed sec. variable as in a light, to realized of related be is (1953). theory can Sciama Sciama’s Before following principle outline statement, Mach’s quantitative shall how I a 2 1.2 in sec. given in motivation brief a After Overview 1.1 Introduction 1 flatness the to explanation alternative correspon an that cosmology. supply term also variable slightly may a proposal calculate to allows distribution S pn h osblt owietettleeg fapril as particle a of energy g total of the tests classical write four to the possibility with the agreement opens the des VSL and equivalent given, an is this In (VSL) scalar-tensor-theories. of precursor a neo l assi h nvre orsodn yohsso S of hypothesis corresponding A universe. the in gravitation masses of constant, all origin the of that ence suggested (1838-1916) Mach Ernst Abstract diinlymthsDrcslrenme yohss(193 hypothesis number large wh Dirac’s (2007a) matches arxiv:0708.3518 additionally in outlined proposal the prefer hl hsi eeyaohrvepito classi- of viewpoint another merely is this While . eedn ntems itiuini ie htrpoue Newton reproduces that given is distribution mass the on depending 1 ontuhl n ogrtefruagvnhr,and here, given formula the longer any uphold not do I c 2 /G ahsPicpeadaVral pe fLight of Speed Variable a and Principle Mach’s = P m i /r i slne oDces(97 rpslo neetoantcoii fg of origin electromagnetic an of proposal (1957) Dicke’s to linked is , e-mail: etlziGmaimMünchenPestalozzi-Gymnasium [email protected] lxne Unzicker Alexander coe ,2007 6, October ich 8). 1 ihrdhf uenveaecmol explained faint commonly relatively rotation the are 1998, flat supernovae Since the high-redshift galaxies. particular of in curves collected, galaxy matter’ been in ‘dark has for mass’ evidence overwhelming Since ‘missing an decades. clusters of past observation the up in Zwicky’s last shown physics have gravitational the riddles in new predict gravity, perihelion, at could mercury Newtonian (GR), the discovery, with relativity its incompatible general observation of of time theory the the gravitational While in questions Open 1.2 are data. consequences observational to 7, respect sec. with accordance In discussed in principle. is that Mach’s given with is distributions mass on rmavariable deduced directly a be from can gravitation of law Newton’s l,i e.6a xlctfruafor formula explicit an 6 sec. princi- in equivalence ple, the for consequences interesting physics rpini em favral pe flight of speed variable a of terms in cription litrcincuddpn ntepres- the on depend could interaction al rnil.Frhroe oml for formula a Furthermore, principle. e nrlrltvt sson Moreover, shown. is relativity eneral rbe n h oio rbe in problem horizon the and problem E c st h ‘constant’ the to ds im 15)o h gravitational the on (1953) ciama hl hsgnrlapoc has approach general this While . slwo gravitation. of law ’s = mc 2 hsncsaiylasto leads necessarily this ; G h present The . c 1 ravitation, hsmass This depending as a manifestation of a new form of matter called approximation valid for m1 >> m2. A slightly dif- ‘dark energy’. As Aguirre et al. (2001) comment, ferent mass dependence2 would remain undetected these new discoveries have been achieved ”at the even by the most precise emphemeris and double expense of simplicity”; it is quite disappointing that pulsar data, since in most cases just the product no counterpart to these forms of matter has been GM is measured. found in the laboratories yet. Thus in the view of In summary, there is quite a big gap between the recent data the following comment of Binney and common belief in the universality of Newton’s law Tremaine (1994), p. 635, is more than justified: of gravitation and the experimental evidence sup- porting it.3 So far there is no evidence at all that it ‘It is worth remembering that all of still works for accelerations below 10−9m/s2. the discussion so far has been based on the premise that Newtonian gravity and general relativity are correct on large 2 Mach’s principle scales. In fact, there is little or no di- rect evidence that conventional theories Besides the observational facts above, Newtonian of gravity are correct on scales much gravity had to face theoretical problems, too. In his larger than a parsec or so. Newtonian famous example of the rotating bucket filled with gravity works extremely well on scales water, Newton deduced the existence of an abso- of ∼ 1014cm (the solar system). (...) It lute, nonrotating space from the observation of the is principally the elegance of general rel- curved surface the water forms. Ernst Mach criti- ativity and its success in solar system cized this ‘non observable’ concept of absolute space tests that lead us to the bold extrap- as follows, suggesting that the water was rotating olation that the gravitational interac- with respect to masses at large distance: tion has the form GM/r2 on the scales 21 26 ‘No one is competent to say how the 10 − 10 cm...’ experiment would turn out if the sides In the meantime, the bold extrapolation seems of the vessel increased in thickness and to have encountered new observational problems. mass till they were ultimately several... The detection of an anomalous acceleration from [miles]... thick.’ the Pioneer missions (Anderson, Laing, Lau, Liu, Mach’s principle is commonly known as follows: The Nieto, and Turyshev 2002; Turyshev et al. 2005) indicate that Newtonian gravity may not even be cor- reason for inertia is that a mass is accelerated with respect to all other masses in the universe, and rect at scales within the solar system. Interestingly, therefore gravitational interaction would be impos- this anomaly occurred at the same dynamic scale - −9 2 sible without the distant masses in the universe. about 10 m/s - as many phenomena indicating A possible solution to the ‘rotating bucket’ part dark matter. Furthermore, the scale 10−9m/s2 is the scale of the problem where the distant masses instead of absolute space define the physical framework has that most of the precision measurements of G ap- been given in a brilliant paper by Lynden-Bell and proach - it is still discussed if the discrepancies between the G measurements in the last decade Katz (1995). However, from Mach’s principle it has been fur- (Gundlach and Merkowitz 2000; Uzan 2003) have ther deduced that the numerical value of the Gravi- a systematic reason. tational G constant must be determined by the mass Besides the distance law, there is another lack of distribution in the universe, while in Newton’s the- experimental data regarding the mass dependence. The Cavendish torsion balance uses masses of about ory it is just an arbitrary constant. Since the square of the speed of light times the radius of the universe 1kg to determine the mass of the earth, since there divided by the mass of the universe is approximately is no precise independent geological measure. The two-body treatment of Newtonian gravity tells us 2See the interesting measurements by Holding et al. that the relative acceleration a12 is proportional to (1986), Ander et. al. (1989) and Zumberge et. al. (1991). 3A more detailed discussion is given in Unzicker (2007c). m1 + m2. It may well be that simple form is just an

2 equal to G, such speculations were first raised by Einstein in the years 1907- 1911. Contrarily to first Dirac (1938), Sciama (1953) and Dicke (1957). intuition, this does not contradict the special theory of relativity (SR). ‘Constancy’ of c, as far as SR is 2.1 Sciama’s implementation of concerned, refers to a limiting velocity with respect Mach’s principle. to Lorentz transformations in one spacetime point; it does not mean c has to be a constant function in Since Mach never gave a quantitative statement of spacetime. Ranada (2004) collected several citations his idea, it has been implemented in various man- of Einstein that put into evidence that the principle ners; the reader interested in an overview is referred of constancy (over spacetime) of the speed of light to Bondi (1952), Barbour and Pfister (eds.) (1995) is not a necessary consequence of the principle of and Graneau and Graneau (2003). An idea of a relativity.4. quantitative statement proposes the following func- Considering variable atomic length and fre- tional dependence of Newton’s constant G (Sciama quency standards λ and f 5 does nothing else but in- 1953; Unzicker 2003): fluence our measurements of distances, i.e. the met- 2 ric of spacetime. A variable metric however mani- c m = i = Φ (1) fests as curvature.6 It will be discussed in sec. 4, G X r i i how the arising curvature can be brought in agree- As in Sciama’s proposal, (1) may be included in the ment with the known tests of GR. definition of the gravitational potential, thus yielding 3.2 Electromagnetic and Brans- m 2 ϕ = −G i = −GΦ= −c . (2) Dicke-theories X r i i Shortly after GR was generally accepted, Wilson Alternatively, one can apply the hypothesis (1) (1921) reconsidered Einstein’s early attempts and directly to Newton’s law. The potential instead can deduced Newton’s law from a spacetime with a then be brought to the form variable refractive index depending on the gravita- m ϕ = −c2 ln i (3) tional field. This and related proposals like Rosen’s X r i i (1940) remained quite unknown until Dicke’s paper from which follows Newton’s law (1957) attracted much attention with the statement that gravitation could be of electromagnetic origin. mi~ri i 3 2 ri mi~ri While the second term in Dicke’s index of refraction −∇ϕ = −c P =: −G(mi, ri) . (4) mi X 3 i r ri (eqn. 5 there) P i i 2GM ǫ =1+ (5) This was already developed by Unzicker (2003), rc2 though I do not support any more the further con- is related to the gravitational potential of the sun, siderations following there. I prefer instead, as (2) Dicke was the first to raise the speculation on the suggests, to relate the (necessarily varying) gravita- first term having ‘its origin in the remainder of tional potential to a variable c. It seems that Sciama the matter in the universe’. While this implemen- did not consider this possibility; an explicit form will tation of Mach’s principle was fascinating, in the be given in sec. 6 which has however similarities with following, more technical development of the theory (3). For systematic reasons however, we should an- (Brans and Dicke 1961), a new parameter ω was alyze the consequences of a variable c at first. introduced that later failed to agree with a reasonable experimental value (Reasenberg 1979). With- 3 A spacetime with variable c. out Mach’s principle, theories with a variable refractive index, and therefore with a variable c, were 3.1 Early attempts of Einstein. developed as ‘polarisable vacuum models’ of GR. 4Further comments are given in Unzicker (2007b). Before further relating c to the gravitational poten- 5 Therefore λf = c is variable, too. tial it is worth remembering that a possible influ- 6A textbook example herefore is given by Feynman et al. ence of the speed of light was already considered by (1963), chap. 42.

3 In particular, Puthoff (2002) showed a far-reaching lower in the gravitational field, seen from outside, agreement with all experimental tests of GR known i.e. at a large distance from the field mass. Equally, so far. frequency scales are lowered, and therefore, time scales raised. That means time runs slower, if we 3.3 Recent VSL theories again observe from outside. According to (7), we must assume c to be low- Recently, Rañada (2004) considered a VSL in the ered by the double amount in the vicinity of masses, context of the Pioneer anomalous acceleration. As if we perform measurements at large distances. This an alternative to inflation models in cosmology, vari- point of view is not that common because we usu- able speed of light (VSL) theories were developed ally do not consider the value of c at a distance, by Moffat (1993; 2002) and Magueijo (2000; 2003). though we do consider λ and f (gravitational red- These attempts had to face some criticism that shift!). For a detailed and elucidating discussion of claimed that considering any variability of c is just this viewpoint in the context of the radar echo delay, a trivial and obsolete transformation. Indeed, this see Will (1986), p. 111 ff. criticism would be correct if spacetime were flat. The lowered c in the vicinity of masses is some- A review article by Magueijo (2003) that focusses times called ‘non-proper speed of light, whereas the on cosmological aspects gives a detailed answer to locally measured ‘proper’ speed of light is a univer- that wrong argument (sec. 2). Unfortunately, a lot sal constant (Ranada 2004). of controversial discussion on VSL theories recently arose. Some clarifying comments on this topic were given by Ellis and Uzan (2003). Since the present 4 Variable speed of light and proposal refers to old established ideas of Einstein, tests of GR. Dirac, Sciama and Dicke, I shall not like to enter the actual discussion. 4.1 Notation 3.4 Time and Length scales In order to fulfill (7) while analyzing variations of c, we are seeking a simple notation. In first order, The only reasonable way to define time and length δx all GR effects involve the relative change x of the scales is by means of the frequency f and wave GM quantity x of the amount − rc2 . δx = xg −x0, where length λ of a certain atomic transition, as it is x0 is the locally measured quantity and xg the quan- done by the CODATA units. Thus by definition, tity in the gravitational field, measured from the ob- m c = fλ = 299792458 s has a fixed value with re- server outside. We shall abbreviate a lowering in the spect to the time and length scales that locally how- 7 gravitational field as ↓, e.g. the well-known slower ever may vary. To analyze such situations in the δf GM running clocks with f = − rc2 as f ↓, and equally following, we get from the definition, δλ GM the shortening of rods λ = − rc2 as λ ↓. ↓↓ stands 2GM dc = λdf + fdλ, (6) for the double amount − rc2 , and ↑ indicates a relative increase, etc.8 Eqn. (7) has to be fulfilled ev- and dc df dλ erywhere. No change is indicated as x →. Only two = + , (7) hypotheses are needed for describing the following c f λ experiments: using the product rule. Since we are interested in analyzing first-order effects, we will use (7) for finite In the gravitational field c ↓↓= f ↓ +λ ↓ holds. differences δ as well. During propagation, the frequency f does not change. 3.5 Spatial variation of c. 8 ↓ − GM Equivalently, stands for a factor 1 rc2 . For factors How can we reasonably speak of a variation of c ? GM 2 2GM close to 1, (1 − 2 ) ≈ 1 − 2 holds, corresponding to ↓↓. Let’s start from eqn. (7). According to GR, λ is rc rc

7Of course, such variation cannot be detected unless we use spacetime curvature as indirect measurement.

4 4.2 The Hafele-Keating experiment. it keeps its (lowered) frequency. The photon does not keep in mind, so to speak, that its origin was a In 1972, two atomic clocks were transported in air- cesium transition - in that case we would not note crafts surrounding the earth eastwards and west- a redshift at the photons arrival - but it maintains wards in an experiment by Hafele and Keating. Be- its frequency. Since this happens at a space posi- sides the SR effect of moving clocks that could be tion outside the gravitational field where c is higher eliminated by the two flight directions, the results by the double amount (see (7)), the photon has to showed an impressive confirmation of the first-order adjust its λ, and raise it with respect to the value general relativistic time delay. We briefly describe ‘at home’. Since the adjustment to c overcompen- the result by f ↓, since the slower rate at which sates the originally lower λ, we detect the photon as clocks run in a gravitational field was measured di- gravitationally redshifted. rectly. The accuracy was very much improved by Vessot et. al. (1980). From this experiment alone it does not follow yet c ↓↓ f ↓ λ ↓, because λ could 4.4 Radar echo delay and light de- even remain unchanged. The following tests however flection. leave no other possibility but the above mentioned. The above results, linked to eqn. (7), already suggest a relative change of the speed of light of the amount 4.3 Gravitational redshift and the c ↓↓ in the gravitational field. During propagation, Pound-Rebka experiment. f again does not change. The first idea of measuring this effect directly was launched by Shapiro (1964). Photons leaving the gravitational field of the earth Data of spectacular precision that agreed on the show a slight decrease in frequency at distances of −3 level of 10 with the theoretical prediction were about 20 m. This high-precision experiment became collected by the Viking lander missions (Reasenberg possible after the recoil-free emission of photons due 1979). Recently, the Cassini mission delivered a still to the Mössbauer effect (Pound and Rebka 1960; better agreement (Williams, Turyshev, and Boggs Pound and Snider 1965). Since the experiment con- 2004). sists in absorption of photons E = hf, this is a Measuring the radar echo delay is only possible true frequency measurement. Again, the frequency because the photon maintains its frequency while change f ↓ was verified. For this point of view it travelling. Since c is lowered by the double amount is important to remember the assumption that the in the vicinity of the star c ↓↓, it has to shorten its photon does not change its frequency while propa- wave length accordingly, and while bypassing the gating in space, but had a lowered rate f already at star the emission. c ↓↓= λ ↓↓ f → (10) The same effect, but measured as an optical wavelength shift, was verified analyzing the emis- holds. There, the photon’s λ is even shorter than sion spectra of the sun (Snider 1972). The situation λ of a photon produced at the star by the same at the emission is: atomic transition (see above, λ ↓). Naturally, the property which the photon has to leave at home for c ↓↓= f ↓ +λ ↓ (8) sure, is ‘its’ c, otherwise it wouldn’t be considered as light any more. As it is outlined by Will (1986), During propagation f = const. holds, and the local p. 111, too, interpreting the radar echo delay as a c at arrival remains per definition unchanged. Thus local modification of c is an uncommon, but correct at the time of detection, interpretation of GR. All the tests are in agreement with (7). c →= f ↓ +λ ↑ (9) A lower c in the vicinity of masses creates light holds, what indeed is observed. In plain words, one deflection just as if one observes the bending of light can imagine the process as follows: consider a pho- rays towards the thicker optical medium. Quantita- ton travelling from its home star with huge gravita- tively, light deflection is equivalent to the radar echo tion to the earth (with approximately zero gravity). delay. A detailed calculation for the total deflection While travelling through different regions in space,

5 ∆ϕ yields outside of the gravitational field. Consequently, the 4GM δϕ = . (11) measurement of masses is influenced as well. Due to rc2 (13), masses are proportional to reciprocal acceler- Again the bending of light appears as curvature, ations and m ↑↑↑ holds. therefore it is justified to describe GR by a spa- Similar considerations lead to v ↓↓ for veloci- tial variation of c, as it was already pointed out by ties (as c), and l → for the angular momentum. Einstein 1911: This last observation will turn out useful for keeping Planck’s constant h spatially constant, since it has ‘From the proposition which has just m2 9 units kg s . been proved, that the velocity of light Table I gives an overview on the change of quan- in the gravitational field is a function of tities. One ↓ abbreviates a relative decrease of the the place, we may easily infer, by means δx GM quantity x = − rc2 in the gravitational field, or a of Huyghens’s principle, that light-rays GM factor (1 − 2 ). propagated across a gravitational field rc undergo deflexion’. Quantity symbol unit rel. change 1 In the same article, Einstein obtained only the Frequency f s ↓ half value for the light deflection, because he con- Time t s ↑ sidered c and f as subject to variation, but not λ. Length λ m ↓ m Independently from describing GR by a VSL the- Velocity v s ↓↓ m ory, in a given spacetime point there is only one rea- Acceleration a s2 ↓↓↓ sonable c, from whatever moving system one tries to Mass m kg ↑↑↑ measure. This, but not more is the content of SR, Force F N → which is not affected by expressing GR by a variable Pot. energy Ep Nm ↓ c. kgm2 Ang. mom. l s → Table I. Overview on the change of quantities. 4.5 Measuring masses and other quantities 4.6 Advance of the perihelion of Not only time and length but every quantity is mea- Mercury sured in units we have to wonder about, once we allow changes of the measuring rods. An interesting In the Kepler problem, the Lagrangian is given by question is how to measure masses. Consider a large m GMm field mass M with gravitational field and at large L = (r ˙2 + r2ϕ˙ 2)+ , (14) distance a test mass m. By Newton’s 3rd law 2 r which after introducing the angular momentum FM = −Fm (12) l = mr2ϕ˙ transforms to holds, thus the ‘force measured at a distance’ is al- 2 1 2 l GMm ways of the same amount as the ‘local force’. How- L = mr˙ + 2 + . (15) 2 2mr r ever, in aM M = −amm (13) Now we have to consider the relative change of the quantities as given above. m ↑↑↑ however refers to am is measured with the unchanged scales, and aM a test particle at rest. The kinetic energy causes an with the scales inside the gravitational field. The additional, special relativistic mass increase ↑ (see measurement of accelerations depends on time and m sec. 5.1 and 5.2 below in detail), thus in eq.(15), length scales, and the unit 2 corresponds to a three s m ↑↑↑↑ holds, while r ↓ and l →. Hence, the middle times lower value for the acceleration in gravita- term changes as ↓↓ in total, which means that m is tional fields, or a ↓↓↓ in the above notation. To remember, this holds if we observe the behavior of 9In agreement, Dicke (1957), p.366, and Puthoff (2002), an accelerated object at a large distance, i.e. from eqns. 8-13, with different arguments.

6 2GM 10 effectively multiplied by a factor (1 − rc2 ). The 5.2 Energy conservation. GMl2 arising − mr3c2 represents the well-known correction A quite interesting consequence of the foregoing cal- which leads to the secular shift culation arises if we consider the quantity E = mc2 6πGM of a test particle moving into a gravitational field. ∆φ = , (16) A(1 − ǫ2)c2 Recall that time and length measurements influence δc2 4GM 2 all quantities, thus c2 = − rc2 or c ↓↓↓↓. This A being the semimajor axis and ǫ the eccentricity of is partly compensated by m ↑↑↑ as outlined above. GM the orbit. There is one relative increase of the amount rc2 left - the part in eqn. (20) due to the increase in kinetic energy! 5 Newton’s law from a vari- This means, the total energy E = mc2 of a test able c. particle appears as a conserved quantity during the motion in a gravitational field. In first order, we may Since time and length measurement effects of GR write can be described by a spatial variation of c, one δ(mc2)= c2δm + mδ(c2)=0. (21) is tempted to try a description of all gravitational Roughly speaking, the left term describes the in- phenomena in the same framework. However, c ↓↓ crease in kinetic energy due to the relativistic δm (see sect. 4.1) requires and the right term the decrease of potential en- 2 4GM ergy in a region with smaller c . To be quantitative, δ(c2)=2cδc = − . (17) r however, the change in m contributes to the change in potential energy such that Ep ↓. The change in This differs by a factor 4 from Sciama’s original idea 2 4GM of the right term, mδ(c ) amounts to − rc2 , four and suggests a Newtonian gravitational potential of times as much as needed to compensate dEk. Three the form parts of it are compensated by an increase of m, ↑↑↑ 1 ϕ = c2. (18) due to mass scales, the remaining 1 md(c2) is con- Newton 4 4 verted into kinetic energy: 5.1 Kinetic energy. 2 2 2 mc → m(c ↓↓↓↓)+ c (m ↑↑↑)+ Ek = Ep + Ek As it is well-known from special relativity (SR), the (22) In summary, there is nothing like a gravitational en- mass increase due to relativistic velocities is, apply- 2 ing Taylor series to the square root, ergy, just E = mc . Kinetic energy is related to a change of m, and potential energy to a change of c2. δm = m(v) − m0 = (19) As it should be, photons do not behave differently, 2 m0 1 v since they conserve their energy hf as well during m0 − 1 ≈ m0( 2 ). propagation. v2 2 c q1 − c2 5.3 Foundation of the equivalence If we apply conventional energy conservation Ek = 1 2 GM principle. Ep and start at r = ∞, v = 0, with 2 v = r follows δm GM While GR provides most elegantly a formalism that = , (20) incorporates the equality of inertial and gravitating m rc2 masses, the question as to the deeper reason of this for the relative mass increase due to kinetic energy, outstanding property of matter (’I was utterly as- corresponding to ↑. tonished about its validity’, Einstein 1930) is still 10As one easily checks, the first and the last terms instead open. Why does the elementary property of mat- do not need a correction. ter, inertial resistance to accelerations, act at the same time as a ‘charge’ of a particular interaction ? Or, equivalently: why does kinetic energy show the

7 same proportionality to m as potential energy in a 6.2 General considerations on the gravitational field ? functional dependence of c. If one supposes that the total energy of a particle can always be written as E = mc2, the differentia- We are therefore seeking a Machian formula which tion in eqn. (21) shows that both terms have to be describes explicitly the dependence of c. The most proportional to the test mass m. general form would be c(mi, ~ri, ~vi,~ai,t), i indicat- One fourth of the first term describes the (rel- ing a single mass point in the universe. Here I do 11 not give detailed reasons why I exclude anisotropy ativistic) increase of kinetic energy, which is usu- 12 1 2 and an explicit dependence on vi,ai and t. The ally approximated as 2 mv . The remainder plus the right term is the change in potential energy. Since resulting hypothesis 2 2 dm is proportional to m, both terms are obviously c = c0 f(mi, ri), (23) proportional to the mass m of the test particle. c0 being the speed of light in some preferred condi- There is no reason any more to wonder why inertial tion of the universe, has however to face the problem and gravitating mass are of the same nature. De- how to make the argument dimensionless. Again, the scribing gravity with a varying speed of light leads only acceptable hypothesis seems to measure m and to the equivalence principle as a necessary conse- i r either in multiples of m , r of elementary parti- quence. The deeper reason for this is that gravita- i p p cles (here protons) or as fraction of the respective tion is strictly speaking not an interaction between quantities of the universe m , r . In both (or con- particles but just a reaction on a changing c. u u sidering combinations, even four) cases, simple al- gebraic relations of mi and ri yield a very large or 6 Dependence of c on the mass very small number. For reasons that will become clear later, I prefer the choice mp, rp. To avoid that distribution in the universe 40 f(mi, ri) in (23) is of the order 10 , one has to in- troduce a function like ln. 6.1 Test masses and field masses. While dealing with the equivalence principle, there 6.3 Proposal for the explicit depen- is a subtle difference between test and field masses. dence of c. The above derivation holds for the former only; all spectacular tests of the EP, starting from Eötvös to Obviously, the mi should enter in an additive way, recently proposed satellite missions, deal with test and the influence of each mass i should decay with mi ri. For these reasons, ln is likely to be a com- masses. P ri An entirely new question is how matter does in- ponent of eqn. (23), as one could suspect immedi- 1 2 ately from Sciama’s original proposal (1). Taking fluence c in its vicinity. Taking the proposal ϕ = 4 c into account that c has to decrease in the vicinity of seriously, there are no more masses creating fields, 13 GMm masses, one of the most simple possibilities is the because once you give up Newton’s F = r2 , the force does not need to depend on a product formula 2 2 of masses. The modification of c could depend in- 2 c0 c0 c (mi, ri)= =: . (24) stead on quantities which are approximately corre- ln mirp ln Σ rimp lated to mass like the baryon number. Eötvös-like P experiments could not detect that, but there could 6.4 Recovery of Newton’s law be material dependencies in torsion balance experiments. For the acceleration of a test mass 11 3 2 mi of it are due to the mass increase m ↑↑↑. 1 c0 r2 4 |a| = | ∇c2| = P i (25) 4 4(ln Σ)2 mi P ri 12This would be unusual, but not a priori senseless. Bar- bour (2002) recently gave an interesting proposal for G depending on vi. 13 2 2 The alternative c = −c0 lnΣ, while yielding similar results, appears less natural since it requires mu instead of mp.

8 holds, yielding the inverse-square law. The gravita- owing to the small eﬀect of local irreg- tional ‘constant’ can be expressed as ularities this coupling is practically con-

2 2 stant over the distances and times avail- c0 1 c 1 able to observation. Because of this con- G = 2 = . (26) 4(ln Σ) mi 4 ln Σ mi stancy, local phenomena appear to be P ri P ri isolated from the rest of the universe.’ which differs by a numerical factor 4 ln Σ from Sciama’s proposal. It should be noted that the term 6.6 Taking away the mystery from mi contained in G is approximately constant. ri PThe contributions from earth, sun, and milky way, Mach 17 19 20 9.4 · 10 , 1.33 · 10 , and 7.5 · 10 (in kg/m), are There is another satisfactory aspect of a gravita- minute compared to the distant masses in the uni- 1 2 14 27 tional potential of the form 4 c . A Machian depen- verse that approximately amount to 1.3·10 . This dence of Newton’s constant G which is determined was first observed by Sciama (1953), p. 39. Thus by the distribution of all masses in the universe slight variations due to motions in the solar system is in a certain sense beautiful, but after all some- are far below the accuracy of current absolute G −4 what mysterious. How does a free falling apple feel measurements (∆G/G =1.5 · 10 ). the mass distribution of the universe ? According to Sciama’s interpretation, its acceleration is much 6.5 Gravitational potential. slower because there are so many galaxies besides the milky way. The gravitational ϕ potential of classical mechanics On the other hand, assumed that the gravita- arises in first order from eqn. (24). Taking the ex- 1 2 ′ GM◦ tional potential is just 4 c , it has been our belief ample ϕs = of the sun, and let Σ be the sum R◦ in the universality of Newton’s law and the some- as defined in (24) without the sun, then, ∆ denoting what naive extrapolation of the potential to Gmi i ri a difference, that created so much surprise while rediscoveringP

2 2 1 1 eqn. (1). Gm ∆c = c0( ′ − ) = (27) Despite the fact that the potential i is ln Σ ln Σ Pi ri 2 2 a purely mathematical tool which cannot be ob- c0 ′ c Σ ′ (ln Σ − ln Σ )= ′ ln ′ = served directly, we have internalized it so much that ln Σ ln Σ ln Σ Σ G seems to depend on the mass distribution. In fact, c2 ′ ln(1 + α), if gravity depends just on c instead, G turns out as ln Σ kind of an artifact. M◦ holds, whereby α is defined as m . The Taylor R◦ i ri series of the ln, after applying (26),P yields in first 7 Discussion of consequences approximation and observational facts 2 2 c 4GM◦ ∆c = ′ α = =2 c∆c =: 4ϕ1, (28) In the meantime, the observational status of cos- ln Σ R◦ mology has improved dramatically. Besides the un- ϕ1 being the first-order approximation of the gravi- resolved problems sketched briefly in sec. 1.2, the tational potential of the sun. increasing precision in measuring the mass of the Though the present proposal differs from universe has revealed further riddles. Sciama’s by a numerical factor, the comment on G given on page 39 in the 1953 paper still applies: 7.1 Visible matter and flatness 2 ‘... then, local phenomena are strongly The approximate coincidence c ≈ mu is usually G ru coupled to the universe as a whole, but expressed as critical density of the universe

14Taking into account that all matter could be visible 2 3H0 (sec. 7.1), the universe term is still dominant (5.4 · 1024). ρ = . (29) c 8πG

9 A universe with ρu > ρc is called closed, and open 7.3 Time evolution of G. in the case ρ >ρ . An analysis of the evolution of c u The increase of the number of m is also necessary the universe tells that if Ω := ρu is about the order i ρc −60 for compensating the increase of the ri. It is at the of 1 at present, it must have been as close as 10 moment not excluded that G in (26) could meet the to 1 at primordial times. For that, many cosmolo- quite restrictive observational evidence for G˙ ≈ 0 gists believe Ω has to be precisely 1, but a cogent (Uzan 2003). theoretical reason is still missing. Moreover, observations clearly indicate that the fraction of visible 7.4 Temporal evolution of c. matter, Ωv is about 0.01, with a large uncertainty. From (26) and the assumption of an homoge- In sec. 4, a spatial variation of c was considered de- neous universe, elementary integration over a spher- 3 scribing effects that occurred whilec ˙ ≈ 0. As it can mi mu 15 ical volume yields ≈ 2 , and therefore P ri ru be deduced from Maxwell’s equations , in this case 2 a photon keeps its frequency f unchanged. On the c ru other hand, if we assumec< ˙ 0 during cosmological mu ≈ (30) 6G ln Σ evolution in an almost isotropic universe with van- −15 ishing gradients of c, Maxwell’s equations require a holds. If one assumes rp = 1.2 · 10 m, mp = −27 constant λ during wave propagation, in the forego- 1.67 · 10 kg (the proton values), and taking the −1 ing notation c ↓↓= f ↓↓ λ →. Since a photon keeps recent measurement of H0 = 13.7 Gyr that leads 26 its original λ, the wavelength appears longer with to r =1.3·10 m, m is in approximate agreement u u respect to the measuring rods that evolve according with the observations, and corresponds to the frac- to c ↓↓= f ↓ λ ↓. Dicke (1957), p. 374, suspected this tion of visible matter Ω = 0.004. Moreover, there v result to be related to the cosmological redshift. is no more reason to wonder about the approximate coincidence (29), since it is a consequence of the 3mu variable speed of light (24), if one uses ρ = 4 3 πru 8 Outlook and ruH0 ≈ c. Newtonian gravity deduced from a variable Galileo’s F = mg, a formula that today could enter speed of light can thus give an alternative expla- the elementary schools, has been a quite remarkable nation for the flatness problem and may not need discovery, though describing effects on earth only. dark forms of matter (DM, DE) to interprete cos- Newton’s generalizing of this formula and describ- mological data. ing celestial mechanics of the solar system was even more difficult. 7.2 The horizon problem I fear that the hierarchy of structures earth - solar system - galaxy - cosmos may require a cor- If we consider the sum in (24) for cosmological time responding hierarchy of physical laws that cannot scales, we know very little about its time evolution. be substituted by introducing new parameters, data In particular, we know about ri(t) only the momen- fitting and numerical simulations. tary Hubble expansionr ˙i(13.7Gyr). Due to satellite technique, improved telescopes It is tempting however to speculate about an in- and the computer-induced revolution in image pro- creasing number of mi dropping into the horizon, cessing we are collecting data of fantastic precision. thus lowering c during the cosmological evolution. In At the moment it is unlikely that our theoretical un- this picture, the big bang could happen at a hori- derstanding on the galactic or cosmologic scale can zon of the size of an elementary particle (Σ = 1) match the amount of data. with infinite c, the logarithmic dependence causing Though the range of the universe we know about however a very rapid decrease to the almost con- since the discovery of GR has increased dramati- stant actual value. A decreasing speed of light was cally, we usually extrapolate conventional theories recently considered by Magueijo (2000) as an alter- of gravity to these scales. Physicists have learned native scenario to inflation. 15Of course, I do not distinguish between an electromagnetic cEM and a spacetime cST , see Ellis and Uzan (2003).

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