Definitions

# Apparent horizon

An apparent horizon is a surface defined in general relativity as the boundary between light rays which are directed outwards and moving outwards, and those which are directed outwards but moving inwards.

Apparent horizons are not invariant properties of a spacetime. They are observer-dependent, and in particular they are distinct from absolute horizons.

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. We take a (compact, orientable, spacelike) surface, and find its outward pointing normal vectors. The basic picture to think of here is a ball with pins sticking out of it; the pins are the normal vectors.

Now we look at light rays that are directed outward, along these normal vectors. The rays will either be diverging (the usual case one would expect) or converging. Intuitively, if the light rays are converging, this means that the light is moving backwards inside of the ball. If all the rays around the entire surface are converging, we say that there is a trapped null surface.

We can take the set of all such trapped surfaces. If we are thinking in terms of a simple Schwarzschild black hole, these surfaces will 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.

## 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 will generally not coincide as long as either horizon is fluctuating.

In the simple picture of stellar collapse leading to formation of a black hole, an event horizon will be formed before an apparent horizon. As the black hole settles down, the two horizons will 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.

## References

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