Does black hole entropy change as a gravitational wave passes it?

Question

The black hole entropy depends on the area of the event horizon.

• Do gravitational waves change this area?
• Does the entropy increase and then decrease as the horizon stretches and contracts?

Answer 1

Gravitational waves might lead to the growth of the black hole, and hence they can make the area increase. However, the area will not decrease again.

A heuristic way of thinking about this is by noticing that if a gravitational wave enters the black hole, it can never come out. Hence, it can bring energy to the black hole, but not take it away.

More technically, there is a result known as the "area theorem" which states that (more details on Hawking & Ellis 1973 The Large Scale Structure of Spacetime, Prop. 9.2.7)

Given a black hole $B$ in a strongly future asymptotically predictable spacetime. Suppose $R_{ab} k^a k^b \geq 0$ holds for all null vector fields $k^a$, which is the case if the Einstein equations hold and the null energy condition is satisfied. Under these conditions, the area of the future horizon of the black hole never decreases.

In plain English, under a few reasonable conditions (reasonable for classical matter, but violated by quantum matter), black holes can never shrink.

Answer 2

Yes, gravitational radiation can change the entropy of a black hole, but it will depend on the behavior of these radiation on the event horizon. One way to study these effects is by considering the example of a shock wave.

Imagine that I have an isolated black-hole which is at a configuration (A), i.e. at some given mass, charge and angular-momentum. It now emits radiation at retarded time $u=u_0$ and then changes to a new configuration (B). This emitted shock wave can be modelled by the following metric $$ds^2=(g_{ab}+\theta (u-u_0)\mathscr{L}_{\zeta}g_{ab})dx^adx^b$$ where $\zeta$ is some vector field. The idea here is that, if we start with the co-ordinate system $x^a$ at $u<u_0$, it will shift abruptly to $x^a\to x^a+\zeta^a$ for $u>u_0$. Also till linear order in $\zeta$, the space-time on either side of the shock wave is vacuum. The shock wave can also have gravitational radiation, for that one have to ensure that there is a non-zero Weyl scalar $\psi_4$ associated to this metric.

This vector-field can be defined arbitrarily, in particular once we choose $\zeta$ to produce symmetry transformation on the horizon (i.e. $\mathscr{L}_{\zeta}g_{ab}|_H=0$ subject to some gauge conditions), we can then comment about conserved charges $Q_{\zeta}$ associated with these vector-fields. As a special case, if we consider super-translation $u\to u+T(\theta,\phi)$ on the horizon, then the charges due to the zero-modes of this transformation actually gives the Bekenstein-Hawking entropy: $$Q^{(B)}_T-Q^{(A)}_T=\frac{T\Delta A}{4G}$$ where $\Delta A$ is change in surface area of event horizon. One possible way to define $\zeta$ and calculation of charges has been considered in this article. Since the charge is essentially a surface term, it only depends on the behavior of $\zeta$ on the event horizon. So the outgoing radiation can have arbitrary behavior in the bulk for a given boundary condition on event horizon.

Although shock wave is an extreme scenario, the qualitative picture of the change in entropy should remain the same in case of a generic outgoing radiation