## 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 |
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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 |
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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 |