Macroscopic backreaction of the trace anomaly on classical vacuum backgrounds

TL;DR

This work probes the macroscopic backreaction of quantum conformal trace anomaly on a classical Schwarzschild background using the RMV-RSET, implemented via an order-reduction scheme to yield a tractable semiclassical system. It reveals that, in vacuum, the backreacted geometry can exhibit horizon-perturbed behavior that is highly sensitive to whether compensatory terms are included to enforce conservation, with potential wormhole-like features emerging in some cases. Comparisons with full RMV-RSET and other approximations suggest some universal qualitative tendencies but also notable scheme-dependent differences, motivating further study of the full RMV-RSET and alternate reduction schemes. Overall, the results provide insight into the robustness of semiclassical backreaction features and guide future investigations into observable consequences of quantum effects in strong gravity.

Abstract

We study the backreaction of quantum fields in the Boulware vacuum state on the Schwarzschild geometry, using the Riegert--Mottola--Vaulin renormalized stress-energy tensor derived from the conformal anomaly. An order-reduction procedure is applied to the first order, paying special attention to the conservation of the resulting stress-energy tensor. The results obtained in these different situations are compared between them, and also to recent works in the literature using other approximations for the renormalized stress-energy tensor.

Figures (10)

• Figure 1: Plots of the metric function $f(r)$ for the semiclassical solutions with (solid blue line) and without (dashed-dotted red line) compensatory terms, as well as the classical Schwarzschild solution (black dashed line). Using numerical values $M=1$, $\hbar=10^{-2}$, and $r_0=10^2$, in the case without compensatory terms our numerical integrations show a tendency of $f(r)$ to diverge at $r \simeq 2.00211$, while introducing compensatory terms makes $f(r)$ tend towards a vanishing value at $r \simeq 2.00112$.
• Figure 2: Plots of the metric function $h(r)$ for the semiclassical solutions with (solid blue line) and without (dashed-dotted red line) compensatory terms, as well as the classical Schwarzschild solution (dashed black line). The qualitative behavior of $h(r)$ is the same as in the classical case, but with $1/h(r)$ vanishing for a slightly larger value of $r \simeq 2.00211$ without compensatory terms and $r \simeq 2.00112$ with compensatory terms (for $M=1$, $\hbar=10^{-2}$, and $r_0=10^2$).
• Figure 3: Plots of the first $r$-derivatives of the auxiliary fields $\varphi(r)$ and $\psi(r)$ for the semiclassical solutions with (solid blue line) and without (dashed-dotted red line) compensatory terms (for $M=1$, $\hbar=10^{-2}$, and $r_0=10^2$), as well as the solution without backreaction on the Schwarzschild spacetime (dashed black line), given by Eq. \ref{['eq:sol-phi-Sch']}. The numerical values obtained by solving the equations with backreaction differ from the analytic expression in Eq. \ref{['eq:sol-phi-Sch']} close to the gravitational radius, which indicates that backreaction effects become important there.
• Figure 4: Kretschmann scalar evaluated for the Schwarzschild metric (dashed black line) and the order-reduced semiclassical Einstein equation with (solid blue line) and without (dashed-dotted red line) compensatory terms. Taking into account that the Kretschmann scalar is positive definite, the observed behavior is compatible with a divergence as the radius approaches $r \simeq 2.00211$ for the solutions without compensatory terms, while this quantity remains finite as the radius approaches $r \simeq 2.00112$ for the solution with compensatory terms (for $M=1$, $\hbar=10^{-2}$, and $r_0=10^2$).
• Figure 5: Plot of the energy density as a function of the radius. The dashed black line indicates any of the approximations discussed in the paper (full RMV-RSET and order-reduced RMV-RSET with and without compensatory terms) evaluated on the Schwarzschild background. The solid lines indicate the order-reduced RMV-RSET in the self-consistent solutions obtained by solving the backreaction problem to first order without compensatory (red) and with compensatory (blue) terms. The remaining dashed lines indicate the full RMV-RSET evaluated on the geometries obtained when the backreaction of first order without compensatory (red) and first order with compensatory (blue) approximations are included. We see that the red (blue) solid and dashed lines have a similar behavior, and that the introduction of compensatory terms induces qualitative deviations from both the order-reduced approximations without compensatory terms and the results for a fixed Schwarzschild background.