# Einstein Equation of State vs Gravity from Entanglement

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1. Jun 24, 2015

### S.Daedalus

There's a somehow related set of issues I find myself pondering time and again:

In 1995, Ted Jacobson derived Einstein's equations from thermodynamics across a horizon. Roughly, he showed that if the horizon's entropy is given by the Bekenstein-Hawking formula, then the second law of thermodynamics implies that Einstein's equations hold.

More recently, it's become clear that there is a deep relationship between entanglement and gravity in the AdS/CFT duality---roughly, the entanglement entropy between a patch of the CFT and its surroundings is given by the area of a minimal AdS-surface that has the same boundary as the patch (Ryu-Takayanagi formula). In fact, it can be shown, again with a thermodynamical argument, that for small perturbations around the vacuum of the CFT, the corresponding AdS obeys the Einstein equations to linear order.

Now these two things seem too close not to be related, in particular if you follow the suggestion that the Bekenstein-Hawking entropy might in fact itself be entanglement entropy, as first proposed by Srednicki and Bombelli et al. (This is not without its own problems, as the entanglement entropy, while it does follow an area law, diverges at the horizon, but there may be ways around that.)

So if this can be made sense of, both approaches seem to relate the Einstein equations to entanglement entropy and its dynamics. But what's puzzling me is that they work on different stages, so to speak: Jacobson's approach gets the Einstein equations of 3+1-dim spacetime by considering horizons within that spacetime; the more modern approach gets them by considering entanglement across boundaries that themselves exist on the boundary of an AdS-spacetime, that is, the entangling surface has (at least) one less dimension. Jacobson uses a surface whose area is directly proportional to the entanglement across that surface, while van Raamsdonk et al. use, via the Ryu-Takayanagi prescription, a surface in the boundary CFT whose entanglement is proportional to the area of a surface in the higher-dimensional AdS-spacetime.

So, where does that difference come from? How can you get, apparently, the Einstein equations within the same space from entanglement across a horizon, and in a space of higher dimensionality? Or is my whole outlook on this just muddled, and the two things really aren't related at all?

2. Jun 24, 2015

### Jimster41

I hope to learn something from this thread.

I have tremendous trouble getting an image of what the regular 4d space-time relationship is to the AdS in the AdS/CFT.
I keep picturing our space-time as being in a container of d>4 dimension and our CFTs and whatnot being an expression, at least in part, of information on the boundary between 4d and 4+nd. I have been ascribing non-locality to the information on this boundary.

Re the question: To me (intuitively at least) this sentence from Jacobson p2 seems relevant:
"In thermodynamics, heat is energy that flows between degrees of freedom that are not macroscopically observable. In spacetime dynamics, we shall define heat as energy that flows across a causal horizon. It can be felt via the gravitational field it generates, but its particular form or nature is unobservable from outside the horizon"

I thought he was proposing the non-zero temperature "Unruh" space-time vacuum as:
• a "causal horizon" through which everything in 4d spacetime has to pass
• the physically real "presence" of the information located on the boundary between Minkowski(4d) and AdS(4+nd) boundary
The question of where the degrees of freedom are that are causing the Unruh "heat", required or implied the other dimension(s). The key thing is that the boundary information get's expressed "causally" in 4d but the boundary itself not so constrained (it is more like a wavy field, pure superposition of states, a pilot wave, etc)

That's what I thought. I'd love to improve on its wrongness.

Last edited: Jun 24, 2015
3. Jun 24, 2015

### atyy

The Jacobson idea is gravity from thermodynamics.

For entanglement in CFTs, there is something one may call "entanglement thermodynamics". Maybe the AdS gravity can be derived from the entanglement thermodynamics. If that is possible, that would instantiate Jacobson's idea in AdS/CFT.

http://arxiv.org/abs/1308.3716
Gravitational Dynamics From Entanglement "Thermodynamics"
Nima Lashkari, Michael B. McDermott, Mark Van Raamsdonk
(Submitted on 16 Aug 2013 (v1), last revised 22 Aug 2013 (this version, v2))
In a general conformal field theory, perturbations to the vacuum state obey the relation \delta S = \delta E, where \delta S is the change in entanglement entropy of an arbitrary ball-shaped region, and \delta E is the change in "hyperbolic" energy of this region. In this note, we show that for holographic conformal field theories, this relation, together with the holographic connection between entanglement entropies and areas of extremal surfaces and the standard connection between the field theory stress tensor and the boundary behavior of the metric, implies that geometry dual to the perturbed state satisfies Einstein's equations expanded to linear order about pure AdS.

4. Jun 24, 2015

### atyy

http://arxiv.org/abs/1505.04753
Entanglement equilibrium and the Einstein equation
Ted Jacobson
(Submitted on 18 May 2015)
We show that the semiclassical Einstein equation holds if and only if the entanglement entropy in small causal diamonds is stationary at constant volume, when varied from a maximally symmetric vacuum state of geometry and quantum fields. The argument hinges on a conjecture about the variation of the conformal boost energy of quantum fields in small diamonds.

5. Jun 25, 2015

### S.Daedalus

Well, the AdS/CFT correspondence as such is probably not directly applicable to our universe: we don't live in an AdS-spacetime. One hope is that there is a more general gauge/gravity or at least dS/CFT correspondence, such that you could extract physics of our 3+1dim gravitational universe from the boundary non-gravitational CFT, but I don't really know how far these ideas have come as of yet.

Jacobson's proposal predates AdS/CFT by a couple years, so he wasn't really concerned with any extra dimensions. Rather, he introduced a causal horizon---such as the Rindler horizon of an accelerated observer in flat 3+1dim (Minkowski) spacetime---which effectively hides anything beyond. He identifies the energy flow across this horizon with the heat flow, and then shows that this together with the Unruh temperature the accelerated observer sees and the Bekenstein-Hawking relation for the entropy implies that Einstein's equations hold.

Yes, but Jacobson himself proposes that the entropy in his argument may be due to entanglement across the horizon.

That's my intuition as well, but my problem is that it doesn't seem to be quite the same thing: in Jacobson's original proposal, the entanglement entropy must follow an area law to agree with the BH-entropy in the space in which the Einstein equations hold; but in the AdS/CFT proposal, the entanglement entropy of the CFT can't follow an area law in the CFT-space, since it is given by the area of a surface in the gravitational AdS. So if you have a 2+1dim CFT, then the entanglement entropy across a 1d boundary is given by the area of a 2-surface in the 3+1dim AdS (say). This gives an entanglement entropy that scales extensively with the volume of the region inside the entangling surface in the CFT, while in Jacobson's proposal, the entanglement entropy, like the BH-entropy, scales with area instead.

One way I could see how to make sense of this is if the entanglement in the gravitational degrees of freedom on the AdS-side follows an area law, roughly by the reasoning that those degrees of freedom really are just copies of the boundary degrees of freedom. Then, the entanglement on the AdS-side is likewise proportional to the area of the Ryu-Takayanagi surface, and one could use Jacobson's argument (maybe). But I'm not sure if this really makes sense.

6. Jun 25, 2015

### Jimster41

So maybe here's my chance to learn something. I literally thought there was a chance we live in an AdS spacetime, but locked like flat-landers in a lower dimensional CFT-Minkowski-Manifold subspace? I have had concern that this was way too naive a picture, maybe downright inverted somehow, and why I'm not quite seeing the way the "cylinder diagrams" work.

I was picturing the "dual" between CFT and AdS as allowing calculations in our bulk to be simplified, by working them through a formalism of the "boundary" that our subspace shares with the complete AdS space, that boundary (because it's a boundary) being of lower dimension than both the AdS and the subspace we inhabit. The surface of a BH is a (the) example of the boundary.

I think I was agreeing with everything you said regarding Jacobson's proposal. I wasn't suggesting he was thinking of extra dimensions aka AdS, just that it was implied by, compatible with his model. Did Jacobson predate Verlinde?

7. Jun 25, 2015

### Jimster41

Is there any sense in which those degrees of freedom on the AdS side could be not just copies but real DOF, ones that can only project onto the boundary dimension(s) a subset or compression of the info they contain - could that in some sense inform the non-local structure we see in CFT entanglement - super-selection rules, SM, SUSY

I mean (trying to say it another way) If the boundary area entropy on the CFT side includes the dimension that connects, with structure, all boundary areas (non-locally) on the CFT side does that fix the discrepancy in scaling (geometry) of the Entropy basis?

(One more try) Suppose SUSY exists, as a coherent rule set encoded on the boundary, connecting spacetime points in the CFT - non-locally - picture that rule set changing. A process in the CFT bulk would never know it. Not saying it might do that, just that it highlights the DOF that seems to be implied there - information defined specifically across the whole non-local SUSY pattern, shining through the boundary everywhere at once.

Yeah, okay, so I'm losing it.