An integral equation for space-time curvature in general relativity. (English) Zbl 1153.83018
Yau, Shing Tung (ed.), Essays in geometry in memory of S. S. Chern. Somerville, MA: International Press (ISBN 978-1-57146-116-2/hbk). Surveys in Differential Geometry 10, 109-146 (2006).
An integral equation for the curvature of vacuum solutions of Einstein’s field equations is derived. It is proved that the curvature depends on the integrals over the past light cone, the initial data surface and the interior of the light cone, where the last one results from the violation of Huygens principle in curved space. These contributions can be reformulated to pure light cone terms and initial data surface terms. Explicit expressions for the frames and the connection one-forms are found. Several remarks to approximate Killing fields and cosmic censorship conjecture are discussed.
For the entire collection see [Zbl 1117.53003].
For the entire collection see [Zbl 1117.53003].
Reviewer: Thoralf Chrobok (Berlin)
MSC:
| 83C05 | Einstein’s equations (general structure, canonical formalism, Cauchy problems) |
| 83C35 | Gravitational waves |
| 83C75 | Space-time singularities, cosmic censorship, etc. |
| 53B21 | Methods of local Riemannian geometry |
| 53Z05 | Applications of differential geometry to physics |
| 53C25 | Special Riemannian manifolds (Einstein, Sasakian, etc.) |
