# The General Relativity from String Theory Point of View

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?

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

The effective gravitational terms of the spacetime action, which can be derived from the Polykov action (gravitons are bosons) are --

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

Where we neglected terms of order $\ell_s^4$ and greater. To first-order in $\ell_s$, the string length --

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

Which is the $n$-dimensional Einstein-Hilbert Action.

The vacuum EFE may also be derived directly from setting the beta functional, which measures the breaking of conformal invariance, to zero: $$\beta^G_{\mu\nu} = \ell_s^2 R_{\mu\nu}+\ell_s^4R_{\mu\nu}R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}+... = 0$$

For weak gravity --

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