Why does gravity ignore the vacuum energy?

Abstract

The equations of motion for matter fields are invariant under the shift of the matter Lagrangian by a constant. Such a shift changes the energy-momentum tensor of matter by T_b^a T_b^a + ρδ_b^a. In the conventional approach, gravity breaks this symmetry and the gravitational field equations are not invariant under such a shift of the energy-momentum tensor. We argue that until this symmetry is restored, one cannot obtain a satisfactory solution to the cosmological constant problem. We describe an alternative perspective to gravity in which the gravitational field equations are [G_ab - κT_ab]n^an^b = 0 for all null vectors n^a. This is obviously invariant under the change T_b^a T_b^a + ρδ_b^a and restores the symmetry under shifting the matter Lagrangian by a constant. These equations are equivalent to G_ab = κT_ab + Cg_ab, where C is now an integration constant so that the role of the cosmological constant is very different in this approach. The cosmological constant now arises as an integration constant, somewhat like the mass M in the Schwarzschild metric, the value of which can be chosen depending on the physical context. These equations can be obtained from a variational principle which uses the null surfaces of space-time as local Rindler horizons and can be given a thermodynamic interpretation. This approach turns out to be quite general and can encompass even the higher order corrections to Einstein's gravity and suggests a principle to determine the form of these corrections in a systematic manner.