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:
Zbl 1171.83001References:
[1] | Alexander, S. B.; Bishop, R. L., Lorentz and semi-Riemannian spaces with Alexandrov curvature bounds, Comm. Anal. Geom., 16, 251-282 (2008) · Zbl 1149.53040 |
[2] | Cheeger, J., Ebin, D. G.: Comparison theorems in Riemannian geometry. North-Holland Mathematical Library, vol. 9. North-Holland Publishing Co., Amsterdam (1975) · Zbl 0309.53035 |
[3] | Cheeger, J.; Gromov, M.; Taylor, M., Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds, J. Differ. Geom., 17, 15-53 (1982) · Zbl 0493.53035 |
[4] | Choquet-Bruhat, Y.; Chruściel, P. T.; Martín-García, J. M., The light-cone theorem, Class. Quantum Gravity, 26, 135011, 22 (2009) · Zbl 1171.83001 |
[5] | Ehrlich, P. E., Kim, S.-B.: Riccati and index comparison methods in Lorentzian and Riemannian geometry. In: Advances in Lorentzian geometry, pp. 63-75. Shaker Verlag, Aachen (2008) · Zbl 1170.53050 |
[6] | Ehrlich, P. E.; Sánchez, M., Some semi-Riemannian volume comparison theorems, Tohoku Math. J, 2, 52, 331-348 (2000) · Zbl 0983.53044 · doi:10.2748/tmj/1178207817 |
[7] | Gromov., M.: Metric Structures for Riemannian and Non-Riemannian Spaces. Progress in Mathematics, vol. 152. Birkhäuser Boston Inc., Boston · Zbl 1113.53001 |
[8] | Harris, S. G., A triangle comparison theorem for Lorentz manifolds, Indiana Univ. Math. J., 31, 289-308 (1982) · Zbl 0496.53042 · doi:10.1512/iumj.1982.31.31026 |
[9] | Hawking, S. W.; Ellis, G. F. R., The Large Scale Structure of Space-Time (1973), London: Cambridge University Press, London · Zbl 0265.53054 |
[10] | Klainerman, S.; Rodnianski, I., Causal geometry of Einstein-vacuum spacetimes with finite curvature flux, Invent. Math., 159, 437-529 (2005) · Zbl 1136.58018 · doi:10.1007/s00222-004-0365-4 |
[11] | Penrose, R., Rindler, W.: Spinors and Space-Time. Cambridge University Press, Cambridge (1987, 1988) · Zbl 0663.53013 |
[12] | Petersen, P., Riemannian Geometry. Graduate Texts in Mathematics, vol. 171 (2006), New York: Springer, New York · Zbl 1220.53002 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.