Two-Dimensional Models of Black Hole Radiation
Diploma thesis, Thomas Lotze
Faculty of physics and astronomy
Friedrich Schiller University, Jena
Contents
| Notation and Conventions | |||
| 1 | Introduction | ||
| Part I | Black Holes & the Hawking Effect in Four Dimensions | ||
| 2 | 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 | ||
| 3 | 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 | ||
| 4 | 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 | ||
| 5 | 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 | ||
| 6 | 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 | ||
| 7 | Conclusion and Outlook | ||
| Bibliography | |||
