Exact massless scalar quasibound states of the Ernst black hole
David Senjaya
Abstract
Abstract In this work, we present a detailed derivation of novel exact massless scalar quasibound state a static magnetized Ernst black hole background. We successfully solve the governing covariant Klein–Gordon equation and discover the exact radial solutions in terms of the Confluent Heun functions. With the exact radial wave solution in hand, applying its polynomial condition leads to the discovery of the quantized energy levels expression that depends on the black hole’s mass M , magnetic field strength $$B_0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>B</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> and also the magnetic and main quantum number $$(m_\ell , n).$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>m</mml:mi> <mml:mi>ℓ</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> <mml:mo>.</mml:mo> </mml:mrow> </mml:math> A massive scalar field around the black hole has complex valued energy levels while massless particle has purely imaginary energy levels. Further investigation in small black hole limit and zero magnetic field, the massless scalar’s purely imaginary energy expression is recovered. We also discover the equivalence between massless scalar field around an Ernst black hole with massive scalar field around a Schwarzschild black hole. In the last section, the Hawking radiation in investigated and applying the Damour–Ruffini method, the Hawking temperature is obtained out of the radiation distribution function.