this refers to the papers: Skakala J., Visser M. – Highly damped quasinormal frequencies for

1. Jozef Skakala, Matt VisserSemi-analytic results for quasi-normal frequenciesSchool of Mathematics, Statistics and Operations ResearchVictoria University of Wellington, New Zealand

2. this refers to the papers: Skakala J., Visser M. – Highly damped quasinormal frequencies for piecewise Eckart potentials, arXiv:1007.4039(published in Phys.Rev.D 81, 125023 (2010) ) Skakala J., Visser M. – Semi-analytic results for quasi-normal frequencies, arXiv:1004.2539(published in JHEP, 1007/08 (2010))

3. Black hole QNM modes are with respect to the boundary conditions physically intuitive perturbations (scalar, spin 1, spin 2) of various types of black holes (ordinary Schwarzschild, SdS, SAdS, Reissner-Nordstrom, Kerr...). • After some time scale they become dominant within arbitrary black hole perturbation. • There are various speculations about the connection between the highly damped QNMs and black hole thermodynamics (entropy).

4. They are given by infinite set of discrete complex QNM frequencies, all with Im(ω) > 0 , hence they describe stable perturbations. • The Re(ω ) part is symmetric with respect to the imaginary axis. • There are various different approaches how to calculate QNM frequencies: analytic approximations, WKB type approximations, monodromy technique, method of continued fractions, Born approximations, phase amplitude methods.

5. The quasinormal modes are specific solutions of the Regge- Wheeler (Schroedinger-like) equation: where V(x) is called the Regge-Wheeler potential and is given by the black hole (background) spacetime and the perturbating field. For example for Schwarzschild asymptotically flat black hole is V(x) given as: where x is a tortoise coordinate given as: and l is angular momentum of the specific wave mode, s is spin number of the perturbating field. The QNMs are such solutions of the Regge-Wheeler equation that they give only outgoing waves (hence on ) .

6. We do not have analytic formulas for QNM frequencies for Regge-Wheeler potentials, but there exist analytically solvable potentials. Particularly the interesting analytically solvable potential which can be used to reconstruct some of the features of Regge-Wheeler potentials is Poeschl-Teller (Eckart) potential. It is defined as: . • The Poeschl-Teller (Eckart) potential was used to estimate the lowest QNMs for Schwarzschild, some Reissner-Nordstrom and Kerr perturbations (Ferrari, Mashoon (1983)). In such case one fits the Poeschl-Teller potential by the real potential peak height and peak curvature.

7. Various results show that the highly damped modes spacing in the Schwarzschild case is given by surface gravity, hence by one tail of the potential (Motl (2003), Medved, Martin, Visser (2003)) . • The asymptotic QNMs are in such case given by the expansion: • Immediate observation is that this is obtained by Poeschl-Teller with exponent fitted by surface gravity, since the general QNMs for Poeschl-Teller potential are:

8. Such Poeschl-Teller exactly recovers behavior of the „left“ tail of the Regge-Wheeler potential. • In Suneeta (2008) we find the observation that for SdS we can fit the high QNMs by continuous Poeschl-Teller potential decreasing on both sides with two different exponents (hence recovering the two tails of SdS potential). • Now turn the logic: take the intuitive presupposition that because of their longer wavelengths (smaller real parts of the QNM frequencies compared to the low damped modes) highly damped QNMs depend mainly on the tails of the potential and confirm the known Schwarzschild results. • The real Regge-Wheeler potential is approximated on its tails by the (generally) discontinuous potential: V(x) hence The solutions of the Regge-Wheeler equation with this potential are on the left side the Bessel functions and on the right side the hypergeometric functions.

9. The continuity + derivative continuity condition at origin gives: where and Being interested only in Im(ω) >> 0 and Im(ω) >> Re(ω) solutions, we obtain (by using some Bessel and hypergeometric function identities + Stirling approximation for Gamma function) the asymptotic equation: implying: confirming qualitatively the old known result.

10. Let us use similar ideas for a much more „mysterious“ case, the Schwarzschild-deSitter space: Take a piecewise continuous Poeschl-Teller: V(x) again In a complete analogy to the previous case we glue the solutions at the origin obtaining: and again

11. again asymptotically (Im(ω) >> 0 , Im(ω) >> Re(ω) ) by Stirling Gamma-function approximation and some algebra: (1) This is an interesting result, since it implies: Theorem 1: For hence being a rational number, all the solutions of (1) are of the form (2) where and for even, for odd. The -s are related in a simple way to a roots of a particular degree polynomial. They are related as and the polynomial is:

12. This means that every QNM belongs to one of maximally families, defined by formula (2) and distinguished by the -s. Another interesting theorem expressing the opposite implication is: Theorem 2 If the ratio is irrational, then there are no periodic families of QNMs . The function in the first theorem implies the interesting statement: Take gap to be as in Theorem 1. If there would be for each only one family of QNMs, all QNMs apart of maximally one QNM would be „strongly“ discontinuous functions of variables at every point .

13. Similar theorems one can prove for most of the results achieved by using monodromy techniques for many various types of black holes (Schwarzschild- deSitter, Reissner-Nordstrom, Reissner Nordstrom – deSitter etc.) This is a result of the fact that the equations are generally given as:

14. There are few interesting cases and some more analytic information we can extract from our model: • Our intuition from the previous Schwarzschild case asymptotic calculation (independence of the results on the quadratically decreasing side of the potential) can be confirmed also by taking deSitter case and „cancelling“ the right side of the potential by setting . • In such case we obtain „Schwarzschild“ QNMs . • The same is (also reflecting the intuition) obtained by infinite exponential decrease on the right side ( ) which has pointwise the limit defined by . • Interestingly the same behavior is recovered also by the limit which has on the right side a pointwise limit given by a general constant function .

15. Conclusions First, we tested our approach of approximating the tails of the real potential by analytically solvable piecewise continuous potentials on the known Schwarzschild asymptotically flat space case. After that we used this given approach to qualitatively investigate the behavior of Schwarzschild-deSitter general perturbations by using the analytically solvable model of piecewise Poeschl-Teller (Eckart) potential. We obtained interesting results saying that the QNMs behavior is periodic when the ratio of survace gravities is being rational, and aperiodic when irrational. The periodic behavior gives in general many periodic families of QNMs, which is different to the Schwarzshild asymptotically flat space case. Our Schwarzschild-deSitter approach also recovers Schwarzschild- flat space case for some limit values of parameters, as intuitively expected.