Volume 107A, number 2 PHYSICS LETTERS 14 January 1985 CLASSICAL THEORY OF A SPACE-TIME SHEET M. PAVgI~ J. Stefan Institute, E. Kardel] University of L/ubl/ana, L/ubl/ana, Yugoslavia Received 21 February 1984 Revised manuscript received 15 October 1984 By embedding the space-time V4 in a higher-dimensional space M N we can formulate a theory of gravity in which the true dynamical variables are the coordinates rla(x) (a = 1,2, ..., N) of V 4 with respect to M N. Before constrained by the variational principle, which gives the equations of the four-surface V4, all the coordinates r~a are independent. This enables the canonical formulation of the theory (without additional constraints except for the initial and boundary condi- tions on r/a and ones due to the reparametrization invariance) which is presented here. When expressed in terms of the metric tensor guy of the space-time four-surface V 4 the theory reduces to the Einstein general relativity. Let us assume that the space-time V 4 is embedded in a higher-dimensional pseudoeucfidean space MN, so that it is a four-surface (or sheet or slice) r/a = ~a(x) in MN. Here we parametrize the events in MN by the coordinates r~ a (a = 1, 2 ..... N) and the events on V 4 by xU (/a = 0, 1, 2, 3). As a working hypothesis we shall assume that the dimension N of the embedding space M N is 10; namely, according to the general theorem [ 1,2] every n-dimen- sional riemannian space can be embedded locally in an N-dimensional pseudoeuclidean space MN with N = n(n + 1)/2, so that for n = 4 it is N = 10. As stated already by Fronsdal [ 1], all space-times which have been tested so far experimentally ,1, like the Schwarzschild solution, the Friedmann cosmological solution, etc., can be embedded in M 6. So it seems reasonable to assume that a 10-dimensional embedding space will suffice. We shall not consider the complications which result from Clarke's work [4] which deals with global embedding of a generic space-time (which moreover is not necessarily a solution of Einstein's equations). ,1 We can hardly consider the Penrose plane wave space- time (see ref. [3]) as a physical one, just because no space- like hypersurface exists for the global specification of Cauchy data. 66 In our approach we do not worry about an embedding of a given space-time; in other words, we do not start from the intrinsic geometry of a V4, and then search for its embedding, but on the contrary, we start from the embedding space MN - with agiven dimension (say 10) - in which there exists a four-surface V 4. The latter satisfies a certain variational principle with respect to MN. Moreover, we consider MN as a physical space and not merely as an auxiliary space; all events (with the coordinates r~ a) of M N are physical, though classically an observer is directly aware only of those events which belong to a certain space-time four-sur- face V 4. A given four-surface V 4 is chosen by initial conditions and by the equation of motion for ~Ta(x). Let guy = aurlaavrla be the intrinsic metric of V4, R the Ricci scalar and GUy -RUV _ ~gUVR the Einstein tensor. Let us consider the action w= f£ d4x, £:X/'-~(R/87r+Lm). (1) The variation yields = f + TtaV)fguv~-Z-gd4x + 0 = o, (2) where 0 is the divergence term, having no influence on the equations of motion. If we require that the variations 6guy[ B are zero 0.375-9601/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)