Four-Dimensional Entropy from Three-Dimensional Gravity

In summary: This paper is about how to use self-dual variables to study loop quantum cosmology. By using these variables, it is possible to get an accurate picture of the dynamics of the universe without having to complicate things with a real Immirzi parameter. This is a significant development because it makes the theory more mathematically sophisticated.
  • #1
kodama
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http://arxiv.org/abs/1503.02981
Four-Dimensional Entropy from Three-Dimensional Gravity
S. Carlip
(Submitted on 10 Mar 2015)
At the horizon of a black hole, the action of (3+1)-dimensional loop quantum gravity acquires a boundary term that is formally identical to an action for three-dimensional gravity. I show how to use this correspondence to obtain the entropy of the (3+1)-dimensional black hole from well-understood conformal field theory computations of the entropy in (2+1)-dimensional de Sitter space.
8 pages
^

is this paper applying ads/cft to LQG?
 
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  • #2
Hi Kodama,
Strictly speaking no. It does not apply any "Anti-deSitter/conformal field theory" theorem or conjecture to studying the LQG black hole.
AdS/CFT is only one of a number of analytic strategies where information in the bulk (e.g. 4d) space is related to information on the boundary (e.g.3d) space.
It's a highly publicized example of this but it is far from being the whole menu of bulk-boundary approaches.

You picked a good author. Steve Carlip is a highly respected longtime expert. I've heard him talk when he gave an invited lecture at UC Berkeley on quantum gravity---the audience included a number of string theorists. He's quite familiar with AdS/CFT and would have said so clearly if he was using it.

But he did not cite any of the main AdS/CFT papers and didn't invoke it, or even mention it in the main body of the paper. He made a brief mention of anti-deSitter at the end as something related that might be explored in future research.
You know anti-deSitter basically means negative cosmological constant, and deSitter on the other hand means positive cosmological constant.
As far as we can tell our universe is NOT anti-deSitter. It seems to have a positive cosmological constant.

Carlip was using a kind of bulk-boundary approach that did not involve AdS/CFT. As I see it that makes his paper all the more innovative, more interesting, and more likely to be in line with the reality of nature. It's a fascinating paper and definitely worth discussing!
 
  • #3
Kodama, for me the thing about Carlip's paper is that he sets γ = i.
He gets the BH area entropy law without complicating matters with a real Immirzi parameter. This is part of an exciting broader development--over the past year or two I've seen a number of LQG papers that use the original and natural Ashtekar variables without having to choose a real number for gamma. They are going back to the original self-dual formulation which uses the complex number i.

Here's what Carlip says about it in the introduction to his paper. He makes it clear that setting γ = i and going through with the BH calculation is the essential thrust of the paper:
==quote==
In the past few years, there have been intriguing hints that the entropy can also be obtained by setting γ = i [5–8]. This is the natural value: it makes the theory self-dual [9], and is the only choice for which the Ashtekar-Barbero-Sen connection (1.1) is a fully diffeomorphism-invariant spacetime connection [10,11]. Unfortunately, with this choice one must impose a reality conditions, a procedure that remains poorly defined. As a consequence, the theory with γ = i is not nearly as mathematically sophisticated as the version with real γ, and far fewer results have been established.

In this paper, I will describe a simple new method for computing black hole entropy in loop quantum gravity with γ = i.
==endquote==

This by itself makes this an important paper and also part of a significant recent trend in LQG.
 
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  • #4
Two papers that are part of this development came out just this past quarter (Jan-March 2015) and are on the "most important paper" Poll:
https://www.physicsforums.com/threa...rter-2015-mip-most-important-qg-paper.806409/

http://arxiv.org/abs/1503.07855
Loop quantum cosmology with self-dual variables
Edward Wilson-Ewing
(Submitted on 26 Mar 2015)
Using the complex-valued self-dual connection variables, the loop quantum cosmology of a closed Friedmann universe coupled to a massless scalar field is studied. It is shown how the reality conditions can be imposed in the quantum theory by choosing a particular measure for the inner product in the kinematical Hilbert space. While holonomies of the self-dual Ashtekar connection are not well-defined in the kinematical Hilbert space, it is possible to introduce a family of generalized holonomy-like operators, some of which are well-defined; these operators in turn are used in the definition of a Hamiltonian constraint operator where the scalar field can be used as a relational clock. The resulting quantum dynamics are similar, although not identical, to standard loop quantum cosmology constructed from the Ashtekar-Barbero variables with a real Immirzi parameter. Effective Friedmann equations are derived, which provide a good approximation to the full quantum dynamics for sharply-peaked states whose volume remains much larger than the Planck volume, and they show that for these states quantum gravity effects resolve the big-bang and big-crunch singularities and replace them by a non-singular bounce. Finally, the loop quantization in self-dual variables of a flat Friedmann space-time is recovered in the limit of zero spatial curvature and is identical to the standard loop quantization in terms of the real-valued Ashtekar-Barbero variables.
10 pages http://inspirehep.net/record/1356275

http://arxiv.org/abs/1503.02981
Four-Dimensional Entropy from Three-Dimensional Gravity
S. Carlip
(Submitted on 10 Mar 2015)
At the horizon of a black hole, the action of (3+1)-dimensional loop quantum gravity acquires a boundary term that is formally identical to an action for three-dimensional gravity. I show how to use this correspondence to obtain the entropy of the (3+1)-dimensional black hole from well-understood conformal field theory computations of the entropy in (2+1)-dimensional de Sitter space.
8 pages http://inspirehep.net/record/1356275

Carlip has an earlier paper on this same theme (developing the original self-dual Ashtekar variables in LQG, simply putting γ = i.)
http://arxiv.org/abs/1410.5763
A Note on Black Hole Entropy in Loop Quantum Gravity
S. Carlip
(Submitted on 21 Oct 2014 (v1), last revised 26 Mar 2015 (this version, v3))
Several recent results have hinted that black hole thermodynamics in loop quantum gravity simplifies if one chooses an imaginary Barbero-Immirzi parameter γ=i. This suggests a connection with SL(2,ℂ) or SL(2,ℝ) conformal field theories at the "boundaries" formed by spin network edges intersecting the horizon. I present a bit of background regarding the relevant conformal field theories, along with some speculations about how they might be used to count black hole states. I show, in particular, that a set of unproven but plausible assumptions can lead to a boundary conformal field theory whose density of states matches the Bekenstein-Hawking entropy.
13 pages.
 
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  • #5
the kodama wave function also was forumlated as y = i.

. "This is the natural value: it makes the theory self-dual [9], and is the only choice for which the Ashtekar-Barbero-Sen connection (1.1) is a fully diffeomorphism-invariant spacetime connection"

for LQG to be a theory of QG doesn't y = i have to be the case for fully diffeomorphism-invariant spacetime connection"
 
  • #6
Good point! there's something happening here...
 
  • #7
marcus said:
Good point! there's something happening here...

y = i makes the theory self-dual.

steve carlip's paper is stating that the entropy can arise regardless of y= real value is
 
  • #8
btw what are the implications for spinfoam if y = i?
 
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  • #9
kodama said:
btw what are the implications for spinfoam if y = i?
That's a really good question. I wish I knew the answer!

I'll keep an eye out for formulations of spinfoam QG that don't use the Immirzi parameter.
 
  • #10
kodama wave function is formulated with y = i.
marcus said:
That's a really good question. I wish I knew the answer!

I'll keep an eye out for formulations of spinfoam QG that don't use the Immirzi parameter.

with y =i can you get twistor space and twistor theory out of complex gr
 
  • #11
I suspect that you can and I think that is the direction in which Wolfgang Wieland is working. I can't say definitely because I don't understand his papers well enough. He has been on a postdoc fellowship at Penn State (Ashtekar's group) and will be taking a postdoc position at Perimeter later this year.
His work seems aimed at a twistorial formulation of spinfoam QG with Hamiltonian.
I would say he's interested in combining advantages of different approaches into a single formulation. (Ordinary spinfoam QG does not have a Hamiltonian, it is a method to calculate transition amplitudes for processes within boundary conditions specified e.g. by initial and final measurements. Wieland reformulates spinfoam QG so that it has a Hamiltonian similar to canonical LQG. He has also tried using the Ashtekar self-dual formulation. Unfortunately I can't say I understand how this works, so may not be able to respond usefully if you have questions about it!)
 
  • #12
BTW Kodama, you see the green B I U...Σ bar at the top of the reply box. If you haven't tried it yet, try clicking on the Σ to get a symbol menu. Then you can get things like hbar ħ which is right at the end of the greek letters just before the arrows.
Also it gives an easy way to use the lower case greek gamma γ and β (both of which are used for the Immirzi parameter in LQG.)
 

1. What is Four-Dimensional Entropy from Three-Dimensional Gravity?

Four-Dimensional Entropy from Three-Dimensional Gravity is a concept in theoretical physics that suggests that the entropy of a four-dimensional black hole can be calculated from the properties of a three-dimensional black hole. This theory was first proposed by physicist Juan Maldacena in 1997 and has since been studied extensively in the field of string theory.

2. How does this theory relate to black holes?

This theory suggests that the entropy, or disorder, of a four-dimensional black hole can be described by the properties of a three-dimensional black hole. This is significant because it allows for a better understanding of the nature of black holes and their relationship to other fundamental forces in the universe.

3. What is the significance of Four-Dimensional Entropy from Three-Dimensional Gravity?

The significance of this theory lies in its potential to bridge the gap between the theories of gravity and quantum mechanics. It also has the potential to provide a deeper understanding of the fundamental nature of space and time.

4. How is this theory supported by evidence?

While there is currently no direct evidence to support this theory, it is consistent with other well-established theories in physics, such as the holographic principle. Additionally, this theory has been tested through various mathematical calculations and simulations, which have shown promising results.

5. What are the implications of Four-Dimensional Entropy from Three-Dimensional Gravity?

If this theory is proven to be true, it could have significant implications for our understanding of the universe and the laws of physics. It could also lead to new developments in fields such as quantum gravity and cosmology.

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