Summary
The problem of the proper formulation of the conservation laws of energy and linear momentum in general relativity is discussed. The Komar expression, taken as a generally covariant conservation law generator, is considered in light of the problem which Møller found with his energy-momentum complex. The conservation laws as formulated here from the Komar generator are shown to be devoid of such difficulties. Also, a generally covariant formulation of the conservation laws, which are differentiated from a covariant conservation-law generator, is effectively achieved without the necessity of introducing a tetrad structure. In this formulation the conservation lawsper se are noncovariant quantities; however, one may always return to the Komar generator and transform in a generally covariant manner any given conservation law, through its underlying symmetry representation, to any space-time co-ordinate system of interest. It is also possible, at least in some simple space-time systems, to formulate the energy and momentum when they are not rigorously conserved entities. Such expressions can have possible application, for example, in radiating gravitational systems.
Riassunto
Si discute il problema della formulazione appropriata delle leggi di conservazione dell’energia e dell’impulso lineare in relatività generale. Si studia l’espressione di Komar, considerata come generatrice di una legge di conservazione generalmente covariante, alla luce del problema trovato da Møller con il suo complesso energia-impulso. Si dimostra che le leggi di conservazione, quali sono formulate qui sulla base dell’espressione di Komar, sono esenti da queste difficoltà. Inoltre si ottiene effettivamente una formulazione generalmente covariante delle leggi di conservazione, che sono differenziate da un generatore covariante delle leggi di conservazione, senza che sia necessario introdurre una struttura tetradica. In questa formulazione le leggi di conservazioneper se sono quantità non covarianti; tuttavia, si può sempre tornare al generatore di Komar e trasformare in modo generalmente covariante ogni data legge di conservazione, tramite la rappresentazione di simmetria che sta alla base, in qualsivoglia sistema di coordinate spazio-temporali. È anche possibile, almeno in alcuni semplici sistemi spazio-temporali, esprimere in formule l’energia e l’impulso quando questi non sono entità rigorosamente conservate. Queste espressioni possono avere applicazione, per esempio, nei sistemi gravitazionali radiativi.
Реэюме
Обсуждается проблема соответствуюшей формулировки эаконов сохранения для знергии и импульса в обшей теории относительности. Выражениe Комара, вэятое как генератор обше ковариантного эакона сохранения, рассматри-вается в свете проблемы, которую обнаружил Меллер с помошью своего знергетически-им пульсного комплекса. Покаэывается, что эаконы сохранения, выведенные эдесь иэ генератора Комара, свободны от таких трудностей. Также беэ необходимости введения тетрадной структуры, получается обше ковариантная формулировка эаконов сохранения, которые отличаются от ковариантного оператора эаконов сохранения. В зтой формулировке эаконы сохранения сами по себе не являются ковариантными величинами, однако, всегда можно вернуться к генератору Комара и преобраэовать, в обшем смысле, ковариантным обраэом любой эаданный эакон сохранения, посредством его основного симметричного представления, к любой интересуюшей нас пространственн о-временной системе координат. Также воэможно, по крайней мере, в некоторых простых пространственн о-временных системах, сформулировать знергию и импульс, когда они не являются строго сохраняюшимися величинами. Такие выражения могут быть испольэованы, например, в иэлучаюших гравитационных системах.
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References
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In particular, see the problem pointed out byH. Bauer (Phys. Zeits.,19, 163 (1918)) and discussed in ref. (3).
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The expressions (11) and (12) have also been obtained byPirani (7) by a completely different approach from that ofKomar. These expressions may also be derived by a variational principle based on the scalar curvature density (W. R. Davis andM. K. Moss:Journ. Math. Phys.,7, 975 (1966)).
In view of the developments considered here, the lack of this dependence upon theξ i would have to be considered a shortcoming of these formulations. Also, seeP. G. Bergmann:Phys. Rev.,112, 287 (1958).
W. R. Davis andM. K. Moss:Nuovo Cimento,38, 1531 (1965);27, 1492 (1963);A. Komar:Phys. Rev.,127, 1411 (1962).
For a more general discussion, seeW. R. Davis andM. K. Moss:Nuovo Cimento,27, 1492 (1963).
This viewpoint is in definite agreement, for example, with that expressed byKomar (A. Komar:Phys. Rev.,127, 1411 (1962)).Komar observes that when one goes to more general Riemannian manifolds that the Killing vectors, when available, appear to be the appropriate choice to take for theξ i in order to have conservation laws which correspond naturally with those of Lorentz covariant theories. Also,Trautman has noted the significance in general relativity of the existence of Killing vectors. (A. Trautman:Lectures on Relativity, Kings College, London, 1958 (unpublished)).
L. P. Eisenhart:Riemannian Geometry, Chap. VI (Princeton, N. J., 1926).
For example, seeW. R. Davis andM. K. Moss:Nuovo Cimento,38 1558 (1965).
For an interesting example involving a quadratic integral of the Friedmann-Lemaitre cosmological solution, which has its basis in geodesic correspondence, seeDavis andMoss (12)W. R. Davis andM. K. Moss:.
J. G. Fletcher:Rev. Mod. Phys.,32, 65 (1960).
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Moss, M.K. Further consideration of energy and momentum in general relativity. Nuovo Cimento B (1965-1970) 57, 257–270 (1968). https://doi.org/10.1007/BF02710199
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DOI: https://doi.org/10.1007/BF02710199