If string theory is a theory of gravity, then
what is the relationship between strings, gravitons and spacetime geometry?
Strings and gravitons
The simplest case to imagine is a single string traveling
in a flat spacetime in d dimensions. As the string moves around in spacetime,
it sweeps out a surface in spacetime called the string
worldsheet, a twodimensional surface with one dimension of space
(s) and one dimension of time
(t).
There are many ways to examine this string theory.
One way is to expand the string coordinates X^{a}(s,t)
into oscillator modes and demand spacetime Lorentz invariance and the
absence of negative norm states. A different way to examine the string
theory is through the field theory defined on the worldsheet, which
is described by the action
where h_{mn} is the metric on the worldsheet, R_{(2)}
is the curvature of the worldsheet, and F
is a scalar field called the dilaton. The consistency condition for
string theory when described in this manner is that the field theory
on the worldsheet satisfy the condition for scale
invariance, also known as conformal
invariance. The
set of functions that describe the scaling properties of quantum fields
are called the beta functions. String worldsheet physics is invariant
under a change in scale if the beta function b^{F}
for the dilaton field F vanishes,
which happens when d=26 for bosonic strings.
(For superstring theories, conformal
invariance is replaced by superconformal invariance, and the required
spacetime dimension is 10.)
The spacetime oscillation
spectrum satisfies Lorentz invariance in 26 dimensions, so that these
string oscillations on the worldsheet can be classified by the spacetime
properties of mass and spin, just like elementary particles. A theory
based on open strings has massless oscillations that are Lorentz vectors,
with spin 1. A closed string theory is like a product of two open string
theories, with an oscillation mode that travels in spacetime as a two
index symmetric tensor, with spin 2.
This mode with spin 2 propagates like as small fluctuation
in the gravitational field propagates according to general relativity.
This string oscillation mode should then be the graviton, the particle
that mediates the gravitational force. The presence of this spin 2 oscillation
mode was the first clue that string theory was not a theory of strong
interactions, but a potential quantum theory of gravity.
Strings and spacetime geometry
In string theory,
if we start with flat spacetime, we see gravitons in the spectrum, and
therefore we deduce that gravity must exist. But if gravity exists,
then spacetime must be curved and not flat. How do the Einstein equations
for the curvature of spacetime come out of string theory?
If a closed string is traveling in a curved
spacetime with metric field g_{ab}(X) , then the string worldsheet
theory looks like
The spacetime metric g_{ab}(X) enters the twodimensional theory
on the string worldsheet as a matrix of nonlinear couplings between
the X^{a}(s,t).
Once again, the goal of conformal invariance
is met by demanding that the beta functions vanish. When the string
coordinates are expanded in a perturbation series in the string scale
a', the terms in the beta functions
that are the lowest order in a'
contain terms proportional to the Ricci curvature R_{ab} of
the spacetime metric field g_{ab}(x) and second derivatives
of the scalar field F(x). The
vanishing of the beta functions ends up being equivalent to satisfying
the Einstein equation for a spacetime with a scalar field
at distance scales large compared to the string scale. Notice this
means that our understanding of spacetime from perturbative string theory
will always be incomplete, except in some special circumstances described
below.
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.
In particular, electric/magnetic duality in string theory has led to
the discovery of many new types of black holes with combinations of
electric and magnetic charge, coupled to both scalar and axion fields.
Also, string theory has motivated an understanding of black holes in
higher dimensions, and of black extended objects such as strings and
branes.
Some of these new stringy extreme black hole
solutions possess unbroken supersymmetries at the event horizon, so
that the physics at the horizon is protected from higher order perturbative
corrections by virtue of supersymmetric nonrenormalization theorems.
These types of black holes have been important for understanding the
origin of black hole entropy in string theory,and that will be described
in the next section.
Is spacetime fundamental?
Note that string theory does not predict that
the Einstein equations are obeyed exactly.
Perturbative string theory adds an infinite series of corrections to
the Einstein equation
So our understanding of spacetime in perturbative string theory is
only valid as long as spacetime curvature is small compared to the string
scale.
However, when these correction terms become
large, there is no spacetime geometry that is guaranteed to describe
the result. Only under very strict symmetry conditions, such as unbroken
supersymmetry, are there known exact solutions to the spacetime geometry
in string theory.
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.
