An Investigation of Quasilocal Systems in General Relativity (2016)
AuthorsUzun, Neziheshow all
We propose a method to define and investigate finite size systems in general relativity in terms of their matter plus gravitational energy content. We achieve this by adopting a generic formulation, that involves the embedding of an arbitrary dimensional timelike worldsheet into an arbitrary dimensional spacetime, to a 2+2 picture. In our case, the closed 2-dimensional spacelike surface S, that is orthogonal to the 2-dimensional timelike worldsheet T at every point, encloses the system in question. The corresponding Raychaudhuri equation of T is interpreted as a thermodynamic relation for spherically symmetric systems in quasilocal thermodynamic equilibrium and leads to a work-energy relation for more generic systems that are in nonequilibrium. In the case of equilibrium, our quasilocal thermodynamic potentials are directly related to standard quasilocal energy definitions given in the literature. Quasilocal thermodynamic equilibrium is obtained by minimizing the Helmholtz free energy written via the mean extrinsic curvature of S. Moreover, without any direct reference to surface gravity, we find that the system comes into quasilocal thermodynamic equilibrium when S is located at a generalized apparent horizon. We present a first law and the corresponding worldsheet–constant temperature. Examples of the Schwarzschild, Friedmann–Lemaître and Lemaître–Tolman geometries are investigated and compared. Conditions for the quasilocal thermodynamic and hydrodynamic equilibrium states to coincide are also discussed, and a quasilocal virial relation is suggested as a potential application of this approach. For the case of nonequilibrium, we first apply a transformation of the formalism of our previous notation so that one may keep track of the quasilocal observables and the null cone observables in tandem. We identify three null tetrad gauge conditions that result from the integrability conditions of T and S. This guarantees that our quasilocal system is well defined. In the Raychaudhuri equation of T, we identify the quasilocal charge densities corresponding to the rotational and nonrotational degrees of freedom, in addition to a relative work density related to tidal fields. We define the corresponding quasilocal charges that appear in our work-energy relation and which can potentially be exchanged with the surroundings. These charges and our tetrad conditions are invariant under the boosting of the observers in the direction orthogonal to S. We apply our construction to a radiating Vaidya spacetime, a C-metric and the interior of a Lanczos-van Stockum dust metric. Delicate issues related to axially symmetric stationary spacetimes and possible extensions to the Kerr geometry are also discussed.