**Reading
course on "Basics of black holes and black holes in string theory"**

Troels
Harmark

Email: harmark at nbi.dk

Office:
FC4

__Presentation:__

This is a
reading course intended for people who already know a little about black holes,
for example from Poul Olesens
General Relativity course, but would like to learn more. The main part of the
course is about basic results on black holes in 4 dimensions, such as a
mathematical description of event horizons and the thermodynamics of black
holes. Later in the course there will also be something about black holes in
string theory and M-theory, specifically their generalizations as charged black
branes. Finally, if time permits, there will be
something on black holes in five dimensions which most of the modern
string-theory inspired black hole research is focused on at the moment.

Black
holes is today at the center stage in the theoretical high-energy research. One
of the main problems in unifying quantum mechanics and General Relativity is
how to explain the quantum nature of a black hole. In string theory black holes
plays a central role in the understanding of non-perturbative
effects. Specifically in the AdS/CFT correspondence
it is possible to relate black hole physics to physics of gauge theories.

The
reading course is largely a self-study course. Each student (master or phd) can either get ECTS credit
for the course, or it can count for their "hovedfagskollokvium".
There will be given 7.5 ECTS for active attendence to
the course and a two hour presentation of part of the subject. If the student
instead chooses to use it for the "hovedfagskollokvium"
then the two hour presentation will count for the "hovedfagskollokvium".

If you
are interested in participating in this course, please send me an email.

__Practical stuff:__

We meet
Friday Nov. 7 at 11:00 in Aud. B (Blegdamsvej 17) to
organize the course, including who should speak when.

Duration:
2008: Week 46,47,48,49,50,51; 2009: Week 2,3,4

Room:
MA-14 (Blegdamsvej 21)

Time:
Thursday 13:15 to 15:00

__Participants:__

__People
attending the course and giving one of the lectures (and getting credit for the
course)__:

__Master students__:

Ask Emil Jensen

Martin Krssak

Esben Mølgaard

Martin
Cramer Pedersen

Lea
Hildebrandt Rossander

Joakim Sandroos

__PHD
students__:

Jay Armas

Pawel Caputa

Mads Toudal Frandsen

__People
attending without giving a lecture (and without getting credit)__:

__Students__:

Laura Gava

Johan Samsing

Andreas Vigand Pedersen

__Postdocs____ and senior people__:

Niels Obers

Marta Orselli

__Literature:__

Notes on
Killing vector fields and other geometric stuff. Download it here

“Black
holes”, Lecture notes by Paul K. Townsend, arXiv:gr-qc/9707012.
Download it here

Notes on causality and the global definitions of the event
horizon.
Download it here

Notes on physical quantities, black hole mechanics and
thermodynamics.
Download it here

“Lecture
Notes on General Relativity” by Sean M. Carroll, arXiv:gr-qc/9712019
(Chapter 7, pages 164 – 216) . Download it here

“Gravity
and strings” by Ortin

My paper
arXiv:hep-th/0408141. Download it here

Other
material: Nice project on black hole physics written by Andreas Vigand Pedersen. Download it here

__Schedule for lectures:__

Lecture
1, 13/11: Martin Krrsak

Lecture
2, 20/11: Ask Emil Jensen

Lecture 3,
26/11: Martin Cramer Pedersen

Lecture
4, 4/12: Joakim Sandroos

Lecture
5, 11/12: Esben Mølgaard

Lecture
6, 18/12: Lea Hildebrandt Rossander

Lecture
7, 8/1: Mads Toudal Frandsen

Lecture
8, 15/1: Jay Armas

Lecture
9, 22/1: Pawel Caputa

__Course plan:__

Lecture
1:

Killing
vector fields, stationary and axisymmetric
space-times

- Vector fields, the Lie derivate
of the metric and Killing vector fields. Killing
vector fields for two-sphere and Schwarzschild metric. (my notes)
- Stationary space-times
(Townsend page 76)
- Axisymmetry (Townsend page 76)

Lecture
2:

Null hypersurfaces, Killing horizons and surface gravity

- Null hypersurfaces
(sec. 2.3.5 of Townsend)
- Killing Horizons and surface gravity
(sec. 2.3.6 of Townsend)
- Rindler space-time (sec. 2.3.7 of
Townsend)
- Acceleration horizons and
surface gravity (sec 2.3.7 of Townsend)

Lecture
3:

Global
characteristics of a space-time and Carter-Penrose diagrams (sec. 2.4 of
Townsend, see also Chapter 7 in Carroll)

- Introduce conformal compactification. Carter-Penrose diagram for Minkowski space. Explain the five different kinds of
“infinity” (Example 1 in sec. 2.4 of Townsend)
- Carter-Penrose diagram for the Rindler space-time (Example 2 in sec. 2.4 of Townsend)
- Carter-Penrose diagram for the
maximally extended Schwarzschild black hole metric (from the Kruskal coordinates) (Example 3 in sec. 2.4 of
Townsend). Mention what a white hole is.
- Explain briefly the
Carter-Penrose diagram for a Collapsing star (page 47 in Townsend)

Lecture
4: Global definition of the Event Horizon (my notes on causality and the global
definitions of the event horizon, plus sec. 2.6 and 2.7 of Townsend).

- Definition of future and past
for a space-time (see section 1 of my notes)
- Asymptotically flat space-times
(see section 2 of my notes)
- Global definition of the Event
horizon (see section 3 of my notes. You can also see sec. 2.6 in
Townsend). Explain in particular the properties of the event horizon and Hawkings theorem.
- If time permits: Explain what a
naked singularity is. Explain what a globally hyperbolic space-time is,
and why a star collapsing into a naked singularity corresponds to a
non-globally hyperbolic space-time. Finally explain what is meant by the
Cosmic Censorship Conjecture. (see Section 4 of
my notes and Section 2.7 of Townsend).

Lecture
5: The black hole Uniqueness theorems and properties of rotating black holes
(Chapter 4 of Townsend)

- Explain the content of the
uniqueness theorems both in pure gravity and with Einstein-Maxwell theory
(Townsend page 77-78).
- Introduce the Kerr-metric
(Section 4.2 of Townsend)
- Angular velocity of the horizon
for the Kerr metric (Section 4.2.1 of Townsend)
- The ergo-sphere (Section 4.3 of
Townsend)
- The Penrose process (Section 4.4
of Townsend)

Lecture
6: Measurement of physical quantities, black hole mechanics and the Penrose
process

- Explain how to measure mass and
angular momentum of a black hole and compute it for the Kerr black hole
(see section 1 of my notes).
- Explain how to measure the area
of the event horizon of a black hole and compute it for the Kerr black
hole (see section 1 of my notes).
- Briefly remind people of how we
found the surface gravity and angular velocity of the Kerr black hole (see
section 1 of my notes and Townsend sections 2.3.6 and 4.2.1)
- Go through the three laws of
black hole mechanics and show they work for the Kerr black hole (see
section 2 of my notes)
- Explain the Penrose process for
the Kerr black hole and explain how this illustrates the second law of
black hole mechanics (see section 4.4 of Townsend (minus sec. 4.4.2))
- If time permits: Discuss the
generalized entropy and Hawking radiation (see
section 3 of my notes).

Lecture
7: Hawking radiation

- The Unruh temperature for accelerated
observers and Hawking radiation (see book by Sean Carroll).
- Derivation of Hawking radiation
(Townsend sec. 7.1 – 7.3)

Lecture
8: Black branes in higher dimensions as
generalizations of black holes in four dimensions (Chapter 18 in Ortins book).

- Classical string as
generalization of a particle. String equations of motion.
- Brane equations of motion and
coupling to generalized Maxwell field.
- Black p-branes
as generalization of Schwarzschild and Reissner-Nordstrøm
solutions.

Lecture
9: Five-dimensional asymptotically flat black holes (Literature: arXiv:0801.3471 and hep-th/0408141)

- Myers-Perry black hole, the
black ring and non-uniqueness given the asymptotic charges
- Rod-structure as classification
of five-dimensional asymptotically flat black holes, uniqueness theorem
- Examples of black holes and
rod-structures