spacetime singularities gary horowitz ucsb gary horowitz ucsb recent progress in understanding

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Spacetime Singularities Gary Horowitz UCSB Recent progress in understanding

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Page 1: Spacetime Singularities Gary Horowitz UCSB Gary Horowitz UCSB Recent progress in understanding

Spacetime SingularitiesSpacetime Singularities

Gary Horowitz

UCSB

Gary Horowitz

UCSB

Recent progress in understanding

Page 2: Spacetime Singularities Gary Horowitz UCSB Gary Horowitz UCSB Recent progress in understanding

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.

Page 3: Spacetime Singularities Gary Horowitz UCSB Gary Horowitz UCSB Recent progress in understanding

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)

Page 4: Spacetime Singularities Gary Horowitz UCSB Gary Horowitz UCSB Recent progress in understanding

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.

Page 5: Spacetime Singularities Gary Horowitz UCSB Gary Horowitz UCSB Recent progress in understanding

Singularities arising from nonsingular initial conditions

• Topology change

• Cosmology

• Black holes

Page 6: Spacetime Singularities Gary Horowitz UCSB Gary Horowitz UCSB Recent progress in understanding

Topology change in Calabi-Yau spaces

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)

Consider M4 x K. Moving within the space of Ricci flat Kahler metrics on K, one can cause spheres to shrink to zero size:

Page 7: Spacetime Singularities Gary Horowitz UCSB Gary Horowitz UCSB Recent progress in understanding

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.

Page 8: Spacetime Singularities Gary Horowitz UCSB Gary Horowitz UCSB Recent progress in understanding

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).

Page 9: Spacetime Singularities Gary Horowitz UCSB Gary Horowitz UCSB Recent progress in understanding

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).

Page 10: Spacetime Singularities Gary Horowitz UCSB Gary Horowitz UCSB Recent progress in understanding

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.

Page 11: Spacetime Singularities Gary Horowitz UCSB Gary Horowitz UCSB Recent progress in understanding

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.

A more general argument:

Page 12: Spacetime Singularities Gary Horowitz UCSB Gary Horowitz UCSB Recent progress in understanding

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.

Page 13: Spacetime Singularities Gary Horowitz UCSB Gary Horowitz UCSB Recent progress in understanding

Cosmological singularities

Page 14: Spacetime Singularities Gary Horowitz UCSB Gary Horowitz UCSB Recent progress in understanding

A simple model of a cosmological singularity is the Milne orbifold: 2D Minkowski spacetime/boost

ds2 = - dt2 + a2 t2 d2

This is clearly an exact solution to string theory since it is flat. It arises in several different contexts.

Page 15: Spacetime Singularities Gary Horowitz UCSB Gary Horowitz UCSB Recent progress in understanding

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.

Page 16: Spacetime Singularities Gary Horowitz UCSB Gary Horowitz UCSB Recent progress in understanding

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.

But backreaction is important

Page 17: Spacetime Singularities Gary Horowitz UCSB Gary Horowitz UCSB Recent progress in understanding

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>

Page 18: Spacetime Singularities Gary Horowitz UCSB Gary Horowitz UCSB Recent progress in understanding

Given ds2 = - dt2 + a2 t2 d2 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).

Page 19: Spacetime Singularities Gary Horowitz UCSB Gary Horowitz UCSB Recent progress in understanding

There is a natural initial state defined by analytic continuation (analog of the Hartle-Hawking 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.

Page 20: Spacetime Singularities Gary Horowitz UCSB Gary Horowitz UCSB Recent progress in understanding

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.

Matrix Big Bang (Craps, Sethi, E. Verlinde, 2005.)

Page 21: Spacetime Singularities Gary Horowitz UCSB Gary Horowitz UCSB Recent progress in understanding

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.

Page 22: Spacetime Singularities Gary Horowitz UCSB Gary Horowitz UCSB Recent progress in understanding

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)

Page 23: Spacetime Singularities Gary Horowitz UCSB Gary Horowitz UCSB Recent progress in understanding

Expanding the metric and 4-form 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)

Page 24: Spacetime Singularities Gary Horowitz UCSB Gary Horowitz UCSB Recent progress in understanding

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.)

Applying AdS/CFT to cosmological singularities (Hertog and G.H. 2005)

Page 25: Spacetime Singularities Gary Horowitz UCSB Gary Horowitz UCSB Recent progress in understanding

Big crunch

Big bang

Time symmetric initial data

Asymptotic AdS

This looks like Schwarzschild AdS, but here conformal infinity is only a finite cylinder.

Page 26: Spacetime Singularities Gary Horowitz UCSB Gary Horowitz UCSB Recent progress in understanding

CFT is like a 3D field theory with potential

=0 is a perturbatively stable vacuum. But nonperturbatively, it decays.

V

Page 27: Spacetime Singularities Gary Horowitz UCSB Gary Horowitz UCSB Recent progress in understanding

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.

Page 28: Spacetime Singularities Gary Horowitz UCSB Gary Horowitz UCSB Recent progress in understanding

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.

Page 29: Spacetime Singularities Gary Horowitz UCSB Gary Horowitz UCSB Recent progress in understanding

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.

Page 30: Spacetime Singularities Gary Horowitz UCSB Gary Horowitz UCSB Recent progress in understanding

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!

This leads to a natural asymmetry between the big bang and big crunch (cf: Penrose)

V

Page 31: Spacetime Singularities Gary Horowitz UCSB Gary Horowitz UCSB Recent progress in understanding

Black hole singularities

Page 32: Spacetime Singularities Gary Horowitz UCSB Gary Horowitz UCSB Recent progress in understanding

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)

Page 33: Spacetime Singularities Gary Horowitz UCSB Gary Horowitz UCSB Recent progress in understanding

For large m, G = < O1 O2> is dominated by spacelike geodesics. Information about the singularity is encoded in properties of (the analytically continued) G.

O1(p) O2(q)

Let O be dual to a bulk scalar field with mass m.

Schwarzschild AdS

Page 34: Spacetime Singularities Gary Horowitz UCSB Gary Horowitz UCSB Recent progress in understanding

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.

Page 35: Spacetime Singularities Gary Horowitz UCSB Gary Horowitz UCSB Recent progress in understanding

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)

Can pure states form black holes?

Page 36: Spacetime Singularities Gary Horowitz UCSB Gary Horowitz UCSB Recent progress in understanding

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

A New Endpoint for Hawking Evaporation (G.H., 2005)

Kaluza-Klein “bubble of nothing”

Page 37: Spacetime Singularities Gary Horowitz UCSB Gary Horowitz UCSB Recent progress in understanding

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.

R3

S1

hole in space

Page 38: Spacetime Singularities Gary Horowitz UCSB Gary Horowitz UCSB Recent progress in understanding

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.

Page 39: Spacetime Singularities Gary Horowitz UCSB Gary Horowitz UCSB Recent progress in understanding

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.

Page 40: Spacetime Singularities Gary Horowitz UCSB Gary Horowitz UCSB Recent progress in understanding

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.

Page 41: Spacetime Singularities Gary Horowitz UCSB Gary Horowitz UCSB Recent progress in understanding

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.

Page 42: Spacetime Singularities Gary Horowitz UCSB Gary Horowitz UCSB Recent progress in understanding

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.

What happens inside the horizon? (Silverstein and G.H., 2006)

Page 43: Spacetime Singularities Gary Horowitz UCSB Gary Horowitz UCSB Recent progress in understanding

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.

Page 44: Spacetime Singularities Gary Horowitz UCSB Gary Horowitz UCSB Recent progress in understanding

flat

AdS5 x S5

shell

AdS bubble x S5

<T>

r =

r = 0

How do particles inside the shell get out?

Page 45: Spacetime Singularities Gary Horowitz UCSB Gary Horowitz UCSB Recent progress in understanding

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)

Page 46: Spacetime Singularities Gary Horowitz UCSB Gary Horowitz UCSB Recent progress in understanding

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…