The local limit of the quasilocal mass and mean quasilocal pressure on rigid spheres and rigid quasilocal frames in a general space-time

Abstract In this paper, general expressions are developed for the Brown-York mass and corresponding mean quasilocal pressure of spatial sections with the topology of 2-spheres in a 4-dimensional space-time, in the limit that the radius of the 2-spheres is small. Two separate cases are considered, one where the unions of the spatial sections coincide with rigid quasilocal frames, in the sense defined in a 2009 paper by Epp, Mann and McGrath, and the other where the spatial sections are geometric 2-spheres. The first situation is referred to as ‘observational rigidity’ (OR), and the second as ‘intrinsic rigidity’ (IR). In the IR case, the mass and mean pressure are linear functionals of the components of the Einstein tensor on the central curve, and on the angle average of the square of the expansion of a timelike vector orthogonal to the spatial sections. In addition, the mean pressure depends on the square of the acceleration scalar of the central curve. The expressions in the OR case are similar, with the addition of further terms proportional to the mean squared vorticity of the congruence of rigid observers. As a specific example, the formalism is applied to the case of a perturbed spatially-flat FLRW space-time in the Newtonian gauge, and the results compared with two other published analyses.