Traversable Wormholes

Juan Maldacena
Institute for Advanced StudyJune, 2018

Based on work in progress with:Alexey Milekhin

Fedor Popov

Related to previous work with Xiaoliang Qi

Inspired by work by Gao, Jafferis and Wall on “Traversable wormholes”

Drawing by John Wheeler, 1966

Charge without charge.
Mass without mass

Spatial geometry. Traversable wormhole

Recall classic results

There are no science fiction wormholes!
• No wormhole allows you to travel faster than
the speed of light in the ambient space.
Friedman Schleich, Witt, Galloway, Woolgar
Gao Wald• Forbidden by:• I) Einstein equations.
• II) The Achronal Average Null Energy
Condition
Not yet proven in a generalspacetime, but believed tohold in QFT

Zdx T0Achronal = points along null line are not timelike separated = fastest line

Longer wormholes?
• What if it takes longer to go through thewormhole ?
• Not possible in classical physics due to the Null
Topological censorship: Friedman Schleich,
Energy Condition.
Witt, Galloway, Woolgar• In classical physics à we can go to the covering
space, which is forbidden by the previous case.
• à We need quantum effects to find a solution.
Casimir-like energy.

•Can we do it in a controllable way ?

Negative null energy in QFT
Eg. Two spacetime dimensions

T<0E/cL

Negative Casimir energy

time

Quantum effect

CircleThe null energy condition does not hold for null lines that are not achronal!

Some necessary elements• We need something looking like a circle tohave negative Casimir energy.
• Large number of bulk fields to enhance thesize of quantum effects.
• We will show how to assemble these elements
in a few steps.

The theoryS=Z4⇥d x RF +i¯ 6 D2⇤

Einstein + U(1) gauge field + massless charged fermion

Could be the Standard Model at very small distances, with the fermionseffectively massless. The U(1) is the hypercharge. SU(3) x SU(2) x U(1).

lPlanck = 1

The first solution: Extremal black holeZ

Magnetic charge q

re ⇠ q

F = q = integer
S2

M =q

AdS2 ⇥ S 2
lPlanck = 1

The next solution: Near Extremal black holeMre ⇠ q

AdS2 ⇥ S32q=q T =q3
2Very small

β is the ‘’length” of the throat. Redshift factor
between the top and the bottom

2horizonlPlanck = 1

Motion of charged fermions• Magnetic field on the sphere.
• There is a Landau level with precisely zero energy.
• Orbital and magnetic dipole energies precisely
cancel.
Ambjorn, Olesen• Explained by an anomaly argument

B

Massless fermions à U(1) chiral symmetry
4d anomaly à 2d anomaly à there should
be massless fermions in 2d.
(Here we view F as non-dynamical).

Motion of charged fermions• Degeneracy = q = flux of the magnetic field onthe sphere. Form a spin j, representation ofSU(2), 2j +1 =q.
• We effectively get q massless two dimensional
fermions along the time and radial direction.
• We can think of each of them as following amagnetic field line.

q massless two dimensional fields, along field lines.

AdS22ds =dt2 + d 2
sin2
Global2ds =(r221)d⌧ +Thermal/Rindler

dr2
(r21)

Nearly AdS2ds2 =dt2 + d 2
sin2

ds2 =

Global(r21)d⌧ 2 +dr2
(r2Thermal/Rindler

Connect them to flat space, so that t is an isometry.
The acquire non-zero energy when the throat has finite length

Mq = q3 T 2 =q321)

Connect a pair black holes

connect and in global AdS2

Connect a pair black holes

Positive magnetic
chargeNot a solution yet.
Not a black hole.

Negative magnetic
chargeNearly AdS2 x S2 wormhole of finite lengthFermion trajectories

Positive magnetic
chargeNegative magnetic
chargeCharged fermion moves along this closed circle.

Casimir energy
Assume: “Length of the throat” is larger than the distance.
LLout > dLoutCasimir energy is of the order ofE/qLFull energy also need to take into
Account the conformal anomaly because
AdS2 has a warp factor.
That just changes the numerical factor.

dLFinding the solution
Solve Einstein equations with the negative quantum stress tensor
Balance the classical curvature + gauge field energy vs the Casimir energy.

M

q3
q= 2
Lq,
L

@M
= 0 ! L ⇠ q2
@L

Now the throat is stabilized. Negative binding energy.

Ebinding = M

q=1=q1rsVery small. Only lowenergy waves can explore itThis is not yet a solution: The two objects attract and would fall on to each otherAdding rotation

drere ⇠ qd⌦=r

red3

Kepler rotation frequency

Throat is fragile• We must make sure not to start sending matter into thethroat that can accumulate there and produce a black hole.
• Rotation à radiation à effective temperature: T ⇠ ⌦• We need that Ω is smaller than the energy gap of the throat
1⌦⌧L• The configuration will only live for some time, until theblack holes get closer..
• These issues could be avoided by going to AdS4 …

Some necessary inequalities
L ⇠ q2

From stabilized throat solutiond⌧L !d⌧qrq1=⌦⌧3d
LBlack holes close enough to that Casmir energy
computation was correct.

253!q ⌧dKepler
rotation frequency

They are compatible

Black holes far enough so thatthey rotate slowly compared to theenergy gap.
Unruh-like temperature less than energy gap

53q ⌧d⌧q

2

Other effects we could think off are also small :can allow small eccentricity, add electromagnetic andgravitational radiation, etc. Has a finite lifetime.

Final solution

Looks like the exterior of two near extremal black holes. But they connected.
But there is no horizon!. Zero entropy solution.
It has a small binding energy.

Rotation à temperature
1T ⇠ ⌦ ⌧ Egap ⇠LTemperature does not create particles
in the throat

Two Black Holes : F =Wormhole : F =T q2
Ebinding =1qWormhole is the stable thermodynamic phase for T < 1/q3
For the solution we described so far: Wormhole is metastable.

Length L as d increases
dddL
11Ec / +4L

1L+d

Conformal
anomaly

Casimir cylinder

Stops beingclassicalq2q2q3d

Wormholes in the Standard ModelIf nature is described by the Standard Model at short distances and d issmaller than the electroweak scale,1 ⌧ q ⌧ 10

8Distance d smaller than electroweak scale.
If the standard model is not valid à similar ingredients might be present in the true theory.

That it can exist, does not mean that it is easily produced by some natural or artificialprocess.

They are connected through a wormhole!
Much smaller than the ones LIGO or the LHC can detect!
Pair of entangled black holes.

Entropy and entanglement• Total spacetime has no entropy and nohorizon.
• If we only look at one object à entanglement
entropy = extremal black hole entropy• Wormhole = two entangled black holes• Total Hamiltonian

H = HL + HR + Hint
Generated by fermions in exterior

Van der Waals interaction
Two neutral atoms exchanging photons.

dd~L .d~RHint /d3

d small enough so that 1/d is larger than the gap between the ground state andthe next states.

Entangle the two atoms.
1/dTwo black holes
Two black holes exchanging fermions.

dψHint /

Entangle the two black holes.

LRSee Xiaoliang Qi’s talkConclusions• We displayed a solution of an Einstein
Maxwell theory with charged fermions.
• It is a traversable wormhole in four
dimensions and with no exotic matter.
• It balances classical and quantum effects.
• It has a non-trivial spacetime topology, which
is forbidden in the classical theory.
• It does not violate causality.
• It has no horizon and no entropy.
• Can be viewed as a pair of entangled blackholes.

Questions• If we start from disconnected near extremal
black holes: Can they be connected quickly
enough ? à topology change.
• Could we turn it into a prediction fromquantum gravity ?