The extended phase space thermodynamics and Ehrenfest scheme for the Kerr–Sen AdS black holes

Abstract In the present work, we numerically investigate the horizon structure of the Kerr–Sen black holes in anti-de Sitter (AdS) spacetime. Further, we investigate the phase transitions and critical phenomena in Kerr–Sen–AdS black holes at the critical points. Such black holes are characterized by its mass ( M ), the dilaton charge ( Q ), and the negative cosmological constant, $$\Lambda (&lt;0)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Λ</mml:mi> <mml:mo>(</mml:mo> <mml:mo>&lt;</mml:mo> <mml:mn>0</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . We define a dimensionless parameter $$\epsilon =\bar{J}/{\bar{Q}^2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ϵ</mml:mi> <mml:mo>=</mml:mo> <mml:mover> <mml:mrow> <mml:mi>J</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>¯</mml:mo> </mml:mrow> </mml:mover> <mml:mo>/</mml:mo> <mml:msup> <mml:mrow> <mml:mover> <mml:mrow> <mml:mi>Q</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>¯</mml:mo> </mml:mrow> </mml:mover> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> and express the mass, temperature, volume, and Gibbs free energy in terms of $$\epsilon $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ϵ</mml:mi> </mml:math> and its polynomials. Moreover, we numerically fit the data for the critical points and find that in the appropriate limit, the expressions for critical points would correspond to the respective critical points of the Kerr–AdS black hole thermodynamics. Such a study involves a systematic analysis of temperature, Gibbs free energy, and volume in the extended phase space. We provide an analytical verification of the nature of the phase transitions at the critical points by introducing the Ehrenfest equations. We find that the Ehrenfest equations are satisfied in accordance with the phase behaviour. We also check that all three quantities, e.g., the specific heat at constant pressure, $$C_P$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>P</mml:mi> </mml:msub> </mml:math> , the volume expansion coefficient, $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> , and the isothermal compressibility, $$\kappa _T$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>κ</mml:mi> <mml:mi>T</mml:mi> </mml:msub> </mml:math> , diverge at the critical points. We find the Prigogine - Defay ratio using the expressions of $$C_P$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>P</mml:mi> </mml:msub> </mml:math> , $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> , and $$\kappa _T$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>κ</mml:mi> <mml:mi>T</mml:mi> </mml:msub> </mml:math> , and find that it identically equals unity. Hence, the phase transition behavior of the Kerr–Sen–AdS black holes at their critical points is of second order. In addition, we propose investigating the energy extraction process via the Penrose process by deriving work efficiency. Later, we calculate the speed of sound and adiabatic compressibility for the rotating Kerr–Sen–AdS black holes. Finally, on a specific note, we calculate the thermodynamic quantities of the boundary conformal field theory (CFT) dual to the extended phase space.