EPJ Plus - A note on Pöschl-Teller black holes

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Published on 10 August 2012

An interesting feature of black holes is the existence of quasi-normal modes, arising because the system has a peak in the wave potential (scalar, electromagnetic, or gravitational waves). The quasi-normal mode is excited when a disturbance is put in the field near but outside the black hole, (like a wave packet roughly in a circular orbit near the peak). The excitation then propagates outward and inward and decays. An excitation “mode” has a definite complex frequency: a given oscillation rate in time, and a corresponding decay rate. For gravitational radiation from a spherical (Schwarzschild) black hole, the least damped mode is: ei 0.747t/tH e -0.178t/tH with tH the time for light to travel a distance equal to the radius of the black hole (S. Chandrasekhar and S. Detweiler, Proc. Roy. Soc. London 344 (1975) 441.). To calculate these modes is typically a computational problem, with attendant difficulties in controlling errors and convergence. A partial step to ameliorate these difficulties has been to substitute the black hole potential (long range, polynomial decay to infinity), with more localized potentials decaying exponentially at infinity. Pöschl and Teller (G. Pöschl and E. Teller, Z. Phys. 83 (1933) 143.) suggested one such potential (in another context): 1/cosh2 α(r - r0 ). This is much simpler – and decays more rapidly – than the correct gravitational potential, but to date even this potential has required numerical/computational treatment. Now, however, Zarrinkamar, Hassanabadi and Eskolaki have found an ingenious analytic transformation of the Pöschl-Teller wave equation with immediate solution in terms of Jacobi polynomials. Jacobi polynomials are well studied and characterized classical “special functions”. Thus questions of accuracy and convergence are now under control, and Zarrinkamar et al. have completely solved the quasi-normal mode problem for the Pöschl-Teller black hole.