The Hamiltonian mass of asymptotically anti-de Sitter space-times
نویسندگان
چکیده
We give a Hamiltonian definition of mass for spacelike hypersurfaces in spacetimes with metrics which are asymptotic to the anti-de Sitter one, or to a class of generalizations thereof. We present the results of another paper (Chruściel P T and Nagy G 2001b, in preparation) which show that our definition provides a geometric invariant for a spacelike hypersurface S embedded in a space-time (M, g). Some further global invariants are also given. PACS numbers: 1130, 0420, 0240M Let S be an n-dimensional spacelike hypersurface in a (n + 1)-dimensional Lorentzian spacetime (M, g). Suppose that M contains an open set U which is covered by a finite number of coordinate charts (t, r, v), with r ∈ [R,∞), and with (v), local coordinates on some compact (n−1)-dimensional manifold M , such that S ∩U = {t = 0}. Assume that the metric g approaches a background metric b of the form b = −a−2(r)dt2 + a(r)dr + rh, h = hAB(v)dvdv, (1) with a(r) = 1/ √ r2/ 2 + k, where h is a Riemannian metric on M , k is a constant, and is a strictly positive constant related to the cosmological constant by the formula 2 = −n(n− 1)/ 2 (somewhat more general metrics are considered in [14]). For example, if h is the standard round metric on S2 and k = 1, then b is the anti-de Sitter metric. To make the approach rates precise it is convenient to introduce an orthonormal frame for b, e0 = a(r)∂t , e1 = 1 a(r) ∂r , eA = 1 r βA, (2) with βA, an h-orthonormal frame on (M, h), so that bab = b(ea, eb) = ηab, the usual Minkowski matrix diag(−1,+1, . . . ,+1). We then require that the frame components gab 1 Supported in part by the Polish Research Council grant KBN 2 P03B 073 15. 2 Supported by a grant from Région Centre. 0264-9381/01/090061+08$30.00 © 2001 IOP Publishing Ltd Printed in the UK L61 L62 Letter to the Editor of g with respect to the frame (2) satisfy3 e = O(r−β), ea(e) = O(r−β), babe = O(r−γ ), (3) where e = g − b, with
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The Hamiltonian mass and asymptotically anti-de Sitter space-times
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