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Black Holes andBlack Holes andHolography inHolography in
String TheoryString Theory
Juan MaldacenaJuan Maldacena
Institute for Advanced StudyInstitute for Advanced Study
Paris 2004
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Why?Why?
Gravitational force decreases with distance. If the velocity is largeGravitational force decreases with distance. If the velocity is largeenoughenough 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
NewtonNewtons Gravitys Gravity
11Km/s !11Km/s !
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Consider an object so massive and so small that its escape velocity isConsider an object so massive and so small that its escape velocity is
larger 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, sunBH, 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 ~ 1800
Black Holes in Newtons Gravity
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Special RelativitySpecial Relativity
The speed of light is the maximum speed at whichThe speed of light is the maximum speed at which
information travels.information travels.
Physics is the same no matter how fast an observer isPhysics is the same no matter how fast an observer is
moving. In particular, all observers measure the same speed ofmoving. In particular, all observers measure the same speed of
light.light.
Space and time are related. How we perceive time dependsSpace and time are related. How we perceive time depends
on how fast we move.on how fast we move.
Time flows differently for observers that are movingTime flows differently for observers that are moving
relative to each other.relative to each other.
Einstein 1905
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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 of
the 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.
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General RelativityGeneral Relativity
NewtonNewtons 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 particleA heavy mass curves space-time and the motion of a light particle
just follows thejust 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
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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 can
flow 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).
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Examples: Redshift factor at:Examples: Redshift factor at:
Surface of the sun: 1Surface of the sun: 12 102 10-6-6
Surface of the earth: 1Surface of the earth: 18 108 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
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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:
RedshiftRedshift
11
Black hole radiusBlack hole radius
? STAR EXTERIOR
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Black HolesBlack Holes
Can 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 was
something unphysical, that maybe objects could never becomesomething unphysical, that maybe objects could never become
that 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 specialAn observer who is falling into the black does not feel anything special
as 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.
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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 comeIf you cross it you do not feel anything special, but you cannot come
back 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 crushed
into ainto a singularity.singularity. This is a region of very high space-time curvatureThis is a region of very high space-time curvaturethat rips you apart.that rips you apart.
singularitysingularity
horizonhorizon
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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 so
massive 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 following
types:types:
1)1) Black holes that collapse from stars with masses of the orderBlack holes that collapse from stars with masses of the order
of a few times the mass of the sunof a few times the mass of the sun(( rrhh ~ 10 km )~ 10 km )
2)2) Black holes in the center of galaxies with masses of the order of aBlack holes in the center of galaxies with masses of the order of a
billion times the mass of the sun.billion times the mass of the sun.(( rrhh ~ 3 10~ 3 10
99 km ~ size of the solar system )km ~ size of the solar system )
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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 gas
and this gas heats up in a characteristic way as it falls inand this gas heats up in a characteristic way as it falls in
and astronomers see the radiation coming from this hot gas.and astronomers see the radiation coming from this hot gas.
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The final shape of a black hole as seen from the outsideThe final shape of a black hole as seen from the outside
is 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 holesTotal 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
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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 thermal
radiation.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 == 3.6 103.6 10-7-7 KK
TTearthearth = 0.1 K= 0.1 K
TTM=10M=10 KgKg = 7000 K (would look white )= 7000 K (would look white )1818
Hawking 1973
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Why?Why? Pair creationPair creation
particleparticle
Anti-particleAnti-particle
In the presence of a horizonIn the presence of a horizon
particleparticleAnti-particleAnti-particle
In flat spaceIn flat space
horizon
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The Life and Death of a Black HoleThe Life and Death of a Black Hole
Black hole emits radiationBlack hole emits radiation looses energylooses energy has a finite lifetime (if nohas a finite lifetime (if no
additional matter is falling in)additional matter is falling in)
Lifetime for various black holes:Lifetime for various black holes:
Mass of the sunMass of the sun much longer than the age of the universemuch longer than the age of the universe
Our mass ( 100 Kg )Our mass ( 100 Kg ) aa millisecond.millisecond.
Mass of 10Mass 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.
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Heat and EntropyHeat and Entropy
HeatHeat
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 systemof the system
First law of thermodynamicsFirst law of thermodynamics relates entropy and the specific heat.relates entropy and the specific heat.the more energy needed to raise the temperaturethe more energy needed to raise the temperature the more entropythe more entropy
How does the heat of the black hole arise? What isHow 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:
hNG
horizontheofAreaS
4=
233 )10( cm
horizontheofAreaS
=
Bekenstein, Hawking
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Black holes and the Structure of Space-timeBlack holes and the Structure of Space-time
Black holes are independent of what forms them. TheirBlack holes are independent of what forms them. Their
thermal properties only depend on gravity and quantumthermal properties only depend on gravity and quantum
mechanics. This should be explained by a theory of quantummechanics. This should be explained by a theory of quantum
gravity.gravity.
Roughly speaking, the black hole entropy should come fromRoughly speaking, the black hole entropy should come from
the motion of thethe motion of the space-time quanta,space-time quanta, from the elementaryfrom the elementary
quanta (or atomsquanta (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.
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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 always
evaporates 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
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Black holes in string theoryBlack holes in string theory
We understand some simple black holes
Collection of D-branes
And strings.
Entropy: Number of ways to arrange the strings and D-branes.
Strominger, Vafa
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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 information
about 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
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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
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Negatively curved space
BoundaryInterior
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Negatively curved space-time
Light trajectory
Particle trajectory
Time
Particles living in the interior attracted to the center
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Interior Boundary
Particles living on the
Boundary describe anobject in the interior
Black holes are described by
a large number of particles on the
boundary
Gravitational physics in the interior Described in an alternative
way by interacting particles living on the boundary.
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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
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ConclusionsConclusions
Black holes are fascinating objects where the effects of space-timeBlack holes are fascinating objects where the effects of space-time
curvature are dramatic.curvature are dramatic.
Black holes combined with quantum mechanics provide veryBlack holes combined with quantum mechanics provide very
interesting challenges to our understanding of space-time.interesting challenges to our understanding of space-time.
String theory is capable of putting together the classical andString theory is capable of putting together the classical and
quantum 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 spaceWe are getting very novel and interesting ways of describing spacetime.time.
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