# Black Hole Event Horizon: Beyond Schwarzschild Radius?

• Andru10
In summary: R = 2GM), and Kerr (non-rotating) black hole solutions which do not have the Schwarzschild radius. Schwarzschild black holes are the only stationary solutions.
Andru10
Are there any known metrics in which black holes do not have the Schwarzschild radius? Specifically, I'm interested in whether it's possible for a black hole to have an event horizon which is not of the form: constant * mass.

Andru10 said:
Are there any known metrics in which black holes do not have the Schwarzschild radius? Specifically, I'm interested in whether it's possible for a black hole to have an event horizon which is not of the form: constant * mass.
The event horizon for a rotating black hole does not only depend on the mass but also on the angular momentum.

Specifically, I'm interested in whether it's possible for a black hole to have an event horizon which is not of the form: constant * mass.
No. The most general black hole is represented by the Kerr-Newman solution, and it has this property.

Kerr-Newman don't have the property that the radius is proportional to the mass times a constant. It depends on both the angular momentum(as passionflower said) and the charge.

Also Kerr-Newman is not the most general black hole. It is only the most general stationary solution to the Einstein-Maxwell equations in four dimensions. There are obviously many more general solutions like for example time dependent ones with matter falling to the horizon.

The Schwarzschild solution is a very very special case.

Finbar said:
Kerr-Newman don't have the property that the radius is proportional to the mass times a constant. It depends on both the angular momentum(as passionflower said) and the charge.

The OP asked if there are black holes which do not have a schwarzchild radius i.e. r = 2GM. The kerr - newman solution is the most general stationary solution that DOES have this property.

WannabeNewton said:
The OP asked if there are black holes which do not have a schwarzchild radius i.e. r = 2GM. The kerr - newman solution is the most general stationary solution that DOES have this property.

No, for a kerr black hole
$r = M + \sqrt{M^2-a^2}$

in units G=1 where a = J/M and J is angular momentum.

Finbar said:
No, for a kerr black hole
$r = M + \sqrt{M^2-a^2}$ in units G=1 where a = J/M and J is angular momentum.

That is a definition of r to find the location of the horizon in the kerr geometry i.e. at $$g_{rr} = \infty$$. This has nothing to with the fact that r = 2M IS the schwarzchild radius for a kerr black hole.

Last edited:
No, for a kerr black hole r = M + √(M2 - a2)
The point is that a is restricted to the range 0 < a < M, so M < r < 2M. The inner horizon is no longer spherical, but its size is proportional to M. In fact this must be the case, since M is the only parameter in the solution that sets the scale.

WannabeNewton said:
That is a definition of r to find the location of the horizon in the kerr geometry i.e. at $$g_{rr} = \infty$$. This has nothing to with the fact that r = 2M IS the schwarzchild radius for a kerr black hole.

r=2M has nothing to do with the event horizon of a Kerr black hole.

The OP asked whether the event horizon of a black hole is proportional to the mass times a constant. This is only the case for schwarzschild black hole.

Bill_K said:
The point is that a is restricted to the range 0 < a < M, so M < r < 2M. The inner horizon is no longer spherical, but its size is proportional to M. In fact this must be the case, since M is the only parameter in the solution that sets the scale.

The inner horizon radius is not proportional to the mass either. Take fix J and take M large and the inner horizon shrinks to zero as J/M goes to zero.

a is not restricted to a<M. a>M is a naked singularity.

Finbar said:
r=2M has nothing to do with the event horizon of a Kerr black hole.

The OP asked whether the event horizon of a black hole is proportional to the mass times a constant. This is only the case for schwarzschild black hole.

Yes excuse me I see what you are saying. $$g_{rr} = \infty$$ and $$g_{tt} = 0$$ for a schwarzchild black hole is r = 2m but for a kerr black hole the actual horizon (not the ergosphere) $$g_{rr} = \infty$$ is when $$\Delta = 0$$. Thanks mate; indeed I was mixing up the terms for the event horizon and the definition of schwarzchild radius.

Andru10 said:
Are there any known metrics in which black holes do not have the Schwarzschild radius? Specifically, I'm interested in whether it's possible for a black hole to have an event horizon which is not of the form: constant * mass.

When "mass" is suitably defined, there are dynamic (non-stationary; no exterior timelike killing vector) black hole spacetimes that don't have this property. For an example that uses the Vaidya metric, see section 5.1.8 on page 132 (pdf page 148) of Eric Poisson's notes,

http://www.physics.uoguelph.ca/poisson/research/agr.pdf,

which evolved into the excellent book, A Relativist's Toolkit: The Mathematics of black hole Mechanics.

In this example, the apparent horizon has this property, but (a portion of) the event doesn't.

Finbar said:
Kerr-Newman don't have the property that the radius is proportional to the mass times a constant. It depends on both the angular momentum(as passionflower said) and the charge.

Also Kerr-Newman is not the most general black hole. It is only the most general stationary solution to the Einstein-Maxwell equations in four dimensions. There are obviously many more general solutions like for example time dependent ones with matter falling to the horizon.

The Schwarzschild solution is a very very special case.

I would appreciate if you had any good links to the subject "Kerr-Newman is not the most general black hole"

I am interested in black holes where a lot of material is falling in from close to the top and bottem of the spinning black hole. Mostly in 3D aspects as I find many references and drawings on the internet really only seem to be 2D in nature (the axis of rotation is not precessing).

Finbar said:
Also Kerr-Newman is not the most general black hole. It is only the most general stationary solution to the Einstein-Maxwell equations in four dimensions. There are obviously many more general solutions like for example time dependent ones with matter falling to the horizon.

edguy99 said:
I would appreciate if you had any good links to the subject "Kerr-Newman is not the most general black hole"

I am interested in black holes where a lot of material is falling in from close to the top and bottem of the spinning black hole. Mostly in 3D aspects as I find many references and drawings on the internet really only seem to be 2D in nature (the axis of rotation is not precessing).

I gave an example for non-spinning black holes in post #12.

## 1. What is a black hole event horizon?

A black hole event horizon is the point at which the gravitational pull of a black hole becomes so strong that nothing, including light, can escape. It is essentially the "point of no return" for anything entering a black hole.

## 2. How is the event horizon related to the Schwarzschild radius?

The Schwarzschild radius is the distance from the center of a black hole at which the event horizon is located. It is named after the German physicist Karl Schwarzschild, who first calculated this value in 1916.

## 3. Can anything escape from beyond the event horizon of a black hole?

No, nothing can escape from beyond the event horizon of a black hole. The intense gravitational pull at this point is strong enough to trap even light, which is the fastest known entity in the universe.

## 4. What happens to matter and energy that enter a black hole's event horizon?

Once matter or energy crosses the event horizon of a black hole, it will be pulled towards the center of the black hole and crushed into an infinitely small point known as the singularity. This is where the laws of physics, as we know them, break down.

## 5. Is there any way to observe the event horizon of a black hole?

While we cannot directly observe the event horizon of a black hole, we can indirectly study the effects of the event horizon on the surrounding matter and energy. For example, we can observe the bending of light and the strong gravitational lensing caused by the event horizon of a black hole.

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