We study the thermodynamics of a shell of self-gravitating radiation, bounded by two spherical surfaces. This system provides a consistent model for a gravitating thermal reservoir for different solutions to vacuum Einstein equations in the shell’s interior. The latter include black holes and flat space, hence, this model allows for the study of black hole phase transitions. Following the analysis of Anastopoulos C and Savvidou N (2012 Class. Quantum Grav. 29 025004), we show that the inclusion of appropriate entropy terms to the spacetime boundaries (including the Bekenstein–Hawking entropy for black hole horizons) leads to a consistent thermodynamic description. The system is characterized by four phases, two black hole phases distinguished by the size of the horizon, a flat space phase and one phase that describes naked singularities. We undertake a detailed analysis of black-hole phase transitions, the non-concave entropy function, the properties of temperature at infinity, and system’s heat capacity.

我们研究由两个球面包围的自引力辐射壳的热力学。该系统为壳内部真空爱因斯坦方程的不同解提供了引力热库的一致模型。后者包括黑洞和平坦空间，因此，该模型允许研究黑洞相变。根据Anastopoulos C 和Savvidou N (2012 Class. Quantum Grav. 29025004），我们表明将适当的熵项包含到时空边界（包括黑洞视界的 Bekenstein-Hawking 熵）会导致一致的热力学描述。该系统的特点是四个阶段，两个以视界大小区分的黑洞阶段，一个平坦的空间阶段和一个描述裸奇点的阶段。我们对黑洞相变、非凹熵函数、无穷远处的温度特性和系统的热容量进行了详细分析。