Areas and volumes for null cones. (English) Zbl 1218.83010
Summary: Motivated by recent work of Y. Choquet-Bruhat, P. T. Chruściel and J. M. Martín-García [Classical Quantum Gravity 26, No. 13, Article ID 135011, 22 p. (2009; Zbl 1171.83001)], we prove monotonicity properties and comparison results for the area of slices of the null cone of a point in a Lorentzian manifold. We also prove volume comparison results for subsets of the null cone analogous to the Bishop-Gromov relative volume monotonicity theorem and Günther’s volume comparison theorem. We briefly discuss how these estimates may be used to control the null second fundamental form of slices of the null cone in Ricci-flat Lorentzian four-manifolds with null curvature bounded above.
MSC:
| 83C05 | Einstein’s equations (general structure, canonical formalism, Cauchy problems) |
| 53Z05 | Applications of differential geometry to physics |
| 83C75 | Space-time singularities, cosmic censorship, etc. |
| 83C10 | Equations of motion in general relativity and gravitational theory |
Citations:
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