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Maldacena J. Black Holes and Holography in String Theory. 2004

Black Holes and Holography in String Theory

Juan Maldacena
Institute for Advanced Study
Paris 2004

Newton’s Gravity
“Everything that goes up comes down again” Only if it goes slower than ve = 11Km/s ! Why? Gravitational force decreases with distance. If the velocity is large enough ? we can escape, e.g. spacecrafts sent to other planets, etc.

v less than ve

v greater than ve

Black Holes in Newton’s Gravity

Laplace ~ 1800

Consider an object so massive and so small that its escape velocity is larger than the speed of light.
?Black hole: no light can come out of it.

The black hole size depends on the mass: RBH, sun = 3 Km RBH, earth = 1 cm RBH, us = smaller than any distance we can measure today.

Special Relativity
? The speed of light is the maximum speed at which information travels.

Einstein 1905

? Physics is the same no matter how fast an observer is moving. In particular, all observers measure the same speed of light. ? Time flows differently for observers that are moving relative to each other. ? Space and time are related. How we perceive time depends on how fast we move.

Equivalence Principle
Aristotle: Heavy objects fall faster. Galileo: All objects fall in the same way once we remove the effects of the air resistance.

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

General Relativity
? Newton’s theory does not obey special relativity. ? Einstein: space-time is curved.

Einstein 1915

? A heavy mass curves space-time and the motion of a light particle just follows the “shortest” line along that space-time. ?Gravity is due to space-time curvature.

Gravity changes the flow of time
If we have two observers in a curved space-time, time can flow differently for each of those observers. Top floor

First floor

Time flows slower
By one part in 1015 (one in one quatrillion).

Redshift Factor
An observer far from a heavy mass sees time going faster than an observer close to it.

Redshift factor =

(flow of time at some position) (flow of time far away )

Examples: Redshift factor at: Surface of the sun: 1 – 2 10-6 Surface of the earth: 1 – 8 10-10 Surface of a neutron star: 0.7

Karl Schwarzschild found the space-time outside a massive object. One finds precisely how time slows down:






Black hole radius

Black Holes
Can an object have a size smaller than the black hole radius? When this was first discovered it was thought that it was something unphysical, that maybe objects could never become that small. Later it was understood that:

Some stars can collapse into a black hole.

2) An observer who is falling into the black does not feel anything special as he crosses the horizon 3) There are some objects in the sky that are probably black holes.

The Horizon and Beyond
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 back out again. Once you cross the horizon, you continue to fall in and you are crushed into a “singularity.” This is a region of very high space-time curvature that rips you apart.


Real Black Holes
Black holes can form in astrophysical processes. Some stars are so massive that collapse into black holes. Black holes produced through these processes are of the following types: 1) Black holes that collapse from stars with masses of the order of a few times the mass of the sun 2) Black holes in the center of galaxies with masses of the order of a billion times the mass of the sun.
( rh ~ 3 109 km ~ size of the solar system ) ( rh ~ 10 km )

How Do We See Them?
In principle we could see them by seeing how they deflect light, etc. In practice these black holes are surrounded by some gas and this gas heats up in a characteristic way as it falls in and astronomers see the radiation coming from this hot gas.


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

Area law:
Total area of the horizon always increases. Total mass of black holes does not necessarily increase.

Gravity waves

White Black Holes!

Hawking 1973

The laws of quantum mechanics imply that black holes emit thermal radiation. Smaller black holes have a higher temperature.
What is this temperature for black holes of different masses? Tsun = 3.6 10-7 K Tearth = 0.1 K TM=10


= 7000 K (would look white )

Why? ? Pair creation
In flat space Anti-particle particle

In the presence of a horizon Anti-particle

particle horizon

The Life and Death of a Black Hole
? Black hole emits radiation ? looses energy ? has a finite lifetime (if no additional matter is falling in) Lifetime for various black holes: ? Mass of the sun ? much longer than the age of the universe ? Our mass ( 100 Kg ) ? a millisecond. ? Mass of 1012 Kg (mass of a mountain) ? age of the universe. Could be observed if it was formed at the big bang.

Theoretical Puzzles with Black Holes
Thermal properties of black holes mix quantum mechanics and gravity. It is hard to mix these two theories. Puzzles: ? Entropy of the black holes. ? Information loss.

Heat and Entropy
Heat ? motion of the microscopic components of the system. We can measure the number of microscopic degrees of freedom by the “entropy” of the system
First law of thermodynamics ? relates entropy and the specific heat. the more energy needed to raise the temperature ?the more entropy

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

Area of the horizon S= 4 GN h

Area of the horizon S= (10 ?33 cm) 2

Black holes and the Structure of Space-time
Black holes are independent of what forms them. Their thermal properties only depend on gravity and quantum mechanics. This should be explained by a theory of quantum gravity. Roughly speaking, the black hole entropy should come from the motion of the “space-time quanta,” from the elementary quanta (or atoms…) that make space and time. Understanding precisely these thermal aspects of black holes, we learn something about the quantum structure of space-time.


Information Loss
We can form a black hole in many different ways, but it always evaporates in the same way.

The principles of quantum mechanics imply that there should be a precise description of black holes. There should be subtle differences in what comes out of a black hole, depending on how we made it.

Black holes in string theory
? We understand some simple black holes
Strominger, Vafa

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

Holography and Black Holes
Entropy = Area = / volume Usual optical hologram: 2d surface encodes the information about 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



Negatively curved space-time

Particle trajectory

Light trajectory

Particles living in the interior ? attracted to the center

Particles living on the Boundary describe an object 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.

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

? Black holes are fascinating objects where the effects of space-time curvature are dramatic. ? Black holes combined with quantum mechanics provide very interesting challenges to our understanding of space-time. ? String theory is capable of putting together the classical and quantum aspects of black holes. There is no information loss. We are getting very novel and interesting ways of describing space time.



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