Thermodynamics of Black Holes from an Entropy Functional: An Other Approach Using Generalized Elasticity

In a series of recent papers Padmanabhan et al. derived Einstein equations for gravity by introducing an entropy functional for space-time viewed as an elastic medium. They showed that the same entropy functional applied to the thermodynamics of horizons yields an entropy that is always proportional to the area of the horizon. Following the same philosophy as theirs we first note that it may be arrived at Einstein equations, with a cosmological constant as an integration constant, using a slightly different route from theirs that also results in the same final expression for the entropy functional. We generalize the fundamental equations of three-dimensional elasticity to four dimensions and propose that the elastic deformation of space-time be constrained by those equations. A general Lagrangian describing the elastic deformation of space-time is deduced. When applied to the special case of a Schwarzschild black hole, viewed as an infinite line defect in space-time, the approach developed here permits to recover the black hole's mass from the elastic deformation it caused to space-time, to reproduce the Hawking temperature, and to yield an entropy that is also in agreement with the Bekenstein-Hawking formula.