Summary
Investigation of the relationship between conservation laws in general relativity and certain types of « generalized symmetry properties » (generated by transformations which map space-times onto other space-times which are not describable as co-ordinate mappings) yields, in contrast to the case for co-ordinate mappings, a very restricted class of possible formal conservation expressions. Here we consider in particular the affine, projective, and conformal correspondence of Riemannian space-times. Particle and field conservation-law generators are formulated as well as the conditions necessary for conservation laws and related symmetry properties to be admitted. The particle case is illustrated by a well-known quadratic first integral of the geodesic equations of the Friedmann-Lemaître cosmological space-time which follows in consequence of a projective symmetry property (geodesic correspondence). The field case is illustrated by formulating a conservation law following in consequence of a conformal symmetry property manifested by the familiar plane gravitational wave solutions which are included by some of the earlier work of Brinkmann.
Riassunto
Lo studio delle relazioni fra le leggi di conservazione nella relatività generale ed alcuni tipi di « proprietà di simmetria generalizzate » (generate da trasformazioni che rappresentano gli spazio-tempo in altri spazio-tempo non descrivibili come rappresentazioni di coordinate) fornisce contrariamente al caso delle rappresentazioni di coordinate, una classe molto ristretta di possibili espressioni formali di conservazione. Qui si considera in particolare la corrispondenza affine, proiettiva e conforme dello spaziotempo di Rieman. Si esprimono sia i generatori delle leggi di conservazione dei campi e delle particelle sia le condizioni necessarie perché siano ammesse le leggi di conservazione e le proprietà di simmetria collegate. Si illustra il caso delle particelle con un integrale primo quadratico ben noto delle equazioni geodesiche dello spazio-tempo cosmologico di Friedmann-Lemaître che deriva da una proprietà proiettiva della simmetria (corrispondenza geodesica). Si illustra il caso dei campi formulando una legge di conservazione derivante da una proprietà conforme della simmetria manifestata dalle familiari soluzioni dell’onda piana gravitazionale; soluzioni queste trattate da alcuni dei lavori precedenti di Brinckmann.
Реэюме
Проводится исследование соотнощения между эаконами сохране-ния в обшей теории относительности и некоторыми типами « свойств обобшенной симметрии » (обраэованными преобраэованиями, которые отображают пространство и время на другие пространство и время, которые не могут быть описаны, как ото-бражение координат), в противоположность случаю отображения координат, очень ограниченного класса воэможных формальных выражений сохранения. В частности, мы эдесь рассматриваем аффинное, проективное и конформное соот-ветствие пространства-времени Римана. Формулируются генераторы эаконов сох-ранения для частиц и для полей, а также условия, необходимые, чтобы допускались эаконы сохранения и свойства свяэанной симметрии. Иллюстрируется случай частиц с помошью хорощо иэвестного квадратичного первого интеграла для геодеэических уравнений космологического пространства-времени Фридмана-Леметра, который вытекает, как следствие свойства проективной симметрии (геодеэическое соответ-ствие). Рассматривается случай поля посредством формулирования эакона сох-ранения, вытекаюшего, как следствие свойства конформной симметрии, которое проявляется благодаря рещениям обычной плоской гравитационной волны, которые учитываются с помошью более ранней работы Бринкмана.
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References
W. R. Davis andM. K. Moss:Nuovo Cimento,38, 1531, 1558 (1965).
W. R. Davis andM. K. Moss:Nuovo Cimento,27, 1492 (1963).
In the previous papers of this series (ref. (1,2)), the term « symmetry property » referred to detailed functional form invariance at the co-ordinate level (described by Lie derivatives). However, in the present work the transformations considered do not involve a change of co-ordinates. Therefore, the question of whether a given quantity is altered as a function of the co-ordinates does not arise.
As a general reference to this Section, seeL. P. Eisenhart:Riemannian Geometry (Princeton, N. Y., 1926);Non-Riemannian Geometry (New York, 1927);J. A. Schouten:Ricci-Calculus, II Ed. (Berlin, 1954).
As a somewhat more general example of such prospective symmetry properties one has the so-called subprojective transformations discussed, for example, byJ. A. Schouten:Ricci-Calculus, II Ed. (Berlin, 1954).
L. P. Eisenhart:Riemannian Geometry (Princeton, N. J., 1926), p. 128.
Here the question of formulating conservation-law generators for higher-order first integrals for general symmetry properties will not be considered; however, it is clear that one could proceed in much the same manner as in II (W. R. Davis andM. K. Moss:Nuovo Cimento,38, 1558 (1965)) where the higher-order first-integral conservation law generators were formulated corresponding to co-ordinate symmetry properties. Also, in this connection seeG. H. Katzin andJ. Levine:Journ. Math. Phys.,9, 8 (1968).
See, for example,R. C. Tolmann:Relativity, Thermodynamics and Cosmology Chap. X (Oxford, 1934).
C. F. Martin: Dissertation, North Carolina State University (Raleigh, N. C., 1962).
L. P. Eisenhart:Riemannian Geometry, Chap. III (Princeton, N. J., 1926).
R. C. Tolman:Relativity, Thermodynamics and Cosmology (Oxford, 1934), p. 384.
H. W. Brinkmann:Math. Ann.,94, 119 (1925).
See, for example, the article ofJ. Ehlers andW. Kundt:Gravitation: An Introduction to Current Research, edited byL. Witten (New York, 1962).
The simplest generally covariant conservation expression\([\sqrt { - g} (\xi ^{i;j} - \xi ^{j;i} )_{;j} ]_{,i} = 0\) was first introduced byA. Komar:Phys. Rev.,113, 934 (1959). The use of this expression as a conservation law generator in terms of the « symmetry method » is discussed in ref. (1,2). Another particularly simpler covariant conservation law generator is provided by [(U [ij] k ξ k )];i,j =0 whereU [ij] k is the tetrad superpotential first employed byC. Møller:Mat. Fys. Dan. Vid. Selsk.,1, No. 10 (1961). For a discussion of covariant tetrad conservation laws seeW. R. Davis andJ. W. York jr.:Nuovo Cimento 61 B, 271 (1969).
For example, one could consider the « curvature symmetry » defined by\(\bar R_{ijk}^m = R_{ijk}^m \), or infinitesimally byδR m ijk =0, which would be a generalization of the co-ordinate symmetry property called a « curvature collineation » defined by\(\mathop \pounds\limits_{\xi ^i } R_{ijk}^m = 0\) (here\(\mathop \pounds\limits_{\xi ^i } \) denotes the Lie derivative with respect to theξ i). For the curvature symmetry not to reduce to a « co-ordinate » symmetry it follows that we must haveδR m ijk =0 where no solution of\(\mathop \pounds\limits_{\xi ^i } R_{ijk}^m = 0\) (orδg ij =ξ i;j +ξ j;i ) exists in terms of anξ i field. For a discussion of curvature collineations seeG. H. Katzin, J. Levine andW. R. Davis:Journ. Math. Phys,10, 617 (1969).
In connection with the question of an extended general invariance group, (associated with a generalized variational formulation of general relativity) leading to the full Bianchi identities seeW. R. Davis:Ann. der Phys.,22, 77 (1968).
For discussions of particle conservation laws in the form ofm-th-order first integrals of the geodesic equations, which follow in consequence of given space-times admitting symmetry properties that can be described in terms of co-ordinate mappings, see, for example,W. R. Davis andM. K. Moss:Nuovo Cimento,38, 1558 (1965);G. H. Katzin andJ. Levine:Journ. Math. Phys.,9, 8 (1968);Tensor,19, 317 (1968);G. H. Katzin, J. Levine andW. R. Davis:Journ. Math. Phys.,10, 617 (1969). Also, it might be mentioned that the following interesting theorem can be stated for spaces of constant curvatureK n : « The geodesics in aK n admitN m linearly independentm-th-order first integrals whereN m =n(n+1)2(n+2)2...(n+m−1)2(n+m)/m!(m+1)! andN m is the maximum number of suchm-th-order first integrals admitted by aK n » (G. H. Katzin andJ. Levine:Tensor,19, 42 (1968)). This theorem is of particular interest in that it probably sets an upper bound for the maximum number of linearly independentm-th-order first integrals that can be realized in any Riemannian spaceV n in that a spaceK n , or a flat spaceS n , presumably admits the maximum number of symmetry groups.
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Davis, W.R., Moss, M.K. & York, J.W. Conservation laws in the general theory of relativity. Nuovo Cimento B (1965-1970) 65, 19–32 (1970). https://doi.org/10.1007/BF02711611
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DOI: https://doi.org/10.1007/BF02711611