Why does Wheeler's bag of gold solution contradict the holographic principle and a black hole does not?

Question

The interior volumen of a typical black hole is greater than the one we expect when measuring its external area and calculating its volume:

https://plus.maths.org/content/dont-judge-black-hole-its-area-2

But Wheeler's "bags of gold" solutions (basically, a new born universe inside a black hole) contradicts the holographic principle:

https://en.wikipedia.org/wiki/Holographic_principle

I guess its due to a great interior volume being enclosed by a small area (exterior black hole area), but does that already happen in typical black holes?

Answer 1

1. The basic idea of the bags of gold paradox is that naively there are too many degrees of freedom in the bulk. For slicings which have a very large volume (e.g. exponentially increasing volumes as in FRW universes), one can naively place many scalar excitations in that volume, and consequently the entropy counting (which goes as volume) can increase the Bekenstein-Hawking entropy of the black hole. Hence the paradox.

2. We will work in AdS as there we best understand quantum gravity in this setup. As you mentioned, one can construct spacelike slicings in the case of AdS black holes which can have increasing volumes. See the worked out case for "maximal volume slicings" in Appendix 1 of this paper. Here the paradox is this, these maximal volume slicings have increasing interior volume with the Kruskal time, and any scalar field will naively have a similar overcounting problem as in Point 1 above.

3. Consequently there is an overcounting problem in the AdS bulk, which arises either because we are treating all the volume excitations independently, or because there is no interior of black holes in quantum gravity and therefore no map from the interior of the black hole to the holographically dual CFT (as the firewall papers suggest). I strongly think that the map exists from the interior to the CFT (this is a possible state-dependent map), and therefore the bags of gold is a question why the degrees of freedom are overcounted here.

4. Note that the holographically dual CFT description doesn't have any paradox. This is because conventional counting of the CFT degrees of freedom have precisely given us the Bekenstein Hawking entropy in numerous cases. The CFT doesn't know about which spacelike slicing we are working on and gives the correct BH entropy.

EDIT: It might be helpful to look at this recent paper which studies the bags of gold in AdS, and proposes a resolution.

Answer 2

1) Black hole entropy is a semi-classical (somewhat quantum) solution. BH entropy $S$ is proportional to the surface area $A$ of the BH, not their volume. $S=A/4$ (in natural units). BH’s can also be considered as 2D objects in a 4D space-time, see e.g. this paper.

A hologram is a representation of a system using fewer dimensions that can still pack in all the information from the original system. E.g. a 2D surface that has all the information in a 4D space-time – a BH. So BH’s are the very definition of the holographic principal.

2) Now, Wheelers ‘bag of gold’ analogy means – curve classical space into a bag and you can stuff a great deal of ‘gold’ (i.e. entropy) into it. So ‘WBG’ solutions in General Relativity allow different volumes (and entropies) contained in the same surface area in 4D space-times. This conflicts with the surface-area : entropy BH relationship. Remember though that GR is non-quantum.

A full quantum gravity theory is expected to resolve this ‘paradox’, and indeed, in AdS (not quite a full quantum gravity theory) there are pointers to this, as @Bruce Lee expands on.

Answer 3

Trying to be as simple as possible:

• Black hole entropy is believed to be proportional to surface area (see Bekenstein/Hawking etc).
• Things falling into a black hole should increase the entropy of the black hole to satisfy the second law of thermodynamics.
• But Wheeler's Bag of Gold solutions show that in curved space, a given surface area can enclose an arbitrarily large volume.
• CONTRADICTION: If a given surface can enclose an arbitrary amount of in-falling material, i.e. an arbitrary amount of entropy, how is it correct to say the entropy contained therein is proportional to the surface area?