Vacuum energy in freely falling frames and spacetime curvature

TL;DR

This paper investigates how quantum vacuum energy density in freely falling frames depends on spacetime curvature and UV scale, revealing a generic coupling that impacts cosmological constant and horizon entropy considerations.

Contribution

It derives a general expression for vacuum energy density in freely falling frames, linking it to spacetime curvature and UV scale, highlighting the non-trivial dependence on derivatives of modes.

Findings

Vacuum energy density scales with curvature and UV cutoff as $ ho_0 o ( ext{ extpi} ext{hbar} c/2) ext{ extsf R} ext{ extsf{l}_ ext{ extsf{UV}}}^{-2}$.

The stress-energy tensor's dependence on derivatives of modes prevents it from reducing to Minkowski form in freely falling frames.

Implications for the cosmological constant and horizon entropy are discussed based on the derived coupling.

Abstract

The structure of quantum vacuum in presence of gravity, and the corresponding vacuum energy density $\rho_{v}$, 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 $U^i(\tau)$. We show that local vacuum modes within a small causal diamond based on $U^i(\tau)$, whose size is set by wavelength of the modes, generically give a contribution $\rho_{0}$ to $\rho_v$ which, to leading order, scales as: $\rho_{0} = \left( \pi \hbar c/2 \right) {\mathrm{\textsf R}} \, {\ell_\mathrm{\textsf{UV}}}^{-2}$, where the curvature term $\mathrm{\textsf R}=\alpha R_{ab} U^a U^b + \beta R$, and $(\alpha, \beta) \in \mathbb{R}$ are constants. The genericness of this result arises from the fact that, although the modes may reduce to Minkowski plane waves along $U^i(\tau)$, the stress-energy tensor $T_{ab}$, since it depends on derivatives of the modes, does not reduce to it's Minkowski value on $U^i(\tau)$. 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.