Question

Why can apparent horizon be computed based on its local geometry?

Answer 1

Another very interesting and natural phenomenon in general relativity is the existence of black holes. Instead of thinking of black holes as singularities in a spacetime, we will think of black holes in terms of their horizons. For example, suppose we are exploring the universe in a spacecraft capable of traveling any speed less than the speed of light. If we are investigating a black hole, we want to make sure that we do not get too close and get trapped by the “gravitational forces” of the black hole. We can imagine a “sphere of no return” beyond which escape from the black hole is impossible. It is called the event horizon of a black hole. However, one limitation of the notion of an event horizon is the difficulty of determining its location. One way is to let daredevil spacecraft see how close they can get to the black hole and still escape from it eventually. The problem with this approach (besides the cost in spacecraft) is that it is hard to know when to stop waiting for a daredevil spacecraft to return. Even if it has been fifty years, it could be that this particular daredevil was not trapped by the black hole but got so close that it will take one thousand or more years to return. Thus, to define the location of an event horizon even mathematically, we need to know the entire evolution of the spacetime. Hence, event horizons cannot be computed based only on the local geometry of the spacetime. This problem is solved (at least for the mathematician) by the notion of apparent horizons of black holes. Given a surface in a spacetime, suppose that it emits an outward shell of light. If the surface area of this shell of light is decreasing everywhere on the surface, then this is called a trapped surface. The outermost boundary of these trapped surfaces is called the apparent horizon of the black hole. Apparent horizons can be computed based on their local geometry, and an apparent horizon always implies the existence of an event horizon outside of it.

Answer 2

Black holes, which are exact solutions in general relativity (GR) (representing, for example, physical objects formed out of the gravitational collapse of fuel-exhausted stars), are characterized by the boundary of the region from where light can still travel to asymptotic null infinity, called the event horizon, which ...