Remarkable power of Einstein’s equation

Remarkable power of Einstein’s equation

Gary Horowitz

UC Santa Barbara

Gary Horowitz

UC Santa Barbara

• Renormalization group flow in QFT• Aspects of QCD• Superconductivity

In addition to describing gravitational phenomena (black holes, gravitational waves, etc.) Einstein’s equation can reproduce other fields of physics:

Surprising Claim:

Gauge/gravity duality

Under certain boundary conditions, string theory (which includes gravity) is completely equivalent to a (nongravitational) gauge theory living on the boundary at infinity.

When string theory is weakly coupled, gauge theory is strongly coupled, and vice versa.

(Maldacena)

Early Evidence

• (Super) symmetries agree

• Gauge theory analogs of all massless string modes were found, including all (super) gravity degrees of freedom

• Gauge theory analogs of some massive string modes were found

• Some interactions were shown to agree

More evidence

Long strings: The gauge theory contains matrix-valued scalar fields X, Y. The long operators Tr(XXXX…YYYY…XXXXX) have properties which exactly reproduce the behavior of long strings in the gravity background.

Much more…

Traditional applications of gauge/gravity duality

Gain new insight into strongly coupled gauge theories, e.g., geometric picture of confinement.

Gain new insight into quantum gravity, e.g., quantum properties of black holes

Black hole evaporation

Hawking showed that when matter is treated quantum mechanically (and gravity classically), black holes emit approximately thermal radiation with temperature T 1/M.

If no matter falls in, a black hole will evaporate.

This led to several puzzles which can be resolved using gauge/gravity duality.

(Note: Planck length Lp = 10-33 cm)

Black holes have enormous entropy S = A/4Lp2

(Bekenstein, Hawking). The gauge theory has enough microstates to reproduce this entropy.

Hawking argued that black hole evaporation would violate quantum mechanics, but this can now be described by Hamiltonian evolution in dual gauge theory.

Puzzles and resolution

In a certain limit, all stringy and quantum effects are suppressed and gravity theory is just general relativity

(in higher dimensions, with Λ < 0 so spacetimes are asymptotically

anti de Sitter).

Renormalization group (RG) flow in a QFT corresponds to obtaining an effective low energy action by integrating out high energy modes.This corresponds to radial dependence on the gravity side:Gauge theory: add mass terms and follow RG flow to low energies to obtain a new field theoryGravity theory: modify the boundary conditions for certain matter fields and solve Einstein’s equation

One finds detailed numerical agreement between the small r behavior of the gravity solution and the endpoint of the RG flow.

Hydrodynamics from gravity

One expects that the long distance dynamics of any strongly interacting field theory is described by (relativistic) hydrodynamics.

Gauge/gravity duality predicts that hydrodynamics can be recovered from Einstein’s equation. Start with:

gauge theory at temp. T

black brane

Boost black brane to give it velocity uα. Consider metrics where the horizon radius and uα vary slowly compared to the temperature T.

Find that Einstein’s equation implies

where

with . Recover equations for fluid with viscosity.

(Bhattacharrya, Hubeny, Minwalla, Rangamani)

But the viscosity is very low. The ratio of the (shear) viscosity η to entropy density s is

This is much smaller than most materials. Conjectured to be a universal lower bound. (Kuvton, Policastro, Son, Starinets)

Something close to this is seen at RHIC!

Relativistic Heavy Ion Collider

at Brookhaven

Gold nuclei collide at

200GeV/nucleon

Gauge/gravity duality and RHIC

The quark/gluon plasma produced at RHIC is strongly coupled and thermalizes quickly. Surprisingly, it is well described by fluid dynamics with a very low viscosity - close to value calculated from gravity.

Gauge/gravity duality currently offers the best explanation of this fact.

New application of gauge/gravity duality

Condensed matter physics: the Hall effect and Nerst effect have been shown to have a dual gravitational description.

(Hartnoll, Herzog, Kovton, Sachdev, Son)

Gravity dual of a superconductor

Gravity Superconductor

Black hole Temperature

Charged scalar field Condensate

Need to find a black hole that has scalar hair at low temperatures, but no hair at high temperatures.

This is not an easy task.

(Hartnoll, Herzog, and G.H.)

Gubser argued that a charged scalar field around a charged black hole would have the desired property. Consider

For an electrically charged black hole, the effective mass of is

But the last term is negative. There is a chance for nontrivial hair.

This doesn’t work in asymptotically flat spacetimes, but it does in asymptotically AdS.

Intuitive picture: If qQ is large enough, even extremal black holes create pairs of charged particles. In AdS, the charged particles can’t escape, and settle outside the horizon.

Qi Qf<QiQf<Qi

Scalar field with charge Qi - Qf

We have two cases depending on the boundary conditions for . Let Oi i = 1,2 denote the charged condensates. They have dimension i.

Tc = .226 1/2 Tc = .118 1/2

Condensate (hair) as a function of T

O1 O2

Curves represent successively lower temperatures. Gap opens up for T < Tc. Last curve has T/Tc=.007 (O1) and .003 (O2)

Frequency dependent conductivity

There is a delta function at = 0 for all T < Tc.

This can be seen by looking for a pole in Im[].

Simple derivation:

If E(t) = Ee-it, where k=ne2/m

So

For superconductors, ,

The energy gap at small T, normalized by the condensate.

The width of the gap is equal to the condensate.

The conductivity is not strictly zero in the gap. The normal (nonsuperconducting) component to the DC conductivity is

We find

where

is the energy gap for charged excitations.

The gap in the frequency dependent conductivity is 2. This suggests that the condensate consists of pairs of quasiparticles.

At T=0 we find

2 = 8.4Tc

BCS superconductors always have 2 = 3.54Tc . Our larger value is what one expects for more deeply bound pairs.

These results were all obtained by solving Einstein’s equation with matter around a black hole.

Gravitational theorists can make contributions to other areas of physics.

Someday we might be testing general relativity by doing high energy or condensed matter experiments!