On the Metric of Space-Time

Home , Harry Bateman

Journal of Modern Physics, 2013, 4, 1511-1518 Published Online November 2013 (http://www.scirp.org/journal/jmp) http://dx.doi.org/10.4236/jmp.2013.411184

Carl E. Wulfman Department of Physics, University of the Pacific, Stockton, USA Email: [email protected]

Received May 26, 2013; revised October 27, 2013; accepted August 28, 2013

Copyright © 2013 Carl E. Wulfman. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

ABSTRACT Maxwell’s equations are obeyed in a one-parameter group of isotropic gravity-free flat space-times whose metric depends upon the value of the group parameter. An experimental determination of this value has been proposed. If it is zero, the metric is Minkowski’s. If it is non-zero, the metric is not Poincare invariant and local frequencies of electromagnetic waves change as they propagate. If the group parameter is positive, velocity-independent red-shifts develop and the group parameter play a role similar to that of Hubble’s constant in determining the relation of these red-shifts to propagation distance. In the resulting space-times, the velocity-dependence of red shifts is a function of propagation distance. If 2c times the group parameter and Hubble’s constant have approximately the same value, observed frequency shifts in radiation received from stellar sources can imply source velocities quite different from those implied in Minkowski space. Electromagnetic waves received from bodies in galactic Kepler orbits undergo frequency shifts which are indistinguishable from shifts currently attributed to dark matter and dark energy in Minkowski space, or to a non-Newtonian physics.

Keywords: Space-Time; Metric; Hubble; Dark Matter; Red Shift; Astrophysics; Electrodynamics

1. Introduction parameter subgroup of the Bateman-Cunningham Lie group define a relation isomorphic to Hubble’s law [3]. Because cosmology so heavily depends upon interpreta- Hill’s discovery was overlooked for 65 years. In the tions of the properties of electromagnetic radiation, any misunderstandings of these properties can have extensive year 2000, unaware of it, Hoyle, Burbidge, and Narlikar consequences. Maxwell’s equations govern the propaga- used the Bateman-Cunningham Lie group in their devel- tion and modification of this radiation and permit rela- opment of a generalized relativistic cosmology [4], and tions between observations of cosmic events to be codi- in 2010, Tomilchik [5] argued that Hubble’s relations are fied in global metrics which determine geometric rela- a consequence of properties of transformations of the tions in space-time. special conformal subgroup of the Bateman-Cunningham These geometric relations provide the framework for group. Hill’s paper is however listed in Kastrup’s recent the analyses in this paper, which is based on work of review of papers dealing with conformal invariance [6], Harry Bateman, Ebenezer Cunningham, and E.L. Hill. which led to the research reported here, and to the reali- We show that Minkowski’s metric cannot be assumed, a zation that the value of Hill’s group parameter can be priori, to be the metric of physical gravitation-free space- determined by experiments proposed by Wulfman [7,8]. time. It is just one member of a one-parameter family of A more recent list of references to research on the phy- allowed metrics. As a consequence, current observations sical role of conformal transformations is contained in an allow Hubble’s relations to have consequential new inter- article on the quantum mechanics of accelerated relati- pretations. Among these are interpretations that eliminate vistic particles written by Calixto, et al. [9]. the current need for dark matter and dark energy. Section 2 of the present paper describes the finite In 1909, Bateman [1] and Cunningham [2] proved that transformations that result from the infinitesimal trans- Maxwell’s equations are invariant under an inversion, formations investigated by Hill. In determining these and the transformations of a fifteen-parameter Lie group. transformations and their effects on electromagnetic Thirty-six years later, just as WW II ended, E. L. Hill waves and Minkowski’s metric, we make use of proper- showed that the infinitesimal transformations of a one- ties of Lie transformation groups that can be found in

Open Access JMP 1512 C. E. WULFMAN recent texts by Cantwell [10], by Hydon [11], by Stepha- The italic superscripts denote locally Cartesian com- ni [12], and by Wulfman [13]. ponents. Here, and in some of the following paragraphs, In Section 3, it is shown that the transformations con- italics are also used to call attention to variables such as r vert the usual plane-wave and spherical-wave solutions and x4 that may take on initial values, final values, and of Maxwell’s equations into one-parameter families of all values in between, as the group parameter varies from solutions with local wave-numbers that evolve as the zero to its final value, except we will use “v” to refer to waves propagate. velocity and “” to refer to frequency. The Lie generator The action of the transformations on Minkowski’s me- of the group can be expressed as tric is determined in Section 4. They convert Minkows- 2 C2 rx2 444 xxrr. (7) ki’s metric into a one parameter family of metrics which 4   define a one parameter family of space-times. It is shown The transformations can be carried out by the oper- that for each value of the group parameter, the trans- ator expγC  . It produces the projective transforma- formed waves propagate in a transformed space-time, 4 tions whose metric is fixed by the value of the group parame- 44 4 ter. The transformed Minkowski metrics are not Poincare  x  rx'r   expγC4  xr  (8) invariant. In Section 5, to develop physical consequences of the  x44rx1 γ r (9) mathematical connection between the metric of the space-times and the behavior of electromagnetic radia- and tion in them, local measurements are expressed in terms 44 4  x  rx'r  expγС4  xr  (10) of coordinates of points in Minkowski space, and in terms of the transformed coordinates produced by the  x44rx1 γ r. (11) action of the group. Locally observed frequency shifts in 4 the one-parameter family of space-times are thereby re- Acting on rx, , expγ C4  produces the transfor- lated to “Doppler shifts” in Minkowski space. The de- mations velopment of velocity-independent frequency shifts in rrr D, (12) the transformed space-times leads to fundamental revisions of current interpretations of Hubble’s relations. xx'x4444  γ xr 2  D (13) Then, in Section 6, it is shown that the transformed space-times have a surprising property: in them, electro- with 2 magnetic waves received from bodies in galactic Kepler 4 2 orbits can undergo frequency shifts which are indistin- D1  x  r (14) guishable from the shifts currently attributed to the hy- 11x44rx r (15) pothesized presence of dark matter and dark energy in    Minkowski space. The inverse transformations are The final section of the present paper contains a brief 44 4 discussion dealing with the relation between the metrics  x'r'xr  exp  C4  xr  (16) considered here, and those applicable in the presence of 44 gravitational fields.  x'r' 1  x'r' (17) 2. The Isotropic Transformations of the and 44 4 Special Conformal Group  x'r'xr  exp C4  xr  (18) Hill’s one-parameter group of isotropic special confor- x'r'441  x'r'. (19) mal transformations inter-converts the vectors X, X’, and    r, r’ with and X x1234,,,xxx , x 4 ct (1,2) rrr'  D, (20)

2 1234 4 44 4 42 X' x' , x', x' , x',' x ct , (3,4) xxxx  -γ -r D' , (21)   and 4 2 2 r  x123,,xx (5) D1'xrγγ  (22) 44 r  x'123 , x', x'  (6) 1 γ x rx1 γ r. (23)

Open Access JMP C. E. WULFMAN 1513

Changing γ to −γ and also interchanging rr,  and the transformations generated by it leave this function 44 1 x , x produces the inverse transformation. invariant. The transformations generated by C4 also The transformations of the Bateman-Cunningham leave invariant the velocity-dependent function group are conformal in the sense that they do not alter 12 I1xrxr44   v1  v . (39) angles between spatial vectors. The transformations gen- 2    rr erated by C4 also do not alter the origin of coordinates. This invariant has implications which will be used in They interconvert Section 7. X rrrsin θ cos   , sin θ sin   , cos θ , x4 (24) Because C4 does not commute with any of the generators  xk , the transformations generated by it 1 and and C4 are not translation invariant. A translation that 4 moves the origins of the vectors X from O to X'xrrrsin θ cos   , sin θcos  , cos θ , . 1234 (25) A  a ,a ,a ,a  (40) If X has its origin at the vertex of a light cone where a converts X to X + A. It moves the origin of the vectors Xʹ source is located, then Xʹ will originate at the same from O to source. If the time coordinates have non-negative values, XXA''A   , (41) we will say the coordinates are those of points on “source  4 4 which is not the same as XA . cones” on which x = r and x’ = r’ A continuously emitting source produces a continuum of light cones with 3. Electromagnetic Waves with Evolving a common time axis. The vertex of any one of these may be considered that of a source cone. Wave Numbers The first extension of C4 is In this section, we first deal with plane waves, the waves that are potentially observable at great cartesian distances CC21vv1 r 2 , (26) 44rr from sources. An electromagnetic plane wave with wave 3 4 number k propagating in the direction of x may be vdd rx. (27) r defined by the equation It generates transformations of radial velocities in Ψ EHcos k xx34  , (42) which:  0  with vvrr'  AvBABv r  r. (28) 33 The inverse of this is EH xx0, 0, (43,44) EH0, E  H  const. (45-47) vrr v Av r - B A - Bvr . . (29) Both curl (E) and curl (H) vanish, and the function In these relations 34 cos k  xx  0 satisfies the wave equation, 2   A1rr   x44  x  γγ x  4  r2 (30-32) which reduces to the requirement that 3434 2  x xxx  1 γγxr4 2 D,2 (33)   annihilate (48)

44 4 34 B2rx   x  r γ r 1 γ x (34-36) cos k xx0 2γrx1 γ 4 D.2 (37) Spherical waves with the same wave number are obtained if the cosine function is replaced by D is the function defined in (15). Because of the sin k rx44  rx . 2 1   1v r  factor in C4 , the effects of these transforma- The inverse of (9) produces the relation tions weaken as vr approaches the speed of light, 1, 34 3 4 3 4 1 xx xx 1 γ xx  (49) which they cannot alter. Because both C4 and C4 are 4 quadratic functions of γr and γx , they generate Consequently, in the X’ system, the wave defined by transformations that are a weak function of both the co- (42) becomes defined by ordinates and the group parameter near sources, terres- xx34 trial as well as celestial. 34  Ψ'x, x  EH '' cos k 0 As C4 annihilates, 1γ x34x   2 I x4 r2 r (38) 1   (50)

Open Access JMP 1514 C. E. WULFMAN with The γ metric is invariant under the transformations of the rotation group SO( ). If γ is not zero, H'' = E , E' E. (51,52) 3 γ and dsγ 2 are not invariant under the transla- One thus need only require that Ψ' continues to sat- tions and Lorentz transformations of the Poincare group. isfy the wave equation. Equation (48) ensures that it does Thus the general γ metric and line element express so everywhere that the denominator does not vanish. the properties of a space-time in which each source of an Furthermore, because curlE' and curlH'  vanish, electromagnetic field reduces Poincare invariance to a no external potentials or currents are required to produce rotational invariance about itself. the EM waves defined by (42) and those defined by (50). The wave that carries the fields can be considered to 5. Interpretation of Observed Spectral Shifts have wave number 1 The C4 transformations relating primed and unprimed kk1γ xx34 . (53) variables define mappings that relate two sets of coordinates, rt,,v and rt, , v , of the same point. 3  r   r  At a stationary point with coordinate x0 the waves Motions of physical points in Minkowski space will be oscillate with a frequency described by changes in the value of the rt,,v 'r  coor- ω kc=kc 1 γ xx34. (54) dinates of the points, and motions in G γ space-times 0  will be described by changes in the value of the

The corresponding spherical waves are defined by rt, , vr  coordinates of the same physical points. A point П fixed on an electro-magnetic wave can be 4 k rx 4  4 4 1γrx assigned the pair of coordinates rx ,  and rx,  . Ψ' rx, 4  sin (55) 4 4 Let П1 and П2 be two such points separated by one rx 1 γrx   wavelength and centered at П0 on a locally monochro- matic EM wave propagating in otherwise field-free They carry electromagnetic fields in which E’ and H’ space-time. Let an observing apparatus at a point P mo- are everywhere orthogonal to the wave-front and to each mentarily on the wave’s source cone have coordinates other. 4 4 rxvob, ob , r ob , and rxv'ob, ob , rob . Let v and v’ be frequen- 4. The One-Parameter Family of Metrics cies, and λ and λ wavelengths measured at P in the two systems. Then measuring frequencies in the two sets 4 Differentiation of the finite C4 transformation (29) of coordinates determines the times txcr  c, produces the relations and tx 4 crc, that it takes one wavelength to dAdBdrrx4 , (56) pass the point P. The observations thereby relate time intervals at P in the two systems to frequency and wave- dBdAdx4rx4. (57) length measurements in the two systems via the equa- These, together with (14-17) imply that the Min- tions kowski line element vv λλ x 44  x. (63,64)

2 222224 2 We shall assume that the group parameter γ is small ds dxrr d sin θ dθ d, (58) enough to ensure that approximating the last ratio by 44 becomes ddx x leads to no detectable error. In this approximation 2 4 2 22 2 2 2  44 ds γ G γ ddxrr sinθ dθ d, λλ ddx x (65) (59) x44 xxrrx   4  d/d. 4 (66) with 4 Here ddrx vr , represents the radial velocity of 2 G γ 1 γrx  441 γ rx   (60) the observing apparatus at r, t, in Minkowski space. Equa- tions (64-66), and (24-30) together imply that 2 λλ  abv , (67) 1 γrx 41 γ rx 4 (61) r with

The metric tensor, γ , is diagonal with elements 2 a1 γγx4 r2 (68) g jj γ G γ ,j  1,2,3;g44 γ =G γ . (62) b2 γrx1 γ 4 . (69) Minkowski’s metric tensor is 0 .  

Open Access JMP C. E. WULFMAN 1515

To simplify notation, define 2γγrr 11 v    r (89) 44 γγx ct,   γx , (70-72) 12γγrr11vr  γr,  γr. (73,74) When ' is <0.1, (89) implies that in the Xʹ system On source cones,    and    . the position and velocity dependence is well approxi- When expressed in terms of the velocity and position mated by coordinates of an observer’s apparatus in G γ space- λλ ref 1Z  2γr 1v r  . (90) times, (67) becomes This is to be compared with the result of setting λλabavb  a–bv. (75) r' r'  r  R , 2γ equal to Hc0 , and vVr  c in (83) which yields These equations state that λλ is a function of the 4 time x as well as r . However on a source cone λλ ref 1Zapp =Hc0 r  vr . (91-92) x4  r, so for observations of any particular source The functional form of this connection between radial b bb 2'' 1   , (76) velocities and frequency shifts is quite different than that of the connection in (90). In the next Section we investi- a1bb  . (77) gate a relation of (90) and (91-92) to observations. We suppose that γ is so small that reference wavelengths, λref produced and observed by local sources, 6. Observations of Distant Keplerian have the same observable value in Minkowski space as in Motions the G γ space-times. Equations (72-77) then relate observables in a G γ space-time. We now consider In 1914, V. Slipher discovered the existence of internal two cases: rotations in nebulae [14]. When F. Zwicky [15] pointed out in 1933 that motions in the Coma cluster of nebulae 1) If γ  0 , and v0r  , then (72-75) predict red shifts that depend solely on  . These shifts are given failed to obey the virial theorem, he suggested that the by failure might be due to unobserved “dunkel Materie”, or to a failure of the known laws of physics. By 1980, Vera. Zλλ 1 λλ1, (78) 000vvref C. Rubin and her coworkers W. Kent Ford, Jr. and N. 21  121    , (79) Thonnard had established that the motions of stars associated with many spiral galaxies demonstrate the 21  O   4 (80) same effect [16].  The failure is now known to be very general, and it is 2O 2 . (81) commonly attributed to the presence of unobserved mass  and energy, and less commonly, to a modification of These relations may be compared with the Hubble Newtonian dynamics. The following paragraphs show relation that the apparent failure to obey the virial theorem can Z λλ 1HRcVc (82,83) arise from interpreting observations in a G γ space- obs ref 0 time as observations in Minkowski space. Here Vc v represents a “peculiar velocity”. If it is We follow Peebles [17] and exploit the fact that for zero, setting r' =R and γ equal to H2c0 in (83) circular orbits the mean values in the virial theorem yields become single values. For a body of mass m and orbital

Z λλobs ref 1HRc20 γr. (84-86) speed v orbiting a center with effective mass M  m , the speed is related to r, the radius of the orbit by the This puts (83) in 1:1 correspondence with the well-known equation Hubble relation.  12 2) If γ  0, and vr' is nonzero, (76-77) define vMG r . (93) frequency shifts that are a function of the radial velocity The failure of nebulae to obey (93) is known to appear vr and the position r , of the observer relative to the source. These shifts are defined by the equations when observations of luminosities and 21 cm micro- wave intensities establish that the luminous mass of a Z1v0  r' Z  (87) nebula lies within a sphere of radius rr0  . An espe- 1vZ r 0 cially clear example of the phenomenon occurs when the rotational velocities are observed “edge-on”. In most 211v    r (88) cases, the Doppler shifts then observed are those of ve- 12 1 1v r locities indistinguishable from the tangential velocities of

Open Access JMP 1516 C. E. WULFMAN bodies moving on circular orbits about the center of mass The authors analyze orbital motions observed in five of the nebulae. These tangential velocities of bodies in galaxies whose distance from earth ranges from 3.2 to Kepler orbits then become radial velocities along the line 9.2 mpc. The observed trajectories of the bodies have of sight of the observer. radii ranging from a few, to 36 kpc, and deduced orbital When the velocities of such motions of massive bodies, velocities, vVc , less than 10−3. The orbital radii are and hydrogen, orbiting spiral galaxies have been deduced such a small fraction of the distance DS, that the distance from observed Doppler shifts, the masses are found to be from the observers to the centers of the orbits, DG, is at moving on arcs with radii, r, less than the value expected most 1% greater than DS. As a consequence, the first from the mass of the nebulae deduced from luminosities. factor, 2ρʹ, in (94), gives Zʹ a linear dependence upon the The velocities also undergo little change as r increases. distance DG from earth to the galaxy, a dependence like The observed Doppler shifts are thus those that would be that of the Hubble relation (87) if one supposes that 2cγ obtained if there is sufficient dunkel Materie within the and H0 have the same order of magnitude. region to keep MG/r nearly constant as r varies over a In the cases being considered, the peculiar velocities wide range. have the same magnitude as the velocities of the center Equations (90) and (91) suggest that when observa- of mass of the nebulae relative to the orbiting source. −4 −3 tions of the motion of bodies in galactic orbit are ex- Consequently, for v values of the order of 10 to 10 , 4 the factors 1v  in (92) differ from unity by less pressed in rx,,v r' coordinate systems a very different  r  physical interpretation of the observed motions is impli- than 0.1%. Thus, for the reason noted at the end of Sec- cated. This may be shown by examining what would be tion 5, the relation of Zʹ to these peculiar velocities is observed in a G() space-time if a distant orbiting body entirely different than that expected from Hubble’s rela- of mass << than M is moving on a circular Kepler orbit tions. centered on the position of the effective mass M, which The resulting effect on values of Zʹ is displayed in is at rest with respect to the observer. Table 1. It contains values of Zʹ that would result from To investigate this question, we consider the orbiting measuring spectra of radiation emitted from bodies with bodies to be origins of source cones, on which the ob- velocities vʹ orbiting centers with masses equal to those served orbital tangential velocities become radial veloci- of NGC 2403 and NGC 2841, the galaxies with the ties of the observer relative to the source. If the observa- smallest, and the largest, mass of the galaxies considered 4 by van Albada and Sancisi. The radii r’ of the orbits, and tions are being made in an rx,,v , system, and the r the orbital velocities, vʹ, of the orbiting bodies, are observer is at a distance r  DS from the source, then labeled rNewt, and vKep in the table. In it, the hundred-fold the spectral shifts Zʹ arising from vr will satisfy the previously noted relation, change in the radii rNewt changes the local orbital velocities vKep by a factor of 10. If the red and blue shifts 11 v Z2  r , (94) in the G() space-time are interpreted as arising from 12 11v r' peculiar velocities in Minkowski space, they produce the values denoted V“Mink”. For NGC 2403, over the whole  with   γr' , and r’ equal to the distance, DS, from the range of rNewt, the value of Zʹ and V“Mink” vary by 0.02%. body of mass m to the observer in G γ space-time. For NGC 2841 they change by 0.23%. These results are a The velocity v’r’ is a radial velocity of the observer with direct consequence of the weak dependence of Zʹ upon respect to the body. Now let rr,  represent radial velocity that was noted in Section 5. When Zʹ becomes as coordinates of the central mass in the source cone system, large as 0.2, increasing the velocity from 0% to 10% of and let v, v represent the value of its tangential velocity the velocity of light makes a 10% change in Zʹ. in the source cone system. Then, vv r and we may The results in Table 1, taken together with the analysis replace (94) with the equation in the previous Section, make it evident that if earth- 12 based observations are actually being made in a G(γ) vMGr r ; (95) space-time with 2cγ having a value roughly equal to H0, in which it is not necessary to modify Newtonian dynamics, or introduce dark matter, to explain the observations: they  1 r expγC4 r, (96) are a consequence of an unexpected weak dependence of Zʹ on vKep. This conclusion applies to observed fre-  1 quency shifts in the radiation received from bodies orbit- vexpr  γCv.4 r (97) ing a large category of distant galaxies. It is revealing to assign values to MG that are those of The vKep → 0 entries in Table 1 illustrate another re- two of the galaxies investigated in the paper “Dark Mat- vealing property of such G γ metrics: observations in ter in Spiral Galaxies” by van Albada and Sancisi [18]. their space-times do not require the presence of dark en-

Open Access JMP C. E. WULFMAN 1517

Table 1. Velocities of bodies orbiting the center of mass of in the resulting value of the group parameter can be ex- spherically distributed masses of galactic magnitude. pected to be less than the uncertainty in the value of Hubble’s constant [7,8]. NGC DG MG rNewt vKep vMink Zʹ Object (Mpc) (1011 km) (1017 km) (10−4 c) (10−4 c) (10−4 c) If the value turns out to be indistinguishable from zero, NGC 2403 3.2 1.176 1.543 8.730 0.741 h 0.7412 h the known range of validity of Minkowski’s metric will be greatly extended, and much existing astrophysics and 3.086 6.173 0.741 h 0.7411 h cosmology will receive further validation. If the experi- 6.172 4.365 0.741 h 0.7409 h ments confirm a non-zero value for γ, they will establish 154.3 0.873 0.741 h 0.7407 h that sources of electromagnetic radiation, in fact, break Poincare and Lorentz invariance in a consequential man- Infinite 0.000 0.741 h 0.7406 h ner that is both subtle and fundamental. NGC 2841 9.0 9.555 1.543 24.89 2.088 h 2.0881 h Such results would have another consequence. Tomil- 3.086 17.60 2.086 h 2.0855 h chik [5] and Wulfman [7,8] have both argued that the magnitudes of the “Pioneer Anomalies” roughly corre- 154.3 2.489 2.083 h 2.0834 h spond to the value of the Hubble constant because the Infinite 0.000 2.083 h 2.0828 h anomalies are primarily due to the assumption that the

The first set of entries apply to a system whose DG and MG values are those space-time is Minkowskian. However, Turyshev, et al. of NGC2403. The second set apply to a system with the DG and MG values [19] argue that the anomalies can be explained by a re- of NGC2841. vKep is the computed classical velocity of a star in a circular orbit of radius rNewt, and center at the center of mass of the indicated galactic coil effect which had not been properly accounted for. system. Zʹ is the spectral shift that would be observed at distance DG from The proposed experiments would determine the extent to the star, measured in G(γ) space-time with 2γ = h H0/c. If the values of Z’ are interpreted as arising from peculiar velocities in Minkowski space, they which each explanation is correct. satisfy the relation V = cZ when V << c. These velocities are denoted vMink. More importantly, the proposed experiments would −25 −1 The assumed value of H0/c is 7.5 × 10 km , and 0.7 < h < 1. effectively establish the metric governing the transmis-

sion of electromagnetic waves over distances approach- ergy, or any other type of mass, to explain the spectra of ing 100AU. radiation received from objects orbiting at very small ve- If it turns out that the value of 2cγ and Hubble’s con- locities in regions arbitrarily distant from centers of stant are of the same order of magnitude, it will be neces- gravitational attraction. sary to revise currently standard interpretations of the 7. Conclusions spectra of light that has propagated over astronomical distances. The analyses set forth in Sections 5 and 6 es- The mere fact that Maxwell’s equations allow a γ tablish that the revisions could make it possible to reduce metric implies that Minkowski’s metric, (0), cannot be the number of hypotheses currently required to relate assumed to be the metric of physical gravitation-free astrophysical observations. space-time. Finally, it should be noted that the relations in (29-37) The analyses in the previous pages establish that ob- imply that two points moving with the same velocity in served shifts in the wavelengths of stellar radiations may the Minkowski coordinate system X have relative accel- well imply a relation of frequency shifts to recessionary erations in the Xʹ system if they are at different distances velocities and peculiar velocities fundamentally different from the source. This property is reflected in the exis- j from those codified in Hubble’s relations as currently tence of non-zero forces with components ∂gk,k/∂x . Hill’s understood. The analyses also establish that the need to approximation does not make it evident that these accel- introduce dark matter and dark energy, or a modified erations are not constant. The penetrating paper “Con- Newtonian physics, could be eliminated if observed fre- formal Invariance in Physics” by Fulton, Rohrlich, and quency shifts are related to velocities in a space-time in Witten investigates accelerations produced by conformal which the γ metric is one with 2cγ approximately transformations [20]. It deals with the mathematics and equal to H0. the physical applications of both conformal function In the past, it has not seemed possible to determine groups and conformal Lie groups. In its final section, the whether Minkowki’s metric is valid over distances rele- authors state their conclusion, “··· conformal transforma- vant to Hubble’s relation: it is not at all possible to tions are a special way of describing certain phenomena transport clocks and meter sticks to astronomical dis- which in general relativity are accounted for by a re- tances and use them to test Einstein’s arguments that lead stricted class of coordinate transformations”. Thus the to Minkowski’s metric. However, the value of the group relation between the (γ) metric and the metrics of gen- parameter γ could be determined by measuring succes- eral relativity can be expected to be different than the sive return times, as well as frequency shifts within radar relation between Minkowski’s metric and the metrics of wave-pulses sent to receding spacecraft. The uncertainty general relativity.

Open Access JMP 1518 C. E. WULFMAN

8. Acknowledgements [10] B. J. Cantwell, “Introduction to Symmetry Analysis,” Cambridge University Press, Cambridge, 2002. It is a pleasure to acknowledge many clarifying conver- [11] P. Hydon, “Symmetry Methods for Differential Equation,” sations with Andrew S. Wulfman and his assistance with Cambridge University Press, Cambridge, 2000. the bibliographic research and preparation of the manu- http://dx.doi.org/10.1017/CBO9780511623967 script. [12] F. H. Stephani, “Differential Equations: Their Solution Using Symmetries,” Cambridge University Press, Cam- REFERENCES bridge, 1989. [13] C. E. Wulfman, “Dynamical Symmetry,” loc cit. [1] H. Bateman, Proceedings London Mathematical Society, [14] V. M. Slipher, Lowell Observatory Bulletin, Vol. 2, 1914, Vol. 8, 1909, pp. 223-264. p. 66. [2] E. Cunningham, Proceedings London Mathematical So- [15] F. Zwicky, Helvetica Physica Acta, Vol. 6, 1933, pp. 110- ciety, Vol. 8, 1909, pp. 77-98. 125. [3] E. L. Hill, Physical Review, Vol. 68, 1945, pp. 232-233. [16] V. C. Rubin, W. K. Ford Jr. and N. Thonnard, Astrophy- http://dx.doi.org/10.1103/PhysRev.68.232.2 sical Journal, Vol. 238, 1980, pp. 471-487. [4] Hoyle, F., et al., “A Different Approach to Cosmology,” http://dx.doi.org/10.1086/158003 Cambridge University Press, Cambridge, 2000. [17] P. J. E. Peebles, “Principles of Physical Cosmology,” Prin- [5] L. M. Tomilchik, AIP Conference Proceedings, Vol. 1205, ceton University Press, Princeton, 1993. 2010, pp. 177-184. http://dx.doi.org/10.1063/1.3382326 [18] T. S. van Albada and R. Sancisi, Philosophical Transac- [6] H. A. Kastrup, Annals of Physics, Vol. 17, 2008, pp. 631- tions of the Royal Society A, Vol. 320, 1986, pp. 447-464. 690. http://dx.doi.org/10.1002/andp.200810324 http://dx.doi.org/10.1098/rsta.1986.0128 [7] C. E. Wulfman, “Dynamical Symmetry,” World Scienti- [19] S. G. Turyshev, et al., Physical Review Letters, Vol. 108, fic Publishing Co., Singapore, 2011, pp. 426-429. 2012, pp. 241101-241104. [8] C. E. Wulfman, AIP Conference Proceedings, Vol. 1323, http://dx.doi.org/10.1103/PhysRevLett.108.241101 2010, pp 323-329. http://dx.doi.org/10.1063/1.3537862 [20] T. Fulton, F. Rohrlich and L. Witten, Reviews of Modern [9] M. Calixto, et al., Journal of Physics A: Mathematical Physics, Vol. 34, 1962, pp. 442-457. and Theoretical, Vol. 45, 2012, pp. 244010-244026. http://dx.doi.org/10.1103/RevModPhys.34.442 http://dx.doi.org/10.1088/1751-8113/45/24/244010

Open Access JMP