REVIEW
Einstein Equation with Quantum Corrections Reduced to Second Order
General Relativity and Quantum Cosmology · Submitted 3 Nov 1992 · 28 references
Pith's verdict: A careful, useful extension of the reduction-of-order program for semiclassical gravity; the math holds up, but the abstract claims more than the body delivers on the 'only physical solutions' point.
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Referee report
The paper
The paper claims
The physically meaningful solutions of the semiclassical Einstein equation with first-order quantum corrections satisfy reduced equations with no derivatives higher than second order.
The reading
What carries the argument
The machinery is iterative reduction of order. Because the correction terms are already order $hbar$, one may replace any curvature or derivative of curvature inside them by its value from the lower-order Einstein equation, $G_{ab} = kappa T_{ab} + O(hbar)$. This turns the fourth-order terms into expressions involving at most second derivatives of the metric and the classical stress-energy tensor. The same move is applied covariantly in the general case through Eqs. (5.4)-(5.6), and in the conformally flat case through the conformal factor $e^{2S}$ and the flat-space operators of Eqs. (4.7)-(4.9).
Extended reading notes
Core claim
The central claim is that the fourth-order semiclassical Einstein equation contains a large set of solutions that are artifacts of treating a perturbative correction as exact. If one keeps only solutions that are asymptotic expansions in $hbar$ to first order, those solutions satisfy a second-order equation in which the quantum correction terms have been simplified using the ordinary Einstein equation. The authors carry out this reduction explicitly: for Friedmann-Lemaitre universes they obtain Eq. (3.11) and its renormalized form Eq. (3.16); for conformally flat metrics they reduce the linearized correction equation to a flat-space second-order partial differential equation, Eqs. (4.5)-(4.9); and for a static spherical fluid they express the correction tensors in terms of the classical density and pressure, Eqs. (5.9)-(5.20). They then exhibit analytic and numerical solutions, including a radiation-filled flat universe, a turn-around or bounce model, and a matched semiclassical Einstein-static hesitation universe.
Load-bearing premise
The load-bearing premise is that 'physical' means exactly 'expandable in powers of $hbar$ to first order'; if a solution that is not analytic in $hbar$ were physically relevant, the reduction would discard it.
Editorial extensions
If this is right
- The full fourth-order semiclassical equations contain runaway solutions; the reduced equations remove them while preserving the perturbative physical content.
- Numerical integration of semiclassical cosmology and stellar interiors becomes feasible with the same amount of initial data as in classical gravity.
- In radiation-filled Friedmann models, most of the quantum correction can be absorbed into renormalized cosmological and gravitational constants, leaving a residual $a^{-8}$ term.
- On unstable backgrounds such as the Einstein static universe, small quantum corrections can grow into large deviations, and matched branches describe a hesitation phase.
- For conformally flat fluid spheres, the $alpha_3$ correction acts like a shift in pressure and density, Eqs. (5.24)-(5.25).
Reading between the lines
- The reduction is a general template for any effective field theory whose higher-derivative corrections are perturbative: discard non-analytic solutions and solve the reduced system instead.
- If the method holds, it could supply stable numerical initial-value formulations for black-hole evaporation or early-universe models where fourth-order terms are retained.
- Matching the two semiclassical branches suggests a way to construct global histories that linger near an unstable static solution without fine-tuning, which could be tested against cosmological perturbation growth.
- A natural extension is to check whether the reduced equations preserve the trace anomaly and stress-energy conservation exactly, or only to the order retained.
Original 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.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We consider the Einstein equation with first order (semiclassical) quantum corrections. | background 0.81 | 0.67 | 0.04 | asserted 0.72 |
| s1 | 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. | result 0.95 | 0.50 | 0.76 | asserted 0.98 |
| s2 | We obtain the reduced equations for a range of stress-energy tensors. | contribution 0.79 | 0.95 | 0.13 | asserted 1.00 |
| s3 | 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. | interpretation 0.83 | 0.63 | 0.31 | asserted 0.51 |
| s4 | 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. | contribution 0.63 | 0.95 | 0.12 | asserted 1.00 |
What was examined, and what was not
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Figures
Showing 3 of the paper's 4 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:gr-qc/9211002 by the authors.
References
How the literature uses it
32 arXiv papers cite this in 45 sentences
Papers citing it · 34 sentences without a confident label
Contests it 1 paper
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The dependence of the stellar properties on $R^2$ terms in the action has been studied by Parker and Simon [ref] , who simply derived the generalized Tolmann–Oppenheimer–Volkoff equation without solving it, and by Barraco and Hamity [ref] who attempted to solve the problem using a perturbation analysis (unfortunately, this last study suffers from a large number of errors).
Probes and Tests of Strong-Field Gravity with Observations in the Electromagnetic Spectrum · The Need for a Theoretical Framework for Strong-Field Gravity Tests · contest 0.88
Uses its method or tool 5 papers
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This will in general lead to unphysical behavior in solutions, but the problem can be sidestepped by treating the higher derivative terms perturbatively in $1/N$, following [ref] .
Semiclassical Dynamics of Hawking Radiation · Introduction · 0.86
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Here we deal with this problem differently, following an order-reduction procedure first introduced in [ref] .
Macroscopic backreaction of the trace anomaly on classical vacuum backgrounds · General spacetimes · 1.00
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Specifically, one can perform a reduction of order procedure on the equations of motion [ref] , substituting the zeroth-order equations of motion into the higher derivative terms in the equations of motion to eliminate the higher derivatives[This is more general than requiring the solutions of the equation of motion to be analytic in the expansion parameter, as advocated by Simon [ref] ; see Ref. [ref] .].
A Class of Effective Field Theory Models of Cosmic Acceleration · Approach and Assumptions · 0.83
Cites it as background 3 papers
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The problems of the spacetime instability have been widely discussed and there are several proposals in the literature [ref] .
Spacetime Instability and Quantum Gravity as Low Energy Effective Field Theory · Numerical estimation for spacetime instability · 0.95
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The central local quantity is the renormalized stress-energy tensor $⟨T^μ_ν⟩_$, which appears as a source in the semiclassical Einstein equations [ref] [equation] where $T^μ_ν$ is the classical stress-energy tensor of the monopole.
Semiclassical Backreaction of Massive Quantum Fields in the Spacetime of a Global Monopole · Introduction · 0.93
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We also discussed semiclassical gravity to order $ħ$, which had also been discussed by Parker and Simon in [ref] and which is intermediate between the $O(ħ^0)$ theory and the theory of the physical solutions of full semiclassical gravity.
On the initial value problem for semiclassical gravity without and with quantum state collapses · Discussion and conclusions · 0.81
Discussion
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