Geometric properties of a certain class of compact dynamical horizons in locally rotationally symmetric class II spacetimes
Abstract
In this paper, we study the geometry of a certain class of compact dynamical horizons with a time-dependent induced metric in locally rotationally symmetric class II spacetimes. We first obtain a compactness condition for embedded 3-manifolds in these spacetimes, satisfying the weak energy condition, with non-negative isotropic pressure p. General conditions for a 3-manifold to be a dynamical horizon are imposed, as well as certain genericity conditions, which in the case of locally rotationally symmetric class II spacetimes reduces to the statement that “the weak energy condition is strictly satisfied or otherwise violated”. The compactness condition is presented as a spatial first-order partial differential equation in the sheet expansion ϕ, in the form ϕ̂ + (3/4)ϕ2 - cK = 0, where K is the Gaussian curvature of 2-surfaces in the spacetime and c is a real number parametrizing the differential equation, where c can take on only two values, 0 and 2. Using geometric arguments, it is shown that the case c = 2 can be ruled out and the 𝕊3 (3-dimensional sphere) geometry of compact dynamical horizons for the case c = 0 is established. Finally, an invariant characterization of this class of compact dynamical horizons is also presented.
- Publication:
-
International Journal of Geometric Methods in Modern Physics
- Pub Date:
- 2021
- DOI:
- 10.1142/S0219887821500109
- arXiv:
- arXiv:2009.09966
- Bibcode:
- 2021IJGMM..1850010S
- Keywords:
-
- Dynamical horizons;
- locally rotationally symmetric class II spacetimes;
- compact Riemannian manifolds;
- Ricci flow;
- General Relativity and Quantum Cosmology
- E-Print:
- 16 pages and 1 figure