Charged Scalar Quasinormal Modes of Dyonic Black Holes

1. Charged Scalar Quasinormal Modes of Dyonic Black Holes Hing Tong Cho Department of Physics, Tamkang University

2. CONTENT I. Introduction II. Scalar Quasinormal Modes III. WKB Approximation IV. Mode Frequencies V. Discussions

3. I. Introduction * Stability of black hole solutions Linear perturbations of black hole solutions (Regge and Wheeler, Zerilli) Evolution of scalar, vector, and tensor fields in the black hole spacetime backgrounds

4. An initial wave burst • Damped “ringing” with characteristic frequencies • 3. A power-law tail due to the backscattering of the • long-range gravitational field

5. * The damped ringing signal in the intermediate time with characteristic frequencies and decay constants are called the “quasinormal modes”. * The quasinormal mode spectrum depends only on parameters, like mass, charge, and angular momentum of the black hole, but not on the initial wave field

6. II. Scalar Quasinormal Modes * Spherically symmetric black hole spacetime * Dyonic Reissner-Nordstrom black hole with both electric and magnetic charges

7. * Extremal condition

8. * Scalar field evolution - Klein-Gordon equation * Charged scalar field

9. * can be rendered singularity-free by dividing the space into two regions (Wu and Yang), Take

10. * and connected by a well-defined gauge transformation Dirac quantization condition

11. * With the presence of magnetic charges, the angular momentum operator with

12. * The corresponding eigenvectors are called monopole harmonics with

13. * Since the spacetime is spherically symmetric the scalar field can be decomposed into where

14. where

15. * Effective potential

16. * Quasinormal modes correspond to wavefunctions with out-going wave boundary conditions:

17. * Quasinormal mode frequency

18. IV. WKB Approximation * A simple and systematic approximation for the tunneling phenomenon in quantum mechanics

20. where

21. where

22. where

23. The wavefunctions in region II can be expressed in terms of parabolic cylinder functions

25. * Quasinormal mode condition: out-going waves only

26. * Quasinormal mode condition to 3rd order in the WKB approximation (Schutz, Will, and Iyer)

28. V. Mode Frequencies

38. VI. Discussions Some recent works on the quasinormal modes • Asymptotic value of the imaginary part of • the quasinormal mode frequency of the • Schwarzschild black hole (Motl and Neitzke)

39. If this is regarded as a fundamental frequency for the quantum system, one can obtain the entropy of the black hole (Hod) Dreyer used this result to fix the Immiriz parameter and the gauge group in loop quantum gravity

40. 2. Quasinormal mode frequency of anti-de Sitter black holes (i) AdS/CFT correspondence

41. (ii) Relation to the critical exponent Choptuik scaling in gravitational collapse