propagation speed of the gravitational signal has no effect whatsoever on the time delay ... other possible effects of an altered speed cg at this order arise from a ...
The Astrophysical Journal, 590:683–690, 2003 June 20 # 2003. The American Astronomical Society. All rights reserved. Printed in U.S.A.
PROPAGATION SPEED OF GRAVITY AND THE RELATIVISTIC TIME DELAY Clifford M. Will Department of Physics, McDonnell Center for the Space Sciences, Washington University, St. Louis, MO 63130 Received 2003 January 9; accepted 2003 March 4
ABSTRACT We calculate the delay in the propagation of a light signal past a massive body that moves with speed v, under the assumption that the speed of propagation of the gravitational interaction cg differs from that of light. Using the post-Newtonian approximation, we consider an expansion in powers of v=c beyond the leading ‘‘ Shapiro ’’ time-delay effect, while working to first order only in Gm=c2 , and show that the altered propagation speed of the gravitational signal has no effect whatsoever on the time delay to first order in v=c beyond the leading term, although it will have an effect to second and higher order. We show that the only other possible effects of an altered speed cg at this order arise from a modification of the parameterized postNewtonian coefficient 1 of the metric from the value 0 predicted by general relativity. Current solar-system measurements already provide tight bounds on such a modification. We conclude that recent measurements of the propagation of radio signals past Jupiter are sensitive to 1 but are not directly sensitive to the speed of propagation of gravity. Subject headings: gravitation — relativity
where
1. INTRODUCTION
The time delay of light, also known as the Shapiro effect, is one of the observational cornerstones of general relativity. For a light ray passing a body of mass ma at location xa and received by an observer at xr , the excess delay is given by (modulo a constant) D¼
2Gma lnðrra k x xra Þ ; c3
1 K ¼ k k µ ðva µ kÞ c
ð4Þ
and where all quantities are to be evaluated at the same moment of time, for instance, the time of reception of the ray. For a light ray passing close to Jupiter, these corrections are small, of the order of picoseconds or less, but may be detectable with the latest VLBI techniques (Kopeikin & Fomalont 2002). Recently, however, Kopeikin has argued that, because of the motion of a body such as Jupiter during the passage of the light ray, the fact that the gravitational interaction is not instantaneous should have an effect on the time delay (Kopeikin 2001). To study this phenomenon, he allowed the speed of gravity to be cg, where cg 6¼ c; this would modify the retardation of the gravitational fields used to calculate the time delay. He argued that equation (3) would be modified by replacing c by cg in the prefactor 1 k x va =c in equation (3) and in the velocity-dependent term in equation (4) (Kopeikin 2002). Kopeikin & Fomalont (2002) then proposed a test of the propagation speed of gravity via VLBI measurements in the fall of 2002. This result goes against intuition. First, while the retardation of gravity surely has effects, one does not expect to find effects odd in powers of v=c until the leading effects of gravitational radiation reaction occur, far beyond the order of the v-dependent effects claimed by Kopeikin. Second, the time delay is controlled dominantly by the Newtonian gravitational potential U; if this is a retarded potential with speed cg, it is given by Z t jx x0 j=cg ; x0 3 0 d x ; Uðt; xÞ ¼ ð5Þ jx x0 j
ð1Þ
where k is a unit vector in the direction of the incoming light ray, xra ¼ xr xa , and rra ¼ jxra j. If the ray passes by the body by a minimum distance d that is small compared to rra, then 2Gma 2rra ln D : ð2Þ c3 d2 The one-way delay amounts to 70 ls for a ray that grazes the Sun and 10 ns for a ray that grazes Jupiter. In addition to tests of the time delay by the Sun (Reasenberg et al. 1979), the time delay has also been studied in a number of binary pulsar systems (see, for example, Stairs et al. 1998). Advances in precision timekeeping and in VLBI have motivated the study of the first relativistic corrections, of the order of v=c, to the basic Shapiro formula. These corrections would be relevant for signals passing near moving bodies, such as bodies in binary pulsar systems or planets such as Jupiter. In a series of papers, Kopeikin and colleagues have analyzed these correction effects in detail (see Kopeikin & Scha¨fer 1999 and references therein). They find that, to first order in va =c, where va is the velocity of body a, the corrected expression takes the form 2Gma k x va 1 D¼ 3 lnðrra K x xra Þ c c Gma v2a þO ; ð3Þ c3 c2
where is the density of matter. Expanding the retardation in the near zone (jx x0 j5 a characteristic wavelength of gravitational waves), where the time delay predominantly 683
684 occurs, we obtain Z Z ðt; x0 Þ 3 0 1 @ ðt; x0 Þ d 3 x0 d x U¼ jx x0 j cg @t Z 1 @2 þ 2 2 ðt; x0 Þjx x0 j d 3 x0 þ . . . : 2cg @t
WILL
ð6Þ
The second term would have been of the order of v=cg relative to the leading term and might have contributed to the time delay, but it vanishes because of the conservation of mass. The effect of cg first appears in the third term, which is of the order of ðv=cg Þ2 relative to the first. This, rather than first order in v=cg , is what intuition dictates would be the first level at which retardation-dependent effects occur. This argument was invoked implicitly in the paper of Asada (2002). Another argument against such v=cg -dependent effects comes from electrodynamics. It is a well-known fact that the electric field of a uniformly moving charge points not toward the retarded position of the charge but toward the present position (see, for example, x 11.10 and exercise 11.17, Jackson 1999). In gravity, because acceleration involves Gm=r v2 , working to first order in v=c is equivalent to assuming a uniformly moving body, and thus a similar conclusion holds: to the order of v=c, retardation should have no observable effects.1 These considerations have motivated us to check Kopeikin’s claim with a careful calculation of the time delay to the same order. We have assumed that, to the post-Newtonian (PN) order required for the calculation, the metric can be described by a matter gravitational potential U and a current gravitational potential Vi, both of which are assumed to be retarded potentials using a speed cg, as in equation (5). The terms in the metric are also multiplied by suitable parameterized post-Newtonian (PPN) parameters to reflect the possibility that, whatever theory one is considering with a different speed of gravity, it will necessarily not be general relativity, and thus may have different PPN parameter values (see Will 1993 for a review of the PPN framework). In such a theory, those PPN parameters may well depend on cg, but in the absence of a specific theory, we leave them general. We then calculate the time delay to the order of ðGma =c3 Þðva =cÞ, for a signal emitted from a distant source and received on Earth, passing through the gravitational field of the solar system. We carefully treat the propagation of the light ray both through the far zone, where the propagation time is long compared to the dynamical time of the system, and through the near zone, where the propagation time is short compared to the dynamical time and where positions and velocities can be expanded about values at some common time, such as the time of reception on Earth. Our result closely parallels equation (3): 2Gma 1 þ k x va 1 ð1 þ Þ D¼ 3 2 c c 2 Gma va lnðrra K x xra Þ þ O ; ð7Þ c3 c 2 with K given by equation (4). 1 A similar conclusion was reached by Carlip (2000) in the context of binary motion, in response to a proposed bound on the speed of gravity by Van Flandern (1998).
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The first important observation is that cg appears nowhere in this expression. Although it appears in various intermediate expressions, it cancels in the end, to the order of approximation shown. In addition to the overall factor of ð1 þ Þ=2, which is standard, the only change comes in the coefficient ð1 þ Þ in front of the velocity-dependent term in the prefactor. Here is a function of the PPN parameters 1 and , given by 1 ¼ : ð8Þ ð2 þ 2Þ In general relativity, ¼ 1, 1 ¼ 0, and we recover Kopeikin’s result, equation (3). Second, the velocity-dependent corrections in equation (7) have a simple physical interpretation. Since we are working to first order in v=c, only Jupiter’s rectilinear motion is relevant; its acceleration generates effects of higher order in Gm=c2 . The vector K is simply the spatial unit vector of the photon’s direction as seen in the rest frame of Jupiter; the correction terms in equation (4) are nothing but the effect of aberration. They depend only on c, which is the velocity relevant for Lorentz transformations and the propagation of light, not on cg. Indeed, from the viewpoint of Jupiter’s rest frame, cg is completely irrelevant, since the gravitational field is static (again, ignoring Jupiter’s acceleration). The velocity-dependent prefactor is a combination of a gravitomagnetic effect (the fact that the gravitational field is not a scalar quantity but contains both a vector and tensorial part) and a simple Doppler effect in transforming from Jupiter’s rest frame to the barycentric frame. We thus conclude that the v=c corrections to the Shapiro time delay are normal 1.5 PN corrections that occur when there are moving bodies but that they have nothing to do with the speed of propagation of gravity, insofar as it affects the retardation of gravitational interactions. Furthermore, as a potential test of alternative gravitational theories, measuring these v=c terms is not promising, because a variety of solar-system measurements already constrain 1 and to such a degree that jj < 4 103 under relatively weak assumptions or jj < 5 105 under assumptions that invoke so-called preferred-frame tests of the parameter 1 (see Will 2001 for the latest bounds on the PPN parameters). In fact, the VLBI measurements are sensitive mainly to the velocity dependence in the logarithmic term, not to the prefactor. Therefore, measurements of the propagation of radio waves past Jupiter do not directly constrain the propagation speed of the gravitational interaction. The remainder of this paper provides details. In x 2 we describe the assumptions and basic equations that go into our calculation. Section 3 carries out the integrations to find the time delay. In x 4 we give concluding remarks.
2. ASSUMPTIONS AND BASIC EQUATIONS
We assume that, whatever theory of gravity is in force, it is a metric theory; that is, it has a spacetime metric gl that governs the interactions and motions of all nongravitational fields and all ‘‘ test ’’ particles. In particular, we assume that light rays move on null geodesics of the spacetime metric. We assume that atomic clocks measure proper time as given by the invariant interval of the spacetime metric. By choosing the units determined by such physical measurements appropriately, we can make the speed of light as measured
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by any freely falling observer unity. Henceforth, we use units in which G ¼ c ¼ 1. We then assume that the theory of gravity has a PN limit that can be written in the following form: g00 ¼ 1 þ 2U þ O 2 ; 1 Vi þ O 5=2 ; g0i ¼ 2 1 þ þ 4 ð9Þ gij ¼ ij ð1 þ 2U Þ þ O 2 ; where U and Vi are retarded gravitational potentials given by Z ðt0 ; x0 Þ jx x0 j 4 0 0 t tþ U¼ d x ; jx x0 j cg Z ðt0 ; x0 Þvi ðt0 ; x0 Þ jx x0 j 4 0 0 Vi ¼ t t þ d x ; ð10Þ jx x0 j cg where and vi are the density and velocity of matter and where we have inserted cg in the retardation expression to reflect the possible difference between the speed of propagation of gravity and that of light (unity). The parameter is a bookkeeping parameter inserted to keep track of ‘‘ orders of smallness ’’ of various quantities, with v 1=2 , U , Vi 3=2 , and so on. To the orders of approximation shown, the metric in equation (9) is general enough to encompass a broad class of metric theories of gravity with a propagation speed of gravity different from unity, with the exception of ‘‘ massive graviton ’’ theories (Visser 1998; Babak & Grishchuk 2002), whose gravitational potentials also include Yukawa-like modifications. In the special case of ¼ 1, 1 ¼ 0, and cg ¼ 1, the metric corresponds precisely to that of linearized general relativity in the harmonic gauge, to the order considered. There are two free PPN coefficients and 1; the coefficient of U in g00 has been chosen to be unity as usual by our choice of the measured value of the gravitational constant. We ignore all preferred-frame terms that could appear in the PPN metric (see Will 1993 for discussion). Note that the terms ignored in each metric component are all OðÞ larger than the terms kept; no correction terms a ‘‘ half-order ’’ higher are present, such as Oð3=2 Þ terms in g00 or gij or Oð2 Þ terms in g0i. This has certainly been true for all those theories analyzed by the present author or by others to date, including several, such as the Rosen bimetric theory, with cg 6¼ 1 (Lee et al. 1976). Such correction terms generically arise from nonlinear terms in the field equations, which are minimally of the order of m=r higher than the leading term, or from Oðv2 Þ OðÞ corrections in the definition of the stress-energy tensor. However, the retardation effects in the potentials could in principle lead to such halforder terms in a slow-motion expansion. It is precisely these terms whose effects we wish to explore. Before proceeding, however, we emphasize that we have assumed a specific theoretical framework for the analysis. Relatively few theories with cg 6¼ c have been analyzed within the PPN framework (the Rosen theory and a few others). Furthermore, it is entirely possible that cg 6¼ c theories may be more fully or consistently realizable in the context of nonmetric theories, for which the PPN framework is not applicable. In the absence of a concrete example, we restrict attention to the PPN approach described above. In a metric theory of gravity, the gravitationally coupled Maxwell equations require that light rays propagate on null
685
geodesics of the metric in equation (9), given by the equations d 2 xi þ N l N il N i 0l ¼ 0 ; 2 dt 2 ds ¼ g00 þ 2g0i N i þ gij N i N j ¼ 0 ; dt
ð11Þ
where N l ¼ dxl =dt (N 0 ¼ 1) along the light ray. Calculation of the time delay follows the method of Will (1993): write the trajectory of the light ray in the form xi ðtÞ ¼ xie þ ki ðt te Þ þ xip ðtÞ ;
ð12Þ
where e denotes the event of emission of the signal, ki is a spatial Cartesian unit vector, and xip ðtÞ denotes the perturbation in the path of the ray, which is clearly of the order of . Then, substituting N i ¼ ki þ x_ ip and gl ¼ l þ hl , where l is the Minkowski metric, into the second of equations (11), we obtain k x x_ p ¼ hl kl k =2 þ Oð2 Þ, where kl ¼ ð1; ki Þ and the overdot denotes d=dt. However, from equation (12) we see that jx xe j ¼ t te þ k x xp þ O½jxp j2 =ðt te Þ; consequently, t te ¼ jx xe j þ Dðt; te Þ ; Z t 1 l Dðt; te Þ ¼ 2 k k hl ½t; xðtÞ dt þ O 2 jx xe j : ð13Þ te
Equation (13) is completely equivalent to equations (20) and (44) of Kopeikin & Scha¨fer (1999) and to equation (18) of Kopeikin (2002). To the required order, the function xðtÞ that is used to evaluate hl in the integral is the unperturbed path of the ray, given by xðtÞ ¼ xe þ kðt te Þ. We now specialize the metric to the physical situation of a set of bodies of mass ma and velocity va (the solar system). We work in a coordinate system P whose origin is at the barycenter of the system, so that a ma va ¼ 0. We parameterize the trajectory of each body by xa ðuÞ, where u is a parameter that is proportional to the coordinate time (with unit proportionality constant), such that xa ðu ¼ t0 Þ is the location of the body at coordinate time t ¼ t0 , xa ðu ¼ 0Þ is the location of the body at coordinate time t ¼ 0, and so on. The ordinary velocity is given by va ðuÞ ¼ dxa ðuÞ=du ¼ dxa ðuÞ=dt. (We introduce this parameterization merely to avoid confusion with other ‘‘ times ’’ to be discussed.) Treating each P body as effectively a ‘‘ point ’’ mass, with ðt0 ; x0 Þ ¼ a ma 3 ½x0 xa ðu ¼ t0 Þ, we find that the potentials U and Vi take the Lie´nard-Wiechert form: X ma Uðt; xÞ ¼ ; jx xa ðsa Þj va ðsa Þ x ½x xa ðsa Þ=cg a X ma via ðsa Þ Vi ðt; xÞ ¼ ; ð14Þ jx xa ðsa Þj va ðsa Þ x ½x xa ðsa Þ=cg a where sa is the ‘‘ retarded ’’ time given by the implicit equation sa ¼ t
jx xa ðsa Þj : cg
ð15Þ
It is important to understand precisely the meaning of the expressions xa ðsa Þ and va ðsa Þ. They do not mean that xa and va have become functions of sa; they are functions of only the parameter u. Rather, the expression xa ðsa Þ means xa ðu ¼ sa Þ
686
WILL
or ‘‘ xa ðuÞ evaluated at u ¼ sa where sa is given by evaluating equation (15) for a chosen field point ðt; xÞ.’’ This is important, for example, when we carry out Taylor expansions of xa ðsa Þ and va ðsa Þ. These issues are addressed in detail in the Appendix. Substituting equations (14) into equations (9) and thence into equation (13), we can write the time delay in the form X ma Dðtr ; te Þ ¼ ð1 þ Þ a
Z
tr
te
½1 ð2 þ Þk x va ðsa Þ dt ; jx xa ðsa Þj va ðsa Þ x ½x xa ðsa Þ=cg ð16Þ
where tr denotes the time of reception of the ray and where ¼ 1 =ð2 þ 2Þ.
ð2Þ
relevant terms in a in powers of 1= 1 . We see that all dependence on 1 cancels to the order considered, as it must. ð2Þ We first evaluate a . We reinitialize the parameter u for body a so that it is 0 at the time of closest approach and expand the variables of body a about that point, so that xa ðuÞ ¼ xa ð0Þ þ va ð0Þu þ OðuÞ. When evaluated at that value of u equal to retarded time sa, we get an analogous result: xa ðu ¼ sa Þ ¼ xa ð0Þ þ va ð0Þsa þ Oðsa Þ. In addition, va ðu ¼ sa Þ ¼ va ð0Þ þ Oð1=2 va Þ. Note that xa ð0Þ and va ð0Þ are evaluated at u ¼ 0. By defining the vector d a ðsa Þ a xa ðsa Þ, which points from the body at retarded time to the point of closest approach, we can write ra ð ; sa Þ ¼ k þ d a ðsa Þ. We note that, to the necessary order, the denominator in equation (19) can be written ra ð ; sa Þ va ðsa Þ x ra ð ; sa Þ=cg ¼ jDj þ Oðra Þ, where D ¼ ra ð ; sa Þ
3. CALCULATION OF THE TIME DELAY TO 1.5 PN ORDER
Equation (16) is a sum of contributions for each body in the system. Since we are working to linear order in m, we can evaluate the integral for each body separately, then multiply each by ma and sum over the bodies. We rewrite the unperturbed trajectory of the light ray in the more convenient form xðtÞ ¼ k þ na ;
Vol. 590
ð17Þ
where is the time t modulo a constant and na is a vector from the barycenter to the light ray at the moment of its closest approach to body a; the parameter is chosen so that ¼ 0 at this point. Because we are working to first order in ma, we can ignore all effects related to the deflection of this ray. We adopt the simplified notation
ra ð ; sa Þva ðsa Þ ; cg
¼ k þ d a ðsa Þ va ðsa Þð sa Þ; ¼ k þ d a ð0Þ va ð0Þ þ ½d a ðsa Þ d a ð0Þ þ va ð0Þsa þ Oðra Þ :
The term in square brackets in equation (20) vanishes to the necessary order. Note that cg has canceled out. Defining the vector zð Þ ¼ k þ d a ð0Þ, with k x d a ð0Þ ¼ 0 (k and d a are orthogonal at the moment of closest approach to body a), we can write D ¼ zð Þ va ð0Þ , and thus, ra ð ; sa Þ
va ðsa Þ x ra ð ; sa Þ ¼ jzð Þ va ð0Þ j þ Oðra Þ cg ¼ 2 ½1 2k x va ð0Þ þ 2 ½k va ð0Þ x d a ð0Þ
1=2 þ da ð0Þ2 þ Oðra Þ : ð21Þ
ra ð ; sa Þ ¼ xð Þ xa ðsa Þ ; ra ð ; sa Þ ¼ jxð Þ xa ðsa Þj ; ra ð ; sa Þ ; sa ¼ cg
ð20Þ
Inverting this expression and integrating with respect to yields the result ð18Þ
where we have dropped the irrelevant arbitrary constant that initializes time. The integral to be evaluated is Z r ½1 ð2 þ Þk x va ðsa Þ d : ð19Þ a ð r ; e Þ ¼ ð ; sa Þ 1=cg va ðsa Þ x ra ð ; sa Þ r a e We now divide the integral R into twoð2Þcontributions R ð1Þ ð2Þ ð1Þ a and a , where a ¼ e1 . . . d , a ¼ 1r . . . d , and the time parameter 1 is chosen to be large and negative (recall we chose ¼ 0 at closest approach) and to satisfy the following inequalities: jxa j5 j 1 j5 ð1= va Þjxa j. Since va 5 1, an appropriate value of 1 clearly ð1Þ exists. Then, in evaluating a , we can expand the integrand in powers of 1= , since j j4j xa j everywhere, and ð2Þ integrate by parts. In evaluating a , since the integration is always within the near zone, we can expand the variables of the bodies about their values at the coordinate time corresponding to closest approach in a slow-motion expansion. We work only to first order in va . Each integral depends on the arbitrarily chosen value of 1. We then match the integrals by expanding the relð1Þ evant terms in a in a slow-motion expansion and the
ð2Þ
a ¼ ½1 ð1 þ Þk x va ð0Þ lnf½zð Þ þ k x zð Þ½1 ^zð Þ x va ð0Þg r1 þOðÞ : ð22Þ At the reception point r, we define zð r Þ ¼ k r þ ~ra and note the fact
xa ð0Þ ¼ xr ð r Þ xa ð0Þ x ~ra Þ2 . At 1 we expand the integral in that da ð0Þ2 ¼ ~r2ra ðk x x powers of da =j 1 j5 1, using the expansions (to the required order) da ð0Þ2 da ð0Þ4 ; þ þ O da ð0Þ6 5 1 3 2 1 8 1 " # da ð0Þ2 da ð0Þ2 zð 1 Þ þ k x zð 1 Þ ¼ þ O da ð0Þ6 5 1 ; 1 2 2 1 4 1 " # d a ð0Þ da ð0Þ2 ^zð 1 Þ ¼ k : þk þ O da ð0Þ3 3 1 2 1 2 1 zð 1 Þ ¼ 1
ð23Þ Substituting these expansions into equation (22) and keep-
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SPEED OF GRAVITY AND TIME DELAY
ing terms through Oð 2 1 Þ, we obtain ~rra k x x ~ra ð2Þ a ¼ ½1 ð1 þ Þk x va ð0Þ ln 2 x ~ra va ð0Þ x k x va ð0Þ ~rra þ ½1 ð1 þ Þk x va ð0Þ ln j 1 j þ
term is OðÞ relative to the leading term. Because in the integral parameterizes the light path and because we are not Taylor-expanding the orbital variables, we must take into account that sa depends on through equation (25), so that d dsa d a ðsa Þ ¼ va ðsa Þ d "d # 1 d? ðsa Þ2 ¼ va ðsa Þ 1 þ cg 2cg 2 þ O ; da ðsa Þ3 3 :
da ð0Þ2 4 21
d a ð0Þ x va ð0Þ 1 da ð0Þ2 k x va ð0Þ þ 1 4 21 : þ O ; da ð0Þ3 3 1
ð24Þ
ð1Þ
Next, we evaluate a . Here, because is integrated over a range that could include many dynamical periods of the masses as the signal propagates from the distant star toward the near zone, we must leave all variables such as xa ðsa Þ unexpanded. Instead, we expand all expressions in advance in powers of 1= . Then, ra ð ; sa Þ ¼ k þ d a ðsa Þ; d? ðsa Þ2 2 k x d a ðsa Þd? ðsa Þ2 þ þ O da ðsa Þ4 3 ; 2 2 1 k x d a ðsa Þ d? ðsa Þ2 sa ¼ 1 þ þ þ cg cg 2cg
ra ð ; sa Þ ¼ k x d a ðsa Þ
k x d a ðsa Þd? ðsa Þ2 þ O da ðsa Þ4 3 ; 2 2cg
ð25Þ
where d? ðsa Þ ¼ da ðsa Þ2 ½k x d a ðsa Þ2 . Calculating ½ra ð ; sa Þ vðsa Þ x ra ð ; sa Þ=cg 1 , expanding it in powers of 1, and then expanding to first order in 1/2, we obtain the integral " Z 1 1 k x va ðsa Þ ð1Þ 1 a ¼ d cg e " 1 d a ðsa Þ x va ðsa Þ 2 k x d a ðsa Þ þ cg # k x d a ðsa Þk x va ðsa Þ 20 cg o 1n da ðsa Þ2 3½k x d a ðsa Þ2 2 o k x va ðsa Þ n 0 da ðsa Þ2 400 ½k x d a ðsa Þ2 cg ! k x d a ðsa Þk x va ðsa Þ 2 cg
1 3
o 1 n þ 4 3da ðsa Þ2 k x d a ðsa Þ 5½k x d a ðsa Þ3 2 þ O ; da ðsa Þ4 4 ;
687
ð27Þ
After integration, we discard terms that depend on negative powers of the emission time e; for light from a distant quasar, these are clearly negligible. The result is ð1Þ x a ¼ 1 ð1 þ Þk va ðs1 Þ ln j 1 j 1 x þ 1 ð1 þ Þk va ðse Þ ln j e j c o 1 1 n k x d a ðs1 Þ 2 da ðs1 Þ2 3½k x d a ðs1 Þ2 1 4 1 n o 1 þ 3 3da ðs1 Þ2 k x d a ðs1 Þ 5½k x d a ðs1 Þ3 6 1 " 1 1 þ 1 d a ðs1 Þ x va ðs1 Þ 2 1 cg # 1 þ 1 þ 2 þ k x d a ðs1 Þk x va ðs1 Þ cg ( 1 1 2 2 1 k x d a ðs1 Þd a ðs1 Þ x va ðs1 Þ cg 4 1 ð1 þ Þda ðs1 Þ2 k x va ðs1 Þ ) 1 2 þ 1 þ 3 þ 2 k x va ðs1 Þ½k x d a ðs1 Þ cg : þ O ; da ðsa Þ4 4 1
ð28Þ
Since 1 is within the near zone, we can use the expansions d a ðs1 Þ ¼ d a ð0Þ va ð0Þs1 þ Oðda Þ " # 1 da ð0Þ2 ¼ d a ð0Þ va ð0Þ 1 þ 1 þ cg 2cg 1 þ O ; da ð0Þ4 3 1 ; va ðs1 Þ ¼ va ð0Þ þ OðÞ ;
ð29Þ
where, again, ð0Þ means u ¼ 0. Recalling that k x d a ð0Þ ¼ 0 and substituting these expressions into equation (28), we obtain 1 ð1Þ a ¼ ½1 ð1 þ Þk x va ðse Þ ln j e j þ k x va ð0Þ 1 þ cg
#
ð26Þ
where ¼ 1 þ ð2 þ Þcg =c, 0 ¼ ð1 þ Þ=2, and 00 ¼ ð5 þ 3Þ=8. We then integrate the various terms by parts,Rusing R the formula ðd = nþ1 Þf ðsa Þ ¼ f ðsa Þ=n n þ ðd = n n Þdf =d . Each -derivative df =d raises the order of that term by 1/2; we stop integrating by parts when the resulting
½1 ð1 þ Þk x va ð0Þ ln j 1 j
da ð0Þ2 4 21
d a ð0Þ x va ð0Þ 1 da ð0Þ2 k x va ð0Þ 1 4 21 þ O ; da ð0Þ3 3 1 : þ
ð30Þ
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WILL ð2Þ
ð1Þ
Combining the expressions (24) and (30) for a and a , we see that all terms involving 1 cancel, as expected. Our final expression for a is ~rra k x x ~ra x a ¼ ½1 ð1 þ Þk va ð0Þ ln 2 ~ra x va ð0Þ k x va ð0Þ x ~rra cg þ ½1 ð1 þ Þk x va ðse Þ lnj e j þ OðÞ :
ð31Þ
Note that the dependence on cg in the term k x va ð0Þ=cg in equation (31) is illusory: to obtain the measured effect, we must multiply a for body a by its mass ma and sum over all the bodies in the system. Although the event corresponding to ¼ 0 (closest approach) is different for each body, the effect of these differences on va is of the order of the acceleration of each body, which is OðÞ; hence, we can evaluate va ð0Þ at the same coordinate time for each body, for instance, the time of reception of the ray. However, because P a ma va ¼ 0 in barycentric coordinates, the net effect of this cg-dependent term vanishes. Similarly, for a given light ray, the time e is a constant, independent of the body. The difference between the various values of se for different bodies has an effect only at order , so again, summing over all the bodies in the system causes the velocity correction to the ln j e j term to vanish. The ln jc e j term, then, is an irrelevant constant. By the same argument, the factor of 2 in the logarithm produces only an irrelevant constant. The final formula for the time delay, correct to 1.5 PN order, is X ma ½1 ð1 þ Þk x va ð0Þ Dðtr ; te Þ ¼ ð1 þ Þ a
~ra x va ð0Þ x ~ra Þ þ lnð~rra k x x ~rra
:
ð32Þ
The speed of propagation of gravity, which initially appeared in the retarded potentials, is nowhere to be found in this final expression. The only consequence of using an alternative theory of gravity is in the PPN parameters and ¼ 1 =ð2 þ 2Þ. ~ra ¼ xð r Þ xa ð0Þ can be converted so that The variable x all times are referred to the moment of reception of the ray, using the expressions ~ra ; ~ra ¼ xra þ va r ¼ xra þ va k x x x va ð0Þ ¼ va þ OðÞ ;
ð33Þ
where xra xð r Þ xa ð r Þ and va va ð r Þ. Substituting into equation (32) and rearranging, we find, to the necessary order, X n ma ½1 ð1 þ Þk x va Dðtr ; te Þ ¼ ð1 þ Þ a
o lnðrra K x xra Þ ;
Vol. 590 4. DISCUSSION
We have shown that the speed of propagation of gravity has no direct influence on the time delay of light to 1.5 PN order. The only effect comes from any modification of the PPN parameters that might arise in a theory with a different propagation speed. This contradicts claims made by Kopeikin (2001, 2002). In addition, existing solar-system experiments already place sufficiently strong bounds on the PPN parameters and 1 that even the potential difference from general relativity in the 1.5 PN correction represented by the parameter in equation (34) must be small. The parameter is known to be unity to 1 part in 1000 from standard time-delay and light-deflection measurements, while j1 j < 2 104 from analyses of lunar laser ranging (Mu¨ller, Nordtvedt, & Vokrouhlicky´ 1996) and binary pulsar data (Bell, Camilo, & Damour 1996). The latter bounds assume that the solar system and the relevant binary pulsar are moving with respect to the cosmic background radiation with known velocities of the order of 300 km s1 and that, in any theory of gravity with 1 6¼ 0, the rest frame of that radiation coincides with a cosmological preferred frame. Such a frame is to be expected in a theory of gravity in which cg 6¼ c for the following reason: in a theory with either interacting dynamical fields or nondynamical ‘‘ prior-geometrical ’’ fields, the speed of gravity is presumably a function of some cosmological values of the fields in the theory. However, because cg 6¼ c, its value depends on the velocity of the frame in which it is measured. There must therefore exist some frame in which cg takes its value directly from the cosmologically induced values of the underlying fields: the only logical frame is the mean rest frame of the universe and hence the rest frame of the cosmic background radiation. Then, if 1 6¼ 0, there will be measurable effects in any system that moves relative to this preferred frame. With these strong bounds, jj < 5 105 . Nevertheless, a weaker bound on 1 can be inferred from experiments without appealing to preferred-frame effects. We assume only that the theory is a Lagrangian-based theory of gravity, so that it possesses suitable conservation laws for total momentum and energy. Then, lunar laser ranging tests of the Nordtvedt effect bound the combination j4 3 10 =3 1 22 =3j < 103 . The perihelion advance of Mercury combined with the existing bounds on yield j 1j < 3 103 ; Earth tide measurements force the bound j j < 103 . One is then left with the relatively generous bound j1 22 =3j < 1:6 102 . All bounds on 2 come from considerations of preferred-frame effects. Thus, without appealing to preferred-frame tests, one would have to have an extraordinary cancellation between 1 and 2 to conform to Nordtvedt-effect measurements, while still making large enough to make a measurable difference in equation (34).
ð34Þ
where K ¼ k k µ ðva µ kÞ :
ð35Þ
In general relativity ( ¼ 1, ¼ 0), this agrees completely with results derived by Kopeikin & Scha¨fer (1999), to 1.5 PN order.
We are grateful to Steve Carlip, Sergei Kopeikin, Gerhard Scha¨fer, Kenneth Nordtvedt, and Hideki Asada for useful discussions. This work is supported in part by the National Science Foundation, under grant PHY 00-96522.
No. 2, 2003
SPEED OF GRAVITY AND TIME DELAY
689
APPENDIX MEANING OF RETARDED TIME Throughout these calculations, quantities such as xa ðsa Þ and va ðsa Þ appear, where sa is the retarded time evaluated using equation (15). In the near zone, we use the expansion xa ðsa Þ ¼ xa ð0Þ þ va ð0Þsa þ . . .. One is tempted to state that the ð0Þ that appears means something other than that the coordinate time equals zero, but this would not be correct. The trajectory of the body xa ðuÞ is a function of only the parameter u, which is chosen to be directly proportional to the coordinate time. By contrast, sa is not an independent variable; instead, it is a two-point dependent variable, determined by both the field point (t, x) and the body’s trajectory. Equation (15) serves to determine (albeit implicitly or iteratively) that value of the parameter u ¼ sa at which to evaluate xa ðuÞ. The equation quoted above is then simply a special case of the Taylor-expanded equation xa ðuÞ ¼ xa ð0Þ þ va ð0Þu þ . . ., evaluated at u ¼ sa . Figure 1 illustrates this explicitly. On the t0 -x0 plane, assuming flat spacetime, the world line of a body moving with fixed velocity v is shown, together with a field point at (t, x). The point on the trajectory that corresponds to the retarded time from the field point is given by the intersection P between the particle trajectory and the past null ray from the field point (we assume for this illustration that the speed of the null ray is unity). This occurs at a time jx xa ðsa Þj earlier than the field time t; therefore, it occurs at a time t0 ¼ t jx xa ðsa Þj sa , which thus corresponds to the value u ¼ sa along the trajectory. By considering the projection of this event onto the x0 -axis, together with the location and velocity of the particle at t0 ¼ 0, it is clear that xðu ¼ sa Þ ¼ xð0Þ þ vð0Þsa .
Fig. 1.—Illustration of retarded time
690
WILL
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