It was customary to represent black hole horizons via stationary solutions of field equations, i.e., solutions which admit a time-translational Killing vector field everywhere, not just in a small neighborhood of the black hole. While this simple idealization was natural as a starting point, it is overly restrictive. Physically, it should be sufficient to impose boundary conditions at the horizon which ensure only that the black hole itself is isolated. That is, it should suffice to demand only that the intrinsic geometry of the horizon be time independent, whereas the geometry outside may be dynamical and admit gravitational and other radiation.
Contents
- Definition of IHs
- Boundary conditions of IHs
- Extension of the on horizon adapted tetrad
- Applications
- References
An advantage of isolated horizons over event horizons is that while one needs the entire spacetime history to locate an event horizon, isolated horizons are defined using local spacetime structures only. The laws of black hole mechanics, initially proved for event horizons, are generalized to isolated horizons.
An isolated horizon
Definition of IHs
A three-dimensional submanifold
(i)
(ii) Along any null normal field
(iii) All field equations hold on
(iv) The commutator
Note: Following the convention set up in refs., "hat" over the equality symbol
Boundary conditions of IHs
The properties of a generic IH manifest themselves as a set of boundary conditions expressed in the language of Newman–Penrose formalism,
In addition, for an electromagnetic IH,
Moreover, in a tetrad adapted to the IH structure, we have
Remark: In fact, these boundary conditions of IHs just inherit those of NEHs.
Extension of the on-horizon adapted tetrad
Full analysis of the geometry and mechanics of an IH relies on the on-horizon adapted tetrad. However, a more comprehensive view of IHs often requires investigation of the near-horizon vicinity and off-horizon exterior. The adapted tetrad on an IH can be smoothly extended to the following form which cover both the horizon and off-horizon regions,
where
Applications
The local nature of the definition of an isolated horizon makes it more convenient for numerical studies.
The local nature makes the Hamiltonian description viable. This framework offers a natural point of departure for non-perturbative quantization and derivation of black hole entropy from microscopic degrees of freedoom.