Can the Spacetime Interval at a Singularity Change Our Understanding of Time?

[Grasshopper]

TL;DR

Trying to better understand the “flipping” of roles of space and time in a black hole.

Tad Williams’ Otherland series has a scene where the characters are drawn to a temple no matter which direction they try to walk, as if space itself is curved.

This is kind of the intuition I get when a physicist talks about the spacetime interval kind of flipping past an event horizon: if you try to flee the singularity, you just end up moving closer to it. This happens no matter what you do. No matter which direction you go, your future destination is the singularity. It’s as if you can ultimately only travel in one direction through space, just like in normal spacetime there is only on direction to travel through in time (to the future).

Is that a fair intuition? If it is, would it also apply to radial motion perpendicular to the gravitstional field lines of the singularity? (if such things have meaning in this situation)

And if that’s right, since the role of time is kind of flipped, does that mean you can move through the local past and future at will? Or is it something more mundane like, if you look “up” (away from the singularity) you see how the universe used to be, but if you look towards the singularity, you see how the universe will be in the future?

Any insight at any level is welcome. I put intermediate for the level, but math for a Schwartzchild black hole is fair game for me. Along with various spacetime diagrams like Penrose diagrams. I may not understand it but it still has value to me.

Thanks as always.

 

[PeterDonis]

Trying to better understand the “flipping” of roles of space and time in a black hole.

Short answer: there is no such thing.

For longer answers, see below.

This is kind of the intuition I get when a physicist talks about the spacetime interval kind of flipping past an event horizon

There is no such thing. The interval along your worldline is timelike above, at, and below the horizon. And you can set up a local inertial frame that looks just like an SR inertial frame in any small patch of spacetime centered anywhere on your worldline. Nothing about this changes at or below the horizon.

What scientists in pop science contexts are talking about when they talk about "space and time flipping" below the horizon, or statements along those lines, is a purely coordinate effect that only happens in one particular set of coordinates. It has nothing to do with physics. And if you look at actual textbooks and peer-reviewed papers, instead of pop science sources, you will see no such claims.

No matter which direction you go, your future destination is the singularity.

Yes, that's because the singularity is not a place in space; it's a moment of time, which is in your future. You can't avoid it because you can't avoid your future. It's no different from "no matter which direction you go, your future destination is tomorrow".

It’s as if you can ultimately only travel in one direction through space

Nope. Space is 3-dimensional inside the horizon just like it is outside, and you can travel in any spatial direction you like. It's no different from being able to travel in any spatial direction you like here on Earth while still not being able to avoid tomorrow.

Is that a fair intuition?

No. See above.

 

[Ibix]

There's no "flipping". That's pop sci nonsense. What is true is that the $tt$ component of the metric in interior Schwarzschild coordinates happens to have the same form as the $rr$ component in exterior Schwarzschild coordinates, and vice versa. That's just an artefact of the coordinates.

My favourite visualisation is the Kruskal diagram. The version on Wikipedia (drawn by DrGreg) includes lines of constant Schwarzschild $r$ and $t$, and shows how they fail at the horizon. You can also immediately see why you can't escape - because the event horizon is the future lightcone of an event in your causal past once you enter.

 

[Nugatory]

@Grasshopper This paper https://arxiv.org/abs/0804.3619 discusses why the “flipping“ is a misstatement of what’s going on, provides much good background on Kruskal coordinates, and addresses a few other common misconceptions.

 

[Grasshopper]

Thank you for clarifying that. What I am gathering here based on these replies is the only thing that even comes close to the pop science claim is that the singularity is a moment rather only than a location. “It is in your future,” as PeterDonis put it.

I was previously under the impression that the sign signature of the spacetime interval flipped however, but I suppose that is also mistaken (and even if it’s not, that’s just a convention anyway, right?).

 

[PeroK]

I was previously under the impression that the sign signature of the spacetime interval flipped however, but I suppose that is also mistaken (and even if it’s not, that’s just a convention anyway, right?).

Yes, Schwarzschild coordinates have a coordinate singularity at $r = 2M$, the event horizon. Essentially, therefore, you have two solutions: one for $r > 2M$ and one for $r < 2M$. For the latter, $r$ is a timelike coordinate. You could argue that perhaps we should change the coordinate labels in the case of $r < 2M$.

Although other coordinate systems have no singularity at the event horizon, the nature of a black hole below the event horizon is fundamentally different from anything that we find in classical physics.

 

[Grasshopper]

Yes, Schwarzschild coordinates have a coordinate singularity at $r = 2M$, the event horizon. Essentially, therefore, you have two solutions: one for $r > 2M$ and one for $r < 2M$. For the latter, $r$ is a timelike coordinate. You could argue that perhaps we should change the coordinate labels in the case of $r < 2M$.

Although other coordinate systems have no singularity at the event horizon, the nature of a black hole below the event horizon is fundamentally different from anything that we find in classical physics.

But these things are mere coordinate dependent things, correct?

 

[PeroK]

But these things are mere coordinate dependent things, correct?

The event horizon is a null surface that defines a one-way causal structure that splits spacetime into two regions: events inside the horizon and events outside the horizon. That's not a coordinate effect.

The singularity at $r = 0$ itself is a physical (not a coordinate) singularity.

You can't remove either of these things by a change of coordinates.

 

[Grasshopper]

Ah, well, my question has been answered, so digging further here is unlikely to be productive for me given that I have more basic math to learn. Thanks for the replies as always.

 

[Nugatory]

You could argue that perhaps we should change the coordinate labels in the case of $r\lt 2m$.

That’s what the Krasnikov paper linked above does.