Equivalence of Gravity & Thermodynamics: Einstein Eqs & $\delta Q = T \delta S$

[MTd2]

http://arxiv.org/abs/0903.0823

The Einstein equations for generalized theories of gravity and the thermodynamic relation $ \delta Q = T \delta S$ are equivalent
Authors: Ram Brustein, Merav Hadad

Abstract: We show that the equations of motion of generalized theories of gravity are equivalent to the thermodynamic relation $\delta Q = T \delta S$. Our proof relies on extending previous arguments by using a more general definition of the Noether charge entropy. We have thus completed the implementation of Jacobson's proposal to express Einstein's equations as a thermodynamic equation of state. Additionally, we find that the Noether charge entropy obeys the second law of thermodynamics if the matter energy momentum tensor obeys the null energy condition. Our results support the idea that gravitation on a macroscopic scale is a manifestation of the thermodynamics of the vacuum.

Thanks to Marco Frasca for pointing that out:

http://marcofrasca.wordpress.com/2009/03/05/ted-jacobsons-deep-understanding/

 

[MTd2]

This Ram Brunstein is really a great guy. You guys should check his articles. I selected some related with the subject of this thread.

http://arxiv.org/abs/0901.2191
The sound damping constant for generalized theories of gravity
Authors: Ram Brustein, A.J.M. Medved

Abstract: The near-horizon metric for a black brane in Anti-de Sitter (AdS) space and the metric near the AdS boundary both exhibit hydrodynamic behavior. We demonstrate the equivalence of this pair of hydrodynamic systems for the sound mode of a conformal theory. This is first established for Einstein's gravity, but we then show how the sound damping constant will be modified, from its Einstein form, for a generalized theory. The modified damping constant is expressible as the ratio of a pair of gravitational couplings that are indicative of the sound-channel class of gravitons. This ratio of couplings differs from both that of the shear diffusion coefficient and the shear viscosity to entropy ratio. Our analysis is mostly limited to conformal theories but suggestions are made as to how this restriction might eventually be lifted.

http://arxiv.org/abs/0810.2193
The shear diffusion coefficient for generalized theories of gravity
Authors: Ram Brustein, A.J.M. Medved

Abstract: Near the horizon of a black brane in Anti-de Sitter (AdS) space and near the AdS boundary, the long-wavelength fluctuations of the metric exhibit hydrodynamic behaviour. The gauge-gravity duality then relates the boundary hydrodynamics for generalized gravity to that of gauge theories with large finite values of 't Hooft coupling. We discuss, for this framework, the hydrodynamics of the shear mode in generalized theories of gravity in d+1 dimensions. It is shown that the shear diffusion coefficients of the near-horizon and boundary hydrodynamics are equal and can be expressed in a form that is purely local to the horizon. We find that the Einstein-theory relation between the shear diffusion coefficient and the shear viscosity to entropy ratio is modified for generalized gravity theories: Both can be explicitly written as the ratio of a pair of polarization-specific gravitational couplings but implicate differently polarized gravitons. Our analysis is restricted to the shear-mode fluctuations for simplicity and clarity; however, our methods can be applied to the hydrodynamics of all gravitational and matter fluctuation modes.


http://arxiv.org/abs/0808.3498
The ratio of shear viscosity to entropy density in generalized theories of gravity
Authors: Ram Brustein, A.J.M. Medved
(Submitted on 26 Aug 2008)

Abstract: Near the horizon of a black brane solution in Anti-de Sitter space, the long-wavelength fluctuations of the metric exhibit hydrodynamic behaviour. For Einstein's theory, the ratio of the shear viscosity of near-horizon metric fluctuations eta to the entropy per unit of transverse volume s is eta/s=1/4 pi. We propose that, in generalized theories of gravity, this ratio is given by the ratio of two effective gravitational couplings and can be different than 1/4 pi. Our proposal implies that eta/s is equal for any pair of gravity theories that can be transformed into each other by a field redefinition. In particular, the ratio is 1/4 pi for any theory that can be transformed into Einstein's theory; such as F(R) gravity. Our proposal also implies that matter interactions -- except those including explicit or implicit factors of the Riemann tensor -- will not modify eta/s. The proposed formula reproduces, in a very simple manner, some recently found results for Gauss-Bonnet gravity. We also make a prediction for eta/s in Lovelock theories of any order or dimensionality. http://arxiv.org/abs/0712.3206
Wald's entropy is equal to a quarter of the horizon area in units of the effective gravitational coupling
Authors: Ram Brustein, Dan Gorbonos, Merav Hadad
(Submitted on 19 Dec 2007 (v1), last revised 2 Mar 2009 (this version, v3))

Abstract: The Bekenstein-Hawking entropy of black holes in Einstein's theory of gravity is equal to a quarter of the horizon area in units of Newton's constant. Wald has proposed that in general theories of gravity the entropy of stationary black holes with bifurcate Killing horizons is a Noether charge which is in general different from the Bekenstein-Hawking entropy. We show that the Noether charge entropy is equal to a quarter of the horizon area in units of the effective gravitational coupling on the horizon defined by the coefficient of the kinetic term of specific graviton polarizations on the horizon. We present several explicit examples of static spherically symmetric black holes.


http://arxiv.org/abs/hep-th/0702108
Cosmological Entropy Bounds
Authors: Ram Brustein
(Submitted on 14 Feb 2007)

Abstract: I review some basic facts about entropy bounds in general and about cosmological entropy bounds. Then I review the Causal Entropy Bound, the conditions for its validity and its application to the study of cosmological singularities. This article is based on joint work with Gabriele Veneziano and subsequent related research. http://arxiv.org/abs/hep-th/0401081
Area-scaling of quantum fluctuations
Authors: A. Yarom, R. Brustein

Abstract: We show that fluctuations of bulk operators that are restricted to some region of space scale as the surface area of the region, independently of its geometry. Specifically, we consider two point functions of operators that are integrals over local operator densities whose two point functions falls off rapidly at large distances, and does not diverge too strongly at short distances. We show that the two point function of such bulk operators is proportional to the area of the common boundary of the two spatial regions. Consequences of this, relevant to the holographic principle and to area-scaling of Unruh radiation are briefly discussed.


http://arxiv.org/abs/hep-th/0311029
Thermodynamics and area in Minkowski space: Heat capacity of entanglement
Authors: Ram Brustein, Amos Yarom

Abstract: Tracing over the degrees of freedom inside (or outside) a sub-volume V of Minkowski space in a given quantum state |psi>, results in a statistical ensemble described by a density matrix rho. This enables one to relate quantum fluctuations in V when in the state |psi>, to statistical fluctuations in the ensemble described by rho. These fluctuations scale linearly with the surface area of V. If V is half of space, then rho is the density matrix of a canonical ensemble in Rindler space. This enables us to `derive' area scaling of thermodynamic quantities in Rindler space from area scaling of quantum fluctuations in half of Minkowski space. When considering shapes other than half of Minkowski space, even though area scaling persists, rho does not have an interpretation as a density matrix of a canonical ensemble in a curved, or geometrically non-trivial, background.


http://arxiv.org/abs/hep-th/0401081
Area-scaling of quantum fluctuations
Authors: A. Yarom, R. Brustein

Abstract: We show that fluctuations of bulk operators that are restricted to some region of space scale as the surface area of the region, independently of its geometry. Specifically, we consider two point functions of operators that are integrals over local operator densities whose two point functions falls off rapidly at large distances, and does not diverge too strongly at short distances. We show that the two point function of such bulk operators is proportional to the area of the common boundary of the two spatial regions. Consequences of this, relevant to the holographic principle and to area-scaling of Unruh radiation are briefly discussed.


http://arxiv.org/abs/hep-th/0108098
Causal Entropy Bound for Non-Singular Cosmologies
Authors: Ram Brustein, Stefano Foffa, Avraham E. Mayo

Abstract: The conditions for validity of the Causal Entropy Bound (CEB) are verified in the context of non-singular cosmologies with classical sources. It is shown that they are the same conditions that were previously found to guarantee validity of the CEB: the energy density of each dynamical component of the cosmic fluid needs to be sub-Planckian and not too negative, and its equation of state needs to be causal. In the examples we consider, the CEB is able to discriminate cosmologies which suffer from potential physical problems more reliably than the energy conditions appearing in singularity theorems.


http://arxiv.org/abs/hep-th/0101083
CFT, Holography, and Causal Entropy Bound
Authors: R. Brustein, S. Foffa, G. Veneziano

Abstract: The causal entropy bound (CEB) is confronted with recent explicit entropy calculations in weakly and strongly coupled conformal field theories (CFTs) in arbitrary dimension $D$. For CFT's with a large number of fields, $N$, the CEB is found to be valid for temperatures not exceeding a value of order $M_P/N^{{1\over D-2}}$, in agreement with large $N$ bounds in generic cut-off theories of gravity, and with the generalized second law. It is also shown that for a large class of models including high-temperature weakly coupled CFT's and strongly coupled CFT's with AdS duals, the CEB, despite the fact that it relates extensive quantities, is equivalent to (a generalization of) a purely holographic entropy bound proposed by E. Verlinde. http://arxiv.org/abs/hep-th/0009063
The Shortest Scale of Quantum Field Theory
Authors: Ram Brustein, David Eichler, Stefano Foffa, David H. Oaknin

AbstractIt is suggested that the Minkowski vacuum of quantum field theories of a large number of fields N would be gravitationally unstable due to strong vacuum energy fluctuations unless an N dependent sub-Planckian ultraviolet momentum cutoff is introduced. We estimate this implied cutoff using an effective quantum theory of massless fields that couple to semi-classical gravity and find it (assuming that the cosmological constant vanishes) to be bounded by $M_Planck/N^1/4$. Our bound can be made consistent with entropy bounds and holography, but does not seem to be equivalent to either, and it relaxes but does not eliminate the implied bound on N inherent in entropy bounds.


http://arxiv.org/abs/hep-th/0005266
Causal Boundary Entropy From Horizon Conformal Field Theory
Authors: Ram Brustein

Abstract: The quantum theory of near horizon regions of spacetimes with classical spatially flat, homogeneous and isotropic Friedman-Robertson-Walker geometry can be approximately described by a two dimensional conformal field theory. The central charge of this theory and expectation value of its Hamiltonian are both proportional to the horizon area in units of Newton's constant. The statistical entropy of horizon states, which can be calculated using two dimensional state counting methods, is proportional to the horizon area and depends on a numerical constant of order unity which is determined by Planck scale physics. This constant can be fixed such that the entropy is equal to a quarter of the horizon area in units of Newton's constant, in agreement with thermodynamic considerations.


http://arxiv.org/abs/hep-th/9912055
A Causal Entropy Bound
Authors: R. Brustein, G. Veneziano

Abstract: The identification of a causal-connection scale motivates us to propose a new covariant bound on entropy within a generic space-like region. This "causal entropy bound", scaling as the square root of EV, and thus lying around the geometric mean of Bekenstein's S/ER and holographic S/A bounds, is checked in various "critical" situations. In the case of limited gravity, Bekenstein's bound is the strongest while naive holography is the weakest. In the case of strong gravity, our bound and Bousso's holographic bound are stronger than Bekenstein's, while naive holography is too tight, and hence typically wrong. http://arxiv.org/abs/gr-qc/9904061
The Generalized Second Law of Thermodynamics in Cosmology
Authors: Ram Brustein
Abstract: A classical and quantum mechanical generalized second law of thermodynamics in cosmology implies constraints on the effective equation of state of the universe in the form of energy conditions, obeyed by many known cosmological solutions, and is compatible with entropy bounds which forbid certain cosmological singularities. In string cosmology the second law provides new information about the existence of non-singular solutions, and the nature of the graceful exit transition from dilaton-driven inflation.

 

[Lester]

Daniel,

the right link is

http://marcofrasca.wordpress.com/2009/03/05/ted-jacobsons-deep-understanding/

and the right name is Marco Frasca.

Jon

 

[MTd2]

Thank you. I corrected.

 

[Fra]

Thanks for bringing this up. This is a good topic. I'll check that out. The title sounds like Ted Jacobssons reasoning.

"Thermodynamics of Spacetime: The Einstein Equation of State"

"..The Einstein equation is derived from the proportionality of entropy and horizon area together with the fundamental relation $\delta Q=TdS$ connecting heat, entropy, and temperature..."
-- http://arxiv.org/abs/gr-qc/9504004

I recall reading Ted's paper some time ago and I'll see how Brunstein reasonig differs.

I'm looking for a simliar agle myself, but instead of heat, I try to consider a more information abstracted where classical temperautre and mass are replaced by information equivalents, currently it looks like temperature can be interpreted as the inverse of an information divergence, and energy is information capacty. I'm hoping to fund a more baggage free abstraction than I found in Ted's paper. I'll if this guys leaves me some clues!

Thanks for the links.

/Fredrik

 

[MTd2]

You should look for each of his papers, he has several DEEP insights about which are probably valid for any quantum gravity. I was very surprised how I didn't find Ram Brunstein before. I didn't find him within the search option of this forum. It's like not finding Frank Wilczek!

 

[Fra]

http://arxiv.org/abs/0903.0823

The Einstein equations for generalized theories of gravity and the thermodynamic relation $ \delta Q = T \delta S$ are equivalent
Authors: Ram Brustein, Merav Hadad

I skimmed this first paper you listed last night and the general reasoning is to my liking. The question posed and the way of reasoning is a little different than what I'm doing, but there is one striking similarity where I fully share the vision.

But identifying terms and choosing measures, they compare the dynamical equations of generalized gravity theories, and note that this is very analogous to thermodynamics and the the second law.

Set aside the details and choices they make, I think the spirit of this is more or less right in line with my reasoning about constructing measures for disorder and how these measures evolutions translates into a kind of action measures. There is a fundamental unity between the ME principle and the principle of least action. This is as I see it the general basis for how it makes sense that an equation of motion (which in action formulation can be seen as a minimum action principle) and a generalized law of maximum entropy. I think they are not only analogous, it is if you abstract it one level, when you analyse how the measures are constructed the SAME logic. They should follow from each other, if only the relation between the measures are seen.

So if you see the dynamical equations as a state equation, and if you take the deeper reasoning seriously, my personal conclusion is that even THIS this is bound to evolve. This is where I hope to see a lot more. I.e to take this equivalence betwee the ME principle and the principle of least action, find it's common denominator, and then put it into an evolutionary context of interacting observers. Then I think lagrangians and all the stuff that's usually put in by hand, are a result of an evolutionary process. The state equations themselves are evolving.

I'll try to skim some of the other papers.

/Fredrk

 

[Fra]

So I think it should, in line with this reasoning, be possible to argue not only that there is a strong analogy between thermodynamics and GR equations, but that in the context of evolving measures, the very action of GR should emerge. Then it would be much more than just a constructed analogy.

To get there, I think the good reasoning in that paper, needs to be put into an evolving context. And the evolution would similarly be driven by the SAME logic as thermodynamical arrows of time, least action principles or principle of minimum speculation (Principle of Minimum Discrimination Information).

The general core of that logic should be independent of specific microstructurs of lagrangians, I think it's a general evolutionary feature that repeats in nature at several levels, as we already hinter. It remains to see how this is part of nature construction.

/Fredrik

 

[thomheg]

If you apply a 'Wick rotation' to the gravitational force you end up at thermodynamics.
That is like multiply with i, or to shift a relation by 90 degree in a spacetime diagram.
Since gravity connects matter over space in a statistical way (because it connects all matter to all matter), gravity is a connection of parallel lines. Now shift that by 90 degree and you turn a 'vertical' statistic into a horizontal statistic. ('Vertical' is the orientation of the timelike axis and horizontal is spacelike.) Since in the spacelike direction you have a connection over a field, the statistical distribution of a field appears as heat.

 

[Fra]

The various kind of analogies between QM, feymann action amplitudes and statistical mechanics is what has been feeding a lot of curiousty for a long time, but they are rarely understood in a deeper sense. It's more a matter of toying around with expressions and noting that if you put an i in there then it looks like that...

I guess it's the explanation of these analogies that many feel aren't conincidences that are still missing. In feynmanns path integral formulation of QM is where the connection is cleanest, but still there exists no clear logic as to why it is the way it is. That it seems to roughly describe nature is more like an empirical observation. Also aside from the postulated forms of the amplitudes, the normalization procedure and the meaning of action isn't understood.

What exactly is the physical meaning of action or lagrangians? I think the problem is that there exists no clean general abstraction of this as it tend to always come as baggage based on the classical mechanic models. I think in a model based only on communication, this baggage is not acceptable - it should emerge from communication processes.

My personal thinking is that the structure of the action (or equivalently the lagrangian) will be emergent in an evolving process. This implies both GR actions and the actions for the SM of particle physics. I think once we understand how this process evolves, we will also seen in clear, the various conincidental connections that up to this date are mostly empirically observed analogies with our mathematical expressions.

/Fredrik

 

[Fra]

An interesting reflection already in Ted Jacobson's paper is since

- he views Einsteins equation as a an equation of state
- valid only for a special kind of equilibrium condition
- the choice of entropy functional implies another equation of state and thus modification to Einsteins equation

Ted expects future understanding to uncover the laws of non-equilibrium conditions, and that Einsteins equation is then only emergent as an equilibrium condition (at some level).

Then a reformulation of this, that is closely related to evolving measures, is that this vision seems more or less equivalent to understanding an evolving entropy (information) measures.

It's also interesting that Newtons constant of gravitation relates to the proportionality constant between area and information. Ie. it's a relation between two measures, entropy vs area. Again we keep seeing relations between measures.

And I think that the quest Ted suggest, to understand the non-equilibrium and more general case of these equations of state (not that this again, intermixes states and laws! known from smolins ponderings; since up until this analysis, GR equations is a law, and not a state) is the quest to understand how measures interact and evolve. And measure are IMO the abstraction of a measurement operator, that takes into account also how the information not only flows, but also stores.

/Fredrik

 

[Demystifier]

The recent Padmanabhan's contribution:
http://xxx.lanl.gov/abs/0903.1254

 

[MTd2]

Not only that, 2 other people posted on arxiv a similar article in the day following the one from Ram Brunstein and Merav Hadad. It is even cited by the one you posted above.

http://lanl.arxiv.org/abs/0903.1176

They criticize the article that began this thread by saying in the last paragraphy of their paper:

"While this paper was being prepared, the preprint http://arxiv.org/abs/0903.0823 appeared, claiming similar conclusions;unfortunately, among other things, their starting formula for entropy (equation 9 in http://arxiv.org/abs/0903.0823 ) is manifestly incorrect, leaving the result in doubt."

 

[Fra]

People seem to start to see this connection from different angles to gain intuition which is interesting. I am still looking forward to somone what will take yet a different angle, and ask how the measures are connected at a deeper level.

If you loosely associate a causal horizon with an observers communication channel with it's environment, then one might ask how to quantify and measure the properties of such a channel.

I think there must be a connection between a sort of unit of probability, and a sort of unit of space (length), via the area-entropy connection. Area is a measure of the causal connection between two regions, and entropy is a measure of the information on one side presumable hidden to the other side of a communication channel.

Now unless an ad hoc microstructure is to be pulled out of thin air, the microstructure itself, and thus the construction of thte entropy must emerge as a result of information flowing throw the horizon. The entropy measure should then IMO be a result of a self-organization taking place on one side of the horizon. Ie. the construction of the entropy measure, must be a physical process occurring on one side, driven by the flow of information.

I think the most interesting part is isn't Einsteins euqations itself, it's how the general reasoning connects the spacetime degrees of freedom with general properties of a communication channel.

Thus I wouldn't expect to assume or define the entropy in a particular way, I think there should be a more first principle explanation as to why certain measures of missing information is manifested and chosen by nature, presumably as a result of an evolution process.

Shouldn't the cosmological constant and the dimensionality also come out as determined by some interesting connection in this context, when this is done right? It's certainly what I expect.

/Fredrik