Einstein Equation with Quantum Corrections Reduced to Second Order

TL;DR

This paper derives second-order differential equations from the Einstein equation with quantum corrections, simplifying numerical analysis and focusing on physically relevant solutions in various cosmological and gravitational scenarios.

Contribution

It introduces a method to reduce higher-order quantum-corrected Einstein equations to second order, improving numerical stability and physical relevance.

Findings

Reduced equations are suitable for numerical solutions.

Solutions for cosmological and gravitational models are obtained.

Reduced equations contain fewer numerical instabilities.

Abstract

We consider the Einstein equation with first order (semiclassical) quantum corrections. Although the quantum corrections contain up to fourth order derivatives of the metric, the solutions which are physically relevant satisfy a reduced equations which contain derivatives no higher than second order. We obtain the reduced equations for a range of stress-energy tensors. These reduced equations are suitable for numerical solution, are expected to contain fewer numerical instabilities than the original fourth order equations, and yield only physically relevant solutions. We give analytic and numerical solutions or reduced equations for particular examples, including Friedmann-Lema\^\i tre universes with cosmological constant, a spherical body of constant density, and more general conformally flat metrics.