What is the metric for a bag-of-gold spacetime?

[Onyx]

TL;DR Summary

What is the metric for a bag-of-gold spacetime?

What is the metric for a bag-of-gold spacetime?

 

[PeterDonis]

What is the metric for a bag-of-gold spacetime?

There is no simple expression for the metric for this spacetime, since it consists of (at least) two regions with very different properties that are "glued" together. However, a good mathematical treatment can be found in section III (b) of this paper:

https://arxiv.org/abs/0803.4212

Note, though, that this paper requires "A" level background knowledge to properly understand. A "B" level discussion of the "bag of gold" spacetime and its implications is not really possible.

 

[Onyx]

There is no simple expression for the metric for this spacetime, since it consists of (at least) two regions with very different properties that are "glued" together. However, a good mathematical treatment can be found in section III (b) of this paper:

https://arxiv.org/abs/0803.4212

Note, though, that this paper requires "A" level background knowledge to properly understand. A "B" level discussion of the "bag of gold" spacetime and its implications is not really possible.

Is that because the gluing process is so complicated?

 

[Demystifier]

Is that because the gluing process is so complicated?

Not really, the black hole interior can be interpreted as a "bag of gold" even without gluing. See https://arxiv.org/abs/1411.2854

 

[Onyx]

Not really, the black hole interior can be interpreted as a "bag of gold" even without gluing. See https://arxiv.org/abs/1411.2854

Am I correct in assuming that it is just like the OS collapse case, where there has to be two different metrics for two different manifolds? Also, if not a bag of gold, what is this person describing (not a reliable source)?

 

[Demystifier]

Am I correct in assuming that it is just like the OS collapse case, where there has to be two different metrics for two different manifolds?

What's OS?

 

[vanhees71]

Openheimer Snyder I guess.

https://doi.org/10.1103/PhysRev.56.455

 

[PeterDonis]

Not really, the black hole interior can be interpreted as a "bag of gold" even without gluing. See https://arxiv.org/abs/1411.2854

This interpretation is rather contrived, though. I expressed my reservations about it in this previous thread:

https://www.physicsforums.com/threads/interior-volume-of-a-black-hole.962341/

 

[PeterDonis]

Is that because the gluing process is so complicated?

Not really, no. The diagrams in Fig. 3 of the paper are fairly straightforward to understand. But the mathematical details required to verify that everything actually can fit together the way those diagrams depict while satisfying the Einstein Field Equation are not.

 

[PeterDonis]

Am I correct in assuming that it is just like the OS collapse case, where there has to be two different metrics for two different manifolds?

The "gluing" process at the boundary between two regions follows the same general rules in both cases, yes. But the relationship between the regions is not the same in the two cases.

In the OS case, we have a Schwarzschild region extending in from infinity, with ##r## (the areal radius) decreasing monotonically, to a boundary with a closed FRW region, and the closed FRW region occupies the rest of the spacetime, with ##r## continuing to decrease to ##r = 0##.

In the "bag of gold" case described in Fig. 3 of the paper, we have a Schwarzschild region extending in from infinity all the way through an Einstein-Rosen bridge, where ##r## (the areal radius) reaches a minimum, and then ##r## increasing again to a boundary with a closed FRW region, in which ##r## continues to increase to some maximum value (given as ##a_{12}## in the diagram) and then decreases again. (The spacetime could end with that first FRW region decreasing to ##r = 0##, but in Fig. 3 of the paper, it doesn't, there is a second Schwarzschild region going through a second Einstein-Rosen bridge and connecting to a second FRW region that ends by decreasing to ##r = 0##. In principle you could have any number of such transitions added on.)