- Email: [email protected]
Coordinate approach to the energy conservation problem in general relativity
Volume 126, number
8,9
PHYSICS
LETTERS
25 January
A
1988
COORDINATE APPROACH TO THE ENERGY CONSERVATION PROBLEM IN GENERAL RELATIVITY Noah NISSAN1 Centerfor Theoretical Physics, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel
and Elhanan
LEIBOWITZ
Department of Mathematics and Department of Physics, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel Received 16 June 1987; revised manuscript Communicated by J.P. Vigier
received
24 November
1987; accepted
for publication
25 November
1987
A coordinate analysis is proposed for the problem of energy-momentum conservation in general relativity. It is shown that there exists a sub-class of geodesic coordinates systems, in which the energy-momentum tensor satisfies a global continuity equation. The physical implication of these frames are discussed.
From the very beginning general relativity bears with difftculties concerning the energy-momentum conservation and the definition of gravitational energy [ 11. The common feature of the conventional approach to this day is the insistence on requiring general covariance of the conservation law [ 2-41, i.e., the search for a set of four continuity equations valid in all coordinate systems. The point of departure of our approach is the recognition that one has to be satisfied with the global conservation of the energy-momentum components only in a preferred class of reference frames. These systems will be called in the following the non-rotating systems, since in them the components of the energy-momentum affme 4-vector (see eq. ( 8)) are conserved separately, that is, they are not mixed with the passage of time. The idea of the physical significance of global preferred systems of coordinates is in line with the approach proposed by Fock [ 5 1, who assigns special physical meaning to harmonic coordinate systems, to which our non-rotating systems are closely related. In searching for such preferred coordinate systems, one considers the covariant divergencelessness condition (implied by the Einstein field equations): 03759601/88/$ (North-Holland
03.50 0 Elsevier Science Publishers Physics Publishing Division)
(1) Then one looks for coordinates (x’) in which the “non-conservative” term will vanish: r’;~(x’)TvP(X’)=o. In this new coordinate gencelessness condition tinuity equation: (J-g
T"),p=O.
(2) system the covariant diver(1) transforms into a con-
(3)
This leads to the following differential equation for the transformation functions for the non-rotating coordinates:
=o
(4)
where G is the Einstein tensor. Notice that this equation has the same form as the one defining the harmonic coordinates, in terms of which the metric tensor density has a vanishing ordinary divergence. The internal group of this preferred class of systems is defined by B.V.
447
Volume 126, number
8.9
PHYSICS
0 Fig.
PU(l,)=
=6”
lr’
=
I
a2x’a axfiax7 =o.
____
,A
To check the compatibility of the above boundary condition with eq. (4)) notice that it satisfies eq. (4) on the line A. Furthermore, eq. (6) shows that the non-rotating-geodesic coordinates obtained thereby differ from the given arbitrary geodesic system only in a third order term in the distance from the worldline of the observer. Obviously, in an empty region of space eq. (4) reduces to an identity. Accordingly, outside of matter we can choose the coordinate system in an arbitrary manner subject to continuity conditions. Having established the existence of these preferred geodesic systems, we turn our attention to a region of 3-space surrounded by a surface on which the energy-momentum tensor vanishes. Between the times t, and t2 this region traces a 4-volume D in space-time (see fig. 1). Applying the Gauss theorem in the usual manner one obtains in a non-rotatinggeodesic frame, where (y/-g
448
Tafl),o d4x= 0,
25 January
1988
j \ ‘-g \!rg
Tno d’x
T”” d3x=Pa(t2).
t1
As should be expected on physical grounds, it contains the group of affne transformations. Insight into the physical meaning of these new coordinates is gained by realizing that among the geodesic systems, with respect to the world-line of an observer, there exists a sub-class of non-rotating-geodesic coordinates. The transformation function from any ordinary geodesic system to a non-rotating-geodesic system is obtained [ 41 by solving eq. (4) with the following boundary condition imposed along the world-line A of the observer:
axp 4
A
that
t2
(5)
ax’s
LETTERS
(7)
Hence, one finds four conserved integrals that are interpreted as the energy-momentum aftine 4-vector content of the 3-space region. Notice that the last equation is the expression of a conservation law not subject to the restriction of being local only, as is the case in generic geodesic systems. It is to be stressed here that construing these integrals as the energy-momentum content is not an arbitrary identification concocted in order to establish energy conservation. Since the non-rotating frame employed is not only geodesic but can be chosen identical (up to third order) with the most “physical” coordinate system conceivable, the components of P” in eq. (8) are, for all intents and purspecial relativistic energyposes, the familiar momentum quantities as measured in the laboratory system. To analyze the physical implication of the existence of non-rotating-geodesic systems, whereby the energy-momentum is globally conserved, consider a Weber antenna tracing a geodesic path in a totally empty region of space. At a certain moment a gravitational wave reaches the antenna. For the sake of this discussion, assume, as is generally done in the conventional approach, that the gravitational energy is not contained in the energy-momentum tensor. Thus the energy-momentum tensor vanishes on a Gauss surface surrounding the antenna even while the gravitational wave transverses it. In the non-rotating-geodesic systems of coordinates, under this condition, as the foregoing discussion has shown, the energy-momentum 4-vector of the antenna is strictly conserved (eq. ( 8)). Of course this conclusion holds only under the assumption that no part of the gravitational energy is included in the energy-momentum tensor. Notice that such an argument does not apply m generic geodesic systems during the passage of the wave. The reason is that since the dimension of the antenna ought to be of the same order of magnitude as the wave length of the incident radiation, the in-
Volume
126, number
8,9
PHYSICS
homogeneity of the gravitational field precludes extending the local conservation law to a volume containing the antenna. It is instructive to compare this situation with another experimental setting, whereby an electromagnetic, instead of gravitational, wave is detected. Naturally, in this case the Weber antenna is replaced by a Hertz antenna, but otherwise the circumstances are identical. Under the present conditions, in a nonrotating-geodesic system, eq. (3) still holds (exactly) at all times and everywhere. Now, however, instead of eq. (8), the Gauss theorem leads to r rTao d3x-g Tao d3x JJ
JJ -g IZ
LETTERS
A
25 January
1988
An example of non-rotating and non-rotating-geodesic systems where an exact explicit solution of eq. (4) can be written down is given elsewhere [ 41. That is the case of a cloud of dust with the energy-momentum tensor given by T~=puW’,
(10)
but otherwise the metric is not restricted. The comoving coordinates are shown to be non-rotating and the construction of non-rotating-geodesic frames is carried out. The foregoing analysis demonstrates that the existence of non-rotating systems has physical implications, and in particular sheds light on the physical conservation of energy without resort to extraneous pseudo-tensors.
(9)
=-
,I .? References i.e., the net change in the energy-momentum content of the antenna is equal to the flux of the electromagnetic energy-momentum through the Gauss surface. This flux term has no counterpart in the gravitational wave experiment since it is being assumed that the energy-momentum tensor does not include any contribution due to gravitation.
A. Einstein, Ann. Phys. (Leipzig) 49 (1916) 769. C. Moller,Ann. Phys. 12 (1961) 118. R. Penrose, Proc. R. Sot. A 38 1 (1982) 53. N. Nissani and E. Leibowitz, Ben-Gurion University, preprint 28 (1987). [5] V. Fock, The theory of space time and gravitation (Pergamon, New York, 1959) p. XV and section 93. [ 1] [2] [ 31 [4]
449
8,9
PHYSICS
LETTERS
25 January
A
1988
COORDINATE APPROACH TO THE ENERGY CONSERVATION PROBLEM IN GENERAL RELATIVITY Noah NISSAN1 Centerfor Theoretical Physics, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel
and Elhanan
LEIBOWITZ
Department of Mathematics and Department of Physics, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel Received 16 June 1987; revised manuscript Communicated by J.P. Vigier
received
24 November
1987; accepted
for publication
25 November
1987
A coordinate analysis is proposed for the problem of energy-momentum conservation in general relativity. It is shown that there exists a sub-class of geodesic coordinates systems, in which the energy-momentum tensor satisfies a global continuity equation. The physical implication of these frames are discussed.
From the very beginning general relativity bears with difftculties concerning the energy-momentum conservation and the definition of gravitational energy [ 11. The common feature of the conventional approach to this day is the insistence on requiring general covariance of the conservation law [ 2-41, i.e., the search for a set of four continuity equations valid in all coordinate systems. The point of departure of our approach is the recognition that one has to be satisfied with the global conservation of the energy-momentum components only in a preferred class of reference frames. These systems will be called in the following the non-rotating systems, since in them the components of the energy-momentum affme 4-vector (see eq. ( 8)) are conserved separately, that is, they are not mixed with the passage of time. The idea of the physical significance of global preferred systems of coordinates is in line with the approach proposed by Fock [ 5 1, who assigns special physical meaning to harmonic coordinate systems, to which our non-rotating systems are closely related. In searching for such preferred coordinate systems, one considers the covariant divergencelessness condition (implied by the Einstein field equations): 03759601/88/$ (North-Holland
03.50 0 Elsevier Science Publishers Physics Publishing Division)
(1) Then one looks for coordinates (x’) in which the “non-conservative” term will vanish: r’;~(x’)TvP(X’)=o. In this new coordinate gencelessness condition tinuity equation: (J-g
T"),p=O.
(2) system the covariant diver(1) transforms into a con-
(3)
This leads to the following differential equation for the transformation functions for the non-rotating coordinates:
=o
(4)
where G is the Einstein tensor. Notice that this equation has the same form as the one defining the harmonic coordinates, in terms of which the metric tensor density has a vanishing ordinary divergence. The internal group of this preferred class of systems is defined by B.V.
447
Volume 126, number
8.9
PHYSICS
0 Fig.
PU(l,)=
=6”
lr’
=
I
a2x’a axfiax7 =o.
____
,A
To check the compatibility of the above boundary condition with eq. (4)) notice that it satisfies eq. (4) on the line A. Furthermore, eq. (6) shows that the non-rotating-geodesic coordinates obtained thereby differ from the given arbitrary geodesic system only in a third order term in the distance from the worldline of the observer. Obviously, in an empty region of space eq. (4) reduces to an identity. Accordingly, outside of matter we can choose the coordinate system in an arbitrary manner subject to continuity conditions. Having established the existence of these preferred geodesic systems, we turn our attention to a region of 3-space surrounded by a surface on which the energy-momentum tensor vanishes. Between the times t, and t2 this region traces a 4-volume D in space-time (see fig. 1). Applying the Gauss theorem in the usual manner one obtains in a non-rotatinggeodesic frame, where (y/-g
448
Tafl),o d4x= 0,
25 January
1988
j \ ‘-g \!rg
Tno d’x
T”” d3x=Pa(t2).
t1
As should be expected on physical grounds, it contains the group of affne transformations. Insight into the physical meaning of these new coordinates is gained by realizing that among the geodesic systems, with respect to the world-line of an observer, there exists a sub-class of non-rotating-geodesic coordinates. The transformation function from any ordinary geodesic system to a non-rotating-geodesic system is obtained [ 41 by solving eq. (4) with the following boundary condition imposed along the world-line A of the observer:
axp 4
A
that
t2
(5)
ax’s
LETTERS
(7)
Hence, one finds four conserved integrals that are interpreted as the energy-momentum aftine 4-vector content of the 3-space region. Notice that the last equation is the expression of a conservation law not subject to the restriction of being local only, as is the case in generic geodesic systems. It is to be stressed here that construing these integrals as the energy-momentum content is not an arbitrary identification concocted in order to establish energy conservation. Since the non-rotating frame employed is not only geodesic but can be chosen identical (up to third order) with the most “physical” coordinate system conceivable, the components of P” in eq. (8) are, for all intents and purspecial relativistic energyposes, the familiar momentum quantities as measured in the laboratory system. To analyze the physical implication of the existence of non-rotating-geodesic systems, whereby the energy-momentum is globally conserved, consider a Weber antenna tracing a geodesic path in a totally empty region of space. At a certain moment a gravitational wave reaches the antenna. For the sake of this discussion, assume, as is generally done in the conventional approach, that the gravitational energy is not contained in the energy-momentum tensor. Thus the energy-momentum tensor vanishes on a Gauss surface surrounding the antenna even while the gravitational wave transverses it. In the non-rotating-geodesic systems of coordinates, under this condition, as the foregoing discussion has shown, the energy-momentum 4-vector of the antenna is strictly conserved (eq. ( 8)). Of course this conclusion holds only under the assumption that no part of the gravitational energy is included in the energy-momentum tensor. Notice that such an argument does not apply m generic geodesic systems during the passage of the wave. The reason is that since the dimension of the antenna ought to be of the same order of magnitude as the wave length of the incident radiation, the in-
Volume
126, number
8,9
PHYSICS
homogeneity of the gravitational field precludes extending the local conservation law to a volume containing the antenna. It is instructive to compare this situation with another experimental setting, whereby an electromagnetic, instead of gravitational, wave is detected. Naturally, in this case the Weber antenna is replaced by a Hertz antenna, but otherwise the circumstances are identical. Under the present conditions, in a nonrotating-geodesic system, eq. (3) still holds (exactly) at all times and everywhere. Now, however, instead of eq. (8), the Gauss theorem leads to r rTao d3x-g Tao d3x JJ
JJ -g IZ
LETTERS
A
25 January
1988
An example of non-rotating and non-rotating-geodesic systems where an exact explicit solution of eq. (4) can be written down is given elsewhere [ 41. That is the case of a cloud of dust with the energy-momentum tensor given by T~=puW’,
(10)
but otherwise the metric is not restricted. The comoving coordinates are shown to be non-rotating and the construction of non-rotating-geodesic frames is carried out. The foregoing analysis demonstrates that the existence of non-rotating systems has physical implications, and in particular sheds light on the physical conservation of energy without resort to extraneous pseudo-tensors.
(9)
=-
,I .? References i.e., the net change in the energy-momentum content of the antenna is equal to the flux of the electromagnetic energy-momentum through the Gauss surface. This flux term has no counterpart in the gravitational wave experiment since it is being assumed that the energy-momentum tensor does not include any contribution due to gravitation.
A. Einstein, Ann. Phys. (Leipzig) 49 (1916) 769. C. Moller,Ann. Phys. 12 (1961) 118. R. Penrose, Proc. R. Sot. A 38 1 (1982) 53. N. Nissani and E. Leibowitz, Ben-Gurion University, preprint 28 (1987). [5] V. Fock, The theory of space time and gravitation (Pergamon, New York, 1959) p. XV and section 93. [ 1] [2] [ 31 [4]
449