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Strings and Black Holes
David Lowe
Brown University
AAPT/APS Joint Fall Meeting
Introduction
Perturbative string theory wellunderstood
Use this to learn about black holes?
 Usual string states lead to black
holes with singular and/or
strongly coupled horizons
D-branes
 Allow for smooth black holes
Find microstates responsible for
Bekenstein-hawking entropy!
David Lowe, Brown
University
10/21/00
 Same circle of ideas
 proposals for nonperturbative
formulations of string theory
• Matrix theory
• Maldacena conjecture
 large N, SU(N) gauge theory in
various dimensions
 Challenge now: learn about string
theory by studying gauge theory
David Lowe, Brown
University
10/21/00
Black Holes
Mass M, radius r,
space probe mass m
1 2 GMm
mv 
,
2
r
2GM
 need v 2 
.
r
2GM
So if r  2
c
Escape velocity exceeds speed of light!
David Lowe, Brown
University
10/21/00
Why Do We Care About
Black Holes?
 Astrophysical importance:
 supernova remnants
Galactic cores
Binary systems
 Theoretical point of view:
 testing ground for quantum
gravity
 many paradoxes – lead to
important constraints on theory
David Lowe, Brown
University
10/21/00
Paradoxes
Quantum effects  black holes
aren’t black
Hawking radiation
event horizon
Et  
David Lowe, Brown
University
10/21/00
Metric:
2GM
dr 2
2
ds  (1 
) dt 

2
GM
r
(1 
)
r
r 2 (d 2  sin 2  d 2 )
2
Imagine virtual particle starts at
r  2GM  
Proper time to hit event horizon
t  GM


E 
GM
Virtual partner gets redshifted, so
energy at infinity


E  E 

 kT
GM GM
David Lowe, Brown
University
10/21/00
Black Hole Entropy
Finite temperature finite entropy
Compare to usual laws of
thermodynamics
1st law dM   dA  work
2nd law dA  0
Identify
A
S
4G
Famous Bekenstein-Hawking
entropy formula
David Lowe, Brown
University
10/21/00
Challenge
Challenge for last 25 years
Find the microstates
S  k log( # microstate s)
Still unsolved problem in general
One of great successes of string
theory
Can describe microstates for BPS
or near BPS black holes
New nonperturbative formulations
of string theory
May lead to complete
understanding
David Lowe, Brown
University
10/21/00
Black Holes in String
Theory
Strominger and Vafa: find a black
hole that you can describe using
perturbative string theory
G 0
Make sure it becomes a nonsingular
black hole when it becomes
macroscopic
GM   s
Use supersymmetry to prove entropy
doesn’t depend on G when written in
terms of charges
David Lowe, Brown
University
10/21/00
D-branes
Key ingredient: D-branes Polchinski
David Lowe, Brown
University
10/21/00
Counting black hole
microstates
Use D-5branes, D-1branes and
Kaluza-Klein momentum to make a
charged 5d black hole
David Lowe, Brown
University
10/21/00
n5 D - 5branes
n1 D - 1branes
nKK Kaluza - Klein momentum
To count black hole microstatessolve a simple counting problem
Have n1n5 species of massless
particles in 1+1d
Want to count number of states
with total energy nKK
Answer agrees exactly with
Bekenstein-Hawking formula
S  2 n1n5nKK
David Lowe, Brown
University
10/21/00
Nonperturbative String
Theory
Try to formulate nonperturbative
string theory/M-theory by taking N
coincident D-branes and then
adjusting the coupling G so gravity
decouples
Left with QFT on the brane
Argue there is a duality
Large N QFT secretly describes
full string theory that contains
gravity
David Lowe, Brown
University
10/21/00
Maldacena Conjecture
Conjecture: large N SU(N) gauge
theory with 16 SUSY’s is dual to
string theory in a background that is
anti-de Sitter space X sphere
Need to take ‘t Hooft limit
N  , GN fixed, but large
Dimensional analysis
Get Bekenstein-Hawking entropy
right to within an overall factor
To do better requires strong coupling
gauge theory calculations
David Lowe, Brown
University
10/21/00
Large N Quantum
Mechanics
0+1 dimensional version of
Maldacena conjecture
SU(N) gauged quantum
mechanics at large N dual to 9+1
dimensional curved spacetime
Simple enough that the conjecture
can be directly tested by doing
quantum mechanics calculations
Kabat, Lifschytz and Lowe: mean
field approximation valid in large N
limit
David Lowe, Brown
University
10/21/00
Mean field solution
Comparison of mean field solution
to Bekenstein-Hawking result for
free energy
Title:
Creator:
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Works well in limited range of
Hawking temperature
David Lowe, Brown
University
10/21/00
Open Questions
Understanding of microscopic origin
of universal Bekenstein-Hawking
formula still open problem
Have a good chance to understand
this using string theory
Independent of details of vacuum
structure
Dynamical questions
Black hole information problem
David Lowe, Brown
University
10/21/00