Nonlocal stress-energy tensor in time-dependent gravitational backgrounds

TL;DR

This work analyzes the renormalized stress-energy tensor (RSET) for a massless scalar field in time-dependent gravitational backgrounds using a covariant curvature expansion. It derives explicit coordinate-space representations for the nonlocal β(□) operators and develops a multipole expansion of the RSET for time-dependent, spherically symmetric spacetimes, revealing that quantum hair appears already at leading order in time-dependent cases. Although a nonvanishing energy flux exists at null infinity, the total emitted energy at this order vanishes, with the total energy recoverable from conservation up to second order and the first-order RSET. The authors also compute large-distance quantum corrections to the metric, and illustrate the framework with a four-dimensional Newtonian oscillating star, highlighting nonlocality, backreaction, and the connection to particle creation probabilities.

Abstract

We analyze the renormalized stress-energy tensor (RSET) of a massless quantum scalar field in time-dependent gravitational backgrounds. Starting from its formal expression obtained within the covariant perturbative expansion to lowest order in the curvature, we evaluate the RSET in an arbitrary number of dimensions in terms of coordinate-space distributions. For time-dependent spherically symmetric spacetimes, we derive a multipole expansion and determine its asymptotic behavior. We find that the RSET is locally nonvanishing at null infinity and depends on the detailed dynamics of the collapsing body. However, the total emitted energy vanishes at this order, meaning that the leading contribution does not account for the energy density of the created particles. Nevertheless, by enforcing stress-tensor conservation up to second order in the curvature, we show that the total radiated energy can be extracted from the first-order RSET. Finally, we compute the induced quantum corrections to the metric at large distances, which display several interesting features.

Figures (1)

• Figure 1: Penrose diagram of the quasi-flat spacetime considered in the weak-field approximation. The solid gray line represents a spherical shell with a time-dependent radius $R_0(t)$. A perturbation originating at time $t_0$ (circle mark) modifies the RSET according to Eq. \ref{['RSET_4D']}. The resulting disturbance propagates at the speed of light along two null geodesics $t - r = t_0 \pm R_0(t_0)$ (dashed lines), reaching the future null infinity $\mathscr{I}^+$ (square markers). These two trajectories correspond respectively to an outgoing light ray traveling from the shell toward the exterior region (right), and to an ingoing perturbation emitted from the opposite side of the shell and directed toward the center (left), which continues outward after reaching $r = 0$ and eventually also arrives at $\mathscr{I}^+$.