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Strings in Curved Spacetime
---------------------------
First consider how a point particle, x, in the diagram
below, moves on (in) the surface of a sphere. The center
of mass (COM) of the particle is always coincident with
the particle's center and will travel along a geodesic
in the absence of external forces. The energy of the
particle is:
E = (mv^{2}/2)
= p^{2}/2m
The angular momentum is L = r x p
L is quantized so L also equals nh,
The energy equation becomes:
E = p^{2}/2m = L^{2}/2mr^{2} = L^{2}/2I = n^{2}h^{2}/2I
Where I is the moment of inertia (MOM) of the particle.
Now an excited string is no longer a point particle.
Exciting oscillator modes (increasing the energy) causes
the string to stretch. Classically, this corresponds
to:
E = kx^{2}/2
= T^{2}x^{2}/2
Where,
T = √k is the string tension with units of energy
per unit length.
To get some idea about how much an open string stretches
we make use of the formula:
_{∞}
x(σ,τ) = Σx_{n}(τ)cos(nσ)
^{n=0}
_{∞}
= x_{COM} + Σx_{n}(τ)cos(nσ)
^{n=1}
= x_{COM} + Σ_{n}((a_{n}^{-} + a_{n}^{+})/√n)cos(nσ)
The length of the string can be considerd to be the
average deviation of x(σ) from the COM. For convenience
we calculate the quantum mechanical expectation value of
(x(σ) - x_{COM})^{2} as,
<0|(x(σ) - x_{COM})^{2}|0>
= <0|Σ_{nm}((a_{n}^{-} + a_{n}^{+})/√n)((a_{m}^{-} + a_{m}^{+})/√m)cos(nσ)cos(mσ)|0>
= Σ_{nm}((a_{n}^{-}a_{n}^{+})/n)cos^{2}(nσ) for n = m
= (1/2)Σ_{n}(1/n) since cos^{2}(nσ) ~ 1/2
= (1/2)ln(n)
So the distance from the COM increases logarithmically
with n. Although the increase is slow (ln(1000) = 6.9)
the string grows in length and the stretching spreads out
its mass so that its COM is no longer necessariy coincident
with its center. This creates a problem for strings in
curved spacetime. To see why consider the following
diagrams showing a string on the surface of a sphere.
We see that ss the string grows in length its COM gets
closer and closer to e COM of the sphere. r decreases
and the MOM shrinks, ultimately going to 0. At this
point the energy becomes infinite. This is very similar
to the problem of ultraviolet divergences in Feynman loop
diagrams. It turns out that this divergence occurs for
all curved geometries - not just the sphere.
To avoid the situation where we would have impose a
cutoff to the oscillator modes to prevent divergences
we instead look for a background spacetime geometry
that doesn't change as more modes are added to the
string. Since strings have energy (mass) they act as
sources in the spacetime background in which the string
propagates. Therefore, the addition/subtraction of
oscillator modes can change the effective metric tensor
of the background. In addition, we require that the
effective geometry created also has to satisfy Einstein's
field equations. In order meet the requirement for mode
independence, we need to apply the concept of RICCI FLOW.
The Ricci flow equation is written as:
δg_{μν} = -2R_{μν}
For a Riemannian manifold the Ricci flow is a PDE that
evolves the metric tensor over 'time'. It is analogous
to the diffusion of heat. Informally, it can be viewed
as a process that produces an equilibrium geometry for
a manifold for which the Ricci curvature is constant.
Clearly, if we don't want the metric to change as we
add more modes, the RHS must be set to 0. Therefore,
R_{μν} = 0
This geometry is referred to as being RICCI-FLAT.
Ricci-Flat
----------
Consider Einstein's field equations with no source.
R_{μν} - (1/2)g_{μν}R_{α}^{α} = 0
Where R_{α}^{α} is the trace.
R_{μ}_{ν} - (1/2)g_{μ}_{ν}R_{α}^{α} = 0
R_{μ}^{ν} - (1/2)δ_{μ}^{ν}R_{α}^{α} = 0
R_{α}^{α} - 2R_{α}^{α} = 0 since δ_{μ}^{ν} = 4 (4 D Kronecker)
Thus, the curvature scalar disappears and we are left
with:
R_{μν} = 0
R_{μν} represents the part of curvature that derives from
the presence of matter (i.e. that part due to the stress
energy tensor T_{μν}). The remaining components of the
Riemann tensor (called the WEYL tensor) represents the
intrinsic gravitational field in the absence of matter
or non gravitational fields. The intrinsic spacetime
containing only gravitational radiation will satisfy
R_{μν} = 0 but will also have R_{μνσ}^{λ} ≠ 0.
With R_{μν} = 0 we are left with the Weyl tensor, also
called the CONFORMAL TENSOR. If the Weyl tensor
vanishes the manifold is said to be CONFORMALLY FLAT.
Any 2-dimensional (smooth) Riemannian manifold is
conformally flat. What this means is that each point
on the manifold has a neighborhood that can be mapped
to flat space by a conformal transformation.
A conformal transformation can be defined as a special
case of general coordinate transformation, x -> x', such
that x' = f(x) has the following effect on the metric:
g'_{μν} = (∂x^{ρ}/∂x'^{μ})(∂x^{σ}/∂x'^{ν})g_{ρσ}
It acts as a rescaling of the metric, x -> x. This
rescaling of the metric is often referred to as a WEYL
transformation and the two terms are often used
interchangeably, although they are in fact different
things. A Weyl transformation takes us to a coordinate
system where the metric has the same form as the one
we started with, but the the proper distance between
points have been changed by a scale factor while
preserving the angles between all lines. Therefore,
under a Weyl transformation:
g_{μν} = Ω^{2}g_{μν}
It is usual to write Ω^{2} = exp(2f). In 2 dimensions
f is realized by analytic functions, f(z), where z is
complex. Thus, f(z) = u(x,y) + iv(x,y) satisfy the
Cauchy-Riemann equations.
∂u/∂x = ∂v/∂y and ∂u/∂y = -∂v/∂x
Ricci flatness is a requirement for all string theories
but fortunately it turns out there are many many topologies
that are Ricci flat. Of particular interest in our simple
analyses is the Ricci flat torus. However, more advanced
analyses model strings on Ricci flat CALABI-YAU MANIFOLDS.