special relativity in general relativity?

Special relativity in general relativity?D. Hirondel Citation: Journal of Mathematical Physics 15, 1471 (1974); doi: 10.1063/1.1666833 View online: http://dx.doi.org/10.1063/1.1666833 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/15/9?ver=pdfcov Published by the AIP Publishing Articles you may be interested in The Boltzmann equation in special and general relativity AIP Conf. Proc. 1501, 160 (2012); 10.1063/1.4769495 Relativistic Fluids in Special and General Relativity AIP Conf. Proc. 1312, 3 (2010); 10.1063/1.3533205 Exploring the transition from special to general relativity Am. J. Phys. 77, 434 (2009); 10.1119/1.3088883 Relativity: Special, General, and Cosmological Am. J. Phys. 71, 1085 (2003); 10.1119/1.1622355 General treatment of the Doppler effect in special relativity Am. J. Phys. 52, 374 (1984); 10.1119/1.13676

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Special relativity in general relativity?* D. Hirondelt

Department of Physics, University of Delaware, Newark, Delaware (Received 13 August 1974)

Starting from the metric in hannonic coordinates for a test particle m 1 around a heavy particle m 2(m 2>m 1) at rest, the EIH Lagrangian is recovered by making a Lorentz transformation, followed by a canonical transformation and an appropriate symmetrization in the two masses. This raises the question of a special relativity content in general relativity, a feature not directly implied by the general covariance.

1. INTRODUCTION

The laws of gravitation in Einstein's general relativity are generally covariant with respect to any change of coordinates; in any set of coordinates they will look like Ru - ~guR = 87TGC·4T iJ' where RiJ is a certain function of the gkl and their derivatives, up to second order. ds2

= gu (q) dq'dqJ is postulated to be an invariant.

General covariance was of much value to Einstein, and inspired the name of his theory. But it is now ad­mitted that general covariance in itself is physically empty, to the extent that the equations of any theory could be cast in covariant form. This format invariance gives the equations of motion, for example, the same aspect in any set of coordinates, but the functional de­pendence of the accelerations on their arguments will be a priori different in each set of coordinates.

However, we observe that, for a certain family of coordinates, we do have functional invariance of the accelerations with respect to their arguments, under Lorentz transformations, for the case of two masses, up to order c·2 • Einstein, Infeld, and Hoffman (EIH)l and, later, Fock2 showed there existed a common Lagrangian for the problem of two gravitating particles,

L l~1 vtl...21 ~ = 2ml 1 + 8ml c2 + 2m2vli + 8m2 c2

Gmlm2 [1 1 ( . .2 • .2 r O V;-;oV2)·] + -r-- + 2c2 3vi + 3vi! - 7v1 ' v2 -

G2mlm2(ml +m2) 2c2y2

On the other hand, Currie and HilP have given the conditions on a dynamics:

~rl(t) = r (t) drl(t) dr2 (t)] dt2 al lr , dt ' dt '

guaranteeing that, after a Lorentz transformation, it will look like

~r{(t') ['(t') drs. (t') drg(t')] ---;iii2 = a l r 'dt' 'dt' ,

where the a's are the same functions of their arguments as before the transformation. World line invariance is postulated to obtain these conditions. They put all Lorentz frames on the same footing, and privilege none of them. This special relativistic covariance is a strong reqUirement, while general covariance is not.

U is a fact that the accelerations obtained from the

EIH Lagrangian,

x ~ (-~ +4vlo v2-~ +~ (r;:a)2) +(vl-v)(4rov -3r ov)] +G2mlm2(5ml+4m2)r

2 1 2 C2r4 ,

satisfy Currie-Hill conditions up to order c·2 :

+ (V2V2 - raa) 0 0Y2 al - 2vlal - alvl] ,

as one can check by direct calculation.

The original computations for obtaining the EIH Lagrangian being quite lengthy, the preceding result suggests how to recover it in a simple way by starting from the known metric for a test particle around a heavy particle at rest, in harmonic coordinates (the chOice of this particular set will be justified), and making a Lorentz transformation for setting the heavy body into uniform motion. A canonical transformation will sym­metrize the Lagrangian in the velocities of both parti­cles. Symmetrization in the two masses, to yield a Lagrangian usable when the two masses are comparable, requires a certain care. The end result is the EIH Lagrangian •

2. THE EIH LAGRANGIAN FROM THE CASE OF ONE BODY AT REST

The metric for a test particle ml in the field of a spherically symmetric, heavy particle m 2 (m 2 » ml) is, in harmonic coordinates (ro'" Gm2c·2) ,

ds2=c2dt2 r-ro _ r+ro dr 2

r+ro r-ro

- (r + ro)2(d02 + sin2 9 d(j02)

= c 2 dt2 {r - ro _ ..;. [r +ro (rOrVI) 2 r+ro c r-ro

+( + )2~-(rOVI)2J} r ro y2 •

The Lagrangian for particle 1 is, by expanding ds up to order 1/c2:

1471 Journal of Mathematical Physics, Vol. 15, No.9, September 1974 Copyright © 1974 American Institute of Physics 1471

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1472 D. Hirondel: Special relativity in general relativity?

= (Tr + vi ) + Gmlm2 (1 + ~ ~) m 1 2 8el1 r 2 CZ

Since 1/ ell appears in the two dimensionless ratios v2

/ CZ and Gm2/ CZr, keeping only the lowest powers of 1/ CZ is an approximation valid for slow motion and suffi­ciently large separation. A cutoff at a given level in 1/ ell implies a corresponding cutoff Of the series in G: The 1/ CZ term in L 1 is a polynomial in G containing no higher power than G2. Later on, the expression "up to order G or G2" will refer to the coefficient of 1/CZ in

L· Now, we go into a frame in which the velocity of

particle 2 is v2 • The kinetic energy part of L 1: m1 (~/2 + yt/8el1) is left unchanged by a Lorentz transformation as it corresponds to the invariance of the proper time dTI = (1-VVl!2 dt.

For the remainder, it is enough to use:

_ + (r o va)va r r 2el1 '

dt-dt(1- v,oVa+.!L) CZ 2 ell ,

where, after the Lorentz transformation, we made an instantaneity correction: r 2(t2) = r2(t) - V2 (t1 - t2) to make the two particles simultaneous in the new frame. We obtain

L - (~+-Y1.) + Gm tm a[1 +~ (_ (r·vg )2 I-

ml 2 8CZ r CZ r2

+~(~ - 2v1 • v2 +v~) - VI· v2 +~)J Adding

(!i+..!i.) +E..(Gmlma r

o Va)

m2 2 8el1 dt 2CZ r '

does not change the acceleration of particle 1, but yields a Lagrangian which, up to order G, is not symmetric in the indices of both particles, and can be used for both up to that order:

G2mlma{ma) 2e2r2

This last addition is the same as the one we would make to recover by a Lorentz transformation the Darwin­Breit Lagrangian from the Lagrangian of a test particle in the electromagnetic field of a heavy charged particle at rest:

f[ ( V2)1!2 e e ] o -m1 CZ 1-? -~ dt=O.

In this case, there is no term quadratic in the product of the two charges (contrasting with the gravitational case, where the G2 term was the mark of the nonlinear-

J. Math. Phys., Vol. 15, No.9, September 1974

1472

ity). The only effect of the Lorentz transformation is the appearance of a magnetic force; radiation is absent.

Now, by admitting that there exists a common Lagrangian up to order 1/ ell, symmetric in the indices of the velocities and the mass of both particles,

L =kinetic energy + Gm;ma[1 + ~[A(~+~)+Bvl°V2

(A,B, C,D being numerical coefficients), which we know from the work of Einstein, Infeld, and Hoffman, and Fock, then the only function symmetric in m 1 and m2

which yields m 2 when m 2 » m 1 is unambiguously m 1 + m 2 •

This last step certainly modifies the acceleration of particle 1, but supplies us with a common Lagrangian which can be used for both particles up to order G2, when both masses are comparable, and which coincides with the EIH Lagrangian.

Now, we have to give the reason why making a Lorentz transformation in harmonic coordinates was expected to lead to the correct result. A first practical reason is that Fock worked out the EIH Lagrangian in harmonic coordinates: Thus, if we wished to find the same result, we had to start from the same kind of coordinates (we will see that the harmonicity condition is left unchanged by a Lorentz transformation). Ein­stein, Infeld, and Hoffman defined their system of co­ordinates at each step of approximation, but their result is the same.

Another reason why, at least up to order 1/ CZ, Lorentz transformations were expected to play an im­portant role is that the field equations in first approxi­mation, and in harmonic coo rdinates, are

where 0 2 is the d' Alembertian in flat space. Thus the field propagates along light cones, as in special rela­tivity. At higher orders in e·2

, the characteristics for the propagation of the field will be curved, not straight, and their form will be codetermined with the configura~ tion of the masses.

However, in spite of the loss of isotropy and uni­formity, a theoretical reason to guess that Lorentz transformation will still playa major role to all orders in e·2 in Fock's proposition2 that it is unlikely that there exists any system of coordinates, other than the har­monic set, determined uniquely apart from a Lorentz transformation, because they are characterized by the fact that they satisfy a linear, generally covariant equation:

rl=~ 03 (Igll!2gfl);; 02Xl =0. Igl oX'

Harmonic coordinates exclude all fictitious gravitational fields, and, in a way, they can be called the most inertiallike coordinates. Is that enough to entail world line invariance and functional invariance under Lorentz transformation to all orders?

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1473 D. Hirondel: Special relativity in general relativity?

CONCLUSION

We have recovered the ElH Lagrangian by a procedure whose simplicity contrasts with the lengthiness of the original computations. It raises the question of a Special Relativity content in General Relativity, apparently restricted to harmonic coordinates. As a test, it would be worth computing the accelerations up to order c·4 to see if they satisfy the Currie-Hill conditions.

ACKNOWLEDGMENTS

It is a pleasure to thank Dr. Kerner and Dr. Hill for discussions. Financial support from the Physics De-

J. Math. Phys., Vol. 15, No.9, September 1974

1473

partment of the University of Delaware is acknowledged.

'* Abstracted from Ph. D. Dissertation "C anonica! Formulation of Instantaneous Electrodynamics. GraVitation", University of Delaware, submitted June 1973.

t Present address: 4, Rue Slldillot, Paris 7, France. 1 L. Infeld and J. Peblanski, Motion and Relativity (Pergamon,

New York, 1960). A. Einstein, L. Infeld, and B. Hoffman, Ann. Math. 39, 65 (1938).

2V. Fock, The Theory of Space, Time, and Gravitation (Pergamon, New York, 1959), p. 288. Fock's remarks on the harmonicity condition, p. 350.

3R. N. Hill, J. Math. Phys. 8, 201 (1967); D. G. Currie, Phys. Rev. 142, 817 (1966).

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