On quasinormal modes in 4D black hole solutions in the model with anisotropic fluid
S. V. Bolokhov, В. Д. Иващук
Abstract
Abstract We consider a family of four-dimensional black hole solutions from Dehnen et al. (Grav Cosmol 9:153 arXiv:gr-qc/0211049 , 2003) governed by natural number $$q= 1, 2, 3 , \dots $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>q</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn><mml:mo>,</mml:mo><mml:mo>⋯</mml:mo></mml:mrow></mml:math> , which appear in the model with anisotropic fluid and the equations of state: $$p_r = -\rho (2q-1)^{-1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>p</mml:mi><mml:mi>r</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mo>-</mml:mo><mml:mi>ρ</mml:mi><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mn>2</mml:mn><mml:mi>q</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math> , $$p_t = - p_r$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>p</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mo>-</mml:mo><mml:msub><mml:mi>p</mml:mi><mml:mi>r</mml:mi></mml:msub></mml:mrow></mml:math> , where $$p_r$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>p</mml:mi><mml:mi>r</mml:mi></mml:msub></mml:math> and $$p_t$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>p</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:math> are pressures in radial and transverse directions, respectively, and $$\rho > 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>ρ</mml:mi><mml:mo>></mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math> is the density. These equations of state obey weak, strong and dominant energy conditions. For $$q = 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>q</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math> the metric of the solution coincides with that of the Reissner–Nordström one. The global structure of solutions is outlined, giving rise to Carter–Penrose diagram of Reissner–Nordström or Schwarzschild types for odd $$q = 2k + 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>q</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math> or even $$q = 2k$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>q</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:math> , respectively. Certain physical parameters corresponding to BH solutions (gravitational mass, PPN parameters, Hawking temperature and entropy) are calculated. We obtain and analyse the quasinormal modes for a test massless scalar field in the eikonal approximation. For limiting case $$q = + \infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>q</mml:mi><mml:mo>=</mml:mo><mml:mo>+</mml:mo><mml:mi>∞</mml:mi></mml:mrow></mml:math> , they coincide with the well-known results for the Schwarzschild solution. We show that the Hod conjecture which connect the Hawking temperature and the damping rate is obeyed for all $$q \ge 2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>q</mml:mi><mml:mo>≥</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math> and all (allowed) values of parameters.