@article{AIHPA_1974__21_2_175_0, author = {Nutku, Yavuz}, title = {Geometry of dynamics in general relativity}, journal = {Annales de l'I.H.P. Physique th\'eorique}, publisher = {Gauthier-Villars}, volume = {21}, number = {2}, year = {1974}, pages = {175-183}, zbl = {0295.53033}, mrnumber = {408724}, language = {en}, url = {http://www.numdam.org/item/AIHPA_1974__21_2_175_0} }

Nutku, Yavuz. Geometry of dynamics in general relativity. Annales de l'I.H.P. Physique théorique, Volume 21 (1974) no. 2, pp. 175-183. http://www.numdam.org/item/AIHPA_1974__21_2_175_0/

[1] Thesis, Göttingen, 1918. J. W. York has kindly informed me that the has also been thinking along these lines. A standard text on this subject is H. RUND, Hamilton-Jacobi Theory in the Calculus of Variations, D. Van Nostrand Co., 1966.

,[2] 86, 1964, p. 109, I am indebted to C. W. MISNER for this reference. | MR 164306 | Zbl 0122.40102

and , Amer. J. Math., t.[3] 17, 1972, p. 472.

, Bull. Amer. Phys. Soc., t.[4] 246, 1958, p. 333. | MR 94206 | Zbl 0080.41403

, Proc. Roy. Soc. (London), t. A[5] Gravitation, and Introduction to Current Research, edited by L. WITTEN, Wiley, New York, 1962, chap. 7. | MR 143629

, and , in[6] 160, 1967, p. 5; in Battelle Rencontres, edited by C. M. De Witt and J. A. Wheeler, W. A. Benjamin, Inc., 1968. | MR 232631

, Phys. Rev., t.[7] Here and in the following the adjective « Riemannian » will always be understood to occur with the prefix « pseudo », denoting that the signature is not necessarily positive definite.

[8] See e. g., 1, 1948, p. 74. | MR 28662 | Zbl 0041.30303

, Math. Japonicae, t.[9] With only one parameter the invariance of the action requires that the Lagrangean be homogeneous of degree one in the velocities and the Euler identities follow. Now the 4-dimensional space-time manifold is the range of the parameters and we have the Noether identities. They consist of the necessary and sufficient conditions for defining a globally invariant volume element for Riemannian geometry, Ricci's lemma, contracted Bianchi identities and the identically vanishing Hamiltonian tensor. It is this last identity that plays a fundamental role here.

[10] 54, 1917, p. 117; , Rend. Accad. dei Lincei, 1917-1919. The generalization which includes rotation is due to many authors, see e. g., , Thesis, Hamburg; , Ann. Phys., t. 12, 1953, p. 309.

, Ann. Phys., t.[11] This was first recognized by 154, 1967, p. 1229, but as they first imposed a coordinate condition, their metric analogous to our Equation (3) is not the full metric.

and , Phys. Rev., t.[12] One possible generalization, cf. ref. 2, is to take for the two 2-covariant tensors the metric g and the Ricci tensor R on M, whereupon g, R will be Einstein's Lagrangean, but in this case we cannot introduce a metric such as g'.

[13] 1, 1970, p. 29 (Torun). | Zbl 0204.29802

, Reports on Math. Phys., t.[14] 14, n° 1, 1964, p. 61-70. | Numdam | MR 172310 | Zbl 0123.38703

. and , Ann. Inst. Fourier, Grenoble, t.[15] 18, 1970, p. 667. | Zbl 0203.54701

, Bull. Acad. Pol., t.[16] 24, 1969, p. 62. I. am indebted to D. Brill for this reference. These authors exploit the isometries of Equation (2) in deriving Equation (3) which shows that Equation (3) is a metric of the type discussed by Trautman in the previous reference. | MR 261931

and , Ann. Phys., t.[17] 10, 1958, p. 338. | MR 112152 | Zbl 0086.15003

, Tohuku Math. J., t.[18] 101, 1956, p. 1597. | MR 78223 | Zbl 0070.22102

, Phys. Rev., t.[19] 3, 1962, p. 566. | Zbl 0108.40905

and , J. math. Phys., t.[20] 9, 1968, p. 1739. | Zbl 0165.29402

, J. Math. Phys., t.