# cosmological constant

03 Mar 2021Why is the field equation in relativity homogeneous when exactly the cosmological constant term makes the Poisson-equation in classical gravity inhomogeneous?

This curious incidence is a matter of the weak field approximation. In fact, the Lagrange-density of general relativity is \(S = \int\mathrm{d}^4x\sqrt{-\mathrm{det}(g)}\:(R-2\Lambda)\) i.e. the cosmological constant term comes in through the variation of the covolume, leading to a homogeneous equation. In classical gravity, \(S = \int\mathrm{d}^3x\:\delta^{ij}\partial_i\Phi\partial_j\Phi - 2\lambda\Phi\) for the gravitational potential \(\Phi\), where the classical cosmological constant \(\lambda\) is part of a inhomogeneous differential equation after variation by \(\Phi\). The covolume \(\mathrm{det}g\) in weak field gravity is \((1+2\Phi/c^2)(1-2\Phi/c^2)^3\simeq 1-4\Phi^2/c^4\), leading to higher-order terms associated with \(\lambda\) that are usually neglected.