Diplomarbeit in Physik

English version

Two-Dimensional Models of Black Hole Radiation

Diplomarbeit, Thomas Lotze

Physikalisch-Astronomische Fakultät
Friedrich-Schiller-Universität Jena


Notation and Conventions
Part I  Black Holes & the Hawking Effect in Four Dimensions
The Spacetime of Black Holes
2.1  The exterior Schwarzschild solution
2.1.1  The Schwarzschild metric as a solution to Einstein's equations
2.1.2  Kruskal coordinates
2.2  Penrose diagrams
2.2.1  A simple example: Minkowski spacetime
2.2.2  Black Holes
Particle Creation by Non-Minkowskian Spacetimes
3.1  Quantum fields and the particle concept
3.1.1  Quantum fields and the wave equation
3.1.2  Symmetries and the particle concept
3.1.3  Asymptotic regions and Bogoliubov transformations
3.2  An example of particle creation: the sudden expansion of the »universe«
3.2.1  The metric
3.2.2  The Klein-Gordon equation
3.2.3  Solutions
3.2.4  Comparison to the results from the literature
The Hawking Effect
4.1  The virtual particle picture
4.1.1  Separation of virtual particle pairs
4.1.2  Estimating the Hawking temperature
4.2  The curvature of spacetime as a scattering potential
4.2.1  The s-wave equation for four-dimensional Black Holes
4.2.2  Reduction to a scattering problem
4.2.3  Discussion
4.3  Hawking's derivation of Black Hole radiation
4.3.1  Gravitational collapse
4.3.2  Black Hole radiation
Part II  Two-Dimensional Models & the Effective Action Approach
Two-Dimensional Effective Action Models
5.1  Why consider two-dimensional models?
5.2  Action principle and conformal trace anomaly
5.2.1  The action principle
5.2.2  Conformal invariance and its breaking
5.3  Two-dimensional gravitational action & dilaton gravity
5.3.1  The naive reduction: ignoring two dimensions
5.3.2  The more physical case: dilaton gravity
5.4  Matter in two dimensions and effective action
5.4.1  The concept of effective action
5.4.2  Genuinely two-dimensional matter and Polyakov action
5.4.3  Spherically symmetric matter in four dimensions
5.4.4  The controversy about the anomaly induced effective action
A Conformally Invariant Correction to the Effective Action
6.1  The contribution to the effective action
6.1.1  The proposal made by Gusev and Zelnikov
6.1.2  The retarded Green function
6.2  The second order correction
6.2.1  Variation
6.2.2  The stress tensor
6.3  The third order correction
6.3.1  Variation
6.3.2  The stress tensor
6.4  The tangential pressure
6.4.1  Variation with respect to the dilaton
6.4.2  The tangential pressure
Conclusion and Outlook