General relativity from string theory

[martinbn]

This came up in another thread, but I have seen the statement many times in various places. The statement is that string theory implies general relativity i.e Einstein's field equations in some kind of a classical limit. So my question is how does that go? I am curious to see the details.

 

[Haelfix]

It's done in almost all the textbooks out there, in particular GSW. Online I think David Tong has some lecture notes: (see here: http://www.damtp.cam.ac.uk/user/tong/string/seven.pdf). The Einstein Hilbert action emerges on page 168. Alternatively I believe Susskind goes over it in his lectures on youtube (this will be at the level of Zweibach)

Actually calculating the full one loop beta functions is a bit of a chore, and I have never done it, but you will get the picture.

 

[marcus]

...The statement is that string theory implies general relativity i.e Einstein's field equations in some kind of a classical limit...

I doubt you would find the actual EFE derived though. In that picture, geometry is fully interactive with matter---there is no dependence on any particular fixed prior geometry.

So I suspect you might have to be satisfied with seeing the EH action derived in a perturbative setup with prior geometry.

(I could be wrong though, some one of the others may know of a reference where the actual Einstein Field Equations are derived from some non-perturbative version of string. That would indeed be interesting to hear about! )

 

[Haelfix]

Stop spreading FUD Marcus! Seriously, this is textbook material of which I just linked a derivation and I'm tired of reading your elementary misunderstandings.

 

[marcus]

We were pointed to page 168 and I see there not the Einstein equations but an action on a fixed 26 dimensional background. The discussion is about gravitions, i.e. perturbative.
It does not quite correspond to what Martin asked for.

He may be satisfied with it. Something, after all. But it isn't what I had in mind in post #3.

 

[suprised]

(I could be wrong though, some one of the others may know of a reference where the actual Einstein Field Equations are derived from some non-perturbative version of string. That would indeed be interesting to hear about! )

The Einstein equations from GR arise already at tree level, as the conditions for conformal invariance on the world-sheet (vanishing beta functions). This has been a main motivation for studying string theory in the first place. And that's not just involving "single" gravitons, rather a classical background can be viewed as coherent superposition of infinitely many gravitons; and the Einstein eqs govern their collective dynamical behavior (within limits for which they are accurate).

And of course, when writing equations of GR the metric enters there, how can it be otherwise? In case LQG ever gets to what you denounce as "something", ie generating the Einstein eqs out of some black box, it will also require to write a metric down in order to even formulate these equations!

 

[Haelfix]

No Marcus! You clearly don't get the derivation, or the fact that there is nothing fixed in the effective action at all. There is nothing perturbative about the field equation results, it is a result that must be true by *consistency*, stemming from the Weyl invariance on the world sheet.

After compactification, and integrating out the matter modes and taking the hbar --> 0 limit, the result is 4 dimensional Einstein Hilbert lagrangian. Solving the Euler-Lagrange equations yields Einsteins field equations in vacuum exactly.

This is completely analogous to the derivation in MTW where a spin 2 field and a weak field expansion is shown to reproduce the EFE exactly. Here though, the consisteny criteria are already staring at you in the face on the worldsheet. There is nothing else that string theory can limit too, it always must have the EFE's in the IR exactly!

 

[martinbn]

Thank you for the link. I would have to read it carefully, but on first glance I get the same impression as Marcus. May be I didn't phrase my question well, or I just have a very different (and may be naive) expectation what a derivation should be. In any case as I said I need to go through it carefully.

 

[Haelfix]

Tong explains it perfectly. You start by fixing a background on the worldsheet, and demanding that the quantum theory be conformally invariant (eg that the beta functions vanish). After a calculation you find a set of equations or requirements that must vanish.

Up to this point, everything is perturbative to a given order and fixed.

Now you switch perspectives, and ask, what is the low energy effective lagrangian over spacetime (as opposed to the worldsheet) that gives those beta functions as equations of motion.

And you are led to the EH lagrangian. This last step is decidedly not perturbative, it is not fixed, it is simply a statement that in the hbar --> 0 limit (which takes care of all the 2+ loop corrections from the worldsheet), that the EFE's are the only possible equations of motion that reproduces that lagrangian classically. All you then need to do is show that it is unique. Which is a classical theorem by Hilbert, and you are done.

The bottomline is that there is no controversy that string theory gives GR in the low energy limit. It is basic textbook material!

(edit: The action here is indeed 26 dimensional, and strictly speaking this is the Bosonic string. The real calculation would involve compactification on the Superstring (eg 10 dimensional), and obviously it is a little more subtle with a lot more notation. But the actual proof goes through in a completely analogous manner, except there you won't derive pure GR, but rather supergravity (and then you have to worry about how to break supersymmetry))

 

[martinbn]

Haelfix, I didn't mean to imply that there was controversy. I just need time to absorb it. For example in (7.5) it says "we require... " it is not obvious to me that this is a consequence, or the only possibility, and not an ad hoc assumption, which was made so that one can get the vacuum equations. And many other details that I still need to clarify for myself.

 

[martinbn]

(edit: The action here is indeed 26 dimensional, and strictly speaking this is the Bosonic string. The real calculation would involve compactification on the Superstring (eg 10 dimensional), and obviously it is a little more subtle with a lot more notation. But the actual proof goes through in a completely analogous manner, except there you won't derive pure GR, but rather supergravity (and then you have to worry about how to break supersymmetry))

This part is ok. When I understand the case in the notes I can look for this.

 

[Haelfix]

Yea, that renormalization condition is actually a feature of nonlinear sigma models. It's more clearly explained in Green Schwartz Witten (p 169).

But yea, the details are not necessarily easy here, for instance calculating the quantum corrections to the beta functionals is decidedly lengthy.

What I think is important though is to see how the sketch of the proof is conducted, and indeed proofs like this are indeed ubiquitous in string theory where you constantly switch pictures (from worldsheet to spacetime back to worldsheet) to make statements that are in a sense more general than what you would naively think.

 

[martinbn]

Green Schwartz Witten is a little easier to read for me, but I still don't understand it, hopefully in time. I also have problem understanding it on a level of ideas. I expected to see a formulation of a theory, from which one can derive Einstein's equations in some sort of a classical low energy limit. What I see is the process of developing the theory and along the process one _needs_ to impose a condition that happens to be Ricci flatness. That doesn't look (to me) as a derivation. It is just a requirement needed to formulate the theory. It also seems accidental. I am sure there is a deep reason for it, but as it goes in the text it could have been something else. Also what about the full equations? If Ricci flatness is a must, does that mean that we cannot get non-empty space equations?

I looked at a few other textbooks and the expositions are very similar. They all skip the details. I am guessing that I am ahead of myself and these are things that one learns before reading these texts or every physicist can do them on his own, but are there text that give all the details?

One more question. As the analysis goes there many choices made. But there are no remarks (probably because this should be clear to the reader) whether the choices are unique and if not whether they effect the results.

As I said before, probably I need more time and knowledge before I can understand, but as of now I am left with a feeling of unsatisfaction.

 

[negru]

I had some of the same issues when I was learning this stuff. It seems that some things are derived (eg scattering amplitudes), and some stuff you get by requiring consistency (eg number of dimensions). But is there a real difference between these things? Is it any different than in QFT for example? Requiring the conformal symmetry on the worldsheet is not a random property you want - you can trace it to Lorentz invariance.

It also seems a bit just like an issue about semantics:
if string is true => derived result
vs
if string theory is true - you need some consistency

Alternatively you can think of the consistency you impose as your guiding principle. In gauge theory your guiding principle is gauge symmetry, right? Well here you could just say that "conformally invariant string theory" is your principle.

These are some of the ways I think about it, maybe someone has a better way to explain it.

 

[fzero]

Green Schwartz Witten is a little easier to read for me, but I still don't understand it, hopefully in time. I also have problem understanding it on a level of ideas. I expected to see a formulation of a theory, from which one can derive Einstein's equations in some sort of a classical low energy limit.

You could do this too. You would compute graviton scattering amplitudes in string theory and then take the low-energy limit of the expressions. You can then ask what effective field theory of gravitons can reproduce those results. You'd find out that the Einstein-Hilbert theory does. These calculations were done in the early 70s and probably tedious, so it's not surprising that textbooks don't go into all of the details. The 1975 review by Scherk http://inspirebeta.net/record/838?ln=en might contain details.

 

[martinbn]

I had some of the same issues when I was learning this stuff. It seems that some things are derived (eg scattering amplitudes), and some stuff you get by requiring consistency (eg number of dimensions). But is there a real difference between these things? Is it any different than in QFT for example? Requiring the conformal symmetry on the worldsheet is not a random property you want - you can trace it to Lorentz invariance.

yes, I guess my discomfort is of that nature.

It also seems a bit just like an issue about semantics:
if string is true => derived result
vs
if string theory is true - you need some consistency

My impression is that it is more like

impose conditions - in order to be able to begin to formulate ST on curved background

Alternatively you can think of the consistency you impose as your guiding principle. In gauge theory your guiding principle is gauge symmetry, right? Well here you could just say that "conformally invariant string theory" is your principle.

These are some of the ways I think about it, maybe someone has a better way to explain it.

I have no problem with this. It is just that talking about ST containing GR as a limiting case I understand something else than this.

 

[martinbn]

You could do this too. You would compute graviton scattering amplitudes in string theory and then take the low-energy limit of the expressions. You can then ask what effective field theory of gravitons can reproduce those results. You'd find out that the Einstein-Hilbert theory does. These calculations were done in the early 70s and probably tedious, so it's not surprising that textbooks don't go into all of the details. The 1975 review by Scherk http://inspirebeta.net/record/838?ln=en might contain details.

I will try to look at this too, but one thing at a time. I want to understand first the explanation given above.

 

[Haelfix]

The matter part of Einstein's equations is the really hard part. Glancing at the Bosonic string spectrum, you see a photon like state, a couple scalars, a tachyon... However, in truth, the real low energy matter terms come from exciting the stringy spectrum in various ways during the compactification process. This is far from unique, and is typically done in different model dependant calculations much later in the textbooks. The universality of gravity comes from the fact that it arises from closed strings. There is no way to not have closed strings in string theory.

And yes, the derivation of Einstein's equations are unusual, as they show up in the last place that you would think. But its actually rather beautiful if you think about it. Imposing this conformal symmetry on the worldsheet, and demanding that it embeds into a particular spacetime consistently actually fixes a foundational requirement for the entire low energy theory! As to why you need this condition.. String theory without the conformal symmetry has violent pathologies, but in particular when it is imposed, it becomes UV finite.

 

[Sardano]

The low energy limit of ST is a Supergravity theory. The supersymmetry and the matter content of this 4 dimensional Supergravity depends on the compactification. So the matter content of the theory depends on the compactification. Since Supergravity is something like GR + matter fields, and there you have the matter content.

 

[suprised]

I sense some confusion to clarify. There are two things: the Einstein eqs from requiring conformal invariance, and scattering amplitudes with "single" graviton vertex operators.

The Einstein equations arise from the condition of conformal invarance of the world sheet theory. So indeed they are not computed directly but obtained from imposing a symmetry principle. Roughly conformal invariance implies certain "Ward-identities" on correlation functions that transalate into symmetry properties of the effectice space-time theory. So a generally covariant effective theory follows automatically. For example, the decoupling of the longitudinal modes of the graviton, which is important for gauge symmetry and unitarity, follows from a simple contour argument on the 2d Riemann surface.

This is of course the more clever way to do that, rather than computing amplitudes order by order in the number of graviton vertex operators. That gives an expansion around a classical background which is cumbersome and not illuminating. But it allows to check the validity of the general arguments above. Indeed a few order had been computed explicitly (my knowledge stems from the eighties), and this reproduces the expansion of the Einstein action (plus string corrections) to that order.

 

[atyy]

Is it true that the world sheet theory is perturbative string theory, and isn't UV complete? So imposing conformal invariance on it is still not enough to say there is a consistent theory, and arguments from dualities are needed for that?

 

[fzero]

Is it true that the world sheet theory is perturbative string theory, and isn't UV complete? So imposing conformal invariance on it is still not enough to say there is a consistent theory, and arguments from dualities are needed for that?

Perturbative amplitudes are UV finite (the Berkovits formalism is the best indication at the moment). I think what you might be referring to is that the radius of convergence of the perturbative series is not known. This is a completely separate issue that has nothing to do with conformal invariance, which already applies term by term to the series. It also doesn't have too much to do with consistency. Even before dualities, we would not have said that strongly coupled gauge theories were inconsistent.

 

[Ben Niehoff]

Regarding "fixed prior geometry", these are some silly words that could really stand not to be repeated any longer. A string is nothing more than:

A continuous map from a two-dimensional parameter space into some spacetime manifold.*

This is all you need to fully develop the worldsheet theory. When you quantize it, you get certain conditions which must be satisfied for the quantum theory to be consistent. Those consistency conditions tell you what sort of spacetime manifold in which the string can be consistently embedded. It turns out that the only requirement is that the spacetime manifold be a solution of (super)-gravity with some matter fields.

The reason you often see strings embedded into flat Minkowski space is because it is easy to explain certain concepts this way.

* Technically speaking, a string is something even more general: a 2-dimensional nonlinear sigma model with a collection of scalar fields. It turns out that in some cases, these scalar fields can be interpreted as coordinates in some spacetime manifold. But that needn't be the case.

 

[suprised]

Regarding "fixed prior geometry", these are some silly words that could really stand not to be repeated any longer. A string is nothing more than:

A continuous map from a two-dimensional parameter space into some spacetime manifold.*
...

* Technically speaking, a string is something even more general: a 2-dimensional nonlinear sigma model with a collection of scalar fields. It turns out that in some cases, these scalar fields can be interpreted as coordinates in some spacetime manifold. But that needn't be the case.

Actually naive geometral concepts apply only near a parameter region of "measure zero", namely for weak coupling and large radii. It is there where strings can be characterized in classical geometrical terms (compactification manifolds, gauge bundles, instantons..) and where the sigma model is a good description. There are other phases where these notions do not make much sense and need to be generalized. This is also where GR breaks down as an effective description (eg near strongly curved or singular regions) and the UV completion becomes important.

So in brief, GR arises from strings in the semi-classical regime precisely as necessary, and is otherwise blurred by quantum corrections.

 

[Ben Niehoff]

Actually naive geometral concepts apply only near a parameter region of "measure zero", namely for weak coupling and large radii. It is there where strings can be characterized in classical geometrical terms (compactification manifolds, gauge bundles, instantons..) and where the sigma model is a good description. There are other phases where these notions do not make much sense and need to be generalized. This is also where GR breaks down as an effective description (eg near strongly curved or singular regions) and the UV completion becomes important.

So in brief, GR arises from strings in the semi-classical regime precisely as necessary, and is otherwise blurred by quantum corrections.

Yes, I agree. I should have specified that.

 

[martinbn]

OK, if understand correctly, GR does not arise in ST, but is imposed as a consistency condition. Now my questions is: is this the _only_ condition that leads to a consistent formulation and why? For example why is the choice of a beta function unique, or any of the other choices? I am guessing these are naive or even stupid questions, so I apologize, but as a non-physicist the answers are not obvious to me.

 

[Ben Niehoff]

OK, if understand correctly, GR does not arise in ST, but is imposed as a consistency condition. Now my questions is: is this the _only_ condition that leads to a consistent formulation and why? For example why is the choice of a beta function unique, or any of the other choices? I am guessing these are naive or even stupid questions, so I apologize, but as a non-physicist the answers are not obvious to me.

The consistency condition is not "imposed". It is required in order for the theory to be consistent, quantum mechanically. Basically, certain quantum anomalies have to vanish, or else the theory simply makes no sense.

 

[martinbn]

The consistency condition is not "imposed". It is required in order for the theory to be consistent, quantum mechanically. Basically, certain quantum anomalies have to vanish, or else the theory simply makes no sense.

That's what I meant! What is the difference?

 

[Finbar]

Regarding "fixed prior geometry", these are some silly words that could really stand not to be repeated any longer. A string is nothing more than:

A continuous map from a two-dimensional parameter space into some spacetime manifold.*

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.

 

[atyy]

Perturbative amplitudes are UV finite (the Berkovits formalism is the best indication at the moment). I think what you might be referring to is that the radius of convergence of the perturbative series is not known. This is a completely separate issue that has nothing to do with conformal invariance, which already applies term by term to the series. It also doesn't have too much to do with consistency. Even before dualities, we would not have said that strongly coupled gauge theories were inconsistent.

Yes, I am thinking about the convergence of the series, rather than the finiteness of each term. I remember remarks eg. by Tom Banks that the series is probably only asymptotic.

My understanding in strongly coupled gauge theories, consistency is due to asymptotic freedom or safety.

So if perturbative string theory is only asymptotic, why is it believed that string theory gives a UV complete theory of quantum gravity? Is that due to the dualities?