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«International Journal of Modern Physics D c World Scientific Publishing Company arXiv:gr-qc/0501001 v2 4 Jan 2005 Comments on the paper by S. Samuel ...»

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January 5, 2005 8:32 WSPC/INSTRUCTION FILE samuel-comm

International Journal of Modern Physics D

c World Scientific Publishing Company

arXiv:gr-qc/0501001 v2 4 Jan 2005

Comments on the paper by S. Samuel ”On the speed of gravity and the

Jupiter/Quasar measurement”

Sergei M. Kopeikin

Department of Physics and Astronomy, University of Missouri-Columbia,

Columbia, Missouri 65211, USA

kopeikins@missouri.edu

Recent review article by S. Samuel ”On the speed of gravity and the Jupiter/Quasar measurement” published in the International Journal of Modern Physics D, 13 (2004) 1753, provides the reader with a misleading ”theory” of the relativistic time delay in general theory of relativity. Furthermore, it misquotes original publications by Kopeikin and Fomalont & Kopeikin related to the measurement of the speed of gravity by VLBI.

We summarize the general relativistic principles of the Lorentz-invariant theory of prop- agation of light in time-dependent gravitational field, derive Lorentz-invariant expression for the relativistic time delay, and finally explain why Samuel’s ”theory” is conceptually incorrect and confuses the speed of gravity with the speed of light.

Keywords: general relativity; speed of gravity; speed of light; relativistic time delay

1. Introduction Exact mathematical solution of the light geodesic equation in time-dependent grav- itational field of arbitrary moving bodies has been constructed in a series of our publications 1,2,3. This solution predicts that a light particle (radio wave) is de- flected by the gravitational field of the moving body from its retarded, with respect to observer, position. The retarded position of the light-

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iment mispresented in 7. In the present paper we outline the basic equations of the complete Lorentz-invariant theory of the time delay and show mistakes in Samuel’s linearized ”theory” 7 originating from his first publication 8.

2. Retardation of Gravity and the Lienard-Wiechert Potentials We denote the barycentric coordinates of the solar system as xα = (x0, xi ), where x0 = ct, xi = x, and c is the fundamental speed limit. The metric tensor gµν = ηµν + hµν, where ηµν = diag(−1, 1, 1, 1) is the Minkowski metric, and hαβ describes gravitational field of the solar system in the linearized post-Minkowskian approximationa. We impose the harmonic gauge conditions ∂ν hµν − 1/2∂ µ h = 0, 9

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a Greek indices run from 0 to 3. Roman indices run from 1 to 3. The Greek indices are rised and lowered with the Minkowski metric. Bold letters denote spatial vectors. Repeated indices mean the Einstein summation rule. Euclidean dot and cross products of two vectors are denoted as a · b and a × b respectively.

b Because Eq. (3) is the null cone of the gravitational field of moving Jupiter we used 6,11 a symbol

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Comments on the paper by S. Samuel ”On the speed of gravity and the Jupiter/Quasar measurement” 3 This fundamental conclusion of general relativity can be tested experimentally by observing gravitational interaction of light from a background source (quasar, star) with the time-dependent field of moving Jupiter and/or other solar system bodies 4.

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can not be confused with the null characteristic of the electromagnetic field propagating along the space-time direction defined by the null vector k α because this vector is not parallel to the null vector rα = xα − xα (s) (see Fig. 1). Unfortunately, the four-dimensional Minkowskian diagram of the experiment shown in Fig. 1, was not understood by Asada 12 and Samuel 8,7 who confused the null characteristic of the quasar radio wave and that of the gravitational field of Jupiter.

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Comments on the paper by S. Samuel ”On the speed of gravity and the Jupiter/Quasar measurement” 5 where δθ = −n · δx/r with n = l × (k × l) as the impact vector of the light ray with respect to the retarded (due to the finite speed of gravity) position of Jupiter xJ (s).

Undetectable terms of order vJ /c have been neglected. The quantity δt = t2 − t1 is the measurable VLBI time delay and δx = x2 − x1 ≡ B is a baseline between two VLBI stations. Since VLBI stations measure the same wave front, δϕ = 0. Thus, for sufficiently small angle θ Eq. (8) yileds

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d These quantities appear in Eq. (1.3) in Samuel’s paper 7 along with the ”distance of the closest approach” ξ spontaneously without rigorous mathematical description so that their meaning is fuzzy.

January 5, 2005 8:32 WSPC/INSTRUCTION FILE samuel-comm

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where R = R|, and R = x − xJ (t) is the vector lying on the hypersurface of constant time t and connecting the point of observation, x, and the present position of Jupiter, xJ (t).

The post-Newtonian expansion given in Eq. (12) originates from the retarded nature of the Lienard-Wiechert gravitational potentials and, thus, describes the effect of the retardation of gravity 4. The presence of the retardation of gravity effect through the retarded position of Jupiter in the Lorentz-invariant Eq. (5) and its small-angle approximation Eq. (10) reflects the causal property of gravitational field which is a consequence of its finite speed. Causality and Lorentz-invariance of the gravitational field are tightly connected fundamental concepts and the measurement of the causal nature (retardation) of gravity in the Fomalont-Kopeikin experiment is equivalent to the proof that the gravitational field is Lorentz-invariant and vice versa.





Samuel also uses the post-Newtonian expansion of the retarded coordinate of Jupiter in Eq. (5.3) of his paper 7. He believes that the post-Newtonian expansion ”arises because the position of Jupiter changes as the quasar signals travel from the Jupiter region to Earth”. Jupiter does move as the quasar signal travels towards observer but the distance traveled by Jupiter is proportional not to the difference between the time t∗ of the closest approach of the quasar signal to Jupiter and the time of observation, that is t∗ − t, but to the difference between the retarded time s and the time of observation t, that is s − t, as clearly follows from Eq. (11).

Therefore, the fuzzy concept of ”the Jupiter region”, which is repeatedly used by Samuel without rigorous mathematical definition, is, in effect, the retarded position of Jupiter defined by the Lienard-Wiechert solution of the gravity field equations.

This consideration makes it evident that the origin of the post-Newtonian expansion in Eqs. (5.3)–(5.6) of Samuel’s paper 7 is caused by the speed of gravity which propagates from moving Jupiter towards observer as well as the quasar signal does.

This point was emphasized in our papers 4,16. Samuel overlooked the gravitational physics of the Jupiter-quasar experiment because of his approximate and, hence, insufficient solution of the problem of propagation of light rays in time-dependent gravitational fields. He was able to integrate the light-ray geodesics only for the case of the small-angle approximation (θ ≪ 1) when the time of the closest approach, t∗, of the quasar signal to Jupiter is comparable with the retarded time s along the null characteristic of Jupiter’s non-stationary gravitational field. It is for this reason that Samuel confused the time t∗ and the retarded time s and replaced the concept of the propagation of gravity from the retarded position of Jupiter by the concept of the propagation of light from ”the Jupiter region” e.

Substitution of the post-Newtonian expansion (12) to Eq. (10) yields

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e Similarmistake has been done also by Will in 15 who used insufficiently elaborated cg parametrization of the Einstein gravity field equations.

January 5, 2005 8:32 WSPC/INSTRUCTION FILE samuel-comm Comments on the paper by S. Samuel ”On the speed of gravity and the Jupiter/Quasar measurement” 7

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is the static Shapiro time delay caused by Jupiter’s gravitational field at the time of observation, where N = p × (k × p) is the impact vector of the light ray with respect to the present position of Jupiter xJ (t), and

–  –  –

is the post-Newtonian correction to the Shapiro time delay due to the the finite speed of gravity in the gravity null-cone equation (3). Eq. (15) is the same as Eq. (4) from 6. It describes the first post-Newtonian vJ /c correction to ∆S and can be detected because of the amplifying factor ∼ 1/Θ2. Samuel’s Eq. (5.6) is an approximate form of our Eq. (15). We notice that Samuel’s notation for the angle θ1 ≡ Θ in our notations. Physical origin of the post-Newtonian correction ∆R to the static Shapiro time delay ∆S is due to the Lorentz-invariant nature of the gravitational field caused by its finite speed of gravity as follows from Eq. (11).

In his first paper 8 Samuel incorrectly assumed that the experiment directly compared the radio position of the quasar with the optical position of Jupiter, and that the direction of Jupiter was determined by ”sunlight that has been reflected off of Jupiter” (see the second paragraph in section III of Samuel’s paper 8 describing figure 1 which is similar with figure 2 of Samuel’s paper 7 ). This assumption would correspond to direct measurement of the angle θ and hence no vJ /c terms would be observed since they are not evident in Eq. (10). This explains why Samuel has erroneously decided ”that the v/c effects are too small to have been measured in the recent experiment involving Jupiter and quasar J0842+1845” 8. The experiment, however, monitored the position of the quasar as a function of the atomic time by the arrival of the quasar’s photons at the telescope, while the Jupiter’s position entering the time delay Eq. (7) was determined separately by fitting VLBI data for the quasar to a precise JPL ephemeris, evaluated at the same atomic time as the arrival of a photon via standard transformations from ephemeris time to atomic time 6. The result of our fitting procedure was that Jupiter deflects light from its retarded position xJ (s) but not from its present position xJ (t). Thus, the difference ∆ − ∆S was measured and the vJ /c correction ∆R was determined within precision of 20% 6.

The measurement of the post-Newtonian correction (15) to the Shapiro time delay (14) is direct demonstration that gravity does propagate with the same speed as the speed of light. Samuel’s claim that ”Fomalont and Kopeikin’s announcement that the speed of gravity is the speed of light to within 20% has no content” is based on his inability to distinguish between gravitational and electromagnetic effects in the post-Newtonian expansion of the Lorentz-invariant time delay Eq. (7) predicting that any moving body deflects light by its gravitational field from the retarded position in accordance with the causal (Lorentz-invariant) nature of gravity.

January 5, 2005 8:32 WSPC/INSTRUCTION FILE samuel-comm

8 Sergei M. Kopeikin

6. Discussion and Further Particular Comments

6.1. On the Ideology of the Experiment The paper by Samuel 7 is an attempt to protect his misleading calculations 8 published in 8 which conceptual inconsistency was revealed in 6,11,16,17. Unfortunately, it does not provide either new mathematical details or more deep insight to the problem. By making use of a linearized Lorentz transform of a spherically-symmetric gravitational field Samuel succeeds in calculation of the first few terms of the differential Shapiro time delay caused by moving Jupiter that approximates the original result by Kopeikin 4 (see Eq. (7)) given in terms of the retarded time, s, connecting the point of observation, x ≡ x(t), and the retarded position of a light-ray deflecting body (Jupiter), xJ (s). Our derivation of Eq. (7) makes it clear that the electromagnetic signal from a quasar is observed at the time, t, and Jupiter deects it at the retarded time, s = t − r/c, where r = |x(t) − xJ (s)| is the radial coordinate of Jupiter with respect to observer directed along the null characteristic of the gravitational field defined by the null cone equation ηµν rµ rν = 0. Both the retarded coordinate of Jupiter, xJ (s), and the retarded time, s, originate from the Lienard-Wiechert solution of the linearized Einstein equations which is a hyperbolic (wave-type) D’Alembert equation. Lorentz-invariant theory of the propagation of light rays through the time-dependent field of the gravitational retarded potentials reveals that the light particle (photon) is deflected by moving Jupiter when it is located at the retarded position xJ (s) due to the finite speed of gravity. Experimental testing whether Jupiter deflects light from its orbital position taken at the retarded time s due to the finite speed of gravity or at the time of observation t, is a direct probe of the numerical value of the speed of gravity which must be equal to the speed of light according to Einstein. This is the key idea of the experiment which has been put forward in our publication 4 and practically tested in September 2002 6. Unfortunately, the einsteinian gravitational physics of the experiment is greatly misunderstood and conceptually misrepresented both in Samuel’s papers 8,7 and in 12,15,18 f.

6.2. What Was Observed and Tested in the Experiment Samuel prefers to re-express our original result for time delay (7) in terms of the ”observable angle” θobs ≡ θ in terms of ”the distance of the closest approach ξ”.

The vertex of this angle is at the point of observation, x(t), and it has two legs – one leg is directed in the sky towards the quasar and another one is directed towards retarded position of Jupiter, xJ (s). If both Jupiter and the quasar were observed simultaneously in radio or in optics, then, the legs composing the ”observable angle” f Critical discussion of the formally correct, but conceptually misleading point of view on the gravitational physics of the process of light scattering by the gravitational field of a moving body presented in 18, requires more elaborated mathematical technique than that presented in the present paper, and will be given somewhere else. Some details are available in 19.



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