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Principle of equivalence, general covariance and the demand that the variation of the action functional should be well defined, lead to a generic Lagrangian for semiclassical gravity of the form $$L=Q_a^{\phantom{a}bcd}R^a_{\phantom{a}bcd}$$ with $$\nabla_b\,Q_a^{\phantom{a}bcd}=0$$. The expansion of $$Q_a^{\phantom{a}bcd}$$ in terms of the derivatives of the metric tensor determines the structure of the theory uniquely. The zeroth order term gives the Einstein–Hilbert action and the first order correction is given by the Gauss–Bonnet action. Remarkably, any such Lagrangian can be decomposed into a surface and bulk terms which are related holographically. The equations of motion can be obtained purely from a surface term in the gravity sector and hence gravity does not respond to the changes in the bulk vacuum energy density.

Gravity’s immunity from vacuum: the holographic structure of semiclassical action