Spacetime Singularities 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 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: Orbifolds Branes 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: An S2 can go to zero and re-expand as a topologically different S2. The mirror description is nonsingular (flop). (Aspinwall, Greene, Morrison, 1993) 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 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 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 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.

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.

G is the hyperbolic Kac-Moody group E10 and 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 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)

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)

Let O be dual to a bulk scalar field with mass m. Schwarzschild AdS Let O be dual to a bulk scalar field with mass m. O1(p) O2(q) 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)

Kaluza-Klein “bubble of nothing” 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. S1 hole in space R3

Q is unchanged, so we form charged bubbles 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.

How do particles inside the shell get out? 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…