# Beckenstein Bound

## Main Question or Discussion Point

So I'm reading "Three Roads to Quantum Gravity" by Lee Smolin, and at one point he brings up something called the Beckenstein Bound which is confusing the heck out of me.

The way Smolin basically describes this (this is a popular, not a technical book, so maybe he left out some details...) is that if you have a block of space, the maximum bound on the entropy within that space is given by the area of its boundary, not its volume.

The proof given for this is that it is known that black holes have a specific entropy which is based on the area of its boundary. So therefore it is argued no region of space could have an entropy higher than the equivalent black hole entropy, because there are processes by which that region of space could become a black hole (maybe you drip in energy until a black hole forms, or maybe a black hole is just passing by and it falls in) and it's not possible that the process of becoming a black hole could cause the region's entropy to decrease.

Alright, fine. Here's the part that confuses me: The choice of "boundary" seems to me to be arbitrary. Let's say that we have a region A of space within a certain sphere. Because we have this sphere boundary, we have a bound on the region's entropy. Now let's say that we have another region B of space of the same volume, but instead of its boundary being a sphere the boundary has folds, like a brain. Although the volume of region B is the same as region A, the entropy bound is much higher because it has more surface area.

Okay, now let's say we draw another boundary, just around region B, which is perfectly spherical. This boundary defines a region C which contains region B, but which would have a lower surface area and thus a lower entropy bound. It seems like we've now lowered the amount of possible entropy within that space just by considering a different boundary.

What am I missing here?

## Answers and Replies

xantox
Gold Member
The Bekenstein formula only applies to spherical boundaries in flat spacetime. For boundaries of different shape in curved spacetime, the entropy bound cannot be computed with the same formula, so that all bounds are expected to agree to some generalized form of covariant bound.

The Bekenstein formula only applies to spherical boundaries in flat spacetime.
Okay, that explains a lot.

For boundaries of different shape in curved spacetime, the entropy bound cannot be computed with the same formula, so that all bounds are expected to agree to some generalized form of covariant bound.
So are you saying that it is expected some equivalent of the Beckenstein formula exists in GR universes, but it is not known yet exactly what it is?

Is it possible that the Beckenstein Bound does not exist in a GR universe? (It seems like the "equivalent" would be hard to define, since it kind of seems like in GR there isn't specifically such a thing as a "spherical boundary"??)

xantox
Gold Member
So are you saying that it is expected some equivalent of the Beckenstein formula exists in GR universes..
Do not say "GR universes", but "curved spacetimes", since they are not synonyms at all.

Is it possible that the Beckenstein Bound does not exist in a GR universe? (It seems like the "equivalent" would be hard to define, since it kind of seems like in GR there isn't specifically such a thing as a "spherical boundary"??)
It is in fact difficult to define a (conjectured) generalized entropy bound in terms of radius and volume. So, it is rather defined by constructing the null hypersurfaces (lightsheets) intersecting the surface boundary being studied, and it will be a bound on the entropy on the lightsheets.

That makes sense I think. Thanks!