Generalized free energy of black holes and nucleated bubbles in the gauged Kaluza–Klein theory

Abstract We explore the phase emergence and corresponding phase transitions of charged dilaton $$\text {AdS}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mtext>AdS</mml:mtext> </mml:math> black holes in the gauged Kaluza–Klein theory in the context of the generalized free energy. The black hole solutions are asymptotic to the AdS $$_D$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mmultiscripts> <mml:mrow/> <mml:mi>D</mml:mi> <mml:mrow/> </mml:mmultiscripts> </mml:math> geometry, and one can obtain them from the dimensional reduction of gauged supergravities on the compact internal manifolds, leading to the restriction as $$4\le D\le 7$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>4</mml:mn> <mml:mo>≤</mml:mo> <mml:mi>D</mml:mi> <mml:mo>≤</mml:mo> <mml:mn>7</mml:mn> </mml:mrow> </mml:math> . In the canonical ensemble, we classify the black hole thermodynamics into three different classes as follows: (i) $$D=4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>D</mml:mi> <mml:mo>=</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> , (ii) $$D=5$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>D</mml:mi> <mml:mo>=</mml:mo> <mml:mn>5</mml:mn> </mml:mrow> </mml:math> , and (iii) $$D=6,7$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>D</mml:mi> <mml:mo>=</mml:mo> <mml:mn>6</mml:mn> <mml:mo>,</mml:mo> <mml:mn>7</mml:mn> </mml:mrow> </mml:math> . In the grand canonical ensemble, the black hole thermodynamics is independent on the number of spacetime dimensions and the pressure, but it depends on the chemical potential $$\Phi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Φ</mml:mi> </mml:math> . As a result, we classify the black hole thermodynamics as follows: (i) $$\Phi &lt;1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Φ</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> , (ii) $$\Phi &gt;1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Φ</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> , and (iii) $$\Phi =1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Φ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> . In addition, we show that the asymptotically AdS solutions can provide a scenario where our Universe lives on a nucleated bubble mediating the decay of unstable AdS space. We point out a minimum value for the scale factor in the case of a neutral bubble by which the initial curvature singularity can be removed. We obtain a dS universe corresponding to the late-time evolution of the nucleated bubble, avoiding the constraint of the dS conjecture.