Understanding White Holes to Region III in GR

[etotheipi]

Trying to follow Townsend's notes; section 2.3 is discussing ways of dealing with the co-ordinate singularity at $r=2M$, i.e. either by transforming to ingoing EF which cover I & II, outgoing EF which cover I & III, or KS which cover the entire manifold.

I got a bit preoccupied with the co-ordinate transformations and I realized just now I don't really understand the big picture. How are you supposed to interpret region III, the white-hole interior region? He says "both black and white holes are allowed by GR", but both regions are part of the same spacetime, right? While we're at it, what's the significance of having two exterior regions?

In other words, I think I misunderstand what these diagrams are supposed to mean. Could someone clarify? Thanks

 

[Dale]

I am with you. The maximally extended solution is [understatement] a bit weird [/understatement].

What gets me is that it looks like there is just a single event horizon covering both singularities.

 

[DrGreg]

In the real universe, black holes come into existence by a collapse of matter. Schwarzschild coordinates, or any transformation of them e.g. Kruskal coordinates, are valid only outside, or after, the collapsing matter.

In a Kruskal diagram, the outermost piece of collapsing matter forms a timelike worldline in regions I and II, so the chart is only valid above or to the right of this timelike path. Below and left of the worldline, a different metric would apply, so regions III and IV wouldn't exist.

There's a diagram I once drew here: https://www.physicsforums.com/threads/oppenheimer-snyder-model-of-star-collapse.651362/post-4164435

The maximally extended spacetime represents something that, as far as we can tell, couldn't exist in our universe.

 

[Ibix]

The Kruskal diagram covers the whole of spacetime, and there are four regions. Two are the exterior and interior of the hole. The other two are another exterior and a white hole region, but they are artefacts of taking the notion of an eternal black hole a bit too seriously. It reminds me of modelling the Earth as a point mass in Newtonian gravity and concluding that there's a singularity at the centre - we've taken our idealisation a little too far. Chapter 7 of Carroll's notes has a sketch of spacetime including a collapsing star, which doesn't include those other two regions.

The thing is that maximally extended Schwarzschild spacetime is symmetric under time reversal, so if there's a singularity in the future inside the horizon then there has to be one in the past too. So if there's a black hole then there has to be something like a time reversal of the black hole. Time reversal of something you can only fall into is something you must leave, and that's the white hole. Test particles (and test particles only, because the spacetime is actually vacuum) may materialise out of the white hole singularity into the white hole region, which they must then leave, usually into one of the two exterior regions (although they can also pass directly into the black hole).

Stuff can appear unpredictably out of the white hole singularity because it's a time reversal of stuff falling into the black hole singularity. GR can't describe what's happening there, so there's no explanation for this (except for a working quantum theory of gravity, which will hopefully make a bit more sense...).

Note: labelling of regions I and II is consistent between sources, but III and IV not so much. Take care comparing different descriptions. Naming the regions is probably safer than using the numbers.

 

[PeterDonis]

How are you supposed to interpret region III, the white-hole interior region?

I talked about this some in this Insights article:

https://www.physicsforums.com/insights/schwarzschild-geometry-part-4/

 

[etotheipi]

Interesting! I read the articles now, think it's making a bit more sense. So the collapsing matter case is not time-symmetric and it follows we can pretty much ignore regions III and IV (or, stronger, anything below and left of the timelike worldline of the outer-most piece of the collapsing matter...). That's what this diagram is trying to show, right?

On that note, one thing I didn't notice yesterday is that he actually goes on to discuss (hypothetical) time-symmetric, "eternal" black holes, which is an example of a case where region IV is important. In isotropic co-ordinates $(t,\rho, \theta, \varphi)$ with $\rho$ by $r = \left( 1 + \frac{M}{2\rho} \right)^2 \rho$, the metric $(\mathrm{2.60})$ which includes both regions is left as an exercise but can be derived by writing $\mathrm{d}r = \left( 1 + \frac{M}{2\rho} \right)\left( 1 - \frac{M}{2\rho} \right) \mathrm{d}\rho$; for simplicity we can also re-write $1 - \frac{2M}{\rho \left(1+ \frac{M}{2\rho} \right)^2} = \frac{\rho \left(1+ \frac{M}{2\rho} \right)^2 - 2M}{\rho \left(1+ \frac{M}{2\rho} \right)^2} = \frac{\left(1-\frac{M}{2\rho}\right)^2}{\left(1+\frac{M}{2\rho} \right)^2}$ so that the original metric $(\mathrm{2.32})$ becomes $\mathrm{d}s^2 = -\frac{\left(1-\frac{M}{2\rho}\right)^2}{\left(1+\frac{M}{2\rho} \right)^2} \mathrm{d}t^2 + \frac{\left(1+\frac{M}{2\rho}\right)^2}{\left(1-\frac{M}{2\rho} \right)^2} \left( 1 + \frac{M}{2\rho} \right)^2\left( 1 - \frac{M}{2\rho} \right)^2 \mathrm{d}\rho^2 + \left( 1 + \frac{M}{2\rho} \right)^4 \rho^2 d\Omega^2$ or more simply $\mathrm{d}s^2 = -\frac{\left(1-\frac{M}{2\rho}\right)^2}{\left(1+\frac{M}{2\rho} \right)^2} \mathrm{d}t^2 + \left( 1 + \frac{M}{2\rho} \right)^4 \left( \mathrm{d}\rho^2 + \rho^2 d\Omega^2 \right)$ so the $t=C$ hypersurfaces $U = Ve^{C/2M}$ (which are straight lines through the origin passing through regions I and region IV) look just like $\mathbf{R}^3$, and the 2-sphere with a minimum $\rho = M/2$ corresponds to the middle of the wormhole.

Test particles (and test particles only, because the spacetime is actually vacuum) may materialise out of the white hole singularity into the white hole region, which they must then leave, usually into one of the two exterior regions (although they can also pass directly into the black hole).

And that is presumably what Townsend refers to when he calls the $r=0$ region of the white hole a "naked singularity", in that signals from this region can reach $\mathfrak{J}^+$, right?

 

[PeterDonis]

isotropic co-ordinates

Note that these coordinates only cover the exterior regions, not the interior regions. The range $0 < \rho < M / 2$ covers the second exterior region (to see this, compute the areas of the 2-spheres for this range of $\rho$).

 

[Stavros Kiri]

Nice topic and good work in that great analysis for all! ...
Thanks guys! A learned a few things ...