Linear Stability of Semiclassical Solutions

A semiclassical background is usable only if its physical perturbations remain controlled. The test is not simply whether a truncated higher-derivative equation has an unfamiliar root. One must isolate gauge-invariant retarded modes, compare their frequencies with the effective cutoff, propagate both intrinsic and matter-induced fluctuations, and check sensitivity to the prescription used for cutoff-scale branches.

Required background. Partial Differential Equations and Green Functions supplies mode evolution, retarded poles, and convergence tests.

Helpful background. Quantum Fields in Curved Spacetime supplies the stress response, Einstein Equations supplies the linearized constraints, Black-Hole Geometry and Mechanics supplies horizon boundary conditions, and Semiclassical Gravity supplies the self-consistent background and effective cutoff.

A physical stability criterion

Let $L$ be the renormalized linear-response operator about a self-consistent solution. After solving the linearized constraints and removing pure gauge modes, a retarded mode has time dependence $e^{-i\omega t}$. A pole with $\operatorname{Im}\omega\gt 0$ grows. It is evidence against the approximation only when it is on the physical retarded sheet, occurs in a gauge-invariant observable, respects the state and boundary conditions, and lies within the frequency-curvature regime in which $L$ was derived.

Anderson, Molina-París, and Mottola formulate the criterion in terms of bounded gauge-invariant solutions of the linear-response equation and analyze Minkowski space in Anderson, Molina-París, and Mottola 2003, §§ III–V, PDF. Their test is stronger than inspecting the mean stress tensor: the response includes the causal stress commutator and the renormalized local curvature terms.

Four classes of apparent instability must be separated:

• a physical low-frequency mode that grows within the EFT regime;
• a gauge mode removed by constraints or gauge-invariant observables;
• a cutoff-scale extra branch introduced by treating a derivative expansion nonperturbatively;
• an induced fluctuation driven by the noise kernel even when the homogeneous equation is stable.

The last class is tested by the Einstein–Langevin covariance, not by the homogeneous pole spectrum alone.

Order reduction and the branch test

A simple response polynomial illustrates the distinction. With $z=\omega^2$, consider

$$\epsilon z^2-z+k^2=0, \qquad z_\pm=\frac{1\pm\sqrt{1-4\epsilon k^2}}{2\epsilon}.$$

For $\epsilon k^2\ll1$, the low branch obeys $z_-=k^2+O(\epsilon k^4)$, whereas $z_+=\epsilon^{-1}-k^2+O(\epsilon k^4)$. The second root runs to infinite frequency as the higher-derivative coefficient tends to zero, so it is not continuously connected to the leading EFT solution. Order reduction retains the perturbative effect of the $O(\epsilon)$ term on the low branch while avoiding the independent high-frequency initial data. Parker and Simon give the perturbative logic, examples, and limitations in Parker and Simon 1993, §§ II–VI and Appendix, PDF.

This classification is not a universal license to delete inconvenient roots. If an extra root lies below the declared cutoff or persists under a nonlocal UV-complete response, it requires physical analysis. Conversely, evolving the fourth-order truncation with arbitrary extra initial data tests a different model from the order-reduced EFT.

Teaching benchmark and reproducible setup

The repository benchmark uses $\epsilon=0.03$, $k=1.2$, and final time $t_f=4.3$. The roots are

$$\omega_-=1.2281058581, \qquad \omega_+=5.6413730008.$$

It analytically selects the exact low root $z_-$ of this polynomial, then evolves the second-order equation

$$\ddot h+z_-h=0, \qquad h(0)=1, \qquad \dot h(0)=0,$$

with fixed-step classical fourth-order Runge–Kutta. The exact comparison is $h(t)=\cos(\omega_-t)$, and the monitored invariant is $I=h^2+\dot h^{\,2}/z_-$, whose exact value is one. Selecting the exact low root sums all powers present in that branch of the toy polynomial; strict first-order order reduction would instead use $z=k^2+\epsilon k^4$ and leave an $O(\epsilon^2)$ dispersion residual. The benchmark tests numerical evolution after branch selection, not the accuracy of those additional higher-order terms in a physical EFT. All numerical inputs are dimensionless after choosing a reference time and its inverse frequency.

Numerical method. The algorithm is explicit fixed-step RK4 applied to the first-order state $(h,\dot h)$ after analytic low-root selection; no adaptive tolerance or interpolation is used.

This is a teaching benchmark, not a physical semiclassical calculation. It contains no tensor constraints, gauge projection, renormalized nonlocal kernel, curved-background state, noise source, or physical cutoff value. Its purpose is narrower: verify low-branch selection, fourth-order time convergence, and invariant preservation for a transparent response model.

The run used CPython 3.9.6, NumPy 2.0.2, and SciPy 1.13.1. Download the MIT-licensed benchmark, then use the standalone download-and-run command sequence:

curl -fL https://blackhole.org/code/semiclassical-gravity/semiclassical-gravity-benchmarks.py -o semiclassical-gravity-benchmarks.py
python3 semiclassical-gravity-benchmarks.py --case linear-stability-semiclassical-solutions

Convergence results

The reported state error is the Euclidean norm of $(h,\dot h)$ minus the exact cosine state at $t_f$. “Order” is the base-two observed order under step doubling.

steps	$\Delta t$	state error	maximum invariant drift	observed order
20	2.1500000e-1	2.1998911e-4	9.3308765e-5	—
40	1.0750000e-1	1.4016197e-5	2.9352510e-6	3.972
80	5.3750000e-2	8.8527533e-7	9.1876934e-8	3.985
160	2.6875000e-2	5.5630242e-8	2.8723294e-9	3.992

The state error decreases monotonically by roughly $2^4$ under each doubling, and the measured order approaches four. The exact quadratic root gives a dispersion residual of 6.661e-16. Agreement with the analytic cosine independently checks the integrator, while the invariant drift checks a quantity not used to define the endpoint error.

The high root is printed but not evolved. It is retained only as a cutoff diagnostic: the ratio $\omega_+/\omega_-\simeq4.59$ makes the branch separation visible, but the benchmark deliberately assigns no physical cutoff and therefore makes no claim that this numerical ratio is adequate in a real theory.

Failure modes and interpretation

The benchmark should fail its interpretation if $1-4\epsilon k^2\lt 0$, because the assumed pair of real frequency branches disappears. It is also inconclusive if the high root is not parametrically above the physical scales, if the response has branch cuts or memory that the polynomial omits, or if the chosen step size cannot resolve the retained mode. Passing a Runge–Kutta convergence test cannot detect an incorrect dispersion relation.

For a genuine semiclassical solution, repeat the analysis in gauge-invariant channels and vary the regulator-independent couplings, state, boundary conditions, and admissible order-reduction prescription. Search the retarded analytic structure rather than only real frequencies. Then propagate the noise kernel and compare smeared metric variances with the background scale. Hu and Verdaguer explain why intrinsic stability and induced fluctuations form complementary validity tests in Hu and Verdaguer 2008, §§ 3.3 and 6.5.

A claimed instability is robust only if it survives these variations within the same EFT regime. A claimed stability remains conditional on the perturbation sector tested and on the absence of nonlinear secular growth.

Exercise

Use the first two state errors in the table to compute the observed convergence order. Then expand $z_-$ and $z_+$ through the first nontrivial correction at small $\epsilon$ and explain which branch has a smooth leading-order limit.

Solution

Step doubling gives

$$p=\log_2\!\left(\frac{2.1998911\times10^{-4}}{1.4016197\times10^{-5}}\right) =3.972,$$

matching the table. Expanding $\sqrt{1-4\epsilon k^2}=1-2\epsilon k^2-2\epsilon^2k^4+O(\epsilon^3)$ yields

$$z_-=k^2+\epsilon k^4+O(\epsilon^2), \qquad z_+=\epsilon^{-1}-k^2-\epsilon k^4+O(\epsilon^2).$$

Only $z_-$ tends continuously to the leading dispersion relation $z=k^2$. The $z_+$ frequency diverges as $\epsilon^{-1/2}$ and therefore represents the extra high-frequency branch of the truncated equation. This analytic branch check and the numerical fourth-order check test different parts of the calculation.

References

• Anderson, P. R., Molina-París, C., and Mottola, E. 2003. “Linear Response, Validity of Semi-Classical Gravity, and the Stability of Flat Space.” Physical Review D 67, 024026. DOI. Open PDF.
• Hu, B. L., and Verdaguer, E. 2008. “Stochastic Gravity: Theory and Applications.” Living Reviews in Relativity 11, 3. DOI.
• Parker, L., and Simon, J. Z. 1993. “Einstein Equation with Quantum Corrections Reduced to Second Order.” Physical Review D 47, 1339–1355. DOI. Open PDF.