Abstract
The present work is a brief review of the development of dynamical black holes from the geometric point view. Furthermore, in this context, universal thermodynamics in the FLRW model has been analyzed using the notion of the Kodama vector. Finally, some general conclusions have been drawn.
Highlights
A black hole is a region of space-time from which all future directed null geodesics fail to reach the future null infinity I +
The black hole region B of the space-time manifold M is the set of all events P that do not belong to the causal past of future null infinity, i.e., B = M − J − ( I + )
For a stationary black hole, the apparent horizon coincides with the event horizon, while in the dynamical situation, the apparent horizon always lies inside the black hole region unless the null energy condition (NEC)
Summary
A black hole is a region of space-time from which all future directed null geodesics fail to reach the future null infinity I +. In the black hole region, there are trapped surfaces that are closed 2-surfaces (S) such that both ingoing and outgoing congruences of null geodesics are orthogonal to S, and the expansion scalar is negative everywhere on S. As the geometric structure of null and spatial hypersurfaces is distinct, it is interesting to study the stationary and dynamical regimes of a quasi-local black hole, characterized by a future outer trapping horizon. The geometric structures constructed on a null hypersurface that is foliated by closed outer marginally-trapped surfaces is provided by isolated horizons. A space-like hypersurface foliated by closed future marginally-trapped surfaces is termed a dynamical horizon.
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