# Strings vs. Loops-Black Hole Metrics?

1. Mar 27, 2004

### jgraber

Strings vs. Loops--Black Hole Metrics?

You read a lot about how strings and loops can both derive the Bekenstein-Hawking entropy formula for black holes. String calculations only work for near extremal black holes, but they get the Immirzi-Barbero parameter right. Loop calculations work for all black holes, but need extra work to get the Immirzi Barbero parameter right. Something like that, anyway. Also, it seems that Strings require a dilaton, which leads to scalar-tensor gravity with a Brans-Dicke type metric. Also, you hear a lot about Gauss- Bonnet gravity, especially in connection with brane-worlds. There’s also a BTZ black hole or something like that, too. And of course, string black holes come in all sorts of weird dimesions all the way from 1+1 to 1+9. But I don’t get the bottom line.

For someone who understands metrics but not strings or loops, what is the difference between an old fashioned Kerr or Schwarzschild black hole and a stringy one or a loop one?

In particular, what black hole metric or metrics does String theory lead to? LQG? Are they the same or different?

References are good. TIA for any help. Jim Graber

2. Mar 27, 2004

### clothoid

I only know a bit about LQG, and I think currently in LQG it isn't known whether the metric would reduce to the GR metric in the large scale or not. It's possible that it's the same situation in string theory. The LQG computation of the area (which I don't understand perfectly yet, so please correct me if I'm wrong) only deals with the area of the event horizon, which is easier to deal with, because in LQG there is a result that _area_ must be quantized.

But the large scale behavior of LQG is really pretty unknown at the moment. And I think there are also some versions of LQG that differ in some details (exact form of partition function), and people aren't sure which is the correct one.

To wave my hands a bit, I'd say that it's quite likely that the large scale limit could turn out to be general relativity, because GR is so neat. But it could also be some other theory..I think it would be fun if the Higgs field turned out to be intimately intertwined with the gravity field. I think this could be possible, because the classical version of the Higgs field assumes that it isn't coupled to anything except the particles that it gives mass to, and that forbids a massless Higgs..But for example this is an interesting paper:

http://xxx.lanl.gov/abs/hep-ph/9510201

A Higgs model that uses BF theory! Now what if that was quantized using spin foams..

..That went a bit off topic. Experts, please correct me if I said something that was ridiculously wrong!

3. Mar 27, 2004

### Stingray

A minor correction - the Barbero-Immirzi parameter does not enter string theory at all. It is a product of the particular quantization procedure used in LQG. There is no way of finding this parameter a priori (as far as anyone knows), so it has to be fixed by comparing to the semiclassical limit. The black hole entropy calculation is one way of doing this fixing. I believe that there is now at least one other matching that has been done, and it yields the same number.

That said, I think that the LQG calculation works for all Kerr type black holes, whereas the string one only works for the case of maximal angular momentum (which probably cannot exist even in principle for several reasons). My memory on this is very fuzzy though, so I may be wrong!

4. Mar 28, 2004

### marcus

Confirming what you say, it is an outstanding merit of ST that it gets the exact number 1/4 in the BH entropy law (in the unrealistic cases considered) without using an adjustable parameter. Even a string-skeptic can gladly salute this achievement

as far as I know the other approach to fixing the parameter does not yet yield the same number. it is close but no cigar (roughly like log 3 is close to log 2) and a succession of novel modifications of LQG have been offered. Here at PF the most knowledgeable person about this is Nonunitary. So far, interesting ideas but no fully satisfactory way to resolve the disharmony between log 3 and log 2. It is possible that the right idea has recently come up and I dont know about it yet and that your comment shows that you have heard more recent news than I have. Since you are in touch with an active group of researchers this would not be unlikely

Again to corroborate what you say: one hears that the string approach is in fact limited to extremal and near extremal cases which are not what astronomers suppose is out there---BH so highly charged, or else spinning so fast, that they can barely get it together. By contrast the Loop approach deals with standard varieties, but still has AFAIK this intriguing barberoimmirzi puzzle to solve. No clear winner on this one apparently. Maybe someone else has a different view of it.

5. Mar 28, 2004

### Stingray

I certainly agree.

There was an issue with quasinormal modes, but I think it has been resolved now. I'm definitely not sure about that. I'll try to find out from more knowledgeable people in the next few days.

I've looked up a little bit more about the types of black holes understood in LQG. Type I isolated horizons (the isolated horizon framework is a very nice way of dealing with black holes that is much more powerful than the old event horizon approach) are understood. This includes much more than Schwarzchild even in the vacuum case. It also allows matter and cosmological constant if you want to put it in. Cosmological horizons are also understood.

Very recently (ie the paper isn't written yet), people now understand type II isolated horizons. Interestingly, there appears to be a precise way to map type I into type II using a multipole formalism. This encompasses a very large class of black holes including the kerr-newman ones, but also ones with large distortions etc. They all give the same immirzi parameter.

6. Mar 29, 2004

### nonunitary

I think that there are two separate issues regarding the BI parameter. The first one has to do with the inclusion of general type of Isolated Horizons, form the type I, the original ones, to type II. These clasification has to do with the instrinsic geometry of the horizon. Type I means that the intrinsic geometry of horizon is spherical, whereas type II is axyally symmetric. The other generalization is the type of matter that the theory allows, which can be an Abelian gauge field (Maxwell), a scalar field, dilatonic coupling, Yang Mills nad so on. The result is that in all these cases the BI parameter is the same as in the original T-I IH, that includes Schwarzschild. It is also true that not all the results have been published. There is a document on line by Ashtekar in which he outlines the case of distorted and rotating horizons which is what has not been published yet. Furthermore, one could also consider non-minimal couplings where the entropy is no longer expected to be proportional (with a constant factor) to the area, but depends aldo on the matter fileds at the horizon. This case has been analysed for non-minimaly couples scalar fields for type I horizons and one still gets the entropy right with the same valeo of the BI parameter.
In this regard, the issue of the same value is very rubust and there is no inconsistency.

The other aspect of the problem, namely what hapens with the QNM
is still, IMHO, unresolved. If one sais that LQG should indeed have something to say about the QNM frequencies, even when only for Schwarzshild (given that Kerr and RN do not have simple asymptotics), then one can only conclude that for some reason, the main contribution from entropy counting comes from $j=1$ spin networks (as opposed to $j=1/2$, which is the standard assumption). Nobody knows why nor how those colorings could be suppresed. There a re some proposal around but none is conclusive yet
(Corichi, Swain, Ling, etc.). If one understood whether QNM are really to be taken seriously and wer had a mechanism for explaining the new value of the BI parameter, then this mechanism would have to be valid for all the cases analysed so far. Thus, one would still say that BH of all kinds and flavors fix a unique value of the BI parameter.

7. Mar 29, 2004

### jgraber

Quantum Normal Modes--Spin 2,1 or 1/2

On the one hand, I don't understand why someone doesn't say Hawking is wrong and LQG is right and the IB parameter really is log 2 or whatever is appropriate for spin 1/2 instead of log 3 or whatever is appropriate for spin 1.
(I guess this changes that 1/4 to log 3/(4 log 2) or something like that.)

On the other hand, I would think that LQG which considers only gravity, should only find spin 2 normal modes instead of spin 1 !!
(What IB parameter would that lead to, log 4?)
Furthermore one would assume this counted only gravitational modes and not modes due to other forces.

Anyway, I am more concerned with the metrics than the IB parameter.
Some of the ones I've seen seem to be radically different from Schwarzschild or Kerr. I have read that at least some of them reduce to the extreme Reissner-Nordstrom metric. I would like to understand more about this.
Jim Graber

8. Apr 24, 2004

### marcus

Sauron just posted some questions about entropy and the gravitational field.

I'd like to connect Sauron's post to some pre-existing PF thread---one in this (SB-and-LQG) forum if possible

the last post here is 29 March, and now it is 24 April (almost a month)
I dont want to misdirect this thread off topic
but much of the discussion here has been about
black hole entropy
so it seems like a good place to raise questions about entropy
in the GR context.

Last edited: Apr 24, 2004
9. Apr 24, 2004

### marcus

the main contributors to this thread seem to have been

jgraber (who initiated it)
nonunitary and
stingray

Unless they object, I'm looking for a home for Sauron's question
"how does entropy make sense in a GR context?" and will try to see if it fits here.

Here's an exerpt edited from Sauron's recent post.
-----------
...
I have a few generic questions/reflections about some of the themes LQG is addressing.

Let´s begin by the question of entropy. My deal is whether the concept of entropy makes sense in GR at all. At least in the same sense as in ordinary statistical mechanics.

I know about two main results. The one, of wich i have a reasonable understanding , about the black hole area behaving like entropy. I also have notice about (but no understanding at all) results of Penrose relating the Weyl tensor to entropy, at least in cosmological scenarios.

The question is that in the microcanonical device the entropy is related to the number of micro-states compatible with an energy. But in GR there is no a good (and less local) definition of the energy of the gravitational field....
-----------

Sauron, I am hoping there may be some response from jgraber, nonunitary, stingray and others but I will tell you my immediate reaction:

I think that in a quantum gravity context----especially BH context---one is looking at the number of micro-states compatible with a certain surface area
rather than a certain energy.

the area does reflect the mass (and thus in a sense the energy) of the BH
but whatever basic notion of a macro-state we have seems to
be represented in the surface which the outside observer sees.

I think that this surface does not always have to be an event horizon
but in some problems it is convenient to make it so

because GR is a geometrical theory it seems appropriate that the generic macrostate (to which a number of microstates correspond) should be a geometrical one---a surface area or some macroscopic observables associated with a surface----rather than an energy. Hope this is at least a partial response to the issues raised in your post.

Last edited: Apr 24, 2004
10. Apr 24, 2004

### jgraber

Hi its Jim Graber aka jgraber:

I would like to agree with marcus that for a black hole area is proportional to the square of its energy (or mass), so the link is pretty direct.

As to sauron's concern about the definition of local energy in GR. This is indeed a well known problem, but in my opinion, for all practical purposes, quasilocal energy works. (See wolram's link in the 3rd post in the BH entropy thread, where sauron may get better answers than here)

In my second post in this thread, I tried to raise the question of the relation between minimal spins (or dominant spins) in the quasi normal mode entropy calculation and the log (ln)term in the IB parameter or the quantum of area.
Dreyer's formula (gr-qc/0404055 page 4 equation 4) involves ln(2 j +1), hence spin 1/2 corresponds to ln(2), spin 1 to ln(3) and spin2 to ln(5). (Baez refers to this spin 5 in an s.p.r post). But Ling and Zhang (gr-qc/0309018 page 2 equation 10) use ln(4 j +1) and hence get spin 1/2 corresponds to ln(3), spin 1 corresponding to ln(5) and spin 2 corresponding to ln(9).
Motl's paper gr-qc/0212096 also uses ln(2 j + 1) (Equations 14 and 15 page 1142). I was hoping Lubos or someone else might try to explain this in simple terms.

Finally, I am actually more intersted in the string derived modifications to the Schwarzschild and Kerr metrics than in the entropy issue. The question is whether these modifications will be large enough to be detected (or ruled out) by LISA, like some other theoretically motivated modifications to GR.

I will leave tomorrow on a trip including the APS meeting in Denver and will be posting (if at all) from on the road until after May 4, so don't expect fast responses. Best to sauron, marcus and any others who read this. Jim

11. May 19, 2004

### etera

In the usual case of the group SU(2), the dimension (or number of degrees of freedom) of the representation of spin j is (2j+1), so that you want to include a ln(2j+1) in the entropy calculations.
Ling and Zhag consider the supersymmetric extension of the the theory and use OSp(1|2) instead of SU(2). Now a representation of OSp(1|2) of superspin j is made of two representations of SU(2): one of spin j and one of spin (j-1/2), with the supersymmetric operators allowing to go from one to the other. So the dimension of the representation of superspin j is the sum of the dimensions of the spin j and spin (j-1/2): (2j+1)+(2(j-1/2)+1)=4j+1. That explains the ln(4j+1) in the susy paper...

12. May 19, 2004

### jgraber

Thanks to etera for a clear explanation of where the different log factors come from. Meanwhile, John Baez has a post over on SPR in which he seems to say that this whole approach is in trouble because it does not work well for rotating black holes. Jim Graber