Does space-time form a closed manifold around a black hole?

[Imax]

Mass can curve space-time. Is it possible that space-time around a black hole is so badly curved that it forms a closed 4D manifold?

 

[71STARS]

Mass can curve space-time. Is it possible that space-time around a black hole is so badly curved that it forms a closed 4D manifold?

If a "black hole" is nothing more than a "dead star" remnant, it should do nothing but wait to be dissolved. There should be no implosion of energy consuming function from a black hole, therefore, space exists as within the universe at the same rate of time.

 

[Imax]

If space-time around a black hole forms a closed manifold, then there should be two distinct manifolds:

1) The black hole’s manifold
2) Our space-time manifold surrounding Earth.

Are they orthogonal?

 

[Chalnoth]

Mass can curve space-time. Is it possible that space-time around a black hole is so badly curved that it forms a closed 4D manifold?

No. Now, you can make a closed 4D manifold that includes a black hole within it, but you can also place a black hole within an open 4D manifold. The black hole itself isn't able to have much of an impact on the space-time geometry far from itself.

 

[Imax]

The reason I’m asking this question is because I can imagine having two black holes, A and B. Black hole A is a “classic” black hole, where gravity is so strong that the escape velocity exceeds the speed of light. Nothing can exceed the speed of light, so nothing can escape black hole A. On the other hand, the curvature of space-time around black hole B is so bad that it forms a closed space-time manifold. If you’re inside black hole B, then it doesn’t matter which x, y, z direction you choose or how long you travel in that direction. You can never escape black hole B because there’s no space-time trajectory you can follow to bring you out of black hole B.

If I’m looking at black holes A and B from a safe distance, then I should see exactly the same thing, nothing. If I’m an outside observer, I can’t make a distinction between black holes A and B.

 

[Chalnoth]

The reason I’m asking this question is because I can imagine having two black holes, A and B. Black hole A is a “classic” black hole, where gravity is so strong that the escape velocity exceeds the speed of light. Nothing can exceed the speed of light, so nothing can escape black hole A. On the other hand, the curvature of space-time around black hole B is so bad that it forms a closed space-time manifold. If you’re inside black hole B, then it doesn’t matter which x, y, z direction you choose or how long you travel in that direction. You can never escape black hole B because there’s no space-time trajectory you can follow to bring you out of black hole B.

If I’m looking at black holes A and B from a safe distance, then I should see exactly the same thing, nothing. If I’m an outside observer, I can’t make a distinction between black holes A and B.

The difference is that there would also be no trajectory for anything to enter black hole B.

 

[Imax]

That’s exactly what I’m wondering about. If space-time formed a closed manifold around a black hole, then there would be no trajectory out but there would be no trajectory in. Maybe black holes don’t go around gobbling everything in their path.

 

[Chalnoth]

That’s exactly what I’m wondering about. If space-time formed a closed manifold around a black hole, then there would be no trajectory out but there would be no trajectory in. Maybe black holes don’t go around gobbling everything in their path.

Well, they don't go around gobbling everything in their path. Far away, they have the exact same gravitational field as any other spherically-symmetric object their mass, but a much smaller radius and thus most things end up just missing a black hole.

But that's somewhat besides the point here. If a black hole made its own closed manifold, it would be disconnected from our manifold, and we wouldn't be able to interact with it at all. Thus it wouldn't be a black hole.

 

[Imax]

It may still be a black hole. A black hole is black because it doesn’t transmit info.

A closed 1D manifold can be expressed as a circle, a 2D construct. A closed 2D manifold can be expressed as the surface of a sphere or a torus, a 3D construct. By analogy, a closed 3D manifold can be expressed in 4D. Ugh. A closed 4D space-time manifold could be expressed in 5D. Ugh Ugh.

What’s the fifth dimension?

 

[Kevin_Axion]

A Black hole is black because it doesn't allow light to escape once it has entered the event horizon. It's arbitrary to ask what the fifth dimension is, as it is the same as the preceding onces but is positioned to create a new degree of freedom.

 

[Imax]

but is positioned to create a new degree of freedom.

Maybe that's what we need, a new degree of freedom. If you know mass and you know angular speed, predicting an axis of rotation and whether it's clockwise or anticlockwise is problematic.