Exploring the Effects of Dark Energy on Extremely Large Black Holes

[A Puzzlement]

Ordinarily a black hole’s Schwarzschild radius is linearly proportional to its mass.

However, wouldn’t there be a deviation from this rule for extremely large black holes? Suppose we assume dark energy is due to a cosmological constant, whose value is the same everywhere (including inside the black hole). Since the amount of dark energy inside the black hole grows as the cube of its radius, but the black hole’s own mass only grows linearly with radius, eventually we will get to a point where the amount of dark energy inside the hole is a significant fraction of it’s “regular” mass. But dark energy is repulsive, so in order to ensure we still have an event horizon, a black hole of a given radius would need to have more mass than we would expect it to. Presumably this would be the case with a black hole formed from all the matter in the observable universe. Is this correct?

Also, would dark energy effects allow very large black holes to be super-extremal (or wormholes)?

 

[Arman777]

Ordinarily a black hole’s Schwarzschild radius is linearly proportional to its volume.

Shouldn't be mass ?

 

[A Puzzlement]

Oops, I meant to say mass, not volume. My mistake.

 

[Arman777]

Ordinarily a black hole’s Schwarzschild radius is linearly proportional to its volume.

However, wouldn’t there be a deviation from this rule for extremely large black holes? Suppose we assume dark energy is due to a cosmological constant, whose value is the same everywhere (including inside the black hole). Since the amount of dark energy inside the black hole grows as the cube of its radius, but the black hole’s own mass only grows linearly with radius, eventually we will get to a point where the amount of dark energy inside the hole is a significant fraction of it’s “regular” mass. But dark energy is repulsive, so in order to ensure we still have an event horizon, a black hole of a given radius would need to have more mass than we would expect it to. Presumably this would be the case with a black hole formed from all the matter in the observable universe. Is this correct?

Also, would dark energy effects allow very large black holes to be super-extremal (or wormholes)?

I guess I understand your idea, I don't think we can talk about a cosmological constant inside the black hole but let's assume we can, Even in that case the density of the cosmological constant will not change with time. Why? Because it is a rustic property of space-time, it doesn't depend on how you choose the volume of the region. Let us suppose with a box size $a^3$ and dark energy density $ρ_Λ$ as the reason as I described above even you increase the volume the density would be the same.

The other thing is it's hard to talk about a volume of a black hole.
@PeterDonis might be more helpful on this subject.