emergent gravity

sunu.engineer

Hello, I am new to this forum and my name is Sunu Engineer. I am a cosmologist by profession and a student of Prof. Padmanabhan whose work is being discussed here. I am familiar with the work (over a very long period of time) and its evolution as well as the related work of Prof. Jacobson and Prof. Verlinde. Please feel free to ask any question that you may have. The work as all of you have indicated, while related to earlier work of Prof. Jacobson, has many important and novel aspects to it. It is also complete and consistent.

Regards
sunu

S.Daedalus

I'd love to, however, I'm a bit pressed for time at the moment... So the key idea is, roughly, that the entanglement entropy in QFT scales with the area rather than with the volume, like black hole entropy does. To get a rough idea about this, consider a universe permeated with some scalar field in a pure state, so the entropy is zero. Then, 'hide' some part of the universe from the rest, say a sphere, i.e. integrate out the degrees of freedom in that part. This will generate an entropy, which by direct calculation can be seen to scale with the area. This I think was originally shown by Srednicki. The wiki article gives some more insight on the issue of entanglement entropy.

Now this is already very suggestive of black hole thermodynamics. If it were the case that the QFT entanglement entropy is equal to (or bounded by) the Bekenstein-Hawking entropy for a certain appropriate space-time volume, then one could apply Jacobson's arguments and derive the Einstein equations from there. However, in general, the entanglement entropy diverges at the horizon. What Jacobson now points out is that if you have gravity, then the entanglement entropy is automatically finite.

So the program is roughly to get general relativity from quantum theory, via space-time thermodynamics. I'll be happy to expand once I get more time (view the above only as a very rough, qualitative sketch), in the meantime, maybe you find this talk by Jacobson interesting...

Hello sunu! Thanks for joining in to the discussion. Unfortunately, I don't have the time right now to think off and ask good questions, but if I get a few hours to spare to sit down with Prof. Padmanabhan's paper, I'll jump at the opportunity... Perhaps, seeing how I'm more familiar with Jacobson's work (and there, too, only an interested outside observer), you could point out some differences, and say a few words about what you think the most significant aspects of Padmanabhan's work are (I realize this is a lot to cover, perhaps you could just provide some pointers to get the discussion started). Again, thanks for joining in!

Naty1

Sunu: My compliments on your last name!
When I read "Engineer" as a surname I was harkened back in time to college when a bunch of fellow engineering students and I would make up names for each other often involving 'engineer'....

I'd be interested in what you view as a few of the key 'novel aspects' to Professor Padmanabhan's paper. Not knowing the history of the development of this subject made it
difficult to sort out the 'new ideas'.
Also, Jacobsens 1995 paper was an easy read....but anything you can add to the prior post regarding his 2012 paper would be appreciated. I found that paper opaque, maybe because I don't understand 'entanglement entropy'.

I hope to listen to the Jacobsen talk, referenced above, in the next day or so.

Naty1

In the first slide of Jacobsen's talk he has:
" The entropy scales with the area because the entanglement is dominated
by vacuum correlations which diverege at short distances."


Can someone paraphrase this?? Maybe explain 'entanglement', 'vacuum correlations'
and why they might diverge at short distances....just a few sentences for perspecgtive...

S.Daedalus

I'll try: Let's say you've got two systems, [itex]A[/itex] and [itex]B[/itex], each of which can be in the states [itex]|0\rangle[/itex] or [itex]|1\rangle[/itex] (perhaps think of a fermion with spin up/down states), or of course in any superposition of both, [itex]\alpha|0\rangle + \beta|1\rangle[/itex], with [itex]|\alpha|^2 + |\beta|^2=1[/itex]. The combined system then can be in any of the states [itex]|\psi\rangle_{AB}=\sum_{i,j}c_{ij}|i\rangle_A \otimes |j\rangle_B[/itex]. If this state can be written in the form [itex]|\psi\rangle_A \otimes |\psi\rangle_B[/itex], it is called separable; if not, it is entangled.

An example of an entangled state is [itex]|\Psi^-\rangle = \frac{1}{\sqrt{2}}(|0\rangle_A \otimes |1\rangle_B - |1\rangle_A \otimes |0\rangle_B)[/itex]. Its entangled nature comes to light if we let [itex]A[/itex], conventionally called Alice, perform a measurement. If she obtains the outcome 0 (with probability [itex](\frac{1}{\sqrt{2}})^2 = \frac{1}{2}[/itex], the state collapses to [itex]|0\rangle_A \otimes |1\rangle_B[/itex], and we know with certainty that [itex]B[/itex] (Bob) will obtain 1 upon measuring the system; conversely, if Alice obtains 1, the state afterwards will be [itex]|1\rangle_A \otimes |0\rangle_B[/itex], and thus, Bob's subsequent measurement will yield 0 as a result.

Now, the state [itex]|\Psi^-\rangle[/itex] is what's called pure, which basically means that it can be represented by a unique ray in Hilbert space (i.e. a single ket vector [itex]|\psi\rangle[/itex]). The converse of pure is mixed. A state is mixed if it consists of an ensemble of pure states -- you can picture this as being uncertain about what state the system is actually in. So if you have an apparatus with a randomizing element that prepares you state [itex]|\psi_1\rangle[/itex] with probability [itex]p_1[/itex], and state [itex]|\psi_2\rangle[/itex] with probability [itex]p_2[/itex] (such that [itex]p_1 + p_2 =1[/itex]), you describe whatever comes out of the apparatus by the statistical mixture of these two states.

Unfortunately, the bra-ket formalism is not well suited to the description of mixed states; to do so, one typically turns to the density matrix formalism. For a mixture of states such as the one above, the density matrix is: [itex]\rho = \sum_i p_i|\psi_i\rangle \langle \psi_i|[/itex]; it gives the probability with which the system is found in either of the states [itex]|\psi_i\rangle[/itex].

Now, a consequence of entanglement is that you can't associate to either of the systems [itex]A[/itex] or [itex]B[/itex] a pure state anymore. This is intuitive -- because of the entanglement, the systems considered on their own do not describe the complete state. Rather, the state of the system [itex]A[/itex] is described by the partial trace over [itex]B[/itex] of the density matrix of the whole system: [itex]\rho_A = tr_B \rho_{AB}[/itex] (nevermind the mathematical terminology; this just means 'whatever's left over when I forget about all the degrees of freedom associated to [itex]B[/itex]).

The last little fact we need is that pure states have zero entropy, while the entropy of mixed states is always nonvanishing. So effectively, if I have an entangled state as above, and restrict my attention to one part of it (say one particle of an entangled two-particle system), then I must describe the state of that part as having nonzero entropy, even though the complete entangled state has no entropy. Because the situation is completely symmetric, the entropy of one part is equal to the entropy of the other, if I remove (say, hide behind a horizon) the remaining one. I.e. if [itex]S[/itex] denotes entropy, [itex]S(A)=S(B)[/itex].

This almost directly leads to the 'area law' scaling of entanglement entropy: if I have some volume uniformly filled with some field, and remove a (spherical, for convenience) portion of it, then the entropy of the removed part relatively to the rest must be equal to the entropy of the rest relatively to the removed part (from inside the sphere, effectively the rest of the universe has been hidden behind the 'horizon'); but the area of the sphere's boundary is the only quantity both sectors have in common, so the entropy must end up proportional to it. (Vacuum correlations are just the correlations -- i.e. entanglement -- that are naturally present in the field.) Unfortunately, while the Bekenstein-Hawking entropy has a definite upper bound, given by the Planck area, the entanglement entropy hasn't -- I can always go to smaller and smaller distances and find higher and higher modes that contribute. What Jacobson's now claiming, essentially, is that gravity, which emerges from the thermodynamics of the horizon (recall, what has entropy, also has temperature), serves to regulate this divergence (if I understand correctly).

Does this help?

Naty1

The math is above my paygrade, but the last paragraph helps:

Although I don't really get 'entanglement entropy'......for another time...
Thank you.

Naty1

Jacobson's 1995 paper mentions this on 'entanglement entropy':

Thermodynamics of Spacetime:
The Einstein Equation of State
http://arxiv.org/pdf/gr-qc/9504004v2.pdf

sunu.engineer

Sorry about the delay in responding. I had been traveling and occupied with a variety of other tasks. Please find a simpler explanation of the work in the attached file.

regards
sunu

Attached Files

Simplicity of the Universe.pdf (96.4 KB, 29 views)

Chronos

Very nice, sunu. I think there may be more observational tests than proposed.

sunu.engineer

Thank you, Chronos. The question of observational tests and further generalizations of this work is what is being discussed at present. I hope you have had a look at the evolution of the idea through the different papers (which may be a bit esoteric in its language) of Prof. Jacobson, Prof. Padmanabhan, Prof. Verlinde, Prof. Visser and others who have been working in the field. It is a very entertaining trail of thoughts and explorations.

regards
sunu

Chronos

I'm still reading the other papers. I concur some of them are rather difficult to digest. I am partial to the works of Dr. Padmanabhan. He has a unique ability to render complex ideas comprehensible to mere mortals.

sunu.engineer

Simplicity of the Universe
This article briefly describes a fundamentally new paradigm for analyzing what we have learnt about the Universe over the last three hundred years. While it is clear that gravity is the dominant force (of the four fundamental forces) that shapes the Universe as we observe it today, the precise nature of the way gravity determines the structure of the Universe has been rather obscure.

Gravity as a force (post Newton), usually described by the General Relativity theory and its derivatives or variants, produces the coupling between space-time and matter (energy). This is needed to describe most of the finite systems that we originally encountered historically in the theories of gravity (apple, moon, mercury, solar system, stars, galaxies etc.). This description assumes the a-priori existence of space and time and matter. However the extension of these theories to describe the dynamics of the Universe as a whole, requires that we understand the genesis of space time and matter, and then use this to describe the rest of the structure - a top down description that would involve the emergence of space time and then matter and then the description of how the smaller scale structures such as stars and planets came to be.

In order to arrive at a consistent understanding of this top down model, we must keep in mind that the broad picture which is usually simple and elegant (Occam’s Razor) is often obscured by details, making the scenarios intractably complex. This has, in the past, confounded us considerably and a simpler theory is a must to make progress.

Let us assume that the Universe can be described by a scale factor a(t) and a Hubble Constant, H(t) (which is a function of the scale factor). (FLRW model equations?)

In a simplified sense, the current observational data indicates that the Universe consists of three phases – an initial exponential expansion (with the Hubble radius remaining constant and the scale factor of the universe expanding exponentially), a radiation and matter dominated phase and a third exponential expansion phase. The galaxies and stars and planets are formed in the second phase of the Universe’s history. (Figure needed?)

Traditionally we have studied gravitation as a force experienced on the surface of our planet and have extended it to describe solar systems and galaxies and clusters of galaxies etc. Further extension allowed us to model the Universe as a product of gravity (as a solution to the equations describing gravity) namely the equations of General Relativity.

If you turn this argument on its head, it is possible to envisage a model where gravity is a cosmic force and the finite gravity observed in smaller systems such as super clusters, clusters, galaxies etc. in scales which are very small in comparison with the scale of the Universe is to be treated differently with respect to Gravity at the Cosmological scale.
Modern Gravitational theory is a geometric theory dealing with gravity as the curvature of space-time (in a manner that gives equal importance to space and time). However at the cosmological scale it is possible to envisage gravity as being ‘pre-geometric’ with a preferred time, thus enabling us to describe the emergence of cosmic space as a function of ‘cosmic time’ – the time associated with a class of preferred observers co-expanding with the Universe.

These pre-geometric variables would thus be the ‘atoms of spacetime’. And like any other atomic theory it is possible to use a statistical description leading to a macroscopic thermodynamic description of gravity. It is quite natural to presume that quantum mechanical description of these ‘atoms’ would lead to the much sought after ‘quantum gravity’.

The thermodynamic description of gravity has already elicited much insight into the structure of gravity as described by Einstein equations or higher dimensional theories such as Lanczos-Lovelock theories (Similar to Einstein’s theory of 4 dimensional space time gravity but with higher space time dimensions). An ‘emergent’ picture of gravity has been arrived at through this approach. In this approach we have seen that gravity can be described as ‘emergent’ a la elasticity or fluid mechanics. The similarities are mathematically rigorous and can be derived with a very small set of assumptions. (Links to the appropriate pages here)

Given that gravity is emergent at the level of current descriptions of finite or small scale gravitational fields, it is a logical next step to inquire about the emergent properties of space time itself.

To describe space-time as ‘emergent’, the most consistent and simple approach is to use the preferred time of the co-moving observers and describe the emergence of space itself, like ice emerging from water when the conditions are appropriate. The length scale of relevance that is to be used is obviously the natural length scale of the universe itself, the Hubble scale.

However there is another natural length-scale, which is obtained from the combination of the fundamental constants – gravitational constant (Gravity), Planck’s constant (Quantum mechanics) and light speed (maximum speed of signal propagation according to theory of relativity). This is the small length scale corresponding to Planck scale. Thus all that the Universe is, lies between the Ultraviolet scale (as in small wavelength of light) of the Planck scale and Infra Red (as in the large wavelength of light) of the Hubble scale.

Our mathematical description of such a universe utilizes a well-known model called the De Sitter model (Named afted Willem De Sitter), which is an exponentially expanding model. ( Advanced comment: This model has the interesting property that it is time translation invariant which means that it could have existed since eternity – a sort of steady state which takes care of the question of initial singularity or what happened at the beginning of the universe)

The first de sitter model is associated with Hubble scale equal to the Planck scale, namely when the Universe is very small.

This small (or ultra violet) scale is deeply entrenched in quantum gravity regime and is unstable to quantum fluctuations. These instabilities can cause the universe to make a transition from exponential expansion of inflation to radiation dominated phase. With the transition comes creation of particles and universe become filled with normal matter. In other words both space and matter emerge simultaneously from the small scale state due to an instability.

The radiation dominated phase will continue until the second length-scale (the infra red length-scale or the present day Hubble Scale) starts to dominate. Why does this happen? We believe it is because the dark energy component of the Universe, the one which looks like the ‘cosmological constant’, becomes more dominant than the matter and radiation energy combination. Once this happens the universe will make a transition to a second de sitter phase in which the Hubble radius remains constant at the large-scale value. Observations suggest that this happened rather recently in the expansion history for our universe and can continue to eternity!

How do we describe the precise dynamics? One can count the number of degrees of freedom on a surface with radius equal to the Hubble radius as well as the number of degrees of freedom contained in the bulk volume. The two de-sitter phases correspond to states of "holographic equipartition" in which the number of degrees of frame in the bulk is equal to that on the surface. The transition stage can then be described as an evolution towards equipartition driven by the holographic discrepancy between the degrees of freedom on the surface and the bulk. (This is similar to systems evolving in response to discrepancy in current and extremal energy or entropy states in other dynamical systems in physics).

In Planck units, this is just a combinatorial evolution with the Hubble volume at the nth time step being given by the recurrence relation:
Vn=Vn-1 +(Nsur-Nbulk)
Which connects the volumes at time steps n and n-1 to the difference in the number of degrees of freedom of the surface Nsur and the bulk of the volume Nbulk. (Figure needed?)

Remarkably enough, this description leads to an equation for the expansion of the Universe, which is identical to the standard description in terms of General Relativity.

The above observation allows us to posit a completely new way of looking at cosmic evolution. We concentrate on the Hubble radius and think of it as describing the emergence of space. Its dynamics is governed by an equation which describes evolution towards holographic equipartition. The two equilibrium states of the universe are described by two de-Sitter or exponential expansion phases – one at the small Planck scale and the other at the large dark energy related scale (or present Hubble scale).

The conventional cosmology, which deals with the existence of humans on Earth finally, in many senses, becomes a rather insignificant part [lasting for a tiny fraction of time] sandwiched between the two eternal de-Sitter phases! More to the point, the above description remains very simple and elegant if we ignore a small interval when the Universe was matter dominated rather than radiation dominated.

Scientific methodology requires testability. Hence the question - how can we test such a paradigm? In addition to theoretical consistency, there are the following possible ways of testing.

(i) Boundary conditions - It is possible that when we think of the Universe with two scales we need some non-trivial boundary condition at the transition point from radiation and matter dominated scale to dark energy dominated scale. This in turn will lead to a discreet [though very closely spaced] spectrum for radiation modes. Such a discreet spectrum must violate statistical isotropy of CMBR; possibly at a small level but in principle these holographic equipartition ideas should leave a trace in the cosmic microwave background radiation (CMBR). Which means that as we look at the background in all directions we should find traces of this ‘edge’ or ‘boundary’.

(ii) Mathematical and observational concordance -
o A mathematical analysis indicates that the three phases of the Universe (two de-Sitter phases and a radiation dominated phase sandwiched in between) must expand by nearly equal factor, say, eN where N is the number of e-foldings of the Universe (Given that e = 2.718 one can think of this as approximate number of times the Universe tripled in size) during the expansion phase. This allows us to link the ratio between the large and small length-scales to the number of e-foldings of the initial Planck scale inflation. Putting in the numbers based on our current understanding of Inflationary phase, we arrive at a quantitative measure for the dark energy component, which is in accord with present day observations. (Figure needed?)

o The density perturbations in the energy that seed the later formation of large observed structures like super clusters and voids, cross the Hubble radius three times during the evolution of the Universe in this model. Theoretical consistency requires equal number of modes of perturbations to cross during all the three phases, which is borne out in this model.

Will this description allow us to incorporate Planck scale corrections in a simpler way into cosmology and study the earliest phases of the evolution of the universe with clearer observational signatures for the model? This is the question that is driving the theory forward. And the answer appears to be strongly positive.