Close to an event horizon of a static black hole, the potential is very low. If we send a photon from there, its redshift is infinite. Thus, the form of equipotential surfaces tells us something about the form of the horizon.
Making a nonspherical horizon with two masses?
Suppose that we try to make a nonspherical horizon. The obvious strategy is to put two masses M at some distance s from each other. We increase the masses gradually and watch what happens to the equipotential surfaces.
field lines bend
/ \
| |
● ●
<----- s ----->
M M
Let us describe the gravity field by "field lines". They are like lines of force for an electric field, but must take into account the nonlinear nature of gravity. We do not try to define the field line concept exactly here, but keep it intentionally vague. For weak gravity fields, the field lines are exactly like the lines of force of an electric field.
Each mass M distorts the metric of spacetime and makes the field lines of the other mass to bend toward M. The combined field of the masses is more spherically symmetric than the corresponding field of two electric charges.
As we increase the masses M, the equipotential surfaces become more and more spherical. It is not that surprising that the horizon is spherical when it ultimately forms.
What is the stress-energy tensor of a hypothetical nonspherical horizon?
Let us approach the problem from another direction. Let us assume that we have a static black hole in a vacuum and its horizon is elongated. What is the stress-energy tensor of the metric?
Our trick of putting two masses M at a distance s from each other did not succeed. What could we do to deform the horizon from a sphere?
There is no obvious way. We could try to add masses outside the event horizon, but that would breach the condition that the black hole sits in a vacuum.
The question is analogous to the following: suppose that we have a set of electric charges within a volume V of a diameter s. Far away, their field is essentially spherically symmetric. Can we break the spherical symmetry? Yes, but then we must put sources of the field outside the volume V. It is no longer a vacuum.
Dynamic configurations
The Israel theorem is formulated for a static metric. This leaves open a possibility that a dynamic system would have a nonspherical event horizon for a very long time.
Our previous blog post suggested that merging black holes are radially squeezed in the global pseudo-Schwarzschild metric around their common center of mass. Thus, it is possible that they can quickly fall inside their common Schwarzschild diameter. However, this not a mathematical proof.
Wikipedia states that there exists no mathematical proof for the no-hair theorem. It is a conjecture.
Conclusion
There are various (partially heuristic?) proofs that a static or stationary black hole is either the Schwarzschild black hole or the Kerr black hole, depending on if its angular momentum is zero or not.
Such idealized black holes cannot exist in nature, though.
We have to check what is known about the end state of a black hole which is formed from realistic matter configurations. Can we show that it very quickly converges toward a stationary Kerr black hole?
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