Fuwen Shu

1. Perturbative calculation of quasinormal modes of d-dimensional black holes Fuwen Shu Asia Pacific Center for Theoretical Physics KPS Jeju meeting

2. Outline • Introduction • Definition of QNMs • Highly damped QNMs • Perturbative calculation of QNMs • Conclusions

3. HawkingRadiation In 1974，Hawking found that, once the quantum effects are considered, black holes will radiate

4. Is black hole stable？

5. Perturbation of spacetimes • Schwarzschild metric： • Small perturbation EinsteinEqns • Regge-Wheeler equation： （1957）

6. The evolution of field perturbation on a black hole consists roughly of three stages: • The first one is an initial wave dependent on the initial form of the original field perturbation. • The second one involves the damped oscillations called QNMs, where frequencies and damping times are entirely fixed by the structure of the background spacetime and are independent of the initial perturbation. • The last stage is a power-law tail behavior of the waves at very late time which is caused by the backscattering of the gravitational field.

8. Black hole quasinormal modes A black hole disturbed from its equilibrium will start to oscillate like a bell would when struck. They will propagate to large distances in the form of gravitational waves. At late times the waves will be dominated by damped oscillations at certain frequencies called quasi-normal modes. Since these frequencies only rely on the properties of black hole, it is regarded as the “sound” of the black hole. It also provides a new way to “see” the black hole in the future by using the gravitational wave detector

9. Definition of QNMs Mathematically, QNMs is the solution of the wave equation： It should satisfy the following two boundary conditions Ingoing waves： Outgoing waves：

10. QNMs and LQG • In 1973, Bekenstein conjectured that in a quantum theory of gravity the surface area of a non-extremal black hole should have a discrete eigenvalue spectrum • According to LQG, the spacetime is also discrete

11. How to fix the Immirzi parameter？ • Statistical entropy of black holes from loop quantum gravity by matching the Bekenstein-Hawking entropy • Hod(98’ ) noted that the value of the real part of the highly damped QN frequencies (which are obtained by Nollert using numerical method) can be written as an analytical formula • Dreyer(03’)applied it to the LQG, getting so SU(2) OR SO(3)?

12. Analytical method In 2003,Motl proved the Hod’s conjecture by using an analytical method, the monodromy technique Along the contour in the complex r-plane,we start from A point to near the singularity r=0, then make a Wick rotation of angle of3/2 to B. At last complete thecontour by going back to A. Comparing the coefficients of two point we get the frequency of QNMs

13. Two questions: • What will happen when we apply the monodromy method to other black hole spacetime. Is it still correct of Hod’s conjecture? • What’s the relationship between the low-lying QNMs and the highly damped QNMs? Can we get the QN frequencies of low-lying QNMs analytically? Full survey on variety of black holes is needed, and higher-order perturbative calculation should be done

14. Perturbative calculation for Schwarzschild The wave equation For Schwarzschild The roots are Near the black hole singularitythe potential can be expanded as

15. Zeroth-order • Wave equation becomes have the general solution corresponding to A，then make a Wickrotation to B. The ratio of the coefficient • The monodromy of R_h

16. First-order Expand the wave function to the first order Wave equation becomes The general solution is Comparing with the monodromy of the wave function in horizon, we get The correction term is related to n、j and L---quanta appear in the lowest QN frequencies, learning the relationship betweenthe low-lying QNMs and the highly damped QNMs

17. RNblack hole • Roots of RN metric • Near the horizon • The potential can be expanded as • The wave equation zeroth-order QN frequencies

18. Expand the wave function to the first-order After monodromy calculation, first-order is obtained Main results： • Hod’s conjecture is no longer valid even for the zeroth-order QN frequencies • For d=4，the QN frequencies will be divergent as q approaches 0，and the corrections would blow up this divergence since • the q->0 limit of the 4th RN correction does not yield the 4th Schwarzschild correction due to the topological change of the contour

19. Extremal RN spacetimes Near the singularity, the tortoise coodinator is The potential is expanded as Using the monodromy method, we obtain • The zeroth-order QN frequency • The first-order QN frequency

20. Main results: • The zeroth-order and first-order QN frequencies of extremal RN black hole cannot be recovered by inserting q=m into RN’s results,since the topology is changed • Results for d=4 show that the first-order corrections of QN frequencies are zero • Is it also the same result for other dimensions？If yes, why? • The result also show that the Hod’s conjecture is not correct for this spacetime

21. Schwarzschild dS case The roots of the SdS metric are The monodromy of the wave function at the horizon is The potential is expanded as After perturbative calculation • Zeroth-order • First-order

22. Key points： • The QN frequencies of Schwarzschild cannot be recovered by lettingλ->0 of the SchwarzschilddS • Whenλ<<1， ,and for the other extremal case, i.e., we have So, if it is possible to conjecture that • The result also shows that the Hod’s conjecture is no longer correct.

23. RN dS The roots of the RNdS metric are After monodromy calculation • The zeroth-order frequencies • The first order

24. Main results: • For d=4，the QN frequencies will be divergent as q approaches 0，and the corrections would blow up this divergence • Forλ<<1,one has.For another extremal case the frequencies are So, if it is possible to image that • Again,q=0 of RN dS does not recover the results of Schwarzschild dS. The same conclusion can be drawn for the limit of λapproaches 0 • Hod’s conjecture failure again in this case

25. Schwarzschild AdS After monodromy calculation • We obtain the zeroth-order frequencies • Using the same method one can directly obtain the first-order frequencies • Hod’s conjecture is not true • One cannot recover the frequency of Schwarzschildby letting the cosmological constant to zero • The correction is related closely to the quantum parameters

26. RN AdS We obtain the zeroth-order frequencies • zeroth order • First order Results： • The QN frequencies will be explosive when q approaches 0 • One cannot recover the frequency of RNby letting the cosmological constant to zero • The frequencies don’t satisfy the Hod’s conjecture

27. Conclusions and Outlook • The Hod’s conjecture in Schwarzschild seems to be nothing but some numerical coincidences. • Due to the change of the topological structure, the QN frequencies cannot be recovered by letting the charged spacetimes or the non-asymptotically flat spacetimes to approach their extremal limit • For d=4，the QN frequencies will be divergent as q approaches 0，and the corrections would blow up this divergence. This indicates that the perturbative method in this case is broken • For asymptotically dS spacetimes we guess that • We found a way to relate the highly damped QNMs to the lowest QNMs by potential expansion, and also provide a way ot calculate low-lying QN frequencies analytically

28. Thank you！