- Email: [email protected]
The motion of a particle of finite mass in an external gravitational field
ANNALS
OF
The
PHYSICS:
Motion
30,
168-177
(1964)
of a Particle of Gravitational MOSHE
Department
of Physics,
Israel
Finite
Mass Field
in an
External
CARMELI Institute
of Technology,
Haifa,
Israel
The analogy between the electromagnetic and the Einstein gravitational field theories raises the question as to whether the motion of a particle of finite mass moving in a gravitational field may be described in terms of an external gravitational field. There are many ways of defining the external field, and it is difficult to decide which to choose. It is shown that an inertial mass M can be defined in such a way that the EIH equations are derivable from the geodesic equation in the external field, this being defined as the field obtained from the whole field by putting M = 0. It turns out, however, that the analogy between the electromagnetic and the gravitational cases is not a close one-terms proportional to the square of the mass do not necessarily describe reaction forces; these appear even in the 6th order, in which the problem of gravitational radiation has not yet arisen. Moreover, certain terms defined as external field functions are reduced, on the basis of the equations of motion, to terms proportional to the square of the mass. I. INTRODUCTION
It is well known that in the case of the motion of a charge in an electromagnetic field there is no difficulty in separating the field, on the basis of the Maxwell-Lorentz theory, into an external field and a self field of the charge. One obtains the Lorentz-covariant classical Dirac equations (1) for the motion of the charge in the external field, the terms proportional to the square of the charge describing the radiation reaction forces. The analogy between the electromagnetic and the Einstein gravitational field theories raises the question as to whether the motion of a particle of finite mass moving in a gravitational field can also be described by generally-covariant equations in terms of an external gravitational Jield. Discussing this problem, Schild (2) showed that such equations cannot exist. He argued that such equations, by analogy with the Dirac equations, will include two terms, the first proportional to the mass, the second to the square of the mass. The form of such equations, however, cannot remain invariant under a generally-covariant transformation. One can easily show this by choosing the transformation as an arbitrary translation of the coordinate system by quantities 168
MOTION
IN
rl
GRAVITATIONAL
169
FIELD
proportional to the mass, as a result of which the world line of the particle will be shifted by arbitrary quantities proportional to the mass. Such a shift will then contribute new arbitrary terms proportional to the square of the mass, which may cancel the second term in the equations. One may suppose that the terms proportional to the square of the mass are due to gravitational radiation, as in the electromagnetic case, and that if we confine ourselves to the problem in which we do not include the gravitational radiation, then the motion of a particle of finite mass may be described in terms of an external field in a generally covariant form. In this paper we intend to deal with the concept of an external gravitational field by discussing the following rather simple question: can the EIH equations of motion (3, 4) be written as generally covariant equations in an external gravitational field? It turns out that this is a question of how to define the external field: is it the field obtained from the whole field by formally setting m = 0, as has been done by Schild, or is it the field obtained by letting m tend to zero and taking into account the effect of this on the motion of all the other particles? If we accept the first definition, then do we have to take m as the constant proper massor as the time-dependent inertial mass (5, 6)? Here the two-particle case is discussed; one of the particles being looked upon as moving in an external field defined according to the above possibilities. In Sections II and III we describe the approximation method and solve the field equations. In Section IV we define the external field and describe the motion in it, while other possible definitions for it will be given in Section V. The last section is devoted to the concluding remarks. II.
THE
APPROXIMATION
METHOD
We solve the Einstein gravitational field equations in a harmonic coordinate system (7) by means of the approximation method to be described. It is convenient to write the field equations in the form’
SRao = S?r(g,,gav - >$gaBgpv)Y’“.
(1)
Here 9 = ( -g)l”, g = det. gma, and 3”’ is the energy-momentum tensor density, which, in the case of two particles described as singularities of the field, may be taken in the form (5) 3BY= m1~‘~y61(2s - C;“) + m27jp7jv6z(xs where t(t)
and $(t)
q”),
(2)
are the coordinates of the two particles, the dot denotes
1 We use units in which the velocity of light c and Newton’s gravitational constant G are equal to one. The metric is (+ - - -). Greek indices run from 0 to 3, Latin from 1 to 3, and zr” = t. Ordinary partial differentiation is denoted by a comma.
170
CARMELT
time-differentiation ( io = fro = 1 ), the 6 is the Infeld delta function (5, 8), and ml and m are the inertial masses (functions of time) of the two particles. In Eq. (1) ROp is the Ricci tensor and in a harmonic coordinate system can be written in the form:
We assume that the metric tensor of the space-time differs from the Minkowskian tensor qub by quantities (perturbations), which can be expanded in a power series in the masses of the particles 10
01
20
11
02
gab = q,8 -I- ml gab + m2 sola + m12gd + ml m2 g4 -I- m22ga8+ . . . ,
(4)
where the two indices written as superscripts on the functions appearing in the above expansion denote the powers of the masses ml and r~ respectively. Putting the expansion (4) into the field equations (1) and equating coefficients of equal powers of the masses, we obtain a series of inhomogeneous partial differential equations (wave equations). These correspond to various approximations, the sources in the wave equation for each approximation being expressed in terms of the previous approximations. We further assume that the velocities of the particles are much smaller than the velocity of light c, so that every quantity can be expanded in a power series in c-l, as was done by Einstein, Infeld and Hoffmann (S-5) : (5) m=p+m+m+ where the indices written
3
-es,
4
as lower subscripts
(6)
indicate the order of c-r, and p ( = m) 2
is a constant (5)) hereafter called the proper mass. Thus we have to distinguish between the approximation, determined by the sum of the powers of the masses, and the order, determined by the power of c-l. Note that as ml and m2are functions of time, the Ricci tensor given by Eq. (3) will, in general, contain terms including derivatives of the masseswith respect to the time. Nevertheless, the riz’s are functions of the 5th order (5), and hence will not contribute to the equations of motion up to the 6th order2 being considered here. 2 I.e., the EIH equations. to be the 4th order.
In this
sense,
the
Newton
equations
of motion
are considered
MOTION
III.
IN
A
SOLUTION
GRAVIT;1TIONAL
OF
THE
171
FIELD
FIELD
EQUATIONS
In order to find the equations of motion up to the 6th order, we have to calculate the components of the metric tensor up to the following orders (9) : g,,-4th, gok-3rd and gkl-2nd. In the first approximation we obtain for the field equations: 10
- ?lprgmB.PLl = 1~7d7~44v
-
(7)
?4wd~f"&,
01
and analogous equations
for gcrB. The solutions
for the required
functions
are
10 q,,
=
-
t
611"
10
fOO=O
(8)
10 p
=
2_ '8'S rl F 5
-
10
,
yoz =
z
+
Q,oo
(
>
6";
?.I2 =
(d
-
‘$s)(2"
-
.$").
01
The analogous solutions
for g,, are obtained from (8) by simply replacing r1 and
,$” by rz and 7’ respectively,
ihere
~22= (x8 - 9’) (2” - VJ”). 20
The required
functions
from the second approximation
11
02
are: osoo, p , and go0. 0
Straightforward
calculations
give :
(9) ;o,.w = +),8(k),8
- ++;),
(10)
02
and an analogous equation to (9) for go0. Their
solutions
are :
0
(11) where r2 = (p - v”)(E”
- q”).
172
CARMELI
In the Appendix functions
we express the required
Christoffel
symbols
in terms
of the
il. . n IV.
MOTION
IN
AN
EXTERNAL
GRAVITATIONAL
FIELD
We now proceed to define the external gravitational field (metric) for the particle having the proper-mass ~1 . Using (5) the values of the m’s, we notice that the power series expansion in k
c-l of ml and m2 given by Eq. (6) can also be written proper masses pl and pcLB
as a power
(i = 1,2).
mi = C ah.hkhz; k.z
Putting this in Eq. (4), the metric tensor will then be written in a power series in p1 and pz :3 10 SaS
=
%a
+
/a
01
YolB
+
20
I42 YnS
+
P12 Ye!9
The above expansion can also be written 0 1 SaS
=
YCIB +
YaB
=
YaB
l-42 YaB
as an expansion 02
P2 yal3
+
I422 Yaj3
+
**.
l.12 -rat3
+
. . * ,
+
* * . ;
(14)
n2 +
P22 yolf9
(15)
00
YdJ
(13)
2
al +
I.0
(12)
in the form (p = pI) :
P Ya!3 +
where we have used the notation n 7LO
11 +
series in the
=
r]aB
(n = 0, 1,2, * *I.
,
0
We define the functions ymp as the external metric from the metric tensor by formally putting p = 0: Y&9
(field) ; they are obtained
= [Sablr=0 *
(16)
The definition given by Eq. (16) is also the one accepted by Schild. As a first step in discussing the motion of the first particle in terms of the external metric, let us see whether the geodesic equation in this external metric (17) which
can also be written
in the form (18)
3 We could have carried out the expansion powers of m. However it is more convenient
in the first place in powers to go over from m to fi than
of p instead of in from p to m.
MOTION
IN
A GRAVITATIONAL
FIELD
173
will give the EIH equations. In Eq. ( 18) the zero superscripts on the Christoffel symbols indicate that these quantities are evaulated by means of the external metric G.6 . This means two things: first, terms containing P are ignored and second, in place of the x” the $ are introduced in the Christoffel symbols of the whole field calculated in the Appendix. The comparison between Eqs. (18) and the “general equations of motion” of Infeld (5): {” + (F$
- r$pjf”,$
= 0
(19)
shows that the former differ from the latter only in that the zero superscripts appear in place of the bars. But the bars also mean two things (5) : first, singularities are ignored and second, as before, we have to introduce the $ for xk . It follows that only when all the terms containing P are singular, are the two operations identical. This is the case, for example, for the components of the Christoffel symbols in Eqs. (A.l)-(A.5). It does not hold, however, for the last one given by Eq. (A.6), which contains the two terms 11 ?4ml
m2
The 4th order of this expression ! i/A
P2 ;m
) k
+
01
900 , k +
Mm2
g00 , k.
CJO)
is given by
?4/icL
2~oo,a:+!iu2(Ni’i.+~)~o,~, ;
where use has been made of the relation
(21)
(5) (22)
A simple calculation
then shows that 0 -k roe 4
This shows that the EIH
k
-
roe 4
=
Pl
(23)
1*2
equations are not derivable jrom the geodesic equation in
0
the external metric Y~.J, since a simple calculation shows that they are obtainable from the Infeld equations. Thus the force term given by
00 1
fl”
=
p2p2
;
;
>t
will be lacking in Eqs. (18). But Eqs. (18), up to the 6th order, are identical with those proposed by Schild. This shows that even up to this order, these equations are not valid. The force term jl” is the lower order approximation of what
174
CARMELI
has been called by C. Dewitt and J. L. Ging (10, 11) the ‘%a&” and has been written as an integral of unknown functions. The fact that such a term as flk appears even in the 6th order, in which the problem of gravitational radiation has not yet arisen, prevents a close analogy with the electromagnetic case, since terms proportional to the square of the mass do not necessarily describe reaction forces due to the particle’s radiation; they may also describe interaction terms arising
from the direct interaction and from the corrections included in the inertial masses given in expression (21) by the first and the third terms respectively. V. OTHER
DEFINITIONS
FOR
THE
EXTERNAL
IUETRIC
In the previous section we looked at the metric tensor as a function of the coordinates as well as of the parameter P and defined the external metric by simply putting P = 0 in the former. One may ask, however, if this is the only possibility for defining the external field. Thus, for example, instead of (14), one can write the expansion (4) in the form 0 gai3
=
ga5
1 +
112 hi3
+
m”
id
+
Cm = ml),
. . - ;
where, in anaology with the former case n ?LO gut3 = g&3+ m ;:5 + - . ’ ) 00
(25)
(26)
0
and gas = qo,g. Then the functions g,s may be used as an external metric obtained by putting m = 0 in the expansion (25). This is a modification of our former definition, and it has no analogy in the Maxwell-Lorentz theory. Investigation of the various terms of the Christoffel symbols given by (A.l)(A.6) shows, in this case, that only the first term in the expression (20) will cause a difference between the equations given by (18) and (19), if we now understand 0
that the zero superscripts on I$ will give the force term
0
indicate
calculations
by means of gab. This
(27)
However, one need not confine oneself to the Infeld inertial Eq. (2) ; other definitions are possible (6). Let us define such a mass by the relation T”’ = M,~“&“S, + M27f7jv~2 ,
where TPY is the energy-momentum
mass defined by
(28)
tensor. Ml will then be related to rn! by ml = ~MI,
(29)
MOTION
IN
A
GRAVITATIONAL
175
FIELD
and a simple calculation shows that if we expand Ml in a power as has been done before for ml in Eq. (6), then
series in c-l,
We see that the first relativistic correction to the proper mass is the sum of its kinetic and potential energies. Repeitition of all the above calculations, but using the M’s instead of the m’s, shows that the two equations (18) and (19) will give the same results, i.e., the EIH equations, provided we understand the zero superscripts in Eq. (18) as indicating evaluation by means of the external metric obtained from the whole metric by formally putting M = 0. VI.
co~cm~m~
REMARKS
We have already seenin Section IV that the analogy between the electromagnetic and the gravitational casesis not a close one: terms proportional to the square of the massdo not necessarily describe reaction forces; these appear even in the 6th order, in which the problem of gravitational radiation has not yet arisen. These terms are the self-field terms. What about the external field terms? Has the separation carried out above any physical significance? Consider, for example, the second term in Eq. (A.6)) which is defined as an external field. Its 4th order value will contribute to the equations of motion the expression -4 I@4 ..k T’l’
Here we may introduce for Ilk its Newtonian value ;i” = PlwG7k,
(32)
since the error will affect the equations of motion only in the 8th order. The term (31) will thus give a force term f” = 4L%(Ilr)(Ilr),
t;
(b = c(l).
(33)
But this is identical (except for a numerical coefficient) with the force terms frk and fZk, which are not associated with the external field. This shows that the methods used above cannot give an unambiguous definition of the external field. The difficulty arises from the double role of the self-mass: it is both inertial and gravitational. One can give an unambiguous definition of the external field only by letting the self-masstend to zero, and taking into account the effect of this on the motions of all the particles. In the two-particle case, for example, the external field
CARMELI
176
terms will contribute only three terms out of all the terms in the EIH equations (Is), while the others may be considered as nonexternal field terms. This seems to us an artificial process. Similar conclusions were also reached by Professor P. Havas via the Lorentzinvariant approximation method (13, 14). APPENDIX
We give below the Christoffel symbols needed for calculating the equations of motion up to the sixth order: 10 I?!,
01
=
%rnl
gOO,s +
l20
=
%rnl
g00,0
I?,“,
=
-%ml
rk 3
=
-%m(~0,8
ri0 2
=
?dml
gOO,k +
r!O 4
=
-ml
gkO.0
?4m2
gOO,s
%m2
g00,0
(A.1)
2
01
10
3
+
10
64.2)
10
(gkr,s
+
gks,r
-
L.k)
-
%m,(Z,,8
+
5.,r
-
(A.3)
;,.,,
2
01
10 +
is,0
10
01
%m2(gkO,s
+
gks,O
01 -
(-4.4)
gOs,k)
(A.5)
gO0.k
01 -
m2
+
gkz
10 10
20
gkO,O
+
m12(%gM,k
+
01 01
02 m22&ig00,k
-
01 ?dm2
10
+
-go&k)
go0,z)
10 +
ml
m2
(gkz
gkl
gOO,Z)
01 g00,z
+
01
10
gkz
good
11 +
?4
10 +
%ml
gOO,k -t
(A.61
gOO.k)
01 %m2
gOO,k ,
where in the above equations the expressions on the right-hand side have to be calculated to the same order as the corresponding Christoffel symbols. ACKNOWLEDGMENTS I wish to express by indebtedness and for the many helpful discussions, discussions.
RECEIVED
to Professor Nathan and to Professor
Rosen for suggesting Asher Peres for many
: March 11, 1964 REFERENCES
1. P. A. M. DIRAC, Proc. Roy. Sot. 167A, 148 (1938). 2. A. SCHILD, Bull. Acad. Polon. Sci. 9, 103 (1961). 3. A. EINSTEIN, L. INFELD, AND B. HOFFMANN, Ann. Math. 39, 65 (1938). 4. A. EINSTEIN AND L. INFELD, Can. J. Math. 1, 209 (1949).
the problem stimulating
MOTION
5. 6. 7. 8. 9. 10. 11. fd.
L. A. V. L. A. C. J. J.
IS
A
GRAVITATIONAL
FIELD
177
INFELD, Reu. Mod. Phys. 29, 398 (1957). PERES, Nuouo Ciwento (10) 11, 644 (1959). Pergamon Press, London, 1959. FOCK, “Theory of Space, Time and Gravitation.” INFELD AND J. PLEBANSKI, “Motion and Relativity.” Pergamon Press, Warsaw, 1960. p. 36. King’s College, London, 1958. TRAUTMAN, “Lectures on General Relativity,” DEWITT AND J. L. GING, C. B. Acad. Sci. 261, 1888 (1960). L. GING, Ph.D. Thesis, University of North Carolina, 1960. ?J. GOLDBERG, in “Gravitation, an Introduction to Current Research,” L. Witten, ed. Chap. 3, Eq. (3-3.20). Wiley, New York, 1962. 15. P. HAVAS, Phys. Rea. 108, 1351 (1957). 14. P. HAVAS AND J. N. GOLDBERG, Phys. Rev. 128,398 (1962).
OF
The
PHYSICS:
Motion
30,
168-177
(1964)
of a Particle of Gravitational MOSHE
Department
of Physics,
Israel
Finite
Mass Field
in an
External
CARMELI Institute
of Technology,
Haifa,
Israel
The analogy between the electromagnetic and the Einstein gravitational field theories raises the question as to whether the motion of a particle of finite mass moving in a gravitational field may be described in terms of an external gravitational field. There are many ways of defining the external field, and it is difficult to decide which to choose. It is shown that an inertial mass M can be defined in such a way that the EIH equations are derivable from the geodesic equation in the external field, this being defined as the field obtained from the whole field by putting M = 0. It turns out, however, that the analogy between the electromagnetic and the gravitational cases is not a close one-terms proportional to the square of the mass do not necessarily describe reaction forces; these appear even in the 6th order, in which the problem of gravitational radiation has not yet arisen. Moreover, certain terms defined as external field functions are reduced, on the basis of the equations of motion, to terms proportional to the square of the mass. I. INTRODUCTION
It is well known that in the case of the motion of a charge in an electromagnetic field there is no difficulty in separating the field, on the basis of the Maxwell-Lorentz theory, into an external field and a self field of the charge. One obtains the Lorentz-covariant classical Dirac equations (1) for the motion of the charge in the external field, the terms proportional to the square of the charge describing the radiation reaction forces. The analogy between the electromagnetic and the Einstein gravitational field theories raises the question as to whether the motion of a particle of finite mass moving in a gravitational field can also be described by generally-covariant equations in terms of an external gravitational Jield. Discussing this problem, Schild (2) showed that such equations cannot exist. He argued that such equations, by analogy with the Dirac equations, will include two terms, the first proportional to the mass, the second to the square of the mass. The form of such equations, however, cannot remain invariant under a generally-covariant transformation. One can easily show this by choosing the transformation as an arbitrary translation of the coordinate system by quantities 168
MOTION
IN
rl
GRAVITATIONAL
169
FIELD
proportional to the mass, as a result of which the world line of the particle will be shifted by arbitrary quantities proportional to the mass. Such a shift will then contribute new arbitrary terms proportional to the square of the mass, which may cancel the second term in the equations. One may suppose that the terms proportional to the square of the mass are due to gravitational radiation, as in the electromagnetic case, and that if we confine ourselves to the problem in which we do not include the gravitational radiation, then the motion of a particle of finite mass may be described in terms of an external field in a generally covariant form. In this paper we intend to deal with the concept of an external gravitational field by discussing the following rather simple question: can the EIH equations of motion (3, 4) be written as generally covariant equations in an external gravitational field? It turns out that this is a question of how to define the external field: is it the field obtained from the whole field by formally setting m = 0, as has been done by Schild, or is it the field obtained by letting m tend to zero and taking into account the effect of this on the motion of all the other particles? If we accept the first definition, then do we have to take m as the constant proper massor as the time-dependent inertial mass (5, 6)? Here the two-particle case is discussed; one of the particles being looked upon as moving in an external field defined according to the above possibilities. In Sections II and III we describe the approximation method and solve the field equations. In Section IV we define the external field and describe the motion in it, while other possible definitions for it will be given in Section V. The last section is devoted to the concluding remarks. II.
THE
APPROXIMATION
METHOD
We solve the Einstein gravitational field equations in a harmonic coordinate system (7) by means of the approximation method to be described. It is convenient to write the field equations in the form’
SRao = S?r(g,,gav - >$gaBgpv)Y’“.
(1)
Here 9 = ( -g)l”, g = det. gma, and 3”’ is the energy-momentum tensor density, which, in the case of two particles described as singularities of the field, may be taken in the form (5) 3BY= m1~‘~y61(2s - C;“) + m27jp7jv6z(xs where t(t)
and $(t)
q”),
(2)
are the coordinates of the two particles, the dot denotes
1 We use units in which the velocity of light c and Newton’s gravitational constant G are equal to one. The metric is (+ - - -). Greek indices run from 0 to 3, Latin from 1 to 3, and zr” = t. Ordinary partial differentiation is denoted by a comma.
170
CARMELT
time-differentiation ( io = fro = 1 ), the 6 is the Infeld delta function (5, 8), and ml and m are the inertial masses (functions of time) of the two particles. In Eq. (1) ROp is the Ricci tensor and in a harmonic coordinate system can be written in the form:
We assume that the metric tensor of the space-time differs from the Minkowskian tensor qub by quantities (perturbations), which can be expanded in a power series in the masses of the particles 10
01
20
11
02
gab = q,8 -I- ml gab + m2 sola + m12gd + ml m2 g4 -I- m22ga8+ . . . ,
(4)
where the two indices written as superscripts on the functions appearing in the above expansion denote the powers of the masses ml and r~ respectively. Putting the expansion (4) into the field equations (1) and equating coefficients of equal powers of the masses, we obtain a series of inhomogeneous partial differential equations (wave equations). These correspond to various approximations, the sources in the wave equation for each approximation being expressed in terms of the previous approximations. We further assume that the velocities of the particles are much smaller than the velocity of light c, so that every quantity can be expanded in a power series in c-l, as was done by Einstein, Infeld and Hoffmann (S-5) : (5) m=p+m+m+ where the indices written
3
-es,
4
as lower subscripts
(6)
indicate the order of c-r, and p ( = m) 2
is a constant (5)) hereafter called the proper mass. Thus we have to distinguish between the approximation, determined by the sum of the powers of the masses, and the order, determined by the power of c-l. Note that as ml and m2are functions of time, the Ricci tensor given by Eq. (3) will, in general, contain terms including derivatives of the masseswith respect to the time. Nevertheless, the riz’s are functions of the 5th order (5), and hence will not contribute to the equations of motion up to the 6th order2 being considered here. 2 I.e., the EIH equations. to be the 4th order.
In this
sense,
the
Newton
equations
of motion
are considered
MOTION
III.
IN
A
SOLUTION
GRAVIT;1TIONAL
OF
THE
171
FIELD
FIELD
EQUATIONS
In order to find the equations of motion up to the 6th order, we have to calculate the components of the metric tensor up to the following orders (9) : g,,-4th, gok-3rd and gkl-2nd. In the first approximation we obtain for the field equations: 10
- ?lprgmB.PLl = 1~7d7~44v
-
(7)
?4wd~f"&,
01
and analogous equations
for gcrB. The solutions
for the required
functions
are
10 q,,
=
-
t
611"
10
fOO=O
(8)
10 p
=
2_ '8'S rl F 5
-
10
,
yoz =
z
+
Q,oo
(
>
6";
?.I2 =
(d
-
‘$s)(2"
-
.$").
01
The analogous solutions
for g,, are obtained from (8) by simply replacing r1 and
,$” by rz and 7’ respectively,
ihere
~22= (x8 - 9’) (2” - VJ”). 20
The required
functions
from the second approximation
11
02
are: osoo, p , and go0. 0
Straightforward
calculations
give :
(9) ;o,.w = +),8(k),8
- ++;),
(10)
02
and an analogous equation to (9) for go0. Their
solutions
are :
0
(11) where r2 = (p - v”)(E”
- q”).
172
CARMELI
In the Appendix functions
we express the required
Christoffel
symbols
in terms
of the
il. . n IV.
MOTION
IN
AN
EXTERNAL
GRAVITATIONAL
FIELD
We now proceed to define the external gravitational field (metric) for the particle having the proper-mass ~1 . Using (5) the values of the m’s, we notice that the power series expansion in k
c-l of ml and m2 given by Eq. (6) can also be written proper masses pl and pcLB
as a power
(i = 1,2).
mi = C ah.hkhz; k.z
Putting this in Eq. (4), the metric tensor will then be written in a power series in p1 and pz :3 10 SaS
=
%a
+
/a
01
YolB
+
20
I42 YnS
+
P12 Ye!9
The above expansion can also be written 0 1 SaS
=
YCIB +
YaB
=
YaB
l-42 YaB
as an expansion 02
P2 yal3
+
I422 Yaj3
+
**.
l.12 -rat3
+
. . * ,
+
* * . ;
(14)
n2 +
P22 yolf9
(15)
00
YdJ
(13)
2
al +
I.0
(12)
in the form (p = pI) :
P Ya!3 +
where we have used the notation n 7LO
11 +
series in the
=
r]aB
(n = 0, 1,2, * *I.
,
0
We define the functions ymp as the external metric from the metric tensor by formally putting p = 0: Y&9
(field) ; they are obtained
= [Sablr=0 *
(16)
The definition given by Eq. (16) is also the one accepted by Schild. As a first step in discussing the motion of the first particle in terms of the external metric, let us see whether the geodesic equation in this external metric (17) which
can also be written
in the form (18)
3 We could have carried out the expansion powers of m. However it is more convenient
in the first place in powers to go over from m to fi than
of p instead of in from p to m.
MOTION
IN
A GRAVITATIONAL
FIELD
173
will give the EIH equations. In Eq. ( 18) the zero superscripts on the Christoffel symbols indicate that these quantities are evaulated by means of the external metric G.6 . This means two things: first, terms containing P are ignored and second, in place of the x” the $ are introduced in the Christoffel symbols of the whole field calculated in the Appendix. The comparison between Eqs. (18) and the “general equations of motion” of Infeld (5): {” + (F$
- r$pjf”,$
= 0
(19)
shows that the former differ from the latter only in that the zero superscripts appear in place of the bars. But the bars also mean two things (5) : first, singularities are ignored and second, as before, we have to introduce the $ for xk . It follows that only when all the terms containing P are singular, are the two operations identical. This is the case, for example, for the components of the Christoffel symbols in Eqs. (A.l)-(A.5). It does not hold, however, for the last one given by Eq. (A.6), which contains the two terms 11 ?4ml
m2
The 4th order of this expression ! i/A
P2 ;m
) k
+
01
900 , k +
Mm2
g00 , k.
CJO)
is given by
?4/icL
2~oo,a:+!iu2(Ni’i.+~)~o,~, ;
where use has been made of the relation
(21)
(5) (22)
A simple calculation
then shows that 0 -k roe 4
This shows that the EIH
k
-
roe 4
=
Pl
(23)
1*2
equations are not derivable jrom the geodesic equation in
0
the external metric Y~.J, since a simple calculation shows that they are obtainable from the Infeld equations. Thus the force term given by
00 1
fl”
=
p2p2
;
;
>t
will be lacking in Eqs. (18). But Eqs. (18), up to the 6th order, are identical with those proposed by Schild. This shows that even up to this order, these equations are not valid. The force term jl” is the lower order approximation of what
174
CARMELI
has been called by C. Dewitt and J. L. Ging (10, 11) the ‘%a&” and has been written as an integral of unknown functions. The fact that such a term as flk appears even in the 6th order, in which the problem of gravitational radiation has not yet arisen, prevents a close analogy with the electromagnetic case, since terms proportional to the square of the mass do not necessarily describe reaction forces due to the particle’s radiation; they may also describe interaction terms arising
from the direct interaction and from the corrections included in the inertial masses given in expression (21) by the first and the third terms respectively. V. OTHER
DEFINITIONS
FOR
THE
EXTERNAL
IUETRIC
In the previous section we looked at the metric tensor as a function of the coordinates as well as of the parameter P and defined the external metric by simply putting P = 0 in the former. One may ask, however, if this is the only possibility for defining the external field. Thus, for example, instead of (14), one can write the expansion (4) in the form 0 gai3
=
ga5
1 +
112 hi3
+
m”
id
+
Cm = ml),
. . - ;
where, in anaology with the former case n ?LO gut3 = g&3+ m ;:5 + - . ’ ) 00
(25)
(26)
0
and gas = qo,g. Then the functions g,s may be used as an external metric obtained by putting m = 0 in the expansion (25). This is a modification of our former definition, and it has no analogy in the Maxwell-Lorentz theory. Investigation of the various terms of the Christoffel symbols given by (A.l)(A.6) shows, in this case, that only the first term in the expression (20) will cause a difference between the equations given by (18) and (19), if we now understand 0
that the zero superscripts on I$ will give the force term
0
indicate
calculations
by means of gab. This
(27)
However, one need not confine oneself to the Infeld inertial Eq. (2) ; other definitions are possible (6). Let us define such a mass by the relation T”’ = M,~“&“S, + M27f7jv~2 ,
where TPY is the energy-momentum
mass defined by
(28)
tensor. Ml will then be related to rn! by ml = ~MI,
(29)
MOTION
IN
A
GRAVITATIONAL
175
FIELD
and a simple calculation shows that if we expand Ml in a power as has been done before for ml in Eq. (6), then
series in c-l,
We see that the first relativistic correction to the proper mass is the sum of its kinetic and potential energies. Repeitition of all the above calculations, but using the M’s instead of the m’s, shows that the two equations (18) and (19) will give the same results, i.e., the EIH equations, provided we understand the zero superscripts in Eq. (18) as indicating evaluation by means of the external metric obtained from the whole metric by formally putting M = 0. VI.
co~cm~m~
REMARKS
We have already seenin Section IV that the analogy between the electromagnetic and the gravitational casesis not a close one: terms proportional to the square of the massdo not necessarily describe reaction forces; these appear even in the 6th order, in which the problem of gravitational radiation has not yet arisen. These terms are the self-field terms. What about the external field terms? Has the separation carried out above any physical significance? Consider, for example, the second term in Eq. (A.6)) which is defined as an external field. Its 4th order value will contribute to the equations of motion the expression -4 I@4 ..k T’l’
Here we may introduce for Ilk its Newtonian value ;i” = PlwG7k,
(32)
since the error will affect the equations of motion only in the 8th order. The term (31) will thus give a force term f” = 4L%(Ilr)(Ilr),
t;
(b = c(l).
(33)
But this is identical (except for a numerical coefficient) with the force terms frk and fZk, which are not associated with the external field. This shows that the methods used above cannot give an unambiguous definition of the external field. The difficulty arises from the double role of the self-mass: it is both inertial and gravitational. One can give an unambiguous definition of the external field only by letting the self-masstend to zero, and taking into account the effect of this on the motions of all the particles. In the two-particle case, for example, the external field
CARMELI
176
terms will contribute only three terms out of all the terms in the EIH equations (Is), while the others may be considered as nonexternal field terms. This seems to us an artificial process. Similar conclusions were also reached by Professor P. Havas via the Lorentzinvariant approximation method (13, 14). APPENDIX
We give below the Christoffel symbols needed for calculating the equations of motion up to the sixth order: 10 I?!,
01
=
%rnl
gOO,s +
l20
=
%rnl
g00,0
I?,“,
=
-%ml
rk 3
=
-%m(~0,8
ri0 2
=
?dml
gOO,k +
r!O 4
=
-ml
gkO.0
?4m2
gOO,s
%m2
g00,0
(A.1)
2
01
10
3
+
10
64.2)
10
(gkr,s
+
gks,r
-
L.k)
-
%m,(Z,,8
+
5.,r
-
(A.3)
;,.,,
2
01
10 +
is,0
10
01
%m2(gkO,s
+
gks,O
01 -
(-4.4)
gOs,k)
(A.5)
gO0.k
01 -
m2
+
gkz
10 10
20
gkO,O
+
m12(%gM,k
+
01 01
02 m22&ig00,k
-
01 ?dm2
10
+
-go&k)
go0,z)
10 +
ml
m2
(gkz
gkl
gOO,Z)
01 g00,z
+
01
10
gkz
good
11 +
?4
10 +
%ml
gOO,k -t
(A.61
gOO.k)
01 %m2
gOO,k ,
where in the above equations the expressions on the right-hand side have to be calculated to the same order as the corresponding Christoffel symbols. ACKNOWLEDGMENTS I wish to express by indebtedness and for the many helpful discussions, discussions.
RECEIVED
to Professor Nathan and to Professor
Rosen for suggesting Asher Peres for many
: March 11, 1964 REFERENCES
1. P. A. M. DIRAC, Proc. Roy. Sot. 167A, 148 (1938). 2. A. SCHILD, Bull. Acad. Polon. Sci. 9, 103 (1961). 3. A. EINSTEIN, L. INFELD, AND B. HOFFMANN, Ann. Math. 39, 65 (1938). 4. A. EINSTEIN AND L. INFELD, Can. J. Math. 1, 209 (1949).
the problem stimulating
MOTION
5. 6. 7. 8. 9. 10. 11. fd.
L. A. V. L. A. C. J. J.
IS
A
GRAVITATIONAL
FIELD
177
INFELD, Reu. Mod. Phys. 29, 398 (1957). PERES, Nuouo Ciwento (10) 11, 644 (1959). Pergamon Press, London, 1959. FOCK, “Theory of Space, Time and Gravitation.” INFELD AND J. PLEBANSKI, “Motion and Relativity.” Pergamon Press, Warsaw, 1960. p. 36. King’s College, London, 1958. TRAUTMAN, “Lectures on General Relativity,” DEWITT AND J. L. GING, C. B. Acad. Sci. 261, 1888 (1960). L. GING, Ph.D. Thesis, University of North Carolina, 1960. ?J. GOLDBERG, in “Gravitation, an Introduction to Current Research,” L. Witten, ed. Chap. 3, Eq. (3-3.20). Wiley, New York, 1962. 15. P. HAVAS, Phys. Rea. 108, 1351 (1957). 14. P. HAVAS AND J. N. GOLDBERG, Phys. Rev. 128,398 (1962).