Transcript Singularities in String Theory

Recent progress in understanding
Spacetime Singularities
Gary Horowitz
UCSB
In general relativity, the singularity theorems
show that large classes of solutions must be
singular. General relativity breaks down and
must be replaced by a quantum theory of
gravity such as string theory.
This does NOT imply that string theory
should resolve all singularities.
Timelike singularities describe
singular initial conditions.
Sometimes they represent
unphysical solutions that should
not be allowed in the theory.
Some timelike singularities are indeed
harmless in string theory:
1) Orbifolds
2) Branes
3) Enhancon (Johnson, Peet, Polchinski, 1999)
But others are not
The Schwarzschild AdS solution with M < 0 has a
timelike curvature singularity. If this was resolved,
there would be states of arbitrarily negative
energy. This would contradict AdS/CFT since the
CFT Hamiltonian is bounded from below.
We do not yet know necessary and sufficient
conditions for a timelike singularity to be resolved
in string theory.
Singularities arising from nonsingular
initial conditions
• Topology change
• Cosmology
• Black holes
Topology change in Calabi-Yau spaces
Consider M4 x K. Moving within the space
of Ricci flat Kahler metrics on K, one can
cause spheres to shrink to zero size:
a) An S2 can go to zero and re-expand as
a topologically different S2. The mirror
description is nonsingular (flop).
(Aspinwall, Greene, Morrison, 1993)
b) An S3 can go to zero and re-expand as
an S2 (conifold).
(Strominger, 1995)
Topology change via tachyon
condensation
Tachyons indicate an instability:
V() = - m2 2
Closed string tachyons are expected to
“remove spacetime”:
The tachyon is a relavent perturbation on the
string worldsheet. RG flow decreases the
central charge - removing dimensions of
spacetime.
In general, RG flow is different from time
evolution in spacetime:
RG  GR
1st order
2nd order
They become equivalent in supercritical
theories: D >> 10 with a timelike linear dilaton.
But for localized tachyons, the endpoint is
often the same. This has been shown explicitly
for orbifolds (Adams, Polchinski, Silverstein, 2001).
Consider a circle with radius R that shrinks below
the string scale in a small region. With
antiperiodic fermions, wound strings become
tachyonic (Rohm, 1984):
It was shown last year that the outcome of this
instability is that the circle smoothly pinches off,
changing the topology of space (Adams et al. 2005).
In a T-dual description, the winding strings are
momentum modes. The worldsheet looks like
a sine-Gordon model. It is known that under
RG flow, this theory has a mass gap.
Modes propagating down the cylinder toward
the tachyon region see an exponentially
growing potential and are reflected back.
A more general argument:
In the presence of a tachyon, the worldsheet
action takes the form:
where
is an operator of dimension <2 and
2=2- .
The corresponding deformation of a worldline
action is a spacetime dependent mass squared
which grows exponentially with time.
So a tachyon effectively gives mass to all
string modes. But spacetime describes the
low energy propagation of the string. If all
massless modes are lifted (including the
graviton) then there is no spacetime.
Closed string tachyon condensation should
remove spacetime.
Analogy: Open string tachyon condensation
removes D-branes - the area for open strings
to propagate.
Cosmological
singularities
A simple model of a cosmological singularity
is the Milne orbifold: 2D Minkowski
spacetime/boost
ds2 = - dt2 + a2 t2 d2
This is clearly an exact solution to string
theory since it is flat. It arises in several
different contexts.
Cyclic universe (Steinhardt, Turok)
The big bang is a collision between branes.
The branes move apart and recollide over
and over.
Although the curvature diverges on the
brane, the higher dimensional description is
just a Milne singularity.
Unexcited wrapped strings have a smooth
evolution through the vertex. Some string
amplitudes appear well behaved.
But backreaction is important
There is a big difference between this
orbifold and the usual Euclidean orbifold:
any momentum around the circle becomes
infinitely blue shifted and turns this simple
singularity into a general curvature
singularity. The Milne singularity is unstable.
It may still be possible to go through the
singularity, but it cannot be justified by the
flat Milne example.
With antiperiodic boundary conditions for
fermions, winding strings can become
tachyonic before the curvature becomes
large. The subsequent evolution is no longer
given by supergravity, but rather by the
physics of tachyon condensation. (McGreevy
and Silverstein, 2005)
<T>
Given ds2 = - dt2 + a2 t2 d2 with small a,
winding strings become tachyonic when the
velocity is small. Approximate this by a static
cylinder with radius slightly smaller than the
string scale. Then the tachyon behaves
like
for small .
McGreevy and Silverstein study string
amplitudes in this background using
techniques from Liouville theory (not RG).
There is a natural initial state defined by
analytic continuation (analog of the HartleHawking state). The amplitudes have
support in region T < O(1) 
Spacetime effectively begins when the
tachyon becomes O(1).
They calculate the number of particles
produced by time dependent tachyon. Find
a thermal distribution of particles with
temperature ~ . For small , the total
energy of produced particles is finite and
backreaction is under control.
Matrix Big Bang
(Craps, Sethi, E. Verlinde, 2005.)
Consider a linear dilaton solution  = ku
where u is a null coordinate in flat spacetime.
This is singular in the Einstein frame or M
theory. There is a dual description in terms of
a 2D Yang-Mills theory on the Milne universe.
In this case the Milne universe is a fixed
background so there is no backreaction.
It is not yet clear if you can evolve through the
singularity.
Generic singularities
In GR, generic approach to a spacelike
singularity exhibits BKL behavior: different
spatial points decouple, and the space
undergoes an infinite series of epochs of
anisotropic expansion.
This does not hold for all matter fields and
all spacetime dimension, but it has been
shown to hold in all supergravity theories.
There is evidence that M-theory may be
equivalent (dual?) to a massless particle
moving on the (infinite dimensional)
homogeneous space G/H where
G is the hyperbolic Kac-Moody group E10
and
H=K(E10) is (formally) its maximal compact
subgroup
(Damour, Henneaux, Nicolai)
Expanding the metric and 4form about a worldline. Get
Expanding the (unique) action for a massless
particle on G/H, one finds agreement with
11D supergravity up to level three.
Can naturally include fermions, and R4 terms.
(see Damour’s talk)
Applying AdS/CFT to cosmological
singularities (Hertog and G.H. 2005)
With a slight modification of the usual boundary
conditions, there exists asymptotically AdS initial
data which evolves to a big crunch. One can use
the CFT to study what happens near the
singularity in the full quantum theory.
(Big crunch: a spacelike singularity which
reaches infinity in finite time.)
Big crunch
Time symmetric
initial data
Asymptotic AdS
Big bang
This looks like Schwarzschild AdS, but here
conformal infinity is only a finite cylinder.
CFT is like a 3D field theory with potential
V

=0 is a perturbatively stable vacuum. But
nonperturbatively, it decays.
In a semiclassical approximation,  tunnels
through the barrier and rolls down to infinity in
finite time.
It appears that field theory evolution ends in
finite time, and hence there is no bounce
through the big crunch.
What about the full quantum theory?
If we restrict to homogeneous field configurations,
the field theory reduces to ordinary quantum
mechanics with a potential
There is a one parameter family of Hamiltonians
(Farhi et al). Picking one, evolution continues for
all time. <x> can diverge and come back.
This looks more like a bounce.
This almost never happens in field theory.
The CFT has infinitely many degrees of freedom
which become excited when the field falls down
a steep potential (tachyonic preheating). Like
particles decaying into lower and lower “mass”
particles.
In the QM problem, the different self-adjoint
extensions correspond to different ways to cut
off the potential. If the same is possible in the
CFT, one would expect to form a thermal state.
<> would not bounce.
This leads to a natural asymmetry between the
big bang and big crunch (cf: Penrose)
V

The universe starts in an approximately thermal state with all
the Planck scale degrees of freedom excited. Very rarely there
is a fluctuation in which most of the energy gets put into the
zero mode which goes up the potential. This is the big bang.
This would help explain the origin of the second law of
thermodynamics!
Black hole singularities
AdS/CFT approach
Large black holes are described by a thermal
state in the CFT. Time in CFT is external
Schwarzschild time. To explore the singularity
we have to see inside the horizon.
One can do this by using both asymptotic
regions of the (eternal) black hole.
(Shenker et al., 2003; Festuccia and Liu, 2005)
Schwarzschild AdS
O1(p)
O2(q)
Let O be
dual to a
bulk scalar
field with
mass m.
For large m, G = < O1 O2> is dominated by
spacelike geodesics. Information about the
singularity is encoded in properties of (the
analytically continued) G.
Argument that behavior of G will change
Fluctuations about an AdS black hole have a
continuous spectrum due to the horizon.
SU(N) gauge theory on S3 has a continuous
spectrum only at large N. At any finite N it will
be discrete.
Can pure states form black holes?
The extremal D1-D5 solution has a null
singularity. But there are lots of SUSY
smooth solutions with the same charges one for each microstate.
Is this also true for BPS black holes with
event horizons of nonzero area?
Is this true for non-BPS black holes
(including Schwarzschild)?
Some people think so… (see Mathur’s talk)
A New Endpoint for Hawking Evaporation
(G.H., 2005)
All RR charged black branes wrapped
around a circle have the property that the
circle at the horizon is smaller than at
infinity. As the black hole evaporates, this
circle can reach the string scale when the
curvature is still small. Winding string
tachyons cause this to pinch off forming a
Kaluza-Klein “bubble of nothing”
Review of Kaluza-Klein Bubbles
Witten (1981) showed that a gravitational
instanton mediates a decay of M4 x S1 into a
zero mass bubble where the S1 pinches off at
a finite radius. There is no spacetime inside
this radius. This bubble of nothing rapidly
expands and hits null infinity.
hole in space
S1
R3
Q is unchanged, so we form charged bubbles. But
there is no longer a source for this charge. Q is a
result of flux on a noncontractible sphere. This is a
nonextremal analog of a geometric transition:
branes
flux
These charged bubbles can be static or
expanding, depending on Q and the size of the
circle at infinity.
Consider the 6D black string with D1-D5 charges
Static bubbles with the same charges can be
obtained by analytic continuation: t=iy, x=i
The 3-form (and dilaton) are unchanged.
There is a similar transition even with
supersymmetric boundary conditions at infinity!
(Ross, 2005)
If you start with a rotating charged black string,
then even if fermions are periodic around the S1
at infinity, they can be antiperiodic around
another circle that winds the sphere as well as
the S1.
For certain choices of angular momentum, this
circle can reach the string scale when the
curvature is small, and one has a transition to a
bubble of nothing.
Comments
• We said that closed string tachyon condensation
should remove spacetime and lead to a state of
“nothing”. We have a very clear example of this.
• Kaluza-Klein bubbles of nothing were previously
thought to require a slow nonperturbative
quantum gravitational process. We now have a
much faster way to produce them. Some black
holes catalyze production of bubbles of nothing.
What happens inside the horizon?
(Silverstein and G.H., 2006)
Initially, when the circle is large everywhere
outside the horizon, it still shrinks to zero inside.
The tachyon instability can set in and replace the
black hole singularity. This can happen before
other complications such as large curvature or
large velocities.
But there are still complications coming from
Hawking radiation.
Simpler Model of Black Hole Evaporation
Consider a shell of D3-branes arranged in an S5.
The geometry is AdS5 x S5 outside the shell and
flat inside.
Compactify one direction along the brane and let
the shell slowly contract. When the S1 at the shell
reaches the string scale, tachyon condensation
takes place everywhere inside the shell and the
region outside becomes an AdS bubble.
AdS bubble
x S5
<T>
r=
AdS5 x S5
r=0
flat
shell
How do particles inside the shell get out?
There are indications that all excitations
inside the shell are forced out (either
classically or quantum mechanically).
This supports the idea of a “black hole final
state” (Maldacena and G.H.) but without
imposing final boundary conditions.
(see Silverstein’s talk)
Summary
• Tachyon condensation is a new tool for
analyzing topology change and certain
spacelike singularities.
• We still do not have definitive answers to
basic questions such as whether one can
“pass through” generic spacelike singularities.
(I think not.)
• Progress is being made, but there is still
much work to do…