Geometry of dynamics in general relativity
Annales de l'I.H.P. Physique théorique, Volume 21 (1974) no. 2, p. 175-183
@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] P. Finsler, 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] J. Eells, Jr. and J.H. Sampson, Amer. J. Math., t. 86, 1964, p. 109, I am indebted to C. W. MISNER for this reference. | MR 164306 | Zbl 0122.40102

[3] Y. Nutku, Bull. Amer. Phys. Soc., t. 17, 1972, p. 472.

[4] P.A.M. Dirac, Proc. Roy. Soc. (London), t. A 246, 1958, p. 333. | MR 94206 | Zbl 0080.41403

[5] R. Arnowitt, S. Deser and C.W. Misner, in Gravitation, and Introduction to Current Research, edited by L. WITTEN, Wiley, New York, 1962, chap. 7. | MR 143629

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

[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., H. Iwamoto, Math. Japonicae, t. 1, 1948, p. 74. | MR 28662 | Zbl 0041.30303

[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] H. Weyl, Ann. Phys., t. 54, 1917, p. 117; T. Levi-Civita, Rend. Accad. dei Lincei, 1917-1919. The generalization which includes rotation is due to many authors, see e. g., J. Ehlers, Thesis, Hamburg; A. Papapetrou, Ann. Phys., t. 12, 1953, p. 309.

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

[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] A. Trautman, Reports on Math. Phys., t. 1, 1970, p. 29 (Torun). | Zbl 0204.29802

[14] J. Eells Jr. and J.H. Sampson, Ann. Inst. Fourier, Grenoble, t. 14, n° 1, 1964, p. 61-70. | Numdam | MR 172310 | Zbl 0123.38703

[15] A. Trautman, Bull. Acad. Pol., t. 18, 1970, p. 667. | Zbl 0203.54701

[16] D. Kramer and G. Neugebauer, Ann. Phys., t. 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

[17] S. Sasaki, Tohuku Math. J., t. 10, 1958, p. 338. | MR 112152 | Zbl 0086.15003

[18] R. Utiyama, Phys. Rev., t. 101, 1956, p. 1597. | MR 78223 | Zbl 0070.22102

[19] E.T. Newman and R. Penrose, J. math. Phys., t. 3, 1962, p. 566. | Zbl 0108.40905

[20] R. Geroch, J. Math. Phys., t. 9, 1968, p. 1739. | Zbl 0165.29402