Negative Modes and Thermodynamic Stability

A Euclidean negative mode is an allowed fluctuation that lowers the quadratic action around a saddle. It diagnoses failure of local minimality on the declared integration cycle. In canonical Schwarzschild–AdS, the physical transverse-traceless negative mode disappears precisely where the heat capacity changes sign. That correspondence depends on matching the ensemble, gauge sector, and boundary conditions; it does not turn every Euclidean negative eigenvalue into a Lorentzian dynamical instability.

Required background. Gravitational Actions and Boundary Terms supplies the complete Euclidean action and fixed boundary data; Variational and Symplectic Geometry supplies the physical fluctuation space and boundary charges.

Helpful background. Quantum Fields in Curved Spacetime supplies one-loop determinants; Einstein Equations supplies the saddle and constraints; and Black-Hole Geometry and Mechanics supplies the horizon boundary conditions.

Quadratic action and the physical eigenproblem

Expand a Euclidean Einstein saddle as $g_{ab}+h_{ab}$. After gauge fixing and separating the conformal factor, the physical spin-two sector consists of transverse-traceless perturbations satisfying

$$\nabla^a h_{ab}=0, \qquad h^a{}_a=0.$$

On an Einstein background its quadratic operator is the Lichnerowicz-type operator

$$(\mathcal Gh)_{ab} =-\nabla^2h_{ab}-2R_{acbd}h^{cd}.$$

If an allowed normalized mode obeys $\mathcal Gh=\lambda h$ with $\lambda\lt0$, then its amplitude $q$ contributes $I_E^{(2)}\propto\lambda q^2$ and the real metric saddle is not a local minimum. Gauge modes, the unrotated conformal-factor direction, and perturbations that violate the ensemble’s boundary sources do not count as physical negative modes. Prestidge constructs this gauge-invariant operator and eigenproblem in Prestidge 2000, §3, Eqs. (3.8)–(3.12), PDF.

The inner product on this Euclidean sector is $\langle h_1,h_2\rangle=\int_{\mathcal M}\sqrt g\,h_{1\,ab}^*h_2^{ab}d^4x$. The domain imposes smoothness at the Euclidean horizon, periodicity around the thermal circle, and normalizable falloff preserving the fixed AdS boundary sources. Those conditions remove boundary terms in the symmetric quadratic form. Merely writing $\mathcal G$ without its domain would not define this spectrum.

Whether a negative direction gives an imaginary contribution to $Z$ also depends on how the integration contour passes the saddle. A negative Euclidean eigenvalue supports a nucleation or canonical-instability interpretation only after that contour is specified. A Lorentzian instability instead requires a physical perturbation with growing real-time behavior and the appropriate causal boundary conditions.

Schwarzschild–AdS stability boundary

For four-dimensional Schwarzschild–AdS, define Prestidge’s dimensionless horizon radius

$$\rho_+=\frac{\sqrt3\,r_+}{L}.$$

The canonical heat capacity changes sign at $r_+=L/\sqrt3$, hence at $\rho_+=1$. After imposing regularity at the horizon and normalizability at infinity, the spherically symmetric transverse-traceless radial problem has one negative eigenvalue on the small black-hole branch, reaches zero at $\rho_+=1$, and has no such negative mode on the large branch. The radial equation, singular-point boundary series, and shooting condition are given in Prestidge 2000, §5, Eqs. (5.3)–(5.11), PDF.

In the small-hole limit the result approaches the asymptotically flat Schwarzschild value

$$\lim_{\rho_+\to0}(GM)^2\lambda_{\mathrm{neg}}\simeq-0.192,$$

as reported in Prestidge 2000, Eq. (5.12), PDF. Prestidge sets $G=1$; the factor $GM$ restores the length associated with the physical mass used here. Since $\lambda$ has dimension $\mathrm{length}^{-2}$, the quoted combination is dimensionless. This agreement is a physical limiting check on the sourced calculation. The zero crossing tracks local canonical stability, not global dominance: the large black hole can have positive heat capacity while thermal AdS still has lower free energy.

The ensemble qualification is essential. Fixing cavity areal radius and wall temperature produces a different fluctuation boundary condition from fixing a coordinate radius. Prestidge explains why a mismatched cavity condition need not reproduce the heat-capacity threshold Prestidge 2000, §§2 and 4, PDF. Microcanonical fluctuations also hold energy rather than temperature fixed, so a canonical negative direction can be absent from their admissible tangent space.

Reproducible radial-solver benchmark

The retained computation is deliberately an algorithm test, not a numerical solution of the Schwarzschild–AdS tensor equation. It diagonalizes the exactly solvable Pöschl–Teller operator

$$\mathcal L =-\frac{d^2}{dx^2}-2\,\operatorname{sech}^2x$$

on $x\in[-8,8]$ with Dirichlet endpoints. On the infinite line, $\psi_0(x)=\operatorname{sech}x$ is the unique negative bound state and obeys $\mathcal L\psi_0=-\psi_0$. Thus the exact comparison eigenvalue is $-1$.

The implementation uses $N$ equally spaced interior points, the centered second-order finite-difference Laplacian, and scipy.linalg.eigh_tridiagonal in index-selection mode to return the two lowest eigenpairs. It uses no interpolation or iterative tolerance; check_finite=True is enabled. Arithmetic is IEEE-754 binary64. The assertions require monotone error decrease, final eigenvalue error below $10^{-4}$, final matrix residual below $10^{-11}$, and no unexpected second negative eigenvalue below $-2\times10^{-3}$.

The reference 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 recipe:

curl -fL https://blackhole.org/code/semiclassical-gravity/semiclassical-gravity-benchmarks.py -o semiclassical-gravity-benchmarks.py
python3 semiclassical-gravity-benchmarks.py --case negative-modes-thermodynamic-stability

Convergence and independent checks

The matrix residual is the binary64 Euclidean norm $\lVert L_Nv-\lambda_Nv\rVert_2$. The eigenvalue error compares with the infinite-line analytic result $-1$.

grid $N$	$\Delta x$	lowest eigenvalue	eigenvalue error	matrix residual
64	2.4615385e-01	-1.0023774e+00	2.3774244e-03	3.7915877e-15
128	1.2403101e-01	-1.0005988e+00	5.9875051e-04	2.1062123e-14
256	6.2256809e-02	-1.0001499e+00	1.4991740e-04	6.9758610e-14
512	3.1189084e-02	-1.0000369e+00	3.6934635e-05	2.7398821e-13

The error ratios are approximately $3.97$, $3.99$, and $4.06$, as expected for second-order convergence under grid doubling. The residual grows mildly with matrix size because roundoff is amplified by the $1/\Delta x^2$ entries, but remains below $3\times10^{-13}$ and therefore far beneath the acceptance threshold. The imposed discrete endpoint values are exactly zero. The analytic infinite-line eigenfunction has $\operatorname{sech}(8)=6.7093\times10^{-4}$, which exposes the finite-domain boundary mismatch separately from discretization error.

Applying the differential operator directly gives

$$\left(-\frac{d^2}{dx^2} -2\,\operatorname{sech}^2x\right)\operatorname{sech}x =-\operatorname{sech}x.$$

That analytic eigenpair and the known single negative bound state independently check the eigenvalue and mode count. They validate the tridiagonal radial workflow only. Prestidge’s tensor potential, horizon series, normalizability weight, and thermodynamic threshold remain separate physical evidence and are not reproduced by this proxy.

Failure modes and interpretation

The benchmark can be fooled by too narrow a domain, an incorrect horizon-to- radial coordinate mapping in a physical problem, or a grid that misses a localized mode. A small matrix residual only shows that the computed vector is an eigenvector of the discretized matrix; it does not bound discretization or model error. A spurious negative eigenvalue can also arise from gauge contamination, a singular horizon series, or a boundary condition outside the declared ensemble.

For the physical problem, one must refine the radial grid or shooting tolerance, move the outer boundary, verify horizon regularity and asymptotic normalizability, count nodes, and compare the zero crossing with the Hessian at fixed thermodynamic sources. Renormalized one-loop determinants and the chosen contour then determine how the mode enters the partition function. None of these Euclidean checks alone proves nonlinear or Lorentzian instability.

Exercise

Use the first two eigenvalue errors to estimate the convergence order. Then suppose a physical canonical branch has $C\lt0$ but a computed spectrum contains no negative mode. Give two checks required before claiming that thermodynamic and spectral stability disagree.

Solution

Under grid doubling,

$$p=\log_2\!\left( \frac{2.3774244\times10^{-3}}{5.9875051\times10^{-4}} \right)=1.989,$$

consistent with the second-order stencil.

First verify that the perturbations preserve exactly the canonical boundary sources: fixed proper boundary temperature and induced boundary geometry, not a coordinate surrogate. Second verify that the physical transverse-traceless sector has been isolated and that horizon regularity, outer normalizability, and domain convergence do not remove or hide a mode. One should also compare the action Hessian at fixed sources and test the eigenvalue as the known heat-capacity turning point is crossed. Until these checks agree, the apparent discrepancy is more plausibly an ensemble, gauge, or boundary error than new physics.

References

• Prestidge, T. (2000), “Dynamic and Thermodynamic Stability and Negative Modes in Schwarzschild–Anti-de Sitter,” Physical Review D 61, 084002. DOI. Open PDF.
• York, J. W., Jr. (1986), “Black-Hole Thermodynamics and the Euclidean Einstein Action,” Physical Review D 33, 2092–2099. DOI.