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I have a hard time understand the statement that

When you only look at the classical limit or classical physics, string theory exactly agrees with general relativity

Because from what I know, String Theory assumes a fixed space time background (ie, all the strings and membranes interact in a fix background, and their interaction gives rise to fundamental particles that we observe), but General Relativity assumes that the space time background is influenced by what is in it and the interaction between them.

Given that both have very different assumptions, what do string theorists mean when they say string theory agrees with general relativity in a classical limit? Or more specifically, how does string theory--a fix spacetime background theory-- reconciles with the general relativity on dynamic spacetime background part? I can understand a fix, static spacetime in the context of changing, dynamic spacetime background, but I cannot understand a chanding, dynamic spacetime in the context of a fix, static spacetime background.

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marked as duplicate by Chris White, Dan, Manishearth Jul 7 '13 at 23:57

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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Nice question. Historically, GR and its interpretation were motivated by concepts such as Mach's principle, the equivalence principle, general covariance, and background independence. Many of these concepts have eluded rigorous formulation or have been difficult to identify as being embodied or not embodied in particular theories. For example, Deser, arxiv.org/abs/gr-qc/0411023 , showed in 1970 that GR was mathematically equivalent to a spin-2 field on a background of flat spacetime. A theory can be background-independent even if formulated in a way that appears background-dependent. –  Ben Crowell Jul 5 '13 at 4:38
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It means that the gravitational terms of the effective action of the the Polyakov (yes, Polyakov, gravitons are not fermions) action is the EH Action. –  Dimensio1n0 Jul 5 '13 at 5:01
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@Graviton: You, like your siblings Dilaton and Photon, must be bosons because your parents are Neveu Schwarz and Neveu Schwarz. If it were Ramond and Neveu Schwarz, then you would have been a fermion, like your superpartner Gravitino... –  Dimensio1n0 Jul 5 '13 at 6:14
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@Graviton: one approach to string theory is the perturbative expansion on a flat spacetime. However this does not constitute string theory, just one method of doing calculations with it. –  John Rennie Jul 5 '13 at 6:31
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@Dilaton, Graviton: Sad to hear that your brother "Photon" and all but 1 of your bosonic cousins (the bosons born to Ramond and Ramond) were killed by worldsheet parity. At least, photon is reborn as your second cousin (Born to Neveu-Schwarz the bacterium). One of the Gravitino twins and the Dilatino twins were already killed by the cruel GSO Projection (1 Dilatino/Gravitino was born 2 Ramond and Neveu Schwarz, the other was born to Neveu Schwarz and Ramond). At least some new gauge bosons and gauginos were also born to help you recover from the tragedy : ( . –  Dimensio1n0 Jul 5 '13 at 9:02

2 Answers 2

up vote 3 down vote accepted

First of all, the statement is by design that perturbative string theory reproduces perturbative quantum-gravity+Yang-Mills at low energy, for perturbation about any solution to the supergravity equations of motion (what user "dimension10" mentions is one part of the statement that perturbative string theory around such backgrounds is consistent to start with). Notice that this perturbative nature is not some secret bug, but is so by the very nature of what perturbation theory is, in whichever context. (See also http://ncatlab.org/nlab/show/string+theory+FAQ#BackgroundDependence).

Moreover, the way in which this works in not new to string theory but is the time-honored process of effective quantum field theory (see there for the historic examples): you write down some scattering amplitudes that you are interested in for one reason or another, and then you look for a quantum field theory that reproduces these scattering amplitudes in some low energy regime. Once found, this is the given effective quantum field theory which approximates whatever theory your scattering amplitudes describe at possible high energy.

Next you play this game with the string scattering amplitudes which are defined by summing up correlation functions of some 2d super-conformal field theory of central charge -15 over all possible Riemann surfaces with given insertions (your asymptotically in- and outgoing states). Next you ask if there is an ordinary quantum field theory such that it's perturbative scattering amplitudes coincide with these at low energy. Turns out that this is a higher dimensional locally supersymmetric Einstein-Yang-Mills theory, which is hence the effective field theory that describes the perturbative dynamics of strings at low energy.

See on the nLab at String theory FAQ -- How is string theory related to the theory of gravity?

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+1. Which surprises me is that Quantum laws appears at a "fundamental" level (String theory), but arise also at the effective theory level (Yang-Mills). This is something strange, after all, if Quantum is fundamental, it should appear only at a fundamental level. And if Quantum laws are only effective laws, Quantum should not appear at a fundamental level. –  Trimok Jul 5 '13 at 11:43
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The principle of quantum physics underlies all of this. Within quantum physics there are fundamental and there are effective theories. Think of a simple example of the quantum mechanics of a neutron in a potenial well (say as in arxiv.org/abs/1207.2953). Fundamentally the neutron is a highly complicated QFT bound state, but neverthess its center of mass point motion is described by simple quantum mechanics. –  Urs Schreiber Jul 5 '13 at 11:59

UPDATE: I have written a more complete answer here: How do the Einstein's equations come out of string theory?

If you take the Polyakov (gravitons are bosons) Action:

$$S_P=-\frac{T}{2}\int \sqrt{\pm h}h^{\alpha\beta}\partial_\alpha X^\mu\partial_\beta X^\nu g_{\mu\nu}\mbox{ d}^2\xi$$

And take the gravitational terms of a somewhat "effective" spacetime action, you get

$$S_{G}=\lambda\int\left(R+\ell_s^2R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\right)\mbox{ d}^D x$$

Where we neglected terms of order $\ell_s^4$ and greater. Since $\ell_s$, the string length, is very small, this is approximately, (if we let ${\ell_s\rightarrow0}$)

$$S_{EH}=\lambda\int R\mbox{ d}^D x$$

The generalised (n-dimensional) EH Action. This is what is meant by string theory going to GR at the classical limit, the Polyakov Action simply goes down to the EH Action.

And that is a really great feature of string theory...

Edit: Also see JoshPhysics's answer here: In what limit does string theory reproduce general relativity?. The method I stated here is possible in principle, but is much more commplicated than the JoshPhysics's answer there. In his answer, he simply uses the Beta functional, $$\beta^G_{\mu\nu} = \ell_s^2 R_{\mu\nu}+\ell_s^4R_{\mu\nu}R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}+...$$ Then, setting the LHS to 0 to preserve conformal invariance:

$$R_{\mu\nu}+\ell_s^2R_{\mu\nu}R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}+...=0$$

For weak gravity, all terms except the first vanish, so that

$$R_{\mu\nu}=0$$

$\mbox{The Vaccuum EFE ...}$

The reason why you only obtain the vacuum EFE, is I guess because only the simple terms of $\beta^G_{\mu\nu} $ was taken...

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This is correct, but the main detail missing is the Ward identity--- this is derived in chapter 2 of Green Schwarz Witten--- a variation in the metric is equal to a graviton insertion, and diffeomorphisms don't count, so you get a Ward identity. –  Ron Maimon Aug 22 '13 at 21:58

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