Unified first law and some general prescription: a redefinition of surface gravity

Abstract

The paper contains an extensive study of the unified first law (UFL) in the Friedmann-Robertson-Walker spacetime model. By projecting the UFL along the Kodama vector the second Friedmann equation can be obtained. Also studying the UFL on the event horizon it is found that Clausius relation cannot be obtained from the UFL by projecting it along the tangent to the event horizon as it can be for the trapping horizon. However, it is shown in the present work that Clausius relation can be obtained by projecting the UFL along the Kodama vector on the horizon and the result is found to be true for any horizon. Finally motivated by the Unruh temperature for the Rindler observer, surface gravity is redefined and a Clausius relation is obtained from the UFL by projecting it along a vector analogous to the Kodama vector.