An Interpretation of the Geometry of Space-Time Inside a BH

[dgm7691]

Assume an eternal, static black hole which has an event horizon, a spherical surface at which any object passes a point of no return and is condemned to move toward a mathematical singularity.

One of the predictions of being inside a black hole is that every spatial direction points towards the singularity. If you had a beacon that emitted photons in all directions, they would all head towards the singularity. If they didn't, they would be violating the law of causality, which would prohibit them from traveling backwards towards the event horizon. Essentially, something would have to travel backwards in time to reach and escape the event horizon.

Consider objects with mass that cannot travel at the speed of light/causality. How would they behave inside this strange universe inside a black hole? Perhaps a mass may tend to accelerate towards the singularity, since that is the behaviour of gravity. However, in this universe inside a black hole, every direction points towards the singularity. Every observable bit of matter would tend to accelerate towards it, such that it may give the appearance, from a given observation point, that everything else is accelerating away from that point, as if space itself were expanding as time moved forward.

Another prediction of the space time geometry within the event horizon of an eternal black hole is that time and space switch roles. One hint that it behaves this way is that it would require the apparently impossible task of moving backwards in time to travel back to the event horizon. Another hint is that the singularity cannot be observed, it exists only in the future, any direct observation of it would require photons/matter moving away from singularity towards the event horizon which violates the law of causality and also require moving backwards in time. The direction of time flows from the event horizon towards the singularity, while a dimension of space (if frozen in time) flows towards the infinite future from the infinite past of an eternal black hole, at least as imagined from the universe on the other side of the event horizon. The derivation of this concept may be difficult to comprehend, but it is reflected in a Penrose diagram of space time inside an eternal static black hole.

If time and space have switched roles, then this may imply that the event horizon, where matter and energy enter this black hole universe, appears to be an event rather than a location. It may appear as if everything in this universe came into existence at the time of this event and began accelerating towards the singularity, which happens to be in every possible direction. It may appear as if a big bang happened.

These imagined observations inside a black hole may seem very strange, however it seems as if it is coincidental to what is observed in our own universe.

I am just a layman, and admittedly my understanding of black hole space time geometry may be rudimentary compared to a professional astrophysicist. But I would like to hear about how I may have gone wrong in this interpretation.

References: Hawking, Stephen & Ellis, G. F. R. (1973). The Large Scale Structure of Space-Time. Cambridge: Cambridge University Press. ISBN 0-521-09906-4. Chapter 5.

 

[PeterDonis]

One of the predictions of being inside a black hole is that every spatial direction points towards the singularity.

No, this is not correct. Every future-directed timelike and null direction in spacetime inside the horizon points towards the singularity; that is why any object inside the horizon must eventually hit the singularity. But not every spacelike direction in spacetime inside the horizon points towards the singularity; there are spacelike curves inside the horizon that never hit the singularity.

Essentially, something would have to travel backwards in time to reach and escape the event horizon.

Or along a spacelike direction. There are spacelike curves that lead from inside the horizon to outside. But there are no future-directed timelike or null curves that do.

Consider objects with mass that cannot travel at the speed of light/causality. How would they behave inside this strange universe inside a black hole?

They would fall towards and eventually hit the singularity.

Perhaps a mass may tend to accelerate towards the singularity, since that is the behaviour of gravity.

Not in GR. In GR, gravity is spacetime curvature, not "acceleration". Objects moving solely under gravity have zero proper acceleration; they are in free fall. They might have nonzero coordinate acceleration, but that depends on your choice of coordinates. Proper acceleration does not.

in this universe inside a black hole, every direction points towards the singularity.

Incorrect. See above.

Every observable bit of matter would tend to accelerate towards it, such that it may give the appearatnce, from a given observation point, that everything else is accelerating away from that point, as if space itself were expanding as time moved forward.

This is not correct. Neighboring objects that are all free-falling inward towards the singularity will see each other converging, not diverging.

Another prediction of the space time geometry within the event horizon of an eternal black hole is that time and space switch roles.

This is not correct. The "switching roles" is an illusion caused by a particular choice of coordinates. Along the worldline of any object falling into the hole, there is no such change.

a dimension of space (if frozen in time) flows towards the infinite future from the infinite past of an eternal black hole, at least as imagined from the universe on the other side of the event horizon.

I have no idea what you are referring to here.

The derivation of this concept may be difficult to comprehend, but it is reflected in a Penrose diagram of space time inside an eternal static black hole.

I don't see how what you said here relates to anything in a Penrose diagram.

If time and space have switched roles

They don't. See above.

the event horizon, where matter and energy enter this black hole universe, appears to be an event rather than a location

It's neither. It's an outgoing lightlike surface--a surface made of radially outgoing light rays.

It may appear as if everything in this universe came into existence at the time of this event and began accelerating towards the singularity, which happens to be in every possible direction. It may appear as if a big bang happened.

None of these speculations are correct.

These imagined observations inside a black hole may seem very strange, however it seems as if it is coincidental to what is observed in our own universe.

You have imagined things inside a black hole incorrectly. See above.

I am just a layman, and admittedly my understanding of black hole space time geometry may be rudimentary compared to a professional astrophysicist.

You might want to review the PF rules on personal speculations. This post is treading close to them.

But I would like to hear about how I may have gone wrong in this interpretation.

See above.

 

[Nugatory]

One of the predictions of being inside a black hole is that every spatial direction points towards the singularity.

That's not quite right. It is true that no matter which direction you shine a light or throw an object, it will end up at the singularity, but when you do that the object is traveling in a time-like (light-like for light) direction through spacetime. But this is no different than the situation outside the black hole - everything is moving into the future no matter what direction you throw it. The singularity is in the future of everything inside the horizon, so everything ends up there the same way that no matter what I do on Monday, Tuesday is in my future.

Every observable bit of matter would tend to accelerate towards it, such that it may give the appearance, from a given observation point

No - you are forgetting that movement towards the singularity is movement forward in time not space. (There's another problem in that there is no such thing as gravitational acceleration in general relativity, but that's a digression here).

Another prediction of the space time geometry within the event horizon of an eternal black hole is that time and space switch roles.

That is a very common misconception that stems from a poorly chosen convention for naming the Schwarzschild coordinates - unfortunately the convention is so widespread and has been used for so long that we're stuck with it.

All that's really going on is that outside the horizon we're using the letter $r$ for one of the three spatial coordinates and $t$ for the time coordinate, while inside the horizon we use the letter $r$ for the time coordinate and $t$ for one of the three space coordinates (We use $\theta$ and $\phi$ for the other two spatial coordinates in both regions). However, the two regions do not overlap so we're actually dealing with two separate coordinate patches and we should have used four different letters for space inside the horizon, space outside the horizon, time inside the horizon, and time outside the horizon to avoid giving the impression that time and space are switching roles. There's some discussion of this, and a bunch of other good stuff here: https://arxiv.org/abs/0804.3619

In any case, if you want to better understand the geometry of a black hole, the best thing you can do is to get comfortable with Kruskal coordinate diagrams, as discussed in that paper (and many bother places).

 

[dgm7691]

That's not quite right. It is true that no matter which direction you shine a light or throw an object, it will end up at the singularity, but when you do that the object is traveling in a time-like (light-like for light) direction through spacetime.

That is actually what I should have said or meant to say. The idea is all of these light-like or time-like paths may converge at an unseen singularity from any point between the event horizon and the singularity, but there is no way to witness this. Perhaps an observer might detect the curvature of space time, but what if it were sufficiently large to appear flat?

No - you are forgetting that movement towards the singularity is movement forward in time not space. (There's another problem in that there is no such thing as gravitational acceleration in general relativity, but that's a digression here).

From a given observation point, every light-light or time-like path is going "down" a gravity gradient towards the singularity, might it appear as if space were expanding given that the further down all of these paths (every possible light-like direction) we look, the faster we see objects moving away from us?

Another analogy: consider from a given observation point in relatively flat space, if we make observations towards the cosmic event horizon of our universe, and then along a path that pointed directly towards the singularity of a nearby (static) black hole. Without making any observations at a non-zero angle to the singularity betraying the space time curvature, wouldn't the observations be similar? Wouldn't intervening objects red shift depending on how far they appeared away from us moving towards an apparent infinite distance at the EH?

 

[PeterDonis]

Perhaps an observer might detect the curvature of space time, but what if it were sufficiently large to appear flat?

The spacetime curvature increases without bound as the singularity is approached, regardless of the mass of the hole. A hole with larger mass will have less spacetime curvature at the horizon, but not near the singularity.

From a given observation point, every light-light or time-like path is going "down" a gravity gradient

Not inside the horizon; inside the horizon, the spacetime is not static, so the concept of "potential energy" makes no sense, so the concept of "gravity gradient", which is the gradient of the potential energy, makes no sense either.

might it appear as if space were expanding given that the further down all of these paths (every possible light-like direction) we look, the faster we see objects moving away from us?

That's not what we would see; we would see other falling objects moving towards us, not away from us, as I've already said.

Without making any observations at a non-zero angle to the singularity betraying the space time curvature, wouldn't the observations be similar?

Somewhat, yes, but they wouldn't be observations of anything either inside the black hole's horizon or beyond the cosmological event horizon, so they're irrelevant to the topic of this thread.

I strongly suggest that you spend some time studying a good reference on the actual properties of the spacetime geometry of a black hole. Carroll's online lecture notes on GR give a good, if somewhat brief, discussion. Most GR textbooks treat the subject in some detail.

 

[PAllen]

Hmm. I thought that if an inertial dust ball crossed the horizon, and approached the singularity, it would compress in two directions and stretch without bound in the third. This is a major distinction from running the Big Bang backwards - you have a completely different singularity structure. I am, of course, referring to the pure SC geometry, rather than something more realistic.

 

[PeterDonis]

I thought that if an inertial dust ball crossed the horizon, and approached the singularity, it would compress in two directions and stretch without bound in the third

Yes, that's correct. But that's not a good case to focus on if you want to know if "space expanding" is a reasonable (heuristic) description of the geometry inside the horizon, because the dust ball you are talking about is infinitesimal in size.

The case I was thinking of was a spherically symmetric shell of dust surrounding the entire hole and falling inward. This shell is obviously contracting, not expanding, and individual dust particles in the shell will converge, not diverge. (Think of the shell as one dust particle thick.)

 

[PAllen]

Yes, that's correct. But that's not a good case to focus on if you want to know if "space expanding" is a reasonable (heuristic) description of the geometry inside the horizon, because the dust ball you are talking about is infinitesimal in size.

The case I was thinking of was a spherically symmetric shell of dust surrounding the entire hole and falling inward. This shell is obviously contracting, not expanding, and individual dust particles in the shell will converge, not diverge. (Think of the shell as one dust particle thick.)

Yes, I agree about that case. I certainly wasn't disputing the 'expanding space' is completely wrong as a description of the interior. I was noting that 'nearby timelike goedesics converge' is not a complete statement. It depends on their orientation. Some will converge and some will diverge.