- #1

Chronos

Science Advisor

Gold Member

- 11,429

- 742

You are using an out of date browser. It may not display this or other websites correctly.

You should upgrade or use an alternative browser.

You should upgrade or use an alternative browser.

- Thread starter Chronos
- Start date

- #1

Chronos

Science Advisor

Gold Member

- 11,429

- 742

- #2

Drakkith

Staff Emeritus

Science Advisor

- 21,277

- 5,119

Interesting. Just wish I knew more math and cosmology so I could make an educated opinion on it.

- #3

Chalnoth

Science Advisor

- 6,197

- 447

I didn't look at it in detail, but it sounds like an interesting idea. It seems, naively, that it might be investigated through a better measurement of the polarization of the CMB, which will provide us more detailed knowledge of the nature of inflation.

- #4

- 5,607

- 40

From the abstract:

Emergent perspective of Gravity and Dark Energy

Isn't this the very perspective that drives some in these forums nuts....I mean the idea of 'new space'....instead of merely a metric phenomena?? I just love it when such

protestations, based on a conventional view, get bypassed. Too many here seem to ignore the historical evidence that without new perspectives progress would be severely limited.

Anyway, When I got to equations 30 to 31, I though "HEY this sounds like Eric Verlinde's ideas", and sure enough a quick check in Wikipedia shows:

http://en.wikipedia.org/wiki/Entropic_gravity

[I don't mean plagerism, just that the idea doesn't seem brilliantly original.]

and a few paragraphs later the Padmanabhan paper says:

I can't tell if this is just "one step for man or one giant step for mankind".....Did Verlinde think in terms of cosmological evolution moving to Holographic equipartition..or is this idea a radical/significant extension...

Cosmologists are not going to like this:

This seems on the surface to conflict with the idea expressed in these forums [regarding particle production, I believe] that the Hubble sphere is NOT a horizon.

Even a humorous name included:

Cool name!!!

edit: I see the author references Verlinde among many others...

Emergent perspective of Gravity and Dark Energy

...In the second part, I describe a novel way of studying cosmology in which I interpret the expansion of the universe as equivalent to the emergence of space itself.

Isn't this the very perspective that drives some in these forums nuts....I mean the idea of 'new space'....instead of merely a metric phenomena?? I just love it when such

protestations, based on a conventional view, get bypassed. Too many here seem to ignore the historical evidence that without new perspectives progress would be severely limited.

Anyway, When I got to equations 30 to 31, I though "HEY this sounds like Eric Verlinde's ideas", and sure enough a quick check in Wikipedia shows:

http://en.wikipedia.org/wiki/Entropic_gravity

....His [Verlinde] theory implies that gravity is not a fundamental interaction, but an emergent phenomenon which arises from the statistical behavior of microscopic degrees of freedom encoded on a holographic screen.

[I don't mean plagerism, just that the idea doesn't seem brilliantly original.]

and a few paragraphs later the Padmanabhan paper says:

It is therefore natural to think of the current accelerated expansion

of the universe as an evolution towards holographic equipartition. Treating the expansion

of the universe as conceptually equivalent to the emergence of space we conclude

that the emergence of space itself is being driven towards holographic equipartition.

I can't tell if this is just "one step for man or one giant step for mankind".....Did Verlinde think in terms of cosmological evolution moving to Holographic equipartition..or is this idea a radical/significant extension...

Cosmologists are not going to like this:

in the overall cosmological evolution matter dominated phase is not of much significance since it again quickly gives way to the second de Sitter phase dominated by the cosmological constant. Viewed in this manner, the domain of conventional cosmology merely describes the emergence of matter degrees of freedom along with cosmic space during the time the universe is making a transition from one de Sitter phase to another.

In a way, the problem of the cosmos has now been reduced to understanding one

single number N closely related to the number of modes which cross the Hubble radius

during the three phases of the evolution..

This seems on the surface to conflict with the idea expressed in these forums [regarding particle production, I believe] that the Hubble sphere is NOT a horizon.

Even a humorous name included:

I thank Dr. Sunu Engineer for several discussions and comments...

Cool name!!!

edit: I see the author references Verlinde among many others...

Last edited:

- #5

- 309

- 4

Thanks in anticipation.

Regards,

Noel.

- #6

- 5,607

- 40

you can use the Wikipedia 'EMERGENCE'

Last edited:

- #7

- 309

- 4

Thanks Naty1.

- #8

- 5,607

- 40

Chronos...

What idea is that??

I'm not trying to be smart, far from it. I have not analyzed the math, but in general he seems to make [and perhaps draw upon] clever linkages between Einstein's equations, the FLRW solution, and thermodynamic degrees of freedom.

I am still plowing through the paper and it sure is interesting...I could have hardly found a single document that would better tie together many discussions in these forums.....but for the life of me I am unable to distinguish where the author begins 'new' ideas...In a number of cases he is integrating the work of others but which is which I haven't really figured yet. I have not had my 'socks knocked off' with some new and original insights, although they may well be present, as I can relate most of his explanations to discussions here in the forums.

I'll probably be back with some additional comments and questions on the paper.

Parts are difficult to understand for me because he doesn't consistently make clear some of the underlying logic/relationships. In other areas he makes those linkages very clear. Early on I wondered "Why does he pick the Hubble distance" and he doesn't explain that until much later ....all he had to say was "I pick THAT distance because it relates directly to the FLRW scale factor" and so may show a connection to the Einstein equations. Maybe pro readers are expected to find that association obvious?

Meantime, it seems around page 17 is the core of this paper: [N degrees of freedom, enclosing surface and the interior bulk] He is able to relate thermodynamic type degrees of freedom using an appropriate choice of proper time t and Hubble volume V. Are these new and original and dramatic new relationships...

from which he develops [around page 22] :

This idea looks pretty solid to me.

What idea is that??

I'm not trying to be smart, far from it. I have not analyzed the math, but in general he seems to make [and perhaps draw upon] clever linkages between Einstein's equations, the FLRW solution, and thermodynamic degrees of freedom.

I am still plowing through the paper and it sure is interesting...I could have hardly found a single document that would better tie together many discussions in these forums.....but for the life of me I am unable to distinguish where the author begins 'new' ideas...In a number of cases he is integrating the work of others but which is which I haven't really figured yet. I have not had my 'socks knocked off' with some new and original insights, although they may well be present, as I can relate most of his explanations to discussions here in the forums.

I'll probably be back with some additional comments and questions on the paper.

Parts are difficult to understand for me because he doesn't consistently make clear some of the underlying logic/relationships. In other areas he makes those linkages very clear. Early on I wondered "Why does he pick the Hubble distance" and he doesn't explain that until much later ....all he had to say was "I pick THAT distance because it relates directly to the FLRW scale factor" and so may show a connection to the Einstein equations. Maybe pro readers are expected to find that association obvious?

Meantime, it seems around page 17 is the core of this paper: [N degrees of freedom, enclosing surface and the interior bulk] He is able to relate thermodynamic type degrees of freedom using an appropriate choice of proper time t and Hubble volume V. Are these new and original and dramatic new relationships...

...consider a pure de Sitter universe with a Hubble constant H. Such a universe obeys the holographic principle in the form Nsur = Nbulk The Eq. (29) represents the holographic equipartition and relates the effective degrees of freedom residing in the bulk, determined by the equipartition condition, to the degrees of freedom on the boundary surface. The dynamics of the pure de Sitter universe can thus be obtained directly from the holographic equipartition condition, taken as the starting point.

Our universe, of course, is not pure de Sitter but is evolving towards an asymptotically de Sitter phase. It is therefore natural to think of the current accelerated expansion

of the universe as an evolution towards holographic equipartition…… we can describe the evolution of the accelerating universe entirely in terms of the concept of holographic equipartition.

from which he develops [around page 22] :

the utter simplicity of delta V = delta t (Nsur − Nbulk)

Eq. (32) is striking and it is remarkable that the standard expansion of the universe can be reinterpreted as an evolution towards holographic equipartition. There is some

amount of controversy in the literature regarding the correct choice for this temperature.

One can obtain equations similar to Eq. (32) with other definitions of

the temperature but none of the other choices leads to equations with the compelling

naturalness of Eq. (32). The same is true as regards the volume element

V which we have taken as the Hubble volume; other choices leads to equations

which have no simple interpretation.

Last edited:

- #9

- 5,607

- 40

In section 5.1, page 28 this statement appears: [is charted in Fig 4, but that does not explain the underlying logic]

...The quantum fluctuations generated during the inflationary phase — which act as seeds of structure formation in the universe— can be characterized by their physical wavelength. Consider a perturbation at some given wavelength scale which is stretched with the expansion of the universe as λ ∝ a(t). During the inflationary phase, the Hubble radius remains constant while the wavelength increases....,

So the first two sentences are ok...but can someone explain the underlying logic from which I can understand why the 'Hubble radius remains constant'...yet wavelength is stretched as the scale factor [a] evolves???

I think it's time to quit this for today.

- #10

Chalnoth

Science Advisor

- 6,197

- 447

The Hubble radius is given by:So the first two sentences are ok...but can someone explain the underlying logic from which I can understand why the 'Hubble radius remains constant'...yet wavelength is stretched as the scale factor [a] evolves???

I think it's time to quit this for today.

[tex]r_H = {c \over H}[/tex]

So a constant expansion rate [itex]H[/itex] means a constant Hubble radius. This also leads to an exponential expansion because:

[tex]H = {1 \over a} {da \over dt}[/tex]

With a constant Hubble rate we can substitute [itex]H = H_0[/itex], giving the following differential equaiton:

[tex]{da \over dt} = H_0 a[/tex]

The solution to this differential equation is:

[tex]a(t) = a(t=0) e^{H_0 t}[/tex]

So we have exponential expansion.

Does that help any?

- #11

Chronos

Science Advisor

Gold Member

- 11,429

- 742

- #12

- 5,607

- 40

Chalnoth:

Oh, That must be it..... I never thought about inflation as a constant H.....that explanation is great!!! Thank you.

I thought inflationary expansion slowed at the end analogous to radioactive decay. Perhaps that 'decay' period was immaterially short relative to the overall expansion.

Getting around this kind of mental obstruction is why I so admire people who have studied on their own, outside formal instruction.

I'm about 2/3 of the way through Allan Guth's INFLATIONARY UNIVERSE [which for the first 2/3 of the book disappointingly seems about everything EXCEPT inflation] and I'll be interested to see what if anything Guth says about this and what I hope will be a more interesting last 1/3 of the book.

Chronos:

I'm about 3/4 of the way through, and I think that may well be it. He lists a number of his own papers as references, more than anyone else referenced, so it could also be he has filled in missing theoretical pieces and new perspectives in those. Regardless, this paper does tie in a wonderful range of concepts.

unrelated footnote 1: Here in NJ, USA, we are due for a near 100 degree day today 7/7...so of course last evening my air conditioning quit as I was reading Padmanabhan's paper... I found a diagnostic LED flashing in my furnace/ a/c unit [and had logged the same code from several years ago] which suggested possibly shorted thermostat leads....I called a friend who does a/c mtc...He said "Check the compressor outside and see if a mouse chewed somewires"....not the first time I have fixed one of those!!....So I spent nearly two hours outside in heat and sure enough I had forgotten two thermostat leads [for a/c] go all the way outside to the compressor and two were stripped of insulation and touching....Voila!!! as I type this is air conditioned comfort!!!

note 2: 'Padmanabhan" is tougher to spell than 'Schwarzschild'!!!

So a constant expansion rate H means a constant Hubble radius.

Oh, That must be it..... I never thought about inflation as a constant H.....that explanation is great!!! Thank you.

I thought inflationary expansion slowed at the end analogous to radioactive decay. Perhaps that 'decay' period was immaterially short relative to the overall expansion.

Getting around this kind of mental obstruction is why I so admire people who have studied on their own, outside formal instruction.

I'm about 2/3 of the way through Allan Guth's INFLATIONARY UNIVERSE [which for the first 2/3 of the book disappointingly seems about everything EXCEPT inflation] and I'll be interested to see what if anything Guth says about this and what I hope will be a more interesting last 1/3 of the book.

Chronos:

It's not, imo, so much a bold new idea as it is the synthesis of existing knowledge into a broader perspective.

I'm about 3/4 of the way through, and I think that may well be it. He lists a number of his own papers as references, more than anyone else referenced, so it could also be he has filled in missing theoretical pieces and new perspectives in those. Regardless, this paper does tie in a wonderful range of concepts.

unrelated footnote 1: Here in NJ, USA, we are due for a near 100 degree day today 7/7...so of course last evening my air conditioning quit as I was reading Padmanabhan's paper... I found a diagnostic LED flashing in my furnace/ a/c unit [and had logged the same code from several years ago] which suggested possibly shorted thermostat leads....I called a friend who does a/c mtc...He said "Check the compressor outside and see if a mouse chewed somewires"....not the first time I have fixed one of those!!....So I spent nearly two hours outside in heat and sure enough I had forgotten two thermostat leads [for a/c] go all the way outside to the compressor and two were stripped of insulation and touching....Voila!!! as I type this is air conditioned comfort!!!

note 2: 'Padmanabhan" is tougher to spell than 'Schwarzschild'!!!

Last edited:

- #13

- 221

- 7

Padmanabhan's been at this a lot longer than Verlinde, though; this paper from 2002 was the earliest of his on the subject I've been able to pull up on a quick search. Both Verlinde's and Padmanabhan's ideas, in turn, can be considered as part of the 'horizon thermodynamics' approach to gravity initiated in '95 by Ted Jacobson (see here), which in turn probably owes a debt to Sakharov gravity, proposed originally in 1967 (that's what Jacobson worked on just prior to his seminal 'Einstein Equation of State'-paper). Personally, I found his most recent paper to be quite illuminating.Anyway, When I got to equations 30 to 31, I though "HEY this sounds like Eric Verlinde's ideas", and sure enough a quick check in Wikipedia shows:

http://en.wikipedia.org/wiki/Entropic_gravity

[I don't mean plagerism, just that the idea doesn't seem brilliantly original.]

- #14

- 5,607

- 40

My Summary notes of the paper:

**Emergent perspective of Gravity and Dark Energy**

A static universe is represented by a universe with constant Hubble radius. Some unknown quantum gravitational instability triggers the universe to make a transition from the initial static state, to one of increasing Hubble distance [currently] and eventually in the final evolution of de Sitter space, to another significantly larger static state.

“The precise description of the transition between the two de Sitter phases is the standard domain of conventional cosmology in which.... a radiation dominated phase {gives} way to a very late time matter dominated phase.”

One can insert some ’hbar’s’ into the FLRW solution to Einstein’s classical theory of gravity to get equivalent statements in equal partition theory [a form of thermodynamic equilibrium] . “Interpreting gravitational field equations as emergent allows us to obtain the gravitational field equations by maximizing the entropy density of spacetime.”

This means [in the static case] the degrees of freedom [N] of a surface equals those of the enclosed bulk [volume] and this equality drives cosmological evolution: Nbulk = Nsur

This bulk volume is taken to be the Hubble volume in which the enclosed bulk space is taken to be the cosmic space that has already emerged;The surface is the Hubble sphere. The emergence of matter [degrees of freedom] along with cosmic space occurs during the current expansion era when the universe is making the transition from one de Sitter phase to another.

The emergence {expansion} of cosmic space is driven by the holographic

discrepancy (Nsur + Nm − Nde) between the surface and bulk degrees of freedom where

Nm is contributed by normal matter and Nde {dark energy} is contributed by

the cosmological constant.

“In a way, the problem of the cosmos has now been reduced to understanding one

single number N closely related to the number of modes which cross the Hubble radius

during the three phases of the evolution.”

edit: So IMO my post #8 does seem to appropriately capture a major aspect of the paper....exactly whose idea it is seems unimportant....

"Both Verlinde's and Padmanabhan's ideas, in turn, can be considered as part of the 'horizon thermodynamics' "

That seems like another good way to capture the transition from relativity to thermodynamic cosmological evolution.

Jacobsen puts it this way: "...the Einstein equation is an equation of state..." referring to 'horizon thermodynamics'

A static universe is represented by a universe with constant Hubble radius. Some unknown quantum gravitational instability triggers the universe to make a transition from the initial static state, to one of increasing Hubble distance [currently] and eventually in the final evolution of de Sitter space, to another significantly larger static state.

“The precise description of the transition between the two de Sitter phases is the standard domain of conventional cosmology in which.... a radiation dominated phase {gives} way to a very late time matter dominated phase.”

One can insert some ’hbar’s’ into the FLRW solution to Einstein’s classical theory of gravity to get equivalent statements in equal partition theory [a form of thermodynamic equilibrium] . “Interpreting gravitational field equations as emergent allows us to obtain the gravitational field equations by maximizing the entropy density of spacetime.”

This means [in the static case] the degrees of freedom [N] of a surface equals those of the enclosed bulk [volume] and this equality drives cosmological evolution: Nbulk = Nsur

This bulk volume is taken to be the Hubble volume in which the enclosed bulk space is taken to be the cosmic space that has already emerged;The surface is the Hubble sphere. The emergence of matter [degrees of freedom] along with cosmic space occurs during the current expansion era when the universe is making the transition from one de Sitter phase to another.

The emergence {expansion} of cosmic space is driven by the holographic

discrepancy (Nsur + Nm − Nde) between the surface and bulk degrees of freedom where

Nm is contributed by normal matter and Nde {dark energy} is contributed by

the cosmological constant.

“In a way, the problem of the cosmos has now been reduced to understanding one

single number N closely related to the number of modes which cross the Hubble radius

during the three phases of the evolution.”

edit: So IMO my post #8 does seem to appropriately capture a major aspect of the paper....exactly whose idea it is seems unimportant....

"Both Verlinde's and Padmanabhan's ideas, in turn, can be considered as part of the 'horizon thermodynamics' "

That seems like another good way to capture the transition from relativity to thermodynamic cosmological evolution.

Jacobsen puts it this way: "...the Einstein equation is an equation of state..." referring to 'horizon thermodynamics'

Last edited:

- #15

- 5,607

- 40

Thanks Daedalus for the reference to the Jacobson paper.....here is a quick summary....it

provides some basic relationships utilized in the current Padmanabhan paper...At 9 pages it's a quick read for concepts...

My comments enclosed thus:{}

Thermodynamics of Spacetime:

The Einstein Equation of State

Ted Jacobson

Summary: [mostly quotes patched together]

provides some basic relationships utilized in the current Padmanabhan paper...At 9 pages it's a quick read for concepts...

My comments enclosed thus:{}

Thermodynamics of Spacetime:

The Einstein Equation of State

Ted Jacobson

Summary: [mostly quotes patched together]

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 is not necessary that the horizon be a black hole event horizon. It can be simply the boundary of the past… a null hypersurface. {so a Hubble sphere works.} Can derive the Einstein equation from the proportionality of entropy and [boundary] horizon area together with the fundamental relation _Q = TdS…This thermodynamic equilibrium relationship applies only when a system is in “equilibrium”, not where the horizon is expanding, contracting, or shearing. {Hence the restrictions on a static universe} In the case of gravity, we chose our systems to be defined by local Rindler horizons, which are instantaneously stationary, in order to have systems in local equilibrium. {Hence the choice of Rindler coordinates}

... Classical General Relativity know~~that [the] horizon area would turn out to be a form of entropy, and that surface gravity is a temperature.....~~

Last edited:

- #16

- 5,607

- 40

Just wish I knew more math and cosmology so I could make an educated opinion on it.

It's a tall order for us casual, part time, amateurs to understand the scope of the math and science such full time science people amass over a career. And you would also need a better knowledge of thermodynamics than I to understand all the details of this paper.

I pretty much skip the math..... except to see if I recognize any basics...and look for the logical connections....the theoretical underpinnings.....how theories link together, as I posted from the Jacobson paper, for example. People often don't agree on what the math means, or perhaps it's better to say the math has multiple valid interpretations.

- #17

- 5,607

- 40

Personally, I found his most recent paper to be quite illuminating.

Well if you, or anyone else, can highlight a few central ideas from that paper:

Gravitation and vacuum entanglement entropy

Ted Jacobson

(Submitted on 28 Apr 2012)

http://arxiv.org/abs/1204.6349

I'd sure appreciate it.... I don't think I understood one paragraph....

Abstract:

The vacuum of quantum fields contains correlated fluctuations. When restricted to one side of a surface these have a huge entropy of entanglement that scales with the surface area. If UV physics renders this entropy finite, then a thermodynamic argument implies the existence of gravity. That is, the causal structure of spacetime must be dynamical and governed by the Einstein equation with Newton's constant inversely proportional to the entropy density. Conversely, the existence of gravity makes the entanglement entropy finite. This thermodynamic reasoning is powerful despite the lack of a detailed description of the dynamics at the cutoff scale, but it has its limitations. In particular, we should not expect to understand corrections to Einstein gravity in this way.

- #18

- 6

- 2

Regards

sunu

- #19

- 221

- 7

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.Well if you, or anyone else, can highlight a few central ideas from that paper:

Gravitation and vacuum entanglement entropy

Ted Jacobson

(Submitted on 28 Apr 2012)

http://arxiv.org/abs/1204.6349

I'd sure appreciate it.... I don't think I understood one paragraph....

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!

Regards

sunu

- #20

- 5,607

- 40

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.

- #21

- 5,607

- 40

" 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...

- #22

- 221

- 7

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

Now, the state [itex]|\Psi^-\rangle[/itex] is what's called

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

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?

- #23

- 5,607

- 40

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).

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

Thank you.

- #24

- 5,607

- 40

....That causal horizons should be associated with entropy is suggested by

the observation that they hide information[3]. In fact, the overwhelming

majority of the information that is hidden resides in correlations between

vacuum fluctuations just inside and outside of the horizon[4]. Because of

the infinite number of short wavelength field degrees of freedom near the

horizon, the associated “entanglement entropy” is divergent in continuum

quantum field theory. If, on the other hand, there is a fundamental cutoff

length lc, then the entanglement entropy is finite and proportional to the

horizon area...... {The cutoff seems to be on the order of Planck length}

Thermodynamics of Spacetime:

The Einstein Equation of State

http://arxiv.org/pdf/gr-qc/9504004v2.pdf

- #25

- 6

- 2

Share: