# Does space expansion mean the escape velocity at a black hole event horizon is strictly less than $c$?

The title pretty much sums it up.

I could ask this question slightly differently; does the expansion of space cause the effective event horizon to shrink below what it would be for flat space?

If so, does this mean that with sufficient "dark energy" we could break apart black holes and violate the No Hair theorem if black hole singularities have a finite volume (such that expansion could exceed the speed of light within the singularity)?

Quick answer: No. The escape velocity is the same.

Medium answer: The defining feature of a black hole is that it has an event horizon, a surface that is trapped (no particle can escape it, not even light). Escape velocity is slightly tricky to define in general relativity and normally not used to calculate where horizons are. Adding dark energy alias a cosmological constant does not change black holes much (unless there is a lot of it), but it does shrink the horizon a little bit.

Long answer: Any spherically symmetric solution of the vacuum Einstein equations has the form $$ds^2=-f(r)dt^2 +\frac{dr^2}{f(r)}+r^2(d\theta^2+\sin^2\theta d\phi^2)$$ where $$f(r)=1-\frac{2a}{r}-br^2.$$ This is the de Sitter–Schwarzschild metric, basically describing a black hole sitting in an otherwise empty universe. The constant $$a$$ is the black hole mass and the constant $$b$$ is the cosmological constant.

Note that when $$f(r)=0$$ bad things happen to the $$dr^2$$ term: this is the coordinate singularity that marks the event horizon. This normally happens at $$r=2M$$, but if $$b>0$$ then the horizon will be further in. More dark energy means smaller black holes.

There is also an outer horizon for large values of $$r$$; this one corresponds to a cosmological horizon. However, as you make $$b$$ bigger and bigger the horizons will approach each other in the $$r$$-coordinate, and you end up with a spacetime that looks symmetric between them. But it is not like the black hole "bursts".

So far we have talked about a constant dark energy. There are some cosmological models where it is increasing, eventually producing a "big rip" where everything expands infinitely in finite-time. It turns out that embedding black holes in such models is pretty intricate, see (Faraoni & Faques 2007) for some models where horizons do get stretched, but there are also models where the black hole shrinks (one can view this as accreting phantom energy; however, they do not reveal naked singularities). A lot hinges on the exact model of dark energy.

So my conclusion is that dark energy is not effective in breaking censorship of black hole singularities. But it can make the horizons shift location a bit.