Conditions for flatness in general relativity: R. Arnowitt and S. Deser, Brandeis University, Waltham, Massachusetts

• Publisher: Elsevier Science
• Year: 1963
• Tongue: English
• Weight: 79 KB
• Volume: 22
• Category: Article
• ISSN: 0003-4916
• DOI: 10.1016/0003-4916(63)90301-4

✦ Synopsis

In this note, a number of conditions which are needed to enforce flatness on an empty Einstein space are derived. The theorems fall into two categories: those which involve conditions throughout space-time and those which impose them only on initial Cauchy data. In the former class, it is shown that if the second fundamental form (essentially the first time derivative of the metric) vanishes throughout four space, space is flat. In the latter class, it is shown that if on an initial minimal surface, three space is flat, or if the two pairs of canonical variables of the theory vanish in one of the previously obtained canonical frames, then four space is flat. A generalization of Birkhoff's theorem is also established: if an initial state defined on a three surface is spherically symmetric in the exterior region, the exterior four space is Schwarzschildian.

All theorems assume one is dealing with regular asymptotically flat solutions of the Einstein equations.