Quantum Inequalities on the Energy Density in Static Robertson-Walker Spacetimes

TL;DR

The paper derives quantum inequalities for negative energy densities of a massive, minimally coupled scalar field in globally static, curved spacetimes by mode decomposition in the presence of a timelike Killing vector. The averaged energy-density bound $\Delta\hat{\rho}$ is shown to depend on curvature through scale functions (e.g., $F(\eta)$, $G(z)$, $H(\eta,\mu)$) and reduces to the flat-space results in the short-sampling regime while yielding AWEC-type constraints for long sampling times. Explicit evaluations are carried out for 3D closed and 4D flat/open/closed Robertson–Walker universes, including massless limits and Casimir-energy contributions, with detailed asymptotics and renormalization considerations in the Einstein universe. The results demonstrate how spacetime curvature modulates quantum energy inequalities and reinforce the connection between QIs and AWEC in curved backgrounds, providing curvature-aware bounds on negative energy that are relevant for cosmology and quantum field theory in curved spacetime.

Abstract

Quantum inequality restrictions on the stress-energy tensor for negative energy are developed for three and four-dimensional static spacetimes. We derive a general inequality in terms of a sum of mode functions which constrains the magnitude and duration of negative energy seen by an observer at rest in a static spacetime. This inequality is evaluated explicitly for a minimally coupled scalar field in three and four-dimensional static Robertson-Walker universes. In the limit of vanishing curvature, the flat spacetime inequalities are recovered. More generally, these inequalities contain the effects of spacetime curvature. In the limit of short sampling times, they take the flat space form plus subdominant curvature-dependent corrections.

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