Ngô Quốc Anh

December 11, 2010

The Positive Energy Theorem

Filed under: Riemannian geometry — Ngô Quốc Anh @ 17:01

In general relativity, the positive energy theorem (more commonly known as the positive mass theorem in differential geometry) states that, assuming the dominant energy condition, the mass of an asymptotically flat spacetime is non-negative; furthermore, the mass is zero only for Minkowski spacetime. The theorem is a scalar curvature comparison theorem, with asymptotic boundary conditions, and a corresponding statement of geometric rigidity.

The proof. The original proof of the theorem for ADM mass was provided by Richard Schoen and Shing-Tung Yau in 1979 using variational methods. Edward Witten gave a simpler proof in 1981 based on the use of spinors, inspired by positive energy theorems in the context of supergravity. An extension of the theorem for the Bondi mass was given by Ludvigsen and James Vickers, Gary Horowitz and Malcolm Perry, and Schoen and Yau.

Gary Gibbons, Stephen Hawking, Horowitz and Perry proved extensions of the theorem to asymptotically anti-de Sitter spacetimes and to Einstein–Maxwell theory. The mass of an asymptotically anti-de Sitter spacetime is non-negative and only equal to zero for anti-de Sitter spacetime. In Einstein–Maxwell theory, for a spacetime with electric charge Q and magnetic charge P, the mass of the spacetime satisfies

\displaystyle M \geq \sqrt{Q^2 + P^2},

with equality for the Majumdar–Papapetrou extremal black hole solutions.

The Positive Energy Theorem. One is given a space-time which satisfies Einstein’s equations

\displaystyle R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=8\pi GT_{\mu\nu}.

The only requirement on the energy momentum tensor T_{\mu\nu} is that the local energy density T_{00} is positive (or zero) at each point in space-time and in each local Lorentz frame.

It is assumed, moreover, that in this space- time there exists a space-like hypersurface (which can be regarded as the initial value surface) that is asymptotically Euclidean. More specifically, we suppose that in the vicinity of this space-like hypersurface the metric behaves at spatial infinity as

\displaystyle\begin{gathered} {g_{^{\mu \nu }}} = {\eta _{^{\mu \nu }}} + O\left( {\frac{1}{r}} \right), \hfill \\ \frac{\partial }{{\partial {x^k}}}{g_{^{\mu \nu }}} = O\left( {\frac{1}{{{r^2}}}} \right), \hfill \\ \end{gathered}

where \eta_{\mu\nu} is the flat space metric (signature – + + + ). [The second condition in the above conditions is needed so that the energy integral defined below should converge.] In the proof due to Witten, there is no assumption about the topology of the initial value surface.

The total energy of this system is defined as a surface integral over the asymptotic behavior of the gravitational field,

\displaystyle E = \frac{1}{{16\pi }}\int {\left( {\frac{\partial }{{\partial {x^k}}}{g_{jk}} - \frac{\partial }{{\partial {x^j}}}{g_{kk}}} \right){d^2}{S^j}}

where the integral is evaluated over a bounding surface in the asymptotically flat region of the initial value surface. The problem is to prove that this total energy E is always positive or zero, and zero only for flat Minkowski space.

References:

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