Apparent horizon

In general relativity, an apparent horizon is a surface that is the boundary between light rays that are directed outwards and moving outwards, and those directed outward but moving inward.

Apparent horizons are not invariant properties of a spacetime. They are observer-dependent, and in particular they are distinct from event horizons. Within an apparent horizon, light is not moving away from the black hole, whereas in an event horizon, light cannot escape from the black hole. It is possible for light to be currently moving away from the black hole (and so outside the apparent horizon), but in the future will not be able to escape (e.g. because the mass of the black hole is growing) and therefore inside the event horizon. Thus the apparent horizon can be thought of as the boundary of black hole for light at this instant, whereas the event horizon is the boundary of the black hole for light in the future.

See, however, the articles on ergosphere, Cauchy horizon, the Reissner–Nordström solution, photon sphere, Killing horizon and naked singularity; the notion of a horizon in general relativity is subtle, and depends on fine distinctions.

Definition

The notion of an "apparent horizon" begins with the notion of a trapped null surface. A (compact, orientable, spacelike) surface always has 2 independent forward-in-time pointing, lightlike, normal directions. For example, a (spacelike) sphere in Minkowski space has lightlike vectors pointing inward and outward along the radial direction. The inward-pointing, lightlike normal vectors converge, while the outward-pointing, lightlike normal vectors diverge. It can, however, happen that both inward-pointing and outward-pointing lightlike normal vectors converge. In such a case, the surface is called trapped.^[1]

Consider the set of all such trapped surfaces. In terms of a simple Schwarzschild black hole, these surfaces fill up the black hole. The "apparent horizon" is then defined as the boundary of these surfaces – essentially, it is the outermost surface of the black hole, in this sense. Note, however, that a black hole is defined with respect to the event horizon, which is not always the same as the apparent horizon.

Any apparent horizon is observer-dependent.

Differences from the (absolute) event horizon

In the context of black holes, the term event horizon refers almost exclusively to the notion of the "absolute horizon". Much confusion seems to arise concerning the differences between an apparent horizon (AH) and an event horizon (EH). In general, the two need not be the same. For example, in the case of a perturbed black hole, the EH and the AH generally do not coincide as long as either horizon fluctuates.

In the simple picture of stellar collapse leading to formation of a black hole, an event horizon forms before an apparent horizon.^[2] As the black hole settles down, the two horizons approach each other, and asymptotically become the same surface. If the AH exists, it is necessarily inside of the EH.

Apparent horizons depend on the "slicing" of a spacetime. That is, the location and even existence of an apparent horizon depends on the way spacetime is divided into space and time. For example, it is possible to slice the Schwarzschild geometry in such a way that there is no apparent horizon, ever, despite the fact that there is certainly an event horizon.^[3]

See also

Absolute horizon Black hole

Cosmological horizon Event horizon

Particle horizon Trapped null surface

References

1. Ivan Booth (2005). "Black hole boundaries". Canadian Journal of Physics. 83 (11): 1073–1099. arXiv:gr-qc/0508107. Bibcode:2005CaJPh..83.1073B. doi:10.1139/p05-063.
2. S. W. Hawking & G. F. R. Ellis (1975). The large scale structure of space-time. Cambridge University Press.
3. Wald, Robert M. & Iyer, Vivek (December 1991). "Trapped surfaces in the Schwarzschild geometry and cosmic censorship". Phys. Rev. D. American Physical Society. 44 (12): R3719–R3722. Bibcode:1991PhRvD..44.3719W. doi:10.1103/PhysRevD.44.R3719.