Foliation and the First Law of Black Hole Thermodynamics

There has been lots of interest in exploring the thermodynamic properties at the horizon of a black hole spacetime. It has been shown earlier that for different spacetimes, the Einstein field equations at the horizon can be expressed as the first law of black hole thermodynamics. Using the idea of foliation, we develop a simpler procedure to obtain such results. We consider r= constant slices, for the Schwarzschild and Reissner–Nordstrom black hole spacetimes. The Einstein field equations for the induced 3−dimensional metrics of the hypersurfaces are expressed in thermodynamic quantities under the virtual displacements of the hypersurfaces. As expected, it is found that the field equations of the induced metric corresponding to the horizon can be written as a first law of black hole thermodynamics. It is to be mentioned here that our procedure is much easier, to obtain such results, as here one has to essentially deal with (n-1)−dimensional induced metric for an n-dimensional spacetime.