A few weeks ago I published a post about Ted Jacobson and his deep understanding of general relativity (see here). Jacobson proved in 1995 that Einstein equations can be derived from thermodynamic arguments as an equation of state. To get the proof, Jacobson used Raychaudhuri equation and the proportionality relation between area and entropy holding for all local acceleration horizons. This result implies that exist some fundamental quantum degrees of freedom from which Einstein equations are obtained by properly managing the corresponding partition function. To estabilish such a connection is presently not at all a trivial matter and there are a lot of people around the World trying to achieve this goal even if we lack any experimental result that could lead the way.

Today in arxiv appeared a nice paper by Ram Brustein and Merav Hadad that generalize Jacobson’s result to a wider class of gravitational theories having Einstein equations as a particular case (see here). This result appears relevant in view of the fact that a theory exploiting quantum gravity could have as a low-energy limit some kind of modified Einstein equations, containing at least coupling with matter. Anyhow, we see how vacuum of quantum field theory seems to become even more important in our understanding of behavior of space-time.

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I wasn’t aware of that relation with Ted’s thesis with Raphael Bousso’s General Covariant Entropy Bound in 1999. I just noticed this today because Raychaudhuri equation was also used to derive Bousso’s bound.

Speaking of null surfaces, I wonder if anyone tried a twistor version of these bounds.

The point here is that we can take as an assumption what can be derived looking at the other way round. Indeed, Jacobson just says in his proof that, if holds proportionality between area and entropy, Einstein equations can be obtained. The paper you cite just shifts the question one step behind and criticize using Wald definition of entropy in the proof that should hold only assuming Einstein equations to hold (on shell) making the argument a circular one.

Let me say that I cannot take this point too seriously. The reason is that I can always take thermodynamic equations to hold in some way and obtain some result from them. This is reminiscent of the arguments adopted for black body radiation. Of course, this does not mean that definitions could not be questioned and proof improved to eliminate any concern.

The author indeed claims:

“Of course, one can just postulate it
and ‘derive’ the field equations but that may not provide much insight.”

But this cannot be shared as I have gained an insight: The right definitions produce the right equations and the argument is self-consistent. This means that Wald entropy is the right definition in both directions.

Interesting!

I wasn’t aware of that relation with Ted’s thesis with Raphael Bousso’s General Covariant Entropy Bound in 1999. I just noticed this today because Raychaudhuri equation was also used to derive Bousso’s bound.

Speaking of null surfaces, I wonder if anyone tried a twistor version of these bounds.

These are cute papers (unlike other papers of T.J.).

I wrote about it, too.

http://motls.blogspot.com/2009/03/einsteins-equations-as-equations-of.html

Hi Marco!

Shouldn´t this be the other way around?

Regards, Rafael.

Dear Rafael,

The point here is that we can take as an assumption what can be derived looking at the other way round. Indeed, Jacobson just says in his proof that, if holds proportionality between area and entropy, Einstein equations can be obtained. The paper you cite just shifts the question one step behind and criticize using Wald definition of entropy in the proof that should hold only assuming Einstein equations to hold (on shell) making the argument a circular one.

Let me say that I cannot take this point too seriously. The reason is that I can always take thermodynamic equations to hold in some way and obtain some result from them. This is reminiscent of the arguments adopted for black body radiation. Of course, this does not mean that definitions could not be questioned and proof improved to eliminate any concern.

The author indeed claims:

“Of course, one can just postulate it

and ‘derive’ the field equations but that may not provide much insight.”

But this cannot be shared as I have gained an insight: The right definitions produce the right equations and the argument is self-consistent. This means that Wald entropy is the right definition in both directions.

Regards,

Marco