How Do Horizons Influence the Entropy and Temperature of Spacetime?

[pervect]

Browsing through the "New Springerlink", I came across

Gravity, the Inside Story

It's got some interesting, though non-classical, points about gravity. The author starts out with an argument about the inevitability of horizons.

But—conceptually—strange things happen as soon as: (i) we let the metric to be
dynamical and (ii) allow for arbitrary coordinate transformations or, equivalently,
observers on any timelike curve examining physics. Horizons are inevitable in such a
theory and they are always observer dependent.

This is sort of amusing considering the amount of angst we appear to be having currently at PF about the mere existence of event horizons. But that's not the main point of the paper.

This raises the famous
question first posed by Wheeler to Bekenstein: what happens if you mix cold and hot
tea and pour it down a horizon, erasing all traces of “crime” in increasing the entropy
of the world? The answer to such thought experiments demands that horizons should
have an entropy which should increase when energy flows across it.
With hindsight, this is obvious.

and

This historical sequence raises a some serious issues for which there is no satisfactory
answer in the conventional approach:
1. How can horizons have temperature without the spacetime having a microstructure?

The author argues that it's not possible for horizons to have a temperature without some sort of microstructure in spacetime, even if we don't yet have a theory that gives said microstructure.

The really interesting stuff happens when the author writes expressions for the entorpy S_matter and S_grav, the entropy of the matter fields and the horizons - and derives Einstein's equations as a low order approximation to the maxmization of entropy,

There's also some higher order terms which make the theory depart from the simple action of GR.

 

[bcrowell]

The link, http://download.springer.com/static/pdf/834/art%253A10.1007%252Fs10714-008-0669-6.pdf?auth66=1353968763_5e7bf57514e8cb0ab0ad355560d4bbdd&ext=.pdf , is broken.

 

[pervect]

The link, http://download.springer.com/static/pdf/834/art%253A10.1007%252Fs10714-008-0669-6.pdf?auth66=1353968763_5e7bf57514e8cb0ab0ad355560d4bbdd&ext=.pdf , is broken.

I relinked it through a Doi - hopefully it works now

 

[Mentz114]

The link works now. This paper has been around for a while and I'm glad you brought it to my attention again. It seems impossible to define energy conservation globally in GR, but does the requirement δ[S_matter + S_grav] = extremum mean we have global entropy conservation ? If you'll allow some latitude in my expression.

 

[pervect]

The link works now. This paper has been around for a while and I'm glad you brought it to my attention again. It seems impossible to define energy conservation globally in GR, but does the requirement δ[S_matter + S_grav] = extremum mean we have global entropy conservation ? If you'll allow some latitude in my expression.

The date is 2008, but this is the first I've seen of it. Excuse me while I think out loud on the entorpy isssue.

Because entropy is a scalar density, we might have a possibility of avoiding the usual parallel transport issues. (My one reservation here is that entropy also has a representation as a 4-vector, by the number-flux density, and the vector representation should have the parallel transport issues - so perhaps I'm being overly optimistic and overlooking something in arguing that the scalar density form avoids the issue).

But let's suppose that treating entropy in it's scalar density form gets rid of the parallel transport issue for now. (The idea being that parallel transport rotates vectors, but numbers don't rotate , so we don't get the path dependency issues when we parallel tranpsort numbers and add them together). We still have the problem of the relativity of simultaneity. Conceptually, we can count the total number of states for any given definition of "now", but unless entropy is constant with time, as we choose different notions of "now" we'll get different numbers for the total number of states / total entropy of the universe "now".

So if one now has the hot tea and the cold tea unmixed, and the other now has them mixed and at the equilibrium temperature, the entropy should be different.

So it's not terribly clear why we demand that the change in entropy is zero, I have to agree. Except that it yields equations that look like they might be correct, or at least interesting.

A sub point here is that S_grav and S_mat are both integrals over d^4x, so setting the change to zero doesn't involve any transport issues, it all appears to be local. But I have to agree it's not clear why we set the change to zero,.

 

[Mentz114]

I understand what you're saying. I find entropy difficult in the GR context. If we want to count microstates say in a gas ( collection of partcles), can the states be expressed in terms of the worldlines (which are not observer dependent) of all the involved pieces and some kind of spatial slicing ? Unless the worldlines are dependent on internal states this looks impossible.

I need to think about this for a year or so.

 

[George Jones]

The author, Padmanabhan, writes about related issues in the last chapter of his 2010 book, "Gravitation: Foundations and Frontiers",

https://www.amazon.com/dp/0521882230/?tag=pfamazon01-20

Interesting stuff.