How do we determine whether a surface is an event horizon? Exactly what to calculate to do this?

Question

Event horizons play an important role in relativistic astrophysics, especially for black holes, but also for other spacetimes. It is fundamental to be able to determine whether a given surface is an event horizon, a one-sided trap. Yet in the textbooks and lectures I know of, I cannot find any precise computational recipes that, when done, reveal the horizon character. Usually it is just stated and illustrated, for example by plotting the inclination of light cones.

I will formulate the question precisely using the example of Schwarzschild spacetime: at the Schwarzschild radius, (R=2GM/c^2) the metric diverges. We first examine it in terms of curvature, for which we calculate the corresponding curvature scalars from the Ricci tensor. We get that they are finite, smooth values, so there is no real singularity here, only a coordinate singularity was involved. Then we would have to show that this surface in turn functions as an event horizon. What kind of quantities can we calculate to determine this? Naively, I would think that one would have to admit: any world line that passes through it is such that if the test object is moving inwards, it does so at a velocity less than c, but if it is moving outwards, it would need a velocity v > c. What mathematical tool can be used to calculate this accurately? Or is it enough to look at the light-like motions? If we are at the Schwarzschild radius, then all light signals are such that if the time coordinate increases, the r coordinate cannot increase? How can we see this in a general way, and how can we conclude from this about time-like motions?

The question arises more significantly in the case of Kerr space-time. Here, two horizons and two infinite redshift surfaces result from the analysis of the metric tensor. What calculation proves that the inner boundary of the ergosphere is also an event horizon, but the outer one is not? It is also important to determine these precisely, since for rotating black holes, the possibility of a possible escape from the event horizon arises in the analysis of the maximal geodetic extension. The effective potential becomes repulsive at a point, the geodetic motion reverses and again passes through the r+ surface. So as not to be contradicted, it is usually resolved that this r+ surface is not the same as the r+ surface of our world through which we entered - but belongs to some other part of the world. This involves making a special topological choice. However, it is not clear that then the surface we call the external event horizon does not act as a trap one hundred percent? Could there be, in principle, world lines that we can exceptionally use to get out from under it?

The previous question leads us a far way, so my main question now is: what kind of precise calculations can we use to determine whether a suspicious surface is an event horizon?

Answer 1

"It is fundamental to be able to determine whether a given surface is an event horizon, a one-sided trap."

That part is easy. The test for being "one-way" is simply whether it is null/spacelike. Across every spacelike surface, the traveller can only travel from past to future. You can't ever get from future to past (locally, at least) without moving faster than light.

But this isn't what we mean by a black hole's event horizon. The point here is that it is a spacelike surface enclosing the black hole, so having travelled from the past outside the hole to the future inside it, you would have to travel backwards in time to escape it.

There isn't any local definition. You can have a situation where you are sat outside what you believe to be the event horizon of a black hole, but unknown to you a large mass is about to fall in, increasing its mass, so in fact there is no way you can escape. Without the yet-to-appear falling mass, you're outside the horizon, with it, you're inside.

In thinking about event horizons, it is very much worth learning about the Rindler wedge. A uniformly accelerated observer (e.g. in a rocket undergoing 1 g thrust in empty space) follows a hyperbolic path in flat Minkowski space. The asymptotes of the hyperbola are two light rays passing through a common point. The accelerated observer can never catch up with the future-directed light ray, or observe any event that occurs after it. This line acts as an event horizon in the accelerated frame of reference.

To the stationary (freefall) observer nearby, there is nothing there. It's just flat Minkowski space, such as we pass through all the time. The event horizon (in 2D) is just the lightcone of a particular event. It's one-way because it divides past from future lightcones at every point. It's an event horizon because the accelerated observer never crosses it. It is always in his future. And so no event that happens on the other side of it can ever be in his past. He can never see anything past it, and anything he drops and which crosses that boundary can never return to him.

An observer hovering at a fixed radius outside a black hole is experiencing the same sort of acceleration. The spacetime looks a lot like the Rindler wedge. If we use Kruskal-Szekeres coordinates the black hole diagram looks almost exactly like flat Minkowski spacetime! It's not the same. The dependence on distance is $1/r$ instead of $1/r^2$, and linear acceleration is not the same as spherically-symmetric radial acceleration, but the parallels are considerable, and very useful for understanding the peculiarities of space around a black hole.

In a 2D spacetime diagram of flat Minkowski space, we have a future lightcone, a past lightcone, and two wedges of "elsewhere" that are spacelike separated, both from the origin and from each other. An accelerating observer is confined to one of the wedges. He never reaches the future lightcone; he can never observe any event happening to an object after crossing it. And accelerated observers confined to the other wedge of "elsewhere" are completely isolated from him. Neither can ever see the other, or influence the other, or send any travellers to the other.

The horizon has no special property of the physics there. It's "one-way-ness" is just that it separates past lightcones from future - a property shared by every point in spacetime. It counts as an event horizon because a particular sort of observer can stay forever on one side of it, and thus never be able to observe events after they pass over the horizon - again, a property shared by every point in spacetime. The event horizon of a black hole is such a surface for observers hovering outside it. If you can by any means escape to infinity, you're outside. If you cannot avoid reaching the singularity, you're inside.

Answer 2

Hopefully I have not smuggled something incorrect.

If you have all the data about your spacetime - the topology of the manifold (in coordinate viewpoint: charts and the ranges of coordinates), as well as its geometry (the metric tensor and the resulting spacetime curvature), you can pose the question: do there exist points in your manifold, from which there can travel null geodesics $\gamma: I\rightarrow \mathcal{M}$ that cannot escape to infinity. In Schwarzschild, there's a three-dimensional submanifold ($r<2M$) from which the null geodesics cannot escape - its boundary is called the event horizon.

In layman terms, there's a limiting spacetime region/point, which you cannot make the curve cross, regardless of the value of the curve's parameter ($s \mapsto \gamma(s))$. Schwarzschild serves as a nice example. Try to integrate a radially-pointing null geodesic from a point inside the event horizon (caveat : appropriately chose the coordinates).

While we're at the topic of usefulness of geodesics and curve to analyze the properties of a spacetime, the curvature singularities can (with some caveats) also be identified through so-called inextendible curves, which cannot be extended beyond a certain value of the curve parameter, as the spacetime simply does not contain a point, to which the curve tends (we are assuming a smooth, or at least some $\mathcal{C}^{N}$ differentiable manifold whose geometry also follows these differentiability requirements - the singularity is treated as not a part of the spacetime then). Aforementioned inextendible curves are employed as tools in the singularity theorems.

In numerical relativity, a more local notion (which is a property of a certain 2-dimensional surface on a $t=const$ spatial hypersurface and depends on the slicing) of some trapping is the apparent horizon, defined by the vanishing of expansion scalar. This effectively means that the expansion of out-going and the in-going light curves is stopped at that surface.

We know that if an apparent horizon exists, it must necessarily be inside the event horizon. In commonly employed formulations in numerical relativity and commonly used gauges, an apparent horizon is robustly found and serves a multitude of purposes - ranging from integrating quasi-local quantities (local ADM mass, spin), to tracking the movement of the black hole on the numerical grid.