Relationship Between Strings and Spacetime Geometry

If string theory is a theory of gravity, then how does it compare with Einstein's theory of gravity? What is the relationship between strings and spacetime geometry?

Strings and gravitons

The simplest case to imagine is a single string traveling in a flat spacetime in d dimensions, meaning that it is traveling across space while time is ticking, so to speak. A string is a one-dimensional object, meaning that if you want to travel along a string, you can only go forwards or backwards in the direction of the string, there is no sideways or up and down on a string. The string can move sideways or up and down in spacetime, though, and as the string moves around in spacetime, it sweeps out a surface in spacetime called the string worldsheet, a two-dimensional surface with one dimension of space and one dimension of time.

The string worldsheet is the key to all the physics of the string. A string oscillates as it travels through the d-dimensional spacetime. Those oscillations can be viewed from the two-dimensional string worldsheet point of view as oscillations in a two-dimensional quantum gravity theory. In order to make those quantized oscillations consistent with quantum mechanics and special relativity, the number of spacetime dimensions has to be restricted to 26 in the case of a theory with only forces (bosons), and 10 dimensions if there are both forces and matter (bosons and fermions) in the particle spectrum of the theory.

So where does gravity come in?

If the string traveling through spacetime is a closed string, then the spectrum of oscillations includes a particle with 2 units of spin and zero mass, with the right type of interactions to be the graviton, the particle that is the carrier of the gravitational force.

Where there are gravitons, then there must be gravity. Where is the gravity in string theory?

Strings and spacetime geometry

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. But how do the Einstein equations come out of string theory?

If a closed string is traveling in a curved spacetime, then the coordinates of the string in spacetime feel this curvature as the string propagates.