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Einstein's equations from first law of thermodynamics in AdS

Happy New Year!

First, Nima Arkani-Hamed (IAS) and Jaroslav Trnka (Caltech) have released another amplituhedron preprint, Into the Amplituhedron, in which they expose the amplituhedron origin of various QFT rules applied to cuts – the unitarity double cut, the multicollinear limit (which explains why the amplitude is sometimes the exponential of something simpler), and other cuts (whose properties emerge from the shape of \(d\)-dimensional faces of the amplituhedron).

But there's another intriguing hep-th paper published on the last day of 2013 (even if we overlook the remaining dozens of papers): Thomas Faulkner, Monica Guica, Thomas Hartman, Robert C. Myers, and Mark Van Raamsdonk wrote

Gravitation from Entanglement in Holographic CFTs
which provides us with a much more constructive, microscopically complete, anti de Sitter version of Ted Jacobson's heuristic derivation of Einstein's equations from entropy considerations (see e.g. Event horizons and thermodynamics: more than analogy, TRF 2010).

These five authors try to reformulate the first law of thermodynamics (with not only \(dE\) but also the entropy-related \(T\cdot dS\) term!) that holds within the CFT in terms of the gravitational AdS bulk variables. What they find are nothing else than Einstein's equations – well, they reach these equations in their linearized form expanded around the zeroth approximation, the empty anti de Sitter spacetime geometry.

The old heuristic picture of Jacobson should be considered to be just a localized (i.e. region-by-region attributed) version of Bekenstein's and Hawking's original intuition relating geometric properties of black holes with the thermodynamic quantities. Neither Bekenstein or Hawking in the 1970s nor Jacobson in the 1990s actually had any good idea about the microscopic degrees of freedom representing the entropy.

This paper brings the ideas to a new level. By being rooted in AdS/CFT, it has a well-defined and controllable microscopic description of the entropy built into the whole game – in this sense, it is adding all the visibility that Strominger and Vafa brought into the research of the black hole entropy. Moreover, the new AdS approach seems to work for higher-derivative curvature terms, too: they also show how Wald's formula (calculated from a Noether's current) arises from their derivation.

So far, the analysis has some drawbacks relatively to Jacobson's heuristic analysis: they quantify the properties of global Rindler horizons and not local ones which is what Jacobson did, and that's why they can only derive the linearized approximation of Einstein's equations.

At any rate, Einstein's equations – dynamical laws determining the spacetime curvature – may indeed be derived from the laws of thermodynamics (they are pretty much the same thing!) as long as we also substitute some basic geometric formula for the entanglement entropy. For the simplest, Bekenstein-Hawking form of the entanglement entropy \(S=A/4G\) proposed by Ryu and Takayanagi, they may derive the original, minimal Einstein's equations in their basic linearized form.

I would say that this paper is the newest paper in one of several interacting branches of the modern research that indicates that the "entanglement's being the glue that may attach the surfaces" is perhaps the most fundamental principle that implies many properties of quantum gravity – black hole thermodynamics and even the original classical Einstein's equations of the general theory of relativity.

2013 and 2014

Your humble correspondent doesn't plan to write extensive summaries of the year 2013 in physics and what to expect in 2014. Just a few words. We saw some interesting experiments in 2013 – hints of the dark matter signal that were killed by the super-powerful negative result from LUX. I think that theoretically, the amplituhedron and the entanglement-related research in quantum gravity (ER-EPR correspondence and the Papadodimas-Raju Ansatz for the black hole interior) belonged among the most interesting advances in theoretical physics.

2014 has been declared by the United Nations as the "International Year of Family Farming and Crystallography" because diamonds and organic feces are among the cleanest and most beautiful objects that the apparatchiks know. I may write a text on crystallography later.

It will be the final complete year of the Two Years' Vacation at the LHC. The machine is being revitalized and upgraded and around April 2015, it should start proton-proton collisions at a higher energy \(13\TeV\), much higher than the \(8\TeV\) center-of-mass energy in 2012. Because of this "quantum leap" in the energy, the exclusion limits from 2012 won't mean much: lots of new particles may be discovered almost immediately (within weeks) despite the null results from the previous collisions. It's also possible that the Standard Model will continue to work for these collisions, too.

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reader BobSykes said...

Back in 2008 or so, B. G. Sidharth published a book entitled "The Thermodynamic Universe," which purports to derive various aspects of General Relativity and Quantum Mechanics from thermodynamics. Thumbing through it, it appears to be the work of a crank. Does anyone knowledgeable in physics have an opinion about it.

Happy New Year! I hope your atheist Xmas was joyous, too.

reader NumCracker said...

Dear Lubos, what are the equivalent geometrical / gravitational analogous of the "third law" of thermodynamics to gravity ? Does this also brings interesting consequences as does the 1st law ? Happy new year !

reader Luboš Motl said...

Dear Numcracker, are you sure that you are familiar with the geometric interpretation of the 2nd law which is much more important than the 3rd law? The total horizon area goes up.

The 3rd law says that a crystal at T=0 has S=0. It's important that it only holds for certain environments like crystals. It doesn't hold for a thermal gas - indeed. some thermal gas in the CFT may be dual to a large black hole whose S is nonzero and,in fact, very large.

The cleanest "analog" but not "dual" to the crystal in the bulk is the vacuum - and it's entropy is zero. That's probably the closest version of the 3rd law. But maybe there's something else I am not aware of.

Happy New Year, LM

reader kashyap vasavada said...

Dear Lubos: I would like to see a blog explaining why CFT or conformal transformations (invariance) are useful and what they accomplish. I noticed that Witten used something like that in his talk. Could you write one, when you have time? If you have already written in past, please give a link. Thanks.

Another question : what new things relativistic thermodynamics brigs in to the undergraduate level NR thermodynamics?

reader Luboš Motl said...

See also


Yes, it's a high-school-level nonsense and all the similarities with the deep and correct stuff are largely coincidental.

reader Luboš Motl said...

I was planning to write about global vs local and conformal symmetries for some time.

The plans were slowed down by a particular answer I was giving to Edward Hughes here:


So far, shortly (I have written many texts of this sort but surely not texts exactly answering this particular mixture of questions in this order):

Every symmetry - either transformations commuting with the Hamiltonian or transformations having the commutator with the Hamiltonian that may close to an interesting group - is something we should know about.

Conformal symmetry is one of the rare spacetime symmetries, extending the rotational or Lorentz or Poincare symmetries. It's a symmetry under transformations that preserve the angle but may scale the distances by a space-dependent scalar factor.

The group of conformal symmetries of the D-dimensional Euclidean/Minkowski space is SO(D+1,1) or SO(D,2), respectively. 1+1 dimensions are added. These groups are the same groups as groups of D+1-dimensional hyperboloid manifolds called the Euclidean AdS or AdS itself, respectively, and that's important because it's the first consistency check that makes the AdS/CFT correspondence possible.

In the world sheet 2D CFTs, the conformal symmetry is essential because it decouples all physical polarizations of the world sheet metric tensor (gravity) and makes the theory solvable and accessible to other cute tricks.

Special relativity doesn't generalize thermodynamics in any interesting way. Thermodynamics still holds. Energy plays an important role in thermodynamics. Special relativity says that energy is a component of a momentum 4-vector. So relativity also implies one may define "chemical potentials" analogous to the inverse temperature for the momentum. But the only role of these extra potentials is that they make some matter "mostly move" but one still gets the same "non-relativistic" thermal ensemble in a corresponding rest frame - the potentials just transform this thermal equilibrium to another frame (in which the particles seem to be moving in a direction).

So truly interesting things only begin in general relativity when one considers the black hole thermodynamics etc. But that's what e.g. this very blog entry was about and I am not going to copy and paste it. If you don't understand this blog post and/or the papers linked to here, you will have to fill your gaps in some prerequisites but there may be many possible gaps you may be facing and I can't foresee which of them they are.

reader kashyap vasavada said...

Thanks Lubos. Looks like you want me to work hard!!!

reader Luboš Motl said...

In some ways hard, in some ways less so.

For example, your question about "relativistic extension of thermodynamics" implicitly suggests the existence of some hard work or discipline that doesn't exist at all.

Thermodynamics is a scientific treatment of *any* physical system that may carry energy, temperature, and entropy. It doesn't matter at all whether the energy etc. is calculated through one formula or another.

Special relativity just means that one considers a subset of theories that obey the Lorentz symmetry. So it's subset but thermodynamics may be used for this subset - and the larger set, too. In this sense, special relativity changes "nothing whatever" about thermodynamics.

reader Uncle Al said...

QM obviously has a stronger hand than GR. Quantum gravitations predict nothing observable. GR keeps falling out of things (thermodynamics, non-commutative algebra). Perhaps our conceptualization of math is wrong, the answer being obvious when properly coded. Could we understand the "obvious" answer?

reader lucretius said...

"Non-commutative algebra"? I woud like to see that ;-)
"Geometry" is another matter.

reader Rehbock said...

In what way Is this really different from Verlinde in his recent entropic gravity? Or is it putting his ideas into the Holographic boundary consistently?

reader Dilaton said...

Thanks for this nice nice end of the year post Lumo :-)

I always like it, when known rules and facts find their deeper underlying reasons, such as for example properties of the amplituhetron. Nima should give a new talk about this paper ... :-)
Cool way to derive Einstein's equations, at least the linearized ones.

What is this "International Year of Family Farming and Crystallography" about, is it a joke?

Anyway, Happy New Year to you Lumo and everybody else here :-D


reader Luboš Motl said...

Happy New Year to you, too! No, it is not a joke at all! See 2nd paragraph at


and check the United Nations links. ;-)

reader Umesh said...

517 exhilarating posts to close the year 2013! Please keep up the awesome work and here's wishing you an awesome, happy, prosperous 2014! May we all encounter more stringy breakthroughs this year...

reader lucretius said...

Happy New Year to Lubos and everyone else including even TRF’s regular (and not yet banned) CPs and “progressives” ;-)

(Of course these wishes should not be taken too far. For example, the sort of thing that would make a happy year for Alexander Ač I prefer not to imagine. )

reader cynholt said...

Best wishes for the New Year, Lubos! Your blog has been interesting, informative, and very educational for me over the years. In addition, the blog draws comments from highly thoughtful readers which I also appreciate. Kudos to all!

reader Gordon said...

Yes, all the best for 2014.
Mark does write nice papers.

reader Uncle Al said...


Non-commutative arithmetic, etc., Alain Connes for geometry, The
Euler equation unites analytic geometry with algebra. It is a rich tapestry not a curt reply.

reader kashyap vasavada said...

Happy New year Lubos and readers of this blog! Please keep on educating us about theoretical physics.

reader Peter F. said...

Happy New Year, Luboš!!!
Thanks for the absolutely lovely sounding HNY/ABBA-video!

And, not to be forgotten, Happy New Year to all you fine and interesting people who feel almost as lucky as I do to have the most TRFic of all terrestrial blogs/bloggers to be enriched, provoked and thrilled by!

reader lucretius said...

People who try to impress by juxtaposing randomly mathematical sounding verbiage are prone to fall into traps. “Non-commutative algebra” is a branch of mathematics (in fact just classical ring theory) and General Relativity does not “fall out” of it. “A non-commutative algebra” is a ring with a compatible vector space structure - an object that is studied in non-commutative algebra and other fields. “Arithmetic geometry” is simply algebraic geometry over a field that is not algebraically closed.

On that other hand, Alain Connes’ “non-commutative geometry” generalizes Riemannian geometry and one can indeed derive something like GR from it but of course there is still the problem of the metric being Riemannian rather than pseudo-Riemannian.

reader Giotis said...

Happy new year to all the inhabitants of Lubos’ stringy
universe and to the host of course!

In terms of posts what I wish to see is a review article
on the status and prospects of String theory.

reader Dilaton said...

Yep, that would be a nice one :-)

And I am still waiting for my loooooooooooooong essay about what mathematics theoretical physicists need to know ... ;-)

reader Rami Niemi said...

Any thoughts about this?

reader DvtheDv said...

Happy New Year, Luboš! As a newcomer to TRF, I find myself having strong disagreements with many posts you make, and insufficient maths to understand many more, but I have overall been delighted by the blog! Wishing the best in 2014 to you.

Speaking of my limited maths, I (re)realized whilst reading this post that I currently have only a childlike conception of what thermodynamics is all about. I'm seeking to remedy this situation in 2014, and was wondering if you could suggest a clear and correct textbook on the fundamentals of thermodynamics? I've been suggested the Reif statistical mechanics textbook by several different people, but was wondering if you had any sort of suggestion.