Einstein–Skyrme black hole’s novel first law of Skyrmion’s black hole thermodynamics

Abstract We investigate the thermodynamics and phase structure of an extremely rare exact, static and spherically symmetric black hole solution in the Einstein–Skyrme theory. The solution is characterized by two matter-sector couplings: K , which controls the strength of the Skyrme field and induces a solid-angle-deficit-like factor $$1-8\pi K$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>-</mml:mo> <mml:mn>8</mml:mn> <mml:mi>π</mml:mi> <mml:mi>K</mml:mi> </mml:mrow> </mml:math> in the metric and $$\lambda $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>λ</mml:mi> </mml:math> , the coupling of the quartic Skyrme term, which generates an $$r^{-2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>r</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:math> contribution analogous to an effective charge. We formulate a novel first law of black hole thermodynamics by treating both K and $$\lambda $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>λ</mml:mi> </mml:math> as extensive thermodynamic variables. Within this framework, we derive the Hawking temperature, Gibbs free energy and heat capacity and analyze the resulting phase structure in the G - T and $$C_H$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>H</mml:mi> </mml:msub> </mml:math> - T planes. Using an implicit method, we uncover Van der Waals-like critical behavior and associated phase transitions. We show that the Skyrme couplings substantially modify the thermodynamic behavior, play a crucial role in governing black hole stability and allow for the existence of phase transitions.