Meaning of Time / Space axes swapping (for Time)

[Grinkle]

TL;DR Summary

Does time continue to march forward for an observer inside the event horizon?

I am including a link to a B level discussion of this I found on-line to try and anchor my question, not because I think the below article is good or poor - I am not able to assess that.

https://www.einstein-online.info/en/spotlight/changing_places/#The_analogy

In particular I am asking about this statement where the author is calling the axial co-ordinate the time co-ordinate in his analogy -

"Inside the cylinder, motion in the axial direction is not constrained at all."

I suspect I have some intuitive idea for what it implies to an observer that they move inexorably towards the singularity no matter which direction they attempt to travel inside the EH.

Still, I know what it means to "attempt to travel". My instinct is painting pictures for me based on what attempts to travel along a distance axis look like in my ordinary experience, and seeing the attempts (in my mind's eye) fail to prevent the observer from getting closer to the singularity as time marches forward. I don't know how to talk about "closer" without invoking forward moving time.

Is there such a thing as "attempting to travel" in time, and is movement along the time (axial) axis in the authors description truly unconstrained? My instinct tells me there is no such doable thing as attempting to travel in time, so what does it mean to say that travel along the time axis is unconstrained? What experiment, if any, could a human observer do to assess their unconstrained-in-time condition?

My instinct is that inside the EH, time continues to move inexorably forward for the observer, and in addition, all futures contain a collision with the singularity, which seems different and less exotic than saying that travel along the time axis becomes unconstrained.

 

[mfb]

The local time always moves on normally for everyone.

For an observer inside a black hole the singularity is an event in the future - they'll reach it as inevitably as we will reach Friday.

 

[PeterDonis]

a B level discussion of this I found on-line

The "swapping of space and time" inside the horizon that this article describes is an artifact of a particular choice of coordinates (Schwarzschild coordinates); it's not anything physical.

You might want to read my Insights article series on the Schwarzschild geometry, which discusses a number of common misconceptions including the "space time swapping" one. The first of the four part article series is here:

https://www.physicsforums.com/insights/schwarzschild-geometry-part-1/

this statement where the author is calling the axial co-ordinate the time co-ordinate in his analogy

What he's saying is that the direction he is calling the "axial" direction is spacelike inside the horizon; it's a direction in space, not a direction in time. (Outside the horizon, the same coordinate is timelike, so it's a direction in time.) The timelike direction (in these particular coordinates) is the radial direction inside the horizon; "radially inwards" points towards the future (whereas outside the horizon, "radially inwards" is a spacelike direction).

Is there such a thing as "attempting to travel" in time

Not really, since you have no choice about traveling into the future. But the kind of "travel" you are talking about here is not "travel in time". See above and below.

is movement along the time (axial) axis in the authors description truly unconstrained?

The axial axis inside the horizon is not timelike, it's spacelike. See above. Motion along this axis is "unconstrained" in the same sense as for any spacelike direction; the only limitation is that once you're inside the horizon, you only have a finite amount of time before you reach the singularity, which limits how far you can go in any spacelike direction. But as far as the spacetime geometry is concerned, the axial axis inside the horizon is infinitely long.

 

[Ibix]

Outside the black hole, Schwarzschild's $r$ is related to distance to the horizon. Inside the hole you can define $r$, but it is related to the time until the singularity. Both inside and outside the black hole you are free to move in space but your worldline always leads forward in time, so your personal experience (apart from a feeling of impending doom) is unaffected.

The difference when you get inside the horizon is that we've rather carelessly labelled one of our spatial directions $t$ and our timelike direction $r$ (many sources use $u$ and $v$ for "interior" Schwarzschild coordinates precisely to avoid the preconceptions that come with $r$ and $t$). There is no physical consequence to this poor choice of label - you notice nothing unusual. It's roughly analogous to walking towards the north pole. North is forward, north is forward, north is forward, then suddenly every direction is south, then south is forward and north is backward. You don't notice anything unusual, because nothing unusual happened. You just shouldn't have attached physical significance ("forward") to a coordinate direction ("northward") when going through a place where the coordinates misbehave.

 

[PeterDonis]

Outside the black hole, Schwarzschild's $r$ is related to distance to the horizon. Inside the hole you can define $r$, but it is related to the time until the singularity.

Careful. First, $r$ is only timelike inside the horizon in Schwarzschild coordinates. Second, the invariant definition of $r$ as the "areal radius" of 2-spheres is true inside the horizon in any coordinates, including Schwarzschild coordinates. So even in Schwarzschild coordinates, $r$ inside the horizon is still telling you about the areas of 2-spheres.

The difference when you get inside the horizon is that we've rather carelessly labelled one of our spatial directions $t$ and our timelike direction $r$

Again, this is only true in Schwarzschild coordinates.

(many sources use $u$ and $v$ for "interior" Schwarzschild coordinates precisely to avoid the preconceptions that come with $r$ and $t$)

Can you give a reference? $u$ and $v$ are normally associated with null coordinates such as Eddington-Finkelstein coordinates, not Schwarzschild coordinates.

 

[Ibix]

Careful. First, $r$ is only timelike inside the horizon in Schwarzschild coordinates.

Indeed - I did call it "Schwarzschild's $r$" in the first sentence.

Second, the invariant definition of rr as the "areal radius" of 2-spheres is true inside the horizon in any coordinates, including Schwarzschild coordinates. So even in Schwarzschild coordinates, rr inside the horizon is still telling you about the areas of 2-spheres.

Agreed - but it's still monotonically decreasing towards the singularity in the same way it monotonically decreases towards the event horizon when you are outside.

Can you give a reference?

My faulty memory, apparently. The source I have to hand is Maxima, which turns out to use $t,z,u,v$, with $t$ being the timelike areal radius coordinate and $u$ and $v$ being the equivalents of $\theta$ and $\phi$. $t$ and $z$ I understand, but I don't see why they've chosen to use $u,v$ instead of $\theta,\phi$, which they use in the exterior region.

 

[PAllen]

When using Schwarzschild interior coordinates I like using -t (instead of r), and z (instead of t), and θ,ϕ for the others. Since these particular slices are 2-sphere X R, z captures the idea of an axial coordinate of a hypercylinder. The metric still tells you that area of the 2-sphere is decreasing as t goes from -R to 0. You see $t^2$ multiplying the angular coordinates. Note, I also prefer to have the future time direction increase the t coordinate, thus the minus sign. Making these changes emphasizes, as well, there is no connection between the Scwarzschild interior coordinate patch and the exterior patch.

Of course, I don't do this when I use other charts that cover the interior.