General relativity from string theory

[atyy]

I think one of the things not so evident in David Tong's notes, but that suprised mentioned in post #3, is that a curved background is not only a "consistency condition", but for small background curvatures, it is also a "solution" of string theory in the sense that it is a coherent state of gravitons on a flat background. This is discussed on p27 of Uranga's http://www.ift.uam.es/paginaspersonales/angeluranga/Lect.pdf

The other thing that may be useful is that GR can, in part (I'm not sure how much), be treated as a field theory on flat spacetime. This point of view is given in Straumann's http://arxiv.org/abs/astro-ph/0006423, especially the section on "Perturbation consistency and uniqueness" on p17.

 

[Finbar]

I think one of the things not so evident in David Tong's notes, but that suprised mentioned in post #3, is that a curved background is not only a "consistency condition", but for small background curvatures, it is also a "solution" of string theory in the sense that it is a coherent state of gravitons on a flat background. This is discussed on p27 of Uranga's http://www.ift.uam.es/paginaspersonales/angeluranga/Lect.pdf

The other thing that may be useful is that GR can, in part (I'm not sure how much), be treated as a field theory on flat spacetime. This point of view is given in Straumann's http://arxiv.org/abs/astro-ph/0006423 .

Tong does mention this

"We know that inserting a single copy of V in the path integral corresponds to the introduction of a single graviton state. Inserting eV in the path integral corresponds to a coherent state of gravitons, changing the metric from δμν to δμν + hμν . In this way we see that the background curved metric of (7.1) is indeed built of the quantized gravitons that we first met back in Section 2."


So for example we could think of the Schwarzschild metric as being being a coherent state of gravitons at least up to leading order in the (inverse) radius of curvature in units of the string length. After this string theory will predict perturbative corrections to the metric up until we get close to the singularity and the radius of curvature is equal to the string length then we need to do some non-perturbative physics to resolve the singularity(e.g. string field theory, AdS/CFT, M-theory).

 

[atyy]

Tong does mention this

"We know that inserting a single copy of V in the path integral corresponds to the introduction of a single graviton state. Inserting eV in the path integral corresponds to a coherent state of gravitons, changing the metric from δμν to δμν + hμν . In this way we see that the background curved metric of (7.1) is indeed built of the quantized gravitons that we first met back in Section 2."


So for example we could think of the Schwarzschild metric as being being a coherent state of gravitons at least up to leading order in the (inverse) radius of curvature in units of the string length. After this string theory will predict perturbative corrections to the metric up until we get close to the singularity and the radius of curvature is equal to the string length then we need to do some non-perturbative physics to resolve the singularity(e.g. string field theory, AdS/CFT, M-theory).

It should be possible to do perturbative string theory on a Schwarzschild background since it is Ricci flat, but is it really possible to view the Schwarzschild spacetime as a perturbation of Minkowski? Comparing Eq 13 of http://arxiv.org/abs/0910.2975 and the comments subsequent to Eq 62-64 of http://emis.math.tifr.res.in/journals/LRG/Articles/lrr-2006-3/ , it seems that, at least classically, gravity as a field on flat spacetime is equivalent to GR for spacetimes that can be covered by harmonic coordinates. I don't think harmonic coordinates can penetrate the event horizon, so presumably full Schwarzschild can't be obtained as a perturbation to Minkowski?

Edit: Hmmm, Deser's http://arxiv.org/abs/gr-qc/0411023 does claim "the full theory emerges in closed form with just one added (cubic) ... no special ‘gauge’ ... need be introduced"

 

[marcus]

Exactly a map to a fixed geometry. There is no dynamics associated to this spacetime- manifold as it apprears in the polyakov action( and it's generalisation to curved space).

G_{\mu \nu}(X) is a fixed function of X. There is only a consistency condition that it must be Ricci flat to preserve conformal invariance.

Perturbative string theory as a theory of quantum gravity is just that: a theory of perturbations( e.g. gravitons) around a fixed background that must obey the vacuum Einstein equations. Since these perturbations have a fixed length \alpha_s, the string length, it follows that perturbative string theory must break down when the radius of curvature of the back ground manifold reaches this length. At this point one must go beyond perturbation theory.

And it appears that one has not yet gone beyond it.
Perturbation theory means starting with a fixed geometry which is a solution to classical and then imposing little ripples on it.

No alternative *dynamics* has been established as yet, I gather. If you want dynamics then, so far, it seems you must embed in a prior geometry. Without that there is no length, no tension, no modes of vibration. Otherwise one puts in the missing degrees of freedom disguised in some ad hoc form---additional fields---and does additional handwaving.

But when applied to gravity this approach seems to be fundamentally flawed, because spacetime geometry largely consists of *causality* relations. In the perturbative approach, causal relations between points/events are permanently established by the fixed prior, which logically must not be the case.

The perturbations are imagined to change the geometry but they continue to run on a pre-established web of causality. This contradiction is built into the perturbative approach. Or?

 

[Sardano]

You are completely mistaken, Marcus, as I expected. Of course, if you want to perform the explicit quantization of the string modes, we must fix a background, usually Minkowski. But that's not the whole picture, of course. You can consider a 10 dimensional space time with an arbitrary curved metric, although you are not going to be able to explicitely quantize the string modes. However, you can still do a lot of things. You can compactify in a CY manifold keeping just the massless modes (low energy approximation), and you will see that this low energy approximation is a N=2 or N=1 Supergravity, which if you set the matter content to zero gives the desired General Relativity with just the EH term. And there is no perturbation approximation at all; in fact this low energy action contains A LOT of information about the non perturbative spectrum of string theory. So you see that you obtain GR from ST with no use of perturbation theory, that you starts with a 10 dimensional curved metric and that you obtain a 4 dimensional theory diffeomorphism invariant. Surprise! You can check the papers by Jan Louis for details, although I strongly recommend you to start by the basics of ST, which is obvious that yo ignore.

You should know too, that ST has gone far from perturbation theory with, for example, the microscopic realization of some non perturbative extendend objects in ST: the so called D Branes etc etc

 

[Haelfix]

"The perturbations are imagined to change the geometry but they continue to run on a pre-established web of causality."

No they do not! Consider the case classically.

For a weak field expansion G --> sum (1..n) G0 + H1 + H2 _ ... where each piece of H i, is required to couple to its own stress energy tensor, the lines of causality seem to follow that which is set down by G0.

But this is a mirage! Consider a change in the perturbation H1+ e J1 + ... (where e is a small epsilon) Linearing the action around G0 + H1 + e Ji + ... and throwing out terms with more than 1 powers of H, yields the result that j propagates not on the light cone of G0, but rather on the new light cone G0 + H1.

Every successive order, adds an extra term. In other words the lightcone converges not to the background G0, but to the lightcone fixed by the 'order by order improved action'. Summing up infinitely many of these terms yields the full nonlinear field equations, just like Deser/Weinberg/Feynman et al showed in their classic papers. In fact, I think there is a clever way to terminate the series after only a few terms.

No the more problematic issue is not causality, its that you might have global topological ambiguities with the weak field expansion, and its not at all obvious how you go about deciding which topology you are in when you are working in that formalism.

 

[Sardano]

And you should check Gravity and Strings by Tomas Ortin, chapter 3, where he recovers GR as SRFT.

 

[petergreat]

String theory predicts lots of modifications to the Einstein-Hilbert action, including the presence of the dilaton, the antisymmetric tensor, and higher derivative terms. Why is it the case that gravity in the real world is almost perfectly described by the Einstein-Hilbert action?

 

[Sardano]

In the low energy approximation you get GR. But of course ST predicts corrections away from this limit. The usual GR is not such a "perfect" description of the universe: it has been only through some local tests in the solar system and at cosmological scales is expected to be corrected: in fact, there are plenty of people working on modifications of EH action who are not ST physicist. I am talking about f(R) models, vector -tensor theories of gravity...etc etc some of them with great experimental results:

http://arxiv.org/abs/0904.0433

http://arxiv.org/abs/0905.1245

Besides, there are several "metric theories of gravity" which gives the same results as the GR.
It is clear that the EH term should be corrected. The LQG people are the only ones that do not understand this fact.

 

[atyy]

http://relativity.livingreviews.org/Articles/lrr-2000-5/ , section 3.3.2 and ref. 42

It seems that harmonic coordinates can in fact penetrate the event horizon, contrary to my expectations in post #33, and consistent with finbar's statement in post #32 that the Schwarzschild solution is well approximated as a coherent state of gravitons.

 

[Finbar]

I think the same goes for Schwarzschild dS/AdS(or more generally any Einstein space). Where you can view the gravitons as perturbations around dS/AdS "vacuum". The reason for this is that the Weyl curvature vanishes in dS/AdS and Weyl curvature is the propagating force which gravitons carry. Of coarse in string theory one needs to find solutions that give AdS/dS first. For AdS these seem to be provided by AdS/CFT at least.

 

[tom.stoer]

It is clear that the EH term should be corrected. The LQG people are the only ones that do not understand this fact.

They are perfectly aware of it.

LQG is not simply about quantizing one specific action, it is about a new method of quantizing gravity and there is no reason why this method should be restricted to just the EH action. Of course there are quantum corrections to the EH action which is no longer the central object in the theory but which is replaced by spin networks and which should arise only as a certain semiclassical approximation.

 

[Finbar]

Is there something like an effective action in spin foams/LQG? A generating functional that contains all quantum corrections??

 

[qsa]

GR does not predict the expansion of the universe without the clumsy CC which is an after thought. the same problem with all QG theories which tries to justify CC with all kinds of tricks. That shows that string mostly is not a fundamental theory.

 

[atyy]

Is there something like an effective action in spin foams/LQG? A generating functional that contains all quantum corrections??

I don't think there is an effective action yet. However, there is a full theory which in a suitable limit/corase graining should gve you something like an effective action. The state of the art is http://arxiv.org/abs/0905.4082 , http://arxiv.org/abs/1103.4602 , http://arxiv.org/abs/1105.0216 , and http://arxiv.org/abs/1105.0566 . So far so good, but it is unclear if the full theory is convergent, whether the right limits have been taken, and even if it is the right limit, whether so many terms have been omitted that the divergence is not seen. Also, they don't know whether the correct limit is cos(iSR) or exp(iSR). All the above papers argue for the former, but this one seems to argue for the latter http://arxiv.org/abs/1004.4550

 

[tom.stoer]

Is there something like an effective action in spin foams/LQG? A generating functional that contains all quantum corrections??

The issue when comparing LQG and strings is not the effective action; LQG in spinfoam representation is closed to Kadanoffs blockspin picture (refer to ordinary QM & condensed matter physics), therefore arbitrary f(R) terms originating from the classical action and new couplings at vertices could be taken into account; the theory will be investigated w.r.t. its renormalization properties etc.

The main issue is how to take matter d.o.f. into account and how they will change the quantum properties of spacetime. In string theory geometry and matter are two properties of one single fundamental object whereas in LQG matter does not (yet) emerge from spacetime but is put in on top of it. This is a fundamental different picture and is therefore to a large degree arbitrary.

My expectation is that on long run LQG will only provide a reasonable theory of nature if it allows us to understand matter as an aspect of spacetime, not as something put in on top. This could e.g. be achieved via algebraic extensions (colorings) or via topological properties (braiding, knotting of graphs). Nevertheless LQG has much value as it provides a new way of quantization which is fundamentally non-perturbative, fully dynamical i.e. background independent from the very beginning.

 

[qsa]

The issue when comparing LQG and strings ...QUOTE]


tom, do you disagree with my last post, and if yes, why.

 

[tom.stoer]

GR does predict expansion; it simply does not predict accelerated expansion w/o the cc.

In a general picture like the asymptotic safety approach the cc is nothing else but one (afaik not very special) coupling constant. Therefore I expect that string theory should be able to deal with the cc as well, i.e. that besides AdS dS is required, too. In addition I think that any viable approach to QG should not use tricks for the cc and treat it is something special.

But I can't see how you can conclude that strings are not fundamental b/c of the cc. The fact that it's not perfectly understood in string theory does not spoil the whole concept.

 

[qsa]

GR does predict expansion; it simply does not predict accelerated expansion w/o the cc.

In a general picture like the asymptotic safety approach the cc is nothing else but one (afaik not very special) coupling constant. Therefore I expect that string theory should be able to deal with the cc as well, i.e. that besides AdS dS is required, too. In addition I think that any viable approach to QG should not use tricks for the cc and treat it is something special.

But I can't see how you can conclude that strings are not fundamental b/c of the cc. The fact that it's not perfectly understood in string theory does not spoil the whole concept.

Thanks. Of course, I did mean accelerated expansion. It is one thing for a general fundamental theory not to predict mass but to miss the behavior sound very disappointing. Even AS uses E-H action with modification, don't they all, more or less.

http://arxiv.org/pdf/1012.2680v1

I did not mean to say that string is not usefull, but just treating it as a theory that ends all was unjustified giving these relatively simple but fundamental issues. But your answer was good as always.

 

[Finbar]

qsa makes a good point. If we go by Tong's calculation string theory is only consistent in strictly Ricci flat space-times. Since evidence(accelerated expansion) points to us living in de-sitter space we must need to find some degrees of freedom(modes of the string), other than gravitons, which form a coherent state corresponding to de-sitter space.

Anyone know if this has been achieved??

 

[Finbar]

. Even AS uses E-H action with modification, don't they all, more or less.

To prove the AS conjecture every possible term must be included in the effective action.

So far evidence points to certain RG trajectories which do in fact resolve the cosmological constant problem. See http://arxiv.org/pdf/hep-th/0702051 page 10 for example.

 

[tom.stoer]

If we go by Tong's calculation string theory is only consistent in strictly Ricci flat space-times. Since evidence ... points to us living in de-sitter space we must need to find some degrees of freedom ... which form a coherent state corresponding to de-sitter space.

I guess this is a fundamental problem, namely that in a certain sense background independenca means something different in string theory. One has to prove for a certain background that a consistent quantization can be achieved. And this has to be done for each background seperately. Therefore the background (or let's say the class of backgrounds) changes the d.o.f.

 

[Finbar]

I don't know if i agree. It seems very plausible that one can simply think of minkowski space as the ground state of the theory. Remember that there is still a lot of structure in flat space-time(Unrhuh temperature etc.) and its vey hard to imagine something which could be in some less excited state than minkowski.

You can argue that there should be no preferred background in a generally relativistic theory on the other hand there must be a ground state of the hilbert space. Flat Minkowski is the spacetime with the maximal conformal killing symmetries so it presents it's self as the most natural ground state.

The question then is to build general relativity with a small cosmological constant from string excitations. This would seem to involve not only gravitons but other modes of the superstring.

 

[tom.stoer]

The problem is that one should somehow categorize backgrounds in terms of something like "classes" or "superselection sectors". Different sectors may or may not be "connected" by dynamics. In string theory the specific background can affect the details of the degrees of freedom living on it.

I see the following problems:
- one has to identify the correct d.o.f. for each background (sector)
- there may be backgrounds (sectors) which cannot be equipped with a viable string theory
- dynamically connected backgrounds (sectors) cannot be studied coherently if they have different string d.o.f.

Now the question is how to construct viable string theories for certain classes of backgrounds relevant in GR, especially
- dynamical collaps, e.g. pre-Schwarzschild and pre-Kerr
- FRW, dS, ...

 

[Finbar]

My understanding is that these different "backgrounds" are actually different dynamical solutions to string theory corresponding to unstable minimum of the potential. These "backgrounds" are then themselves made up of some coherent state of stringy degrees of freedom that have been integrated out. It is then the low energy effective degrees of freedom that differ from one minimum to the next.

But for everyday string theory computations most of this is implicit and one just starts from a given background. This is because string theorists are often still working at low energies for which a background geometry should emerge.

 

[tom.stoer]

Think about a similar problem in QCD: you are not able to describe the phase transition between ordinary nucleon-matter and QGP using low-energy effective models chiral perturbtion theory based on pions, vector mesons etc. As long as you are within one "superselection sector" everything is fine, but as soon as you want to study the global picture it becomes difficult ...

... anyway, this is off-topic as the same superselction sectors exist in GR (FRW, dS, AdS, ...) and therefore to have a semiclassical / low-energy limit restricted to a certain sector is certainly sufficient (in the context of this thread). It's not a fundamental issue, it's simply time-consuming to formulate the different low-energy theories and show for each sector that GR does emerges.

 

[haushofer]

How surprised should we be that conformal invariance in ST gives as condition the Einstein vacuum field equations?

 

[suprised]

It's more surprising that this basic but non-trivial feature is apparently not known to many of those "critics".

 

[atyy]

How surprised should we be that conformal invariance in ST gives as condition the Einstein vacuum field equations?

I've read that Calabi's conjecture was not motivated by GR (although Yau's interest in it was). So maybe there's a "pure" reason for this.

 

[suprised]

I've read that Calabi's conjecture was not motivated by GR (although Yau's interest in it was). So maybe there's a "pure" reason for this.

No, "GR out of strings" has almost nothing to do with Calabi-Yau manifolds. It is a deep physical result.