In what limit does string theory reproduce general relativity?

Question

In quantum mechanical systems which have classical counterparts, we can typically recover classical mechanics by letting $\hbar \rightarrow 0$. Is recovering Einstein's field equations (conceptually) that simple in string theory?

This post imported from StackExchange Physics at 2014-08-30 03:59 (UCT), posted by SE-user hwlin

Comments

Possible duplicates: physics.stackexchange.com/q/1073/2451 , physics.stackexchange.com/q/5815/2451 and links therein.

This post imported from StackExchange Physics at 2014-08-30 03:59 (UCT), posted by SE-user Qmechanic

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Possible Duplicate (and Related): How do Einstein's field Equations come out of String theory?

This post imported from StackExchange Physics at 2014-08-30 03:59 (UCT), posted by SE-user Dimensio1n0

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I think that any answer to this question should do an effort to show explicitly the $\hbar$ in string theory equations. 

Answer 1

To recover Einstein's equations (sourceless) in string theory, start with the following world sheet theory (Polchinski vol 1 eq 3.7.2): $$ S = \frac{1}{4\pi \alpha'} \int_M d^2\sigma\, g^{1/2} g^{ab}G_{\mu\nu}(X) \partial_aX^\mu \partial_bX^\nu $$ where $g$ is the worldsheet metric, $G$ is the spacetime metric, and $X$ are the string embedding coordinates. This is an action for strings moving in a curved spacetime. This theory is classically scale-invariant, but after quantization there is a Weyl anomaly measured by the non-vanishing of the beta functional. In fact, one can show that to order $\alpha'$, one has $$ \beta^G_{\mu\nu} = \alpha' R^G_{\mu\nu} $$ where $R^G$ is the spacetime Ricci tensor. Notice that now, if we enforce scale-invariance at the qauntum level, then the beta function must vanish, and we reproduce the vacuum Einstein equations; $$ R_{\mu\nu} = 0 $$ So in summary, the Einstein equations can be recovered in string theory by enforcing scale-invariance of a worldsheet theory at the quantum level!

This post imported from StackExchange Physics at 2014-08-30 03:59 (UCT), posted by SE-user joshphysics

Comments

Cute explanation :-)

This post imported from StackExchange Physics at 2014-08-30 03:59 (UCT), posted by SE-user Dilaton

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It seems that this says that in order to have string theory on curved background without that anomaly, the manifold has to be Ricci flat. But the question is whether Einstein's equations can be obtained from string theory in a classical limit.

This post imported from StackExchange Physics at 2014-08-30 03:59 (UCT), posted by SE-user MBN

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@MBN I basically agree with the first statement (although the non-linear sigma model action I wrote down is just one example of a string theory). The question asked "Is recovering Einstein's field equations (conceptually) that simple in string theory?" The answer, as far as I am aware, is no, and I attempted to include the most relevant version of an answer that I could despite this fact.

This post imported from StackExchange Physics at 2014-08-30 03:59 (UCT), posted by SE-user joshphysics

Answer 2

One recovers Einstein's equations by considering the bosonic closed string theory with a zero B-field, a constant dilaton $\phi$ and by taking the limit $\alpha' \rightarrow 0$, $g_s \rightarrow +\infty$ such that $g_s \sqrt{\alpha'}$ is some constant $l_p$. Here $\alpha'$ is the Regge's slope i.e. the inverse of the string tension up to a normalization, $g_s$ is the string coupling constant related to the dilaton background by $g_s = e^\phi$. In this limit, the only excitation of the theory is a massless symmetric rank 2 tensor field $h_{\mu \nu}$ and the statement is that its $n$-point correlation functions coincide with the $n$-points correlations functions of the graviton computed at tree level from the Einstein-Hilbert action with the Planck length equals to $l_p$.

The original papers for this result are by Yoneya 

http://ptp.oxfordjournals.org/content/51/6/1907.refs

and Scherk, Schwarz 

http://ccdb5fs.kek.jp/cgi-bin/img/allpdf?196800234

The relation between this derivation of the Einstein's equation and the one through the $\beta$-function presented in the answer of  joshphysics is nicely explained in the section 10.1 of the Polyakov's book "Gauge fields and strings". The low energy effective action for the massless excitations of the string can be computed from the correlation functions of the corresponding vertex operators. These vertex operators are constructed form the weight 2 local operators of the world-sheet CFT and the spacetime correlation functions of the massless excitations can be expressed in terms of the correlation functions of these operators in the 2d CFT. But in the other hand, weight 2 local operators define the marginal deformations of the 2d CFT and the beta function with respect to one of the massless field is obtained from the variation of the theory under the corresponding deformation. To compute this variation, one can perturbatively expand the deformation of the action and one has to compute n-point correlations functions of the weight 2 local operators in the 2d CFT, i.e. the same object as for the spacetime amplitudes. One is simply saying that the deformations of the space-time background by turning on massless fields correspond to the marginal deformations of the 2d worldsheet CFT.

Comments

The "meson-fermion" model :-DDDD