Abstract. The time-keeping property of a mode l ' electric ' clock in an Einstein-deSitter universe is ex
amined.The dynamics of such a clock is derived from the Einstein-Maxwell field equations using a modified
version of the Einstein-Infeld-Hoffmann (EIH) surface integral method for obtaining equations of motion
for compact sources of the gravitatio nal and electromagnetic fields. The results show that such a model is
only approx imately an 'ideal' clock and differs from one by terms ofthe order of (clock period/Hubbl etime)? .
1. INTRODUCTION
The original formulation of the general theory of relativity included an assumptionregarding the measurement of time intervals, namely that ideal clocks measure theproper time along their trajectory as computed using the gravitational field tensor gJl V
as a Riemannian metric in the line element
(1)
Unfortunately, this assumption has a kind ofcircularity associated with it: how do youknow if a candidate clock is an ideal clock. Answer, it measures proper time. How doyou know what the proper time is? Answer, it is the time measured by an ideal clock.There is, in fact, no way to tell ab initio if a candidate clock is in fact an ideal clock.Since all real clocks are physical systems their behavior will in general be affected bythe presence of gravitational and other fields. If, for example, tidal forces acting onthe clock are comparable to its internal forces then the timekeeping properties of theclock will be modified. Likewise, the frequency of an 'electric' clock, e.g. a hydrogenatom, will be modified in the presence of an electric or magnetic field. In fact, onedoes not have to introduce the proper time-ideal clock hypothesis any more that onehas to introduce the geodesic hypothesis into general relativity and therefore one candispense with the metric interpretation of gJlv .
In order to determine the properties of clocks and measuring rods in a gravitationalfield without additional assumptions, one makes use of the ideas first developed by(Einstein, Infeld, and Hoffmann (EIH) 1939; Einstein and Infeld 1940, 1949). Laterwork by myself extended the EIH procedure to include radiation (Anderson 1987)and electromagnetic interactions (Anderson 1997) using the methods of mult iple timescales (an extension of the original EIH slow-motion approximation) and the methodof matched asymptotic expansions (Nayfeh 1973). Later these methods were exten-
ded to deal with motion in an expanding universe (Anderson 1995). In the earlierwork s the fields produced by their sources were treated as perturbations on a 'flat'gravitational field and yielded, in the lowest orders of approximation, equations ofmotion containing Newtonian and Coulomb interactions. High er orders of approximation produced radiation reaction forces in these equations. In the last work cited herethe source fields were treated as perturbations on an Einstein-deSitter gravitationalfield. The resulting equations of motion were used to derive the dynamics of simpleclock models consisting of two gravitationally or electromagnetically bound compactsources. Both of these model clocks were shown to measure cosmic time in the lowestorder of approximation used and hence, to this order, behaved as ideal clocks. In thiswork I have extended these calculations to obtain the next order effects of the cosmicexpansion which cause these clocks to deviate from ideal clock behavior.
2. EIH PROCEDURE
General relativity is unique among classical field theories in yielding equations ofmotion for the sources of the various fields in the theory. In classical electrodynamics oneneeds to postulate the form of the Lorentz force as well as the form of the inertial forcein the equations of motion. Neither such postulate is necessary in general relativity.Furthermore, the point singularity problems one encounters in other field theories areabsent in this theory. Given these virtues it is all the more strang e that Einstein 's lastgreat contribution to general relativity, the work he did together with L. Infeld and B.Hoffmann, is so little known or referred to in the literature. For example, a description of the EIH method appears in only one early text on general relativity by PeterBergmann who was an assistant of Einstein's. Bergmann (1942) and only a handfulof papers in the literature make use of it. It is therefore perhap s not out of place tospeculate for a moment on why this is so. A number of factors may have contributed.For one thing the initial paper was published at the beginn ing of the second WorldWar when the scientific community began turning its attention to the military needsof the combatants. Then too, the original paper was publi shed in the Annal s of Mathemati cs and the two subsequent papers by Einstein and Infeld appeared ten years laterin the Canadian Journal of Physics, neither journal being widel y read by physicists.Some readers were probably scared off by the complexity of the calculations. Thecomplete details were never published, the reader being referred to an archive at theInstitute for Advanced Study. In addition, some puzzling procedures were used, e.g.,the introduction of fictitious dipoles, that later proved to be unnece ssary. Perhaps too ,general relativists in those days were somewhat disdainful of the approximate natureof the EIH results, being more concerned with exact results and deriving theorems.Furthermore. after 1949, Einstein did no further work on the probl ems of motion andInfeld , after returning to Poland, abandoned the EIH approach in favor of his "good"delta functions . Also , at the time these papers appeared the attention of the majority ofphysicists had turned from general relativity, with its lack ofexperimental predictions,to the new and fruitful field of quantum mechanics. As a consequence of one or moreof these and possibly other factors, the EIH method is largely unused by modem dayworkers in the field of general relativity.
TIMEKEEPING IN AN EXPANDING UNIVERSE 277
EIH derived their equations of motion by integrating the field equations over atwo dimensional surface surrounding each compact source. Since the integrals neverextended over the sources the question of their exact nature never arose. In particularit was unnecessary to assume that they were point sources . These integrals are mosteasily obtained by using a form of the field equations given by Landau and Lifshitz(1985 ,282) which are
(2)
where UJLVp is a so-called superpotential, antisymmetric in 1/ and p, and a function ofgJLV and its first derivatives and
(3)
In this latter equation 9 = det(g/w ), t~~ is the Landau-Lifshitz energy-stress pseudotensor and TJLV is the energy-stress tensor of any other fields present. Because theintegrals used in EIH are surface integrals surrounding the sources of these fields andthe gravitational field one does not have to include the contributions of these sourcesto TJLv. It is for this reason that one avoids the usual difficulties associated with theintroduction of point sources such as those encountered in Dirac's derivation of theradiation reaction force in electrodynamics (Dirac 1938). One only needs to assumethat the sources are compact so that the results are valid, for example, both for neutronstars and black holes.
Because of the antisymmetry ofUJLVp in its last two indices , U JLT S ,s is a curl whoseintegral over any closed spatial 2-surface vanishes identically. As a consequence,integration of eq. 3 over such a surface surrounding a source gives
(4)
where nr is a unit surface normal. It is this last equation that yields the equations ofmotion of the source contained within the integral.
To actually obtain equations of motion from the above integral one must use solutions of the field equations to evaluate the integrands appearing in them. These solutions however can only be obtained by an approximation scheme. In their originalwork , EIH introduced what they called the slow-motion approximation to obtain thesesolutions. As I discussed in (Anderson 1987), a systematic application of this approach requires the use of the method of multiple time scale analysis . To apply it oneidentifies the different time scales associated with the motion under consideration andexpands the fields in powers (or more complicated functions) of the ratios of thesetime scales. If one is interested in motion in a 'flat' background field gJLV = diag(l,-1, -1, -1) as in the original EIH papers and in (Anderson 1987; Anderson 1997), thentwo time scales enter. One is TL, the light travel time across the system and the otheris Ts , a characteristic time such as a period associated with the system. The ratio es isthen used as the dimensionless parameter in the expansions one constructs - hence theappellation 'slow-motion.' In addition one also introduces, in addition to t , a slow timet s and assumes that the fields are functions of these two times. (Actually, one needs tointroduce even slower time scales such as F: 2 t in higher orders of the approximation.)
278 JAMES L. ANDERSON
If however one is interested in motion in an expanding universe then one must useone or another of the gravitational fields associated with such universes. In this case ,another time scale enters into the problem, namely, the Hubble time ttt and one canconstruct another small parameter en = tL /tIl to use in one's approximation schemeso that one has a double expansion. However, since for most systems en « cs theeffects of the expansion can be separated out from the slow-motion effects without toomuch trouble.
3. EQUATIONS OF MOTION
In this paper I will not attempt to give the details of the derivation of the equationsof motion in an expanding universe using the EIH procedure and its extensions sincethey are somewhat complicated and instead refer to reader to the references below. Tosimplify the results I will use the Einstein-deSitter gravitational field given by
(5)
(6)
where R(t) = (t /tO)2 /3, t is the cosmic time and to is the current age of the universe.In this case tIl = R / R where a dot over a quantity refers to differentiation withrespect to t . Furthermore, in order to simplify the results I have assumed as a clockmodel a classical hydrogen atom and neglected the gravitational interaction betweenits constituents. Had I used a gravitational clock the results would have differed onlyby numerical factors of the order unity. The resulting equation for the' electron' of theclock, with the proton fixed at the origin takes the form
2 ~ 1 ~mXTT + 2CmXtT + e mXtt + 2cmIixT= - R3 x3 X .
where e = en /c s and the subscripts on the electron coordinate x denotes differentiation with respect to t and the fast time T = t / c . These equations are correct toO(es 2 ) , that is, to Newtonian order and to all orders in CH. The inclusion of postNewtonian terms do not change qualitatively the results obtained below.
In order to construct a uniform approximate solution to these equations of motionit is necessary to employ two approximation schemes in conjunction. One of theseinvolves a multiple time expansion in t and T . The other approximation involves themethod ofstretched coordinates. In this approximation one makes a change ofvariablefrom T to s given by
One also expands the position coordinate x(s, t, c) according to
and substitutes into eq. 6. The sum of the lowest order (e") terms in eq. 6 will vanishif
(9)
TIMEK EEPING IN AN EXPANDING UNIVERSE
For simplicity I will assume a circular orbit so that
279
xo = z(t) cos{w(t)s }, Yo = z(t )sin{w(t )s} and Zo = 0 (10)
where w and z satisfye2
R 3z 3W
2 = - .m
and their dependence on t in the next order of approximation.The next order (101) equation has the fonn
(11)
(12)
In order to obtain a uniform expansion in 10 it is necessary to require that xl not containsecular terms, that is, terms that grow in time without bound. The first three terms inthis equation will lead to such growth so we must require that their sum be zero. Thiswill be the case provided that
Wt=O , 0:1 =0 and (R zw)t =O.
Combining this result with eq. lOwe find to this order of accuracy that
w = Wo and z = Zo/ R .
(13)
(14)
where Wo and zo are constants. As far as Xl is concerned, we will take it to be equalto zero to avoid unnecessary complications. Consequently, we can conclude that tothis order of accuracy our electric clock measures the cosmic time t. Also, if we takeRz to be a measure of our clock size, we see that it does not partake of the cosmicexpansion. Thus, to this order, our clock behaves like an ideal clock.
Finally we come to the order (102) equation. Again one must set equal to zero thoseterms that would lead to secular growth in X2. This condition gives
Rt20:2Xoss + 2R XOt = O. (15)
As a consequence we find that
1 Rtt0:2 = 2wf""R (16)
When this result is substituted back into the expression (7) for s and that in tum issubstituted into the zero order solutions (10) one obtains finally the result
Zo c2 Rttxo = R cos[wo{1 - 2wf""R }T] (17)
and similarly for Yo with the cos replaced by the sin. Thus we see that the effectof the expansion on the clock frequency is to alter it by a term of order 102 whosecoefficient varies slowly with the cosmic time t and to this extent our clock will nolonger measure proper time. For a real hydrogen atom this term however is so smallas to be completely negligible.
280 JAMES L. ANDERSON
4. CONCLUSION
There are two positions one can take concerning the role of measuring devices in aphysical theory. One is that such devices lie outside the province of the theory andtheir properties must be postulated. Such a view is common in quantum mechanics,in which measuring devices are held to be classical devices, and leads to a numberof problems associated with the so-called measurement problem. The other positionholds that measuring devices are physical systems whose behavior must be describablewithin the framework of the theory ofthe systems they are designed to measure. Whenthe measurement process is itselftaken to be describable by quantum mechanics manyof the measurement problems go away. In the case of general relativity I have tried toshow how one can calculate the properties ofphysical systems that can serve as clocksdirectly from the field equations of this theory. Such an approach has, I believe, twoimportant consequences. For one, one is able to dispense with the notion of idealclocks and rods as primative objects in the theory.. The other is that it allows one tocalculate to what extent and to what accuracy a putative clock measures time. In theexample given here we see that if its period is small compared with the Hubble timethen the clock behaves as an ideal clock and does not feel the effects of the expansionof the universe.
Stevens Institute ofTechnology
REFERENCES
Anderson , J. L. 1987. "Gravitational radiation damping in systems with compact components." Phys. Rev.D 36 :2301 .
--. 1995. "Multiparticle dynamics in an expanding universe ." Phys. Rev. Lett . 75 :3602 .- - . 1997. "Asymptotic conditions of motion for radiating charged particles ." Phys. Rev. D 56 :4675 .
Bergmann , P. G. 1942./ntroduction to the theory ofrelativity. New York: Prentice Hall.Dirac, P. A. M. 1938. "Classical theory of radiating electrons ." Proc. Roy. So c. (London) AI67 :148.Einstein , A., and L. Infeld. 1940. "Gravitational equations and the problem of motion . II." Ann. Math ., Sa
2 40:455.- -. 1949. "On the motion of particles in general relativity theory." Can. J. Phys. 3:209.Einstein , A., L. Infeld, and B. Hoffmann . 1939. "Gravitational equations and the problem of motion ." Ann.
Math., Ser. 239:65.Landau, L. D., and E. M. Lifshitz. 1985. The Classical Theory ofFields. 4th cd. Oxford : Pergamon Press.Nayfeh, Ali. 1973. Perturbation Methods . New York: John Wiley & Sons .
