Black Holes andBlack Holes andHolography inHolography in String Theory String Theory

Juan MaldacenaJuan MaldacenaInstitute for Advanced StudyInstitute for Advanced Study

Paris 2004

Why?Why?Gravitational force decreases with distance. If the velocity is largeGravitational force decreases with distance. If the velocity is largeenough enough we can escape, we can escape, e.g. spacecrafts sent to other planets, etc.e.g. spacecrafts sent to other planets, etc.

““Everything that goes up comes down againEverything that goes up comes down again””

Only if it goes slower thanOnly if it goes slower than vvee ==

v less than vv less than vee v greater than vv greater than vee

Newton Newton’’s Gravitys Gravity

GmMGmM rr2 2

11Km/s !11Km/s !

Consider an object so massive and so small that its escape velocity is Consider an object so massive and so small that its escape velocity islarger than the speed of light.larger than the speed of light.

Black hole: no light can come out of it.Black hole: no light can come out of it.

The black hole size depends on the mass:The black hole size depends on the mass:

RRBH, sun BH, sun = 3 Km= 3 Km

RRBH, earthBH, earth = 1 cm = 1 cm

RRBH, usBH, us = smaller than any distance we can measure today. = smaller than any distance we can measure today.

Laplace ~ 1800Black Holes in Newton’s Gravity

Special RelativitySpecial Relativity

The speed of light is the maximum speed at which The speed of light is the maximum speed at whichinformation travels.information travels.

Physics is the same no matter how fast an observer isPhysics is the same no matter how fast an observer ismoving. In particular, all observers measure the same speed ofmoving. In particular, all observers measure the same speed oflight.light.

Space and time are related. How we perceive time dependsSpace and time are related. How we perceive time dependson how fast we move.on how fast we move.

Time flows differently for observers that are moving Time flows differently for observers that are movingrelative to each other.relative to each other.

Einstein 1905

Equivalence PrincipleEquivalence Principle

The motion of a particle in a gravitational field is independent ofThe motion of a particle in a gravitational field is independent ofthe mass of the particle.the mass of the particle.

Aristotle: Heavy objects fall faster.Aristotle: Heavy objects fall faster.

Galileo: All objects fall in the same way once we removeGalileo: All objects fall in the same way once we remove the effects of the air resistance. the effects of the air resistance.

General RelativityGeneral Relativity

NewtonNewton’’s theory does not obey special relativity.s theory does not obey special relativity.

Einstein: space-time is curved. Einstein: space-time is curved.

A heavy mass curves space-time and the motion of a light particle A heavy mass curves space-time and the motion of a light particlejust follows the just follows the ““shortestshortest”” line along that space-time. line along that space-time.

Gravity is due to space-time curvature.Gravity is due to space-time curvature.

Einstein 1915

Gravity changes the flow of timeGravity changes the flow of time

If we have two observers in a curved space-time, time canIf we have two observers in a curved space-time, time canflow differently for each of those observers.flow differently for each of those observers.

Top floor

First floor Time flows slower

By one part in 1015 (one in one quatrillion).

Examples: Redshift factor at:Examples: Redshift factor at:

Surface of the sun: 1 Surface of the sun: 1 –– 2 10 2 10-6-6

Surface of the earth: 1 Surface of the earth: 1 –– 8 10 8 10-10-10

Surface of a neutron star: 0.7Surface of a neutron star: 0.7

(flow of time at some position)(flow of time at some position)Redshift factor =Redshift factor =(flow of time far away )(flow of time far away )

An observer far from a heavy mass sees time going faster than anAn observer far from a heavy mass sees time going faster than anobserver close to it.observer close to it.

Redshift FactorRedshift Factor

rr

Karl Schwarzschild found the space-time outside a massive object.Karl Schwarzschild found the space-time outside a massive object.

One finds precisely how time slows down:One finds precisely how time slows down:

RedshiftRedshift11

r

rh−1 22c

MGrh =

Black hole radius Black hole radius

? STAR EXTERIOR

Black HolesBlack HolesCan an object have a size smaller than the black hole radius?Can an object have a size smaller than the black hole radius?

When this was first discovered it was thought that it wasWhen this was first discovered it was thought that it wassomething unphysical, that maybe objects could never becomesomething unphysical, that maybe objects could never becomethat small.that small.

Later it was understood that:Later it was understood that:

1)1) Some stars can collapse into a black hole.Some stars can collapse into a black hole.

2)2) An observer who is falling into the black does not feel anything special An observer who is falling into the black does not feel anything specialas he crosses the horizonas he crosses the horizon

3)3) There are some objects in the sky that are probably black holes. There are some objects in the sky that are probably black holes.

The Horizon and BeyondThe Horizon and Beyond

The surface where time slows to a halt is called a horizon.The surface where time slows to a halt is called a horizon. If you cross it you do not feel anything special, but you cannot come If you cross it you do not feel anything special, but you cannot comeback out again.back out again.

Once you cross the horizon, you continue to fall in and you are crushedOnce you cross the horizon, you continue to fall in and you are crushedinto a into a ““singularity.singularity.”” This is a region of very high space-time curvature This is a region of very high space-time curvaturethat rips you apart.that rips you apart.

singularitysingularity

horizonhorizon

Real Black HolesReal Black Holes

Black holes can form in astrophysical processes. Some stars are soBlack holes can form in astrophysical processes. Some stars are somassive that collapse into black holes.massive that collapse into black holes.Black holes produced through these processes are of the followingBlack holes produced through these processes are of the followingtypes:types:

1)1) Black holes that collapse from stars with masses of the order Black holes that collapse from stars with masses of the orderof a few times the mass of the sunof a few times the mass of the sun

(( r rhh ~ 10 km )~ 10 km )

2)2) Black holes in the center of galaxies with masses of the order of a Black holes in the center of galaxies with masses of the order of abillion times the mass of the sun.billion times the mass of the sun.

( ( rrh h ~ 3 10~ 3 1099 km ~ size of the solar system ) km ~ size of the solar system )

How Do We See Them?How Do We See Them?

In principle we could see them by seeing how they deflectIn principle we could see them by seeing how they deflectlight, etc.light, etc.

In practice these black holes are surrounded by some gasIn practice these black holes are surrounded by some gasand this gas heats up in a characteristic way as it falls inand this gas heats up in a characteristic way as it falls inand astronomers see the radiation coming from this hot gas.and astronomers see the radiation coming from this hot gas.

http://imagine.gsfc.nasa.gov/Images/advanced/bh_diagram.gif

The final shape of a black hole as seen from the outsideThe final shape of a black hole as seen from the outsideis independent of how we make it (up to the total mass,is independent of how we make it (up to the total mass,charge and angular momentum).charge and angular momentum).

Total area of the horizon always increases. Total mass of black holes Total area of the horizon always increases. Total mass of black holesdoes not necessarily increase.does not necessarily increase.

Gravity waves

Universality

Area law:

Gravity waves

White Black Holes!White Black Holes!The laws of quantum mechanics imply that black holes emit thermalThe laws of quantum mechanics imply that black holes emit thermalradiation.radiation.

Smaller black holes have a higher temperature.Smaller black holes have a higher temperature.

What is this temperature for black holes of different masses?What is this temperature for black holes of different masses?

TTsunsun = = 3.6 10 3.6 10-7-7 K K

TTearthearth = 0.1 K= 0.1 K

T TM=10 M=10 Kg Kg = 7000 K (would look white ) = 7000 K (would look white )

hrTk

h=

1818

Hawking 1973

Why? Why? Pair creation Pair creation

particleparticleAnti-particleAnti-particle

In the presence of a horizonIn the presence of a horizon

particleparticleAnti-particleAnti-particle

In flat spaceIn flat space

horizon

The Life and Death of a Black HoleThe Life and Death of a Black Hole

Black hole emits radiation Black hole emits radiation looses energy looses energy has a finite lifetime (if no has a finite lifetime (if noadditional matter is falling in)additional matter is falling in)

Lifetime for various black holes:Lifetime for various black holes:

Mass of the sun Mass of the sun much longer than the age of the universe much longer than the age of the universe

Our mass ( 100 Kg ) Our mass ( 100 Kg ) a a millisecond. millisecond.

Mass of 10 Mass of 101212 Kg (mass of a mountain) Kg (mass of a mountain) age of the universe. age of the universe. Could be observed if it was formed at the big bang. Could be observed if it was formed at the big bang.

Theoretical PuzzlesTheoretical Puzzleswith Black Holeswith Black Holes

Thermal properties of black holes mix quantum mechanics andThermal properties of black holes mix quantum mechanics andgravity. It is hard to mix these two theories.gravity. It is hard to mix these two theories.

Puzzles:Puzzles:

Entropy of the black holes. Entropy of the black holes.

Information loss. Information loss.

Heat and EntropyHeat and EntropyHeat Heat motion of the microscopic components of the system. motion of the microscopic components of the system.

We can measure the number of microscopic degrees of freedom by theWe can measure the number of microscopic degrees of freedom by the““entropyentropy”” of the system of the system

First law of thermodynamics First law of thermodynamics relates entropy and the specific heat. relates entropy and the specific heat. the more energy needed to raise the temperature the more energy needed to raise the temperature the more entropythe more entropy

How does the heat of the black hole arise? What is How does the heat of the black hole arise? What is ““movingmoving”” on a black hole? on a black hole?Compute entropy from the first law:Compute entropy from the first law:

hNGhorizontheofArea

S4

=233 )10( cm

horizontheofAreaS

−=

Bekenstein, Hawking

Black holes and the Structure of Space-timeBlack holes and the Structure of Space-time

Black holes are independent of what forms them. Their Black holes are independent of what forms them. Their thermal properties only depend on gravity and quantum thermal properties only depend on gravity and quantum mechanics. This should be explained by a theory of quantum mechanics. This should be explained by a theory of quantum gravity. gravity.

Roughly speaking, the black hole entropy should come from Roughly speaking, the black hole entropy should come from the motion of the the motion of the ““space-time quanta,space-time quanta,”” from the elementary from the elementary quanta (or atoms quanta (or atoms……) that make space and time. ) that make space and time.

Understanding precisely these thermal aspects of black holes, Understanding precisely these thermal aspects of black holes, we learn something about the quantum structure of space-time. we learn something about the quantum structure of space-time.

Information LossInformation Loss

We can form a black hole in many different ways, but it alwaysWe can form a black hole in many different ways, but it alwaysevaporates in the same way.evaporates in the same way.

The principles of quantum mechanics imply that there should beThe principles of quantum mechanics imply that there should bea precise description of black holes.a precise description of black holes.

There should be subtle differences in what comes out of a black hole, There should be subtle differences in what comes out of a black hole,depending on how we made it.depending on how we made it.

Hawking

Black holes in string theoryBlack holes in string theory

• We understand some simple black holes

Collection of D-branesAnd strings.

Entropy: Number of ways to arrange the strings and D-branes.

Strominger, Vafa

Holography and Black HolesHolography and Black Holes

Entropy = Area = volumeEntropy = Area = volume

Usual optical hologram:Usual optical hologram: 2d surface encodes the information2d surface encodes the informationabout the three dimensional shape of an objectabout the three dimensional shape of an object

/

Holography in quantum gravity:

Number of degrees of freedom to describe a region grows like the area of the region

‘t Hooft, Susskind

Holography in String Theory

We can describe the interior of some space-times in terms of a theory at the boundary.

Theory at the boundary is a relatively simple theory of particles

Negatively curved space

BoundaryInterior

Negatively curved space-time

Light trajectory

Particle trajectory

Time

Particles living in the interior attracted to the center

Interior Boundary

Particles living on theBoundary describe an object in the interior

Black holes are described by a large number of particles on theboundary

Gravitational physics in the interior Described in an alternative way by interacting particles living on the boundary.

Interior

Particles on the boundary could describe very complex objects

Space-time and everything in it emerges dynamically out of the interaction of the particles living on the boundary.

Photo Credit & Copyright: Jason Ware

Conclusions Conclusions

Black holes are fascinating objects where the effects of space-timeBlack holes are fascinating objects where the effects of space-timecurvature are dramatic.curvature are dramatic.

Black holes combined with quantum mechanics provide very Black holes combined with quantum mechanics provide veryinteresting challenges to our understanding of space-time.interesting challenges to our understanding of space-time.

String theory is capable of putting together the classical and String theory is capable of putting together the classical andquantum aspects of black holes. There is no information loss.quantum aspects of black holes. There is no information loss.

We are getting very novel and interesting ways of describing space We are getting very novel and interesting ways of describing spacetime.time.