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Vacuum energy in freely falling frames and spacetime curvature

The structure of quantum vacuum in presence of gravity, and the corresponding vacuum energy density $ρ_{v}$, is expected to depend on the coupling between the UV scale $\ell_\mathrm{\textsf{UV}}$ and spacetime curvature. We determine this coupling in an arbitrary freely falling frame characterised by it's geodesic tangent $U^i(τ)$. We show that local vacuum modes within a small causal diamond based on $U^i(τ)$, whose size is set by wavelength of the modes, generically give a contribution $ρ_{0}$ to $ρ_v$ which, to leading order, scales as: $ρ_{0} = \left( π\hbar c/2 \right) {\mathrm{\textsf R}} \, {\ell_\mathrm{\textsf{UV}}}^{-2}$, where the curvature term $\mathrm{\textsf R}=αR_{ab} U^a U^b + βR$, and $(α, β) \in \mathbb{R}$ are constants. The genericness of this result arises from the fact that, although the modes may reduce to Minkowski plane waves along $U^i(τ)$, the stress-energy tensor $T_{ab}$, since it depends on derivatives of the modes, does not reduce to it's Minkowski value on $U^i(τ)$. We discuss implications of our result for vacuum processes in freely falling frames, particularly in connection with certain aspects of the cosmological constant and horizon entropy.