# How do the Einstein's equations come out of string theory?

The classical theory of spacetime geometry that we call gravity consists of the Einstein equation, which relates the curvature of spacetime to the distribution of matter and energy in spacetime.

for example: $ds^2=g_{\mu\nu}dx^{\mu}dx^{\nu}$ Mathematically, how do the Einstein's equations come out of string theory?

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It's not exactly a duplicate, but Luboš' answer to physics.stackexchange.com/questions/44732/… is as close as you'll get without the answer turning into a book on string theory. – John Rennie Nov 21 '12 at 19:11
@John Rennie for example, how to derive this relation: $ds^2=g_{\mu\nu}x^{\mu}x^{\nu}$ from string theory? – Neo Nov 21 '12 at 19:20
Dear Neo, the question "how to derive $ds^2=g\cdot x\cdot x$" is meaningless because one may always say that it's a definition of $ds^2$, whether one talks about string theory or not. One could ask why this expression is constant under Lorentz transformation, but it's also true by the definition of the Lorentz group, or because of basic maths, or one could ask why string theory is invariant under this group, which is easily checked because its defining objects such as action are nicely contracting the spacetime vector indices. – Luboš Motl Nov 21 '12 at 19:32
As John says, if you click at the previous question, you will learn that Einstein's equations arise either from effective action one may derive from scattering amplitudes, or from the vanishing of the beta-functions for the metric tensor functions which are "infinitely many coupling constants" of the world sheet theory and the world sheet theory must be conformal (scale-invariant). Explaining all these things with everything one needs to technically understand it is pretty much equivalent to teaching you introduction to string theory which is a 1-semester course, not 1 question on Stack Exc. – Luboš Motl Nov 21 '12 at 19:35
@Luboš Motl $g_{\mu\nu}(X^{\alpha})$ this is very similar. what means $(X^{\alpha})$? – Neo Nov 21 '12 at 20:52
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