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
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. Once again, the answer lies on the
string worldsheet. In order for their to be a consistent quantum theory
in this case, the curved space in which the string travels must be a
solution to the Einstein equations.
Now this is really something! This was a very
convincing result for string theorists. Not only does string theory
predict the graviton from flat spacetime physics alone, but string theory
also predicts the Einstein equation will be obeyed by a curved spacetime
in which strings propagate.
What about strings and black holes?
Black holes are solutions to the Einstein equation,
therefore string theories that contain gravity also predict the existence
of black holes. But string theories give rise to more interesting symmetries
and types of matter than are commonly assumed in ordinary Einstein relativity.
So black holes are more interesting to study in the context of string
theory, because there are more kinds to study.
Is spacetime fundamental?
Note that there is a complication in the relationship
between strings and spacetime. String theory does not predict that the
Einstein equations are obeyed exactly.
String theory adds an infinite series of corrections to the theory of
gravity. Under normal circumstances, if we only look at distance scales
much larger than a string, then these corrections are not measurable.
But as the distance scale gets smaller,
these corrections become larger until the Einstein
equation no longer adequately describes the result.
In fact, when these correction terms become
large, there is no spacetime geometry that is guaranteed to describe
the result. The equations for determining the spacetime geometry become
impossible to solve except under very strict symmetry conditions, such
as unbroken supersymmetry, where the large correction terms can be made
to vanish or cancel each other out.
This is a hint that perhaps spacetime geometry
is not something fundamental in string theory, but something that emerges
in the theory at large distance scales or weak coupling. This is an
idea with enormous philosophical implications.