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 On the Metric of Space-Time 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 de- pends 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 electro- magnetic 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 fre- quency 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 expγC . 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 expγC4 xr (8) invariant. In Section 5, to develop physical consequences of the x44rx1 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 expγС4 xr (10) of coordinates of points in Minkowski space, and in terms of the transformed coordinates produced by the x44rx1 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, , expγ 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 revi- sions 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) xxxx -γ -r D' , (21) and 4 2 2 r x123,,xx (5) D1'xrγγ (22) 44 r x'123 , x', x' (6) 1 γ x rx1 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 I1xrxr44 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'xrrrsin θ cos , sin θcos , 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 CC21vv1 r 2 , (26) 44rr from sources.