Saddles, Negative Modes, and Steepest-Descent Cycles

A negative Hessian eigenvalue is not a property of a geometry by itself. It belongs to a specified action, boundary-value problem, gauge quotient, and integration cycle. Only after those data are fixed can one decide whether the direction is physical, whether the saddle lies on the parent contour, and whether the resulting phase has an interpretation as a decay rate.

The worked result on this page is the static spherical spin-two spectrum of Euclidean Schwarzschild–AdS$_4$ in the standard canonical problem at the AdS boundary. The small-black-hole branch has one physical nonconformal negative mode, the eigenvalue crosses zero at $r_+=L/\sqrt3$, and the large branch has none. That local-stability threshold is different from the Hawking–Page dominance threshold $r_+=L$, and neither threshold determines the thimble intersection number.

Required background. Conformal-Factor Problem, Integration Contours, and Resurgent-Completion Proposals fixes the convergent-cycle problem. Negative Modes and Instability Indices supplies the ordinary saddle classification.

Helpful background. Decay Rates, the Negative Mode, and Prefactors explains when one negative mode produces an imaginary part. Gauge Fixing, BRST Constraints, and the Gribov Problem supplies the gauge-theory qualifications.

The fluctuation domain defines the physical Morse index

Consider pure Einstein gravity with cosmological constant, expanded about a smooth Euclidean Einstein saddle $g^{(s)}$ satisfying

$$R_{\mu\nu}=\frac{2\Lambda}{D-2}g_{\mu\nu}.$$

Boundary terms and counterterms are part of the action. More importantly here, their variational principle determines which components of $h_{\mu\nu}$ may fluctuate. Two calculations about the same local metric but with fixed inverse temperature and fixed energy therefore need not have the same physical Hessian.

For $g_{\mu\nu}=g_{\mu\nu}^{(s)}+h_{\mu\nu}$, a York-type decomposition is

$$h_{\mu\nu}=h_{\mu\nu}^{\mathrm{TT}} +2\nabla_{(\mu}\xi^{\mathrm T}_{\nu)} +\left(\nabla_\mu\nabla_\nu-\frac{1}{D}g_{\mu\nu}\nabla^2\right)\sigma +\frac{1}{D}g_{\mu\nu}\phi, \qquad \nabla^\mu\xi^{\mathrm T}_\mu=0,$$

with $\nabla^\mu h^{\mathrm{TT}}_{\mu\nu}=0$ and $g^{\mu\nu}h^{\mathrm{TT}}_{\mu\nu}=0$. This is a decomposition relative to an inner product and a boundary domain, not a pointwise declaration that every TT representative is physical. Boundary-preserving diffeomorphisms must be quotiented; Killing, conformal-Killing, and scalar kernels must be removed; and boundary-changing diffeomorphisms are not gauge redundancies of the fixed-source problem. At a finite Dirichlet wall, the allowed boundary condition can even couple the TT and trace pieces Marolf and Santos 2022, “Canonical Ensemble Reloaded,” §§1 and 3.

The sectors have different meanings and must not be compressed into a single determinant:

Sector	Diagnostic	One-loop treatment	What it does not establish
Boundary-trivial gauge direction	$h_{\mu\nu}=2\nabla_{(\mu}\xi_{\nu)}$ with allowed $\xi$	Quotient the orbit with a compatible gauge and ghost domain	A physical zero or negative mode
Gauge stabilizer	A Killing field leaves $g^{(s)}$ unchanged	Remove the corresponding Faddeev–Popov zero mode and divide by the stabilizer volume	A bosonic collective coordinate
Genuine modulus	A family of inequivalent saddles preserves the same boundary data and action	Prime the Hessian determinant and integrate the modulus with its induced Jacobian	A generic branch-merger zero mode
Positive physical mode	Positive eigenvalue of the reduced quadratic form	Ordinary convergent Gaussian on its descent direction	Global dominance of the saddle
Negative physical mode	Negative eigenvalue after the gauge quotient	Rotate onto an oriented descent direction and retain its phase	A decay rate without an observable and parent contour
Conformal scalar sector	Wrong-sign kinetic structure on the real metric slice	Use the separately declared conformal contour	A finite physical Morse index before that contour is fixed

The physical Morse index $\mu_{\mathrm{phys}}$ is the number of negative eigenvalues of the reduced quadratic form on its stated domain. It excludes boundary-trivial gauge directions and the conformal-factor divergence treated on the preceding page.

The spin-two operator is geometric, not a matter-mixing label

On a pure-Einstein background, the TT block can be written

$$\mathcal O_2 =\Delta_L^{(2)}-\frac{4\Lambda}{D-2},$$

where

$$(\Delta_L^{(2)}h)_{\mu\nu} =-\nabla^2h_{\mu\nu} -2R_{\mu\rho\nu\sigma}h^{\rho\sigma} +2R_{(\mu}{}^\rho h_{\nu)\rho}.$$

Using the Einstein equation, the Ricci term cancels the explicit cosmological-constant shift, so the operator used below is

$$(\mathcal O_2 h)_{\mu\nu} =-\nabla^2h_{\mu\nu} -2R_{\mu\rho\nu\sigma}h^{\rho\sigma}.$$

Its domain consists of regular TT tensors obeying the canonical boundary conditions and the norm requirement

$$\lVert h\rVert^2 =\int d^Dx\sqrt g\,h_{\mu\nu}h^{\mu\nu}<\infty.$$

Normalize its eigenmodes so that

$$\mathcal O_2 h^{(a)}=\lambda_a h^{(a)}, \qquad I_E^{(2)}=\frac12\sum_a\lambda_a q_a^2$$

on the physical TT block. With matter, one must diagonalize a coupled metric–matter Hessian. Matter mixing is not part of the Lichnerowicz operator itself.

At an isolated zero crossing, the Gaussian approximation fails. The zero mode may mark coalescing saddle branches rather than an exact modulus, so it must be treated with the leading nonquadratic terms rather than automatically integrated as a collective coordinate.

A negative Gaussian acquires a phase only on a chosen contour

In a finite-dimensional regulator, Picard–Lefschetz theory gives

$$Z_\Gamma\simeq\sum_s n_s Z_s, \qquad n_s=\langle\Gamma,\mathcal K_s\rangle,$$

where $\Gamma$ is the parent cycle, $\mathcal K_s$ is the upward cycle dual to the saddle thimble, and $n_s$ is their oriented intersection number. A regular solution with $n_s=0$ is absent from this decomposition. In the unregulated gravitational functional integral, the same notation is a formal guide whose definition must be inherited from a regulator or nonperturbative completion.

Suppose one normalized physical coordinate has $\lambda=-|\lambda|$. The real Gaussian diverges, but either oriented descent ray $q=\pm iy$ gives

$$\int_{\mathcal J_\pm}dq\, e^{+|\lambda|q^2/(2\hbar)} =\pm i\sqrt{\frac{2\pi\hbar}{|\lambda|}}.$$

The sign follows from the orientation inherited from $\Gamma$. It is not determined by the spectrum. Nor does this local factor by itself supply the familiar $i/2$ of a false-vacuum rate: the half-thimble, analytic continuation, and persistence boundary conditions belong to the decay observable developed in Decay Rates, the Negative Mode, and Prefactors.

After all sectors are treated, it is safer to display the one-loop bookkeeping schematically as

$$Z_s =n_s e^{-I_s/\hbar} e^{i\vartheta_s(\Gamma)} \mathcal D_s^{\mathrm{nz}} J_{\mathrm{zm}}\int_{\mathcal M_s}d\mu_s .$$

Here $\mathcal D_s^{\mathrm{nz}}$ is the regulated nonzero-mode factor of the complete gauge-fixed system, including ghosts where they have not already been canceled; $\mathcal M_s$ contains genuine moduli; and $J_{\mathrm{zm}}$ is their Jacobian. Stabilizer volumes are divided out separately. This notation avoids simultaneously inserting an explicit ghost determinant and calling the remaining bosonic determinant “physical,” which would mix two different reduction conventions.

The complete phase can differ from the phase of one TT mode. A sharp example is the $S^2\times S^{D-2}$ saddle: it has one negative TT mode, yet a calculation including the scalar and measure sectors gives a real positive empty-saddle path integral Shi and Turiaci 2025, §§3–4. A 2026 preprint finds an imaginary transition amplitude after adding the observer data appropriate to black-hole nucleation Shi, Turiaci, and Wu 2026, §§1–2. The geometry and TT index did not answer the observable question on their own.

Canonical Schwarzschild–AdS₄ separates three questions

Take $\Lambda=-3/L^2$ and

$$ds^2=f(r)d\tau^2+\frac{dr^2}{f(r)}+r^2d\Omega_2^2,$$ $$f(r)=1-\frac{r_+}{r}\left(1+\frac{r_+^2}{L^2}\right) +\frac{r^2}{L^2}.$$

Regularity at $r=r_+$ fixes the period of Euclidean time. With the conventional normalization of the conformal boundary metric,

$$\beta_H=\frac{4\pi r_+L^2}{L^2+3r_+^2}, \qquad M=\frac{r_+}{2G}\left(1+\frac{r_+^2}{L^2}\right), \qquad S=\frac{\pi r_+^2}{G}.$$

Differentiating along the family gives

$$C=\frac{dM}{dT} =\frac{2\pi r_+^2}{G} \frac{L^2+3r_+^2}{3r_+^2-L^2}.$$

The small branch $r_+<L/\sqrt3$ has $C<0$; the large branch has $C>0$. A useful independent diagnostic is the fixed-$\beta$ reduced action on the standard constraint-satisfying spherical family,

$$I_{\mathrm{red}}(r;\beta) =\beta M(r)-S(r) =\frac{\beta r}{2G}\left(1+\frac{r^2}{L^2}\right) -\frac{\pi r^2}{G}.$$

Its stationary-point condition is exactly $\beta=\beta_H$. At that point,

$$I_{\mathrm{red}}'' =\frac{2\pi}{G} \frac{3r_+^2-L^2}{L^2+3r_+^2} =\frac{4\pi^2r_+^2/G^2}{C}.$$

Thus the reduced family supplies a negative variational direction on the small branch. It proves the existence of at least one negative direction if that variation is admissible, but it does not prove uniqueness and a positive curvature along this one family does not exclude negative modes elsewhere. Off shell, this is a constrained thermodynamic reduction; simply changing $r_+$ at fixed $\beta$ inside the on-shell metric would instead create a conical bolt. The full spectral problem below supplies the independent physical-mode result. The relation between the constraint-satisfying reduced action and canonical stability was developed explicitly for finite gravitational cavities York 1986, §§II–III.

The static spherical TT problem reduces to one radial equation

For a general spherical metric with function $f(r)$, write a static mixed-index TT perturbation as Prestidge 2000, §§3–5

$$h^\mu{}_{\nu} =\operatorname{diag}\bigl(\psi(r),\chi(r),k(r),k(r)\bigr), \qquad k=-\frac{\psi+\chi}{2}.$$

Tracelessness fixes $k$. Transversality fixes the second radial amplitude:

$$\psi =\frac{2rf}{rf'-2f}\chi' +\frac{rf'+6f}{rf'-2f}\chi.$$

Substitution into $\mathcal O_2h=\lambda h$ leaves a second-order equation for $\chi$. The overall normalization of $\chi$ is irrelevant; regularity at the horizon, regular continuation through an apparent interior singular point, and normalizability at the AdS boundary quantize $\lambda$.

Reproducible Schwarzschild–AdS shooting problem

Define

$$\rho=\frac{\sqrt3\,r}{L}, \qquad \rho_+=\frac{\sqrt3\,r_+}{L}, \qquad a=\frac{\rho_+^3+3\rho_+}{4}, \qquad \widetilde\lambda=L^2\lambda.$$

The TT equation becomes

$$\begin{aligned} &\rho(\rho-2a)(\rho^3+3\rho-4a)\chi'' \\ &\quad+4\left(2\rho^4-5a\rho^3+3\rho^2-11a\rho+8a^2\right)\chi' \\ &\quad+\left[(10+\widetilde\lambda)\rho^3 -2a(18+\widetilde\lambda)\rho^2-16a\right]\chi=0. \end{aligned}$$

Normalize $\chi(\rho_+)=1$. The regular Frobenius branch at the horizon begins with

$$\chi'(\rho_+) =-\frac{\rho_+^2(18+\widetilde\lambda)+24} {6\rho_+(\rho_+^2+1)}.$$

The parametrization has another regular singular point at $\rho=2a$. Smoothness imposes

$$\chi'(2a)=-\frac{2}{a}\chi(2a).$$

At large $\rho$,

$$\chi(\rho)\sim A(\widetilde\lambda)\rho^{-7/2+\sqrt{9/4-\widetilde\lambda}} +B(\widetilde\lambda)\rho^{-7/2-\sqrt{9/4-\widetilde\lambda}}.$$

Square integrability selects $A(\widetilde\lambda)=0$. Numerically, one starts with the horizon series, continues through $\rho=2a$ using its regular series, and tunes $\widetilde\lambda$ until the nonnormalizable coefficient $A$ vanishes. This is an eigenvalue problem, not a fit to thermodynamic data.

The result is one negative eigenvalue for $0<\rho_+<1$, a zero crossing at $\rho_+=1$, and no negative eigenvalue on the larger branch Prestidge 2000, §5 and fig. 4. Since $\rho_+=1$ means $r_+=L/\sqrt3$, the spectral crossing agrees with the heat-capacity sign change.

The small-hole limit recovers asymptotically flat Euclidean Schwarzschild. If $m=GM$ denotes the geometric Schwarzschild mass, so $r_+=2m$, the lowest eigenvalue approaches

$$m^2\lambda_{\mathrm{GPY}}\simeq-0.192, \qquad r_+^2\lambda_{\mathrm{GPY}}\simeq-0.768.$$

Gross, Perry, and Yaffe found this unique static spherical nonconformal mode in the conventional fixed-$\beta$ Euclidean problem Gross, Perry, and Yaffe 1982, pp. 330–355; an action-first gauge-invariant reduction gives the same single-field spectrum Kol 2008, §3. Hot asymptotically flat space still has an infrared Jeans problem, so this limit is not a stable canonical equilibrium at infinity.

The reproducible check node scripts/benchmark-euclidean-schwarzschild-negative-mode.mjs independently integrates Kol’s gauge-invariant master equation from the bolt and from a large-radius cutoff. Refining both the cutoff and the Runge–Kutta step gives $\kappa=0.8761604020$ and $\lambda r_+^2=-0.7676570501$, with zero nodes and a scale-invariant matching residual below $5\times10^{-15}$.

This Euclidean eigenmode is also not a growing Lorentzian perturbation of the four-dimensional Schwarzschild black hole. Lorentzian Schwarzschild is linearly stable Dafermos, Holzegel, and Rodnianski 2019. After uplifting to a translationally invariant black string, however, the same eigenvalue sets the threshold Gregory–Laflamme wavenumber; that is a different Lorentzian system Reall 2001, §§2–4.

Local stability, dominance, and contour membership are distinct

For Schwarzschild–AdS$_4$, the black-hole free energy relative to thermal AdS is

$$F_{\mathrm{BH}}-F_{\mathrm{AdS}} =\frac{r_+}{4G}\left(1-\frac{r_+^2}{L^2}\right).$$

The resulting regime table keeps three questions separate:

Regime	$C$	$\mu_{\mathrm{phys}}$ in the static spherical TT sector	Classical action relative to thermal AdS	Licensed conclusion
$0<r_+/L<1/\sqrt3$	Negative	$1$	Higher	Locally unstable and classically subdominant
$r_+/L=1/\sqrt3$	Divergent	$0$, with one zero mode	Higher	Branch merger; the Gaussian approximation fails
$1/\sqrt3<r_+/L<1$	Positive	$0$	Higher	Locally stable but classically subdominant
$r_+/L=1$	Positive	$0$	Equal	Hawking–Page exchange between admitted saddles
$r_+/L>1$	Positive	$0$	Lower	Locally stable and classically dominant among these admitted real saddles
Any value of $r_+/L$	—	—	—	Actual contribution still requires $n_s\ne0$ and a defined phase

The dedicated Hawking–Page page develops the global saddle comparison. The present calculation answers a different question: whether the quadratic form has a physical negative direction.

The action crossing and its separation from the minimum-temperature point are the original Hawking–Page result Hawking and Page 1983, §§II–III.

Gauge and ensemble stress tests answer different objections

For the conventional asymptotic TT problem, take the de Donder family

$$F_\mu=\nabla^\nu h_{\mu\nu}-\frac12\nabla_\mu h, \qquad I_{\mathrm{gf}}^{(\alpha)} =\frac{1}{32\pi G\alpha}\int d^Dx\sqrt g\,F_\mu F^\mu, \qquad \alpha>0.$$

The GPY representative is transverse and traceless, so $F_\mu=0$. Repeating the calculation at, for example, $\alpha=1$ and $\alpha=2$ changes longitudinal and ghost blocks but leaves $\mathcal O_2$, $\lambda_{\mathrm{GPY}}$, and $\mu_{\mathrm{phys}}=1$ unchanged. The action-first gauge-invariant reduction provides an independent route to the same eigenvalue. If a candidate mode moves into a boundary-trivial gauge or ghost sector under this comparison, it was never a physical negative mode.

This test has hypotheses: the two gauges must use the same physical boundary data, boundary-preserving gauge group, BRST-compatible ghost domain, contour, and zero-mode prescription. It cannot be exported unchanged to a finite Dirichlet cavity. Earlier analyses already showed that the wall conditions determine whether the Schwarzschild negative mode survives Allen 1984, pp. 1153–1157 and that a fixed induced metric restores agreement with canonical thermodynamics Gregory and Ross 2001, §§2–4. More generally, the wall condition can couple TT and trace modes, the relevant operator may have complex eigenvalues, and a proposed contour criterion then tests $\operatorname{Re}\lambda$ rather than a self-adjoint TT count Marolf and Santos 2022, “Canonical Ensemble Reloaded,” §§1–3.

Changing ensemble is not a gauge test. The fixed-$\beta$ radius variation obeys

$$\delta M=M'(r_+)\delta r_+,$$

so it is not an admissible fixed-energy variation unless additional fluctuations enforce the microcanonical constraint. In a specific off-shell-energy construction and contour prescription, the Euclidean Schwarzschild and Schwarzschild–AdS saddles are quadratically stable in the microcanonical problem Marolf and Santos 2022, “Stability of the Microcanonical Ensemble,” §§1 and 4. The boundary term, allowed domain, and observable changed; this is not a contradiction with the canonical index.

Finally, a gauge-fixing parameter must not be confused with the parameter in a DeWitt metric on field space used to select a contour. In four bulk dimensions, one studied prescription continues to match canonical thermodynamic stability only for the finite interval $-2<\lambda_{\mathrm{DW}}<-1/2$; outside it, failures correlate with gauge degeneration or loss of diagonalizability Liu, Marolf, and Santos 2024, §§1 and 5. This is evidence about that contour proposal, not a universal theorem selecting the gravitational cycle.

Evidence, controls, and claim ceiling

Evidence and proposal status were checked through 29 August 2026.

Claim	Control or evidence	Ceiling
Small Schwarzschild–AdS$_4$ has one static spherical TT negative mode	Explicit regular, normalizable shooting problem and Prestidge’s numerical spectrum	This sector does not exclude negative matter or nonspherical modes in another theory
The mode crosses zero at $r_+=L/\sqrt3$	Spectral result agrees with $C^{-1}=0$ and $I_{\mathrm{red}}''=0$	Agreement is for the stated canonical boundary problem
Large Schwarzschild–AdS$_4$ has no such negative mode	Same TT spectrum	Absence does not prove global dominance, contour membership, or convergence of the topology sum
The physical index is gauge independent	Same boundary data and gauge quotient; TT and action-first reductions agree in the Schwarzschild limit	Raw gauge-fixed spectra need not match sector by sector
Ensemble can change stability	Canonical and microcanonical allowed variations differ; modern constructions realize the difference	Current gravitational contour rules remain proposals
One TT negative mode need not make the full answer imaginary	Complete phase calculations can include compensating scalar, ghost, measure, or observer sectors	The result is observable- and contour-specific

Fixed induced-metric Dirichlet data do not in general define an elliptic Euclidean Einstein boundary problem. Alternative well-posed boundary conditions can alter the relation among thermodynamic, Euclidean, and Lorentzian stability Liu, Santos, and Wiseman 2024, §§1 and 6. Higher-order interactions, extra matter modes, endpoints of moduli space, Stokes jumps, and regulator dependence remain outside a one-loop Morse-index calculation.

A recent wormhole application finds negative canonical overlap saddles and a different microcanonical conclusion in its specific model Barbón and Velasco-Aja 2025, §§3–5. It reinforces the method used here: define the observable and boundary data before interpreting a mode. Apply that checklist next on Euclidean Wormholes and Connected Boundary Amplitudes.

Common pitfalls

“TT” does not automatically mean “physical.” It is a useful representative only after boundary conditions, residual kernels, and the gauge quotient are fixed. Finite walls can mix TT and trace sectors.

Negative heat capacity does not count the entire spectrum. It supplies a negative direction in a constrained thermodynamic family. The full Morse index comes from the operator domain.

One negative mode does not automatically produce a decay rate. The parent cycle fixes the thimble coefficient and orientation; the state or observable fixes whether an imaginary discontinuity has a decay interpretation.

No negative mode does not mean dominant. The interval $L/\sqrt3<r_+<L$ is the clean counterexample: the black hole is locally stable but has higher action than thermal AdS.

A zero crossing is not automatically a modulus. At $r_+=L/\sqrt3$, two thermal branches merge. Cubic or higher terms replace the Gaussian description.

A Euclidean negative mode is not automatically a Lorentzian instability. The four-dimensional Schwarzschild spacetime is linearly stable; the Gregory–Laflamme relation arises only after a black-string uplift.

Exercises

1. Derive the canonical curvature

Starting from $M(r_+)$ and $S(r_+)$ above, derive $\beta_H$, $C$, and $I_{\mathrm{red}}''$.

Solution

The derivatives are

$$M'=\frac{1+3r_+^2/L^2}{2G}, \qquad S'=\frac{2\pi r_+}{G}.$$

Stationarity of $I_{\mathrm{red}}=\beta M-S$ gives

$$\beta_H=\frac{S'}{M'} =\frac{4\pi r_+L^2}{L^2+3r_+^2}.$$

Differentiating $T=1/\beta_H$ gives

$$C=\frac{M'}{T'} =\frac{2\pi r_+^2}{G} \frac{L^2+3r_+^2}{3r_+^2-L^2}.$$

Finally,

$$I_{\mathrm{red}}''\big|_{\beta_H} =\beta_H M''-S'' =\frac{2\pi}{G} \frac{3r_+^2-L^2}{L^2+3r_+^2}.$$

2. Check the two radial regularity conditions

Use the TT relation for $\psi$ to show that regularity at the horizon requires $\psi(r_+)=\chi(r_+)$. Then explain why finiteness at a root $r_s$ of $rf'-2f$ imposes

$$\frac{\chi'(r_s)}{\chi(r_s)} =-\frac{rf'+6f}{2rf}\bigg|_{r_s}.$$

Solution

Near a nonextremal horizon, $f\simeq f'(r_+)(r-r_+)$. The regular Frobenius branch of the radial equation makes the derivative term in the numerator of $\psi$ vanish at the same rate as $f$, leaving $\psi(r_+)=\chi(r_+)$. At $r_s$, the denominator $rf'-2f$ vanishes. Finiteness requires the numerator to vanish as well:

$$2r_sf(r_s)\chi'(r_s) +\bigl[r_sf'(r_s)+6f(r_s)\bigr]\chi(r_s)=0,$$

which is the stated condition. In the dimensionless Schwarzschild–AdS variables, $r_s$ becomes $\rho=2a$ and this reduces to $\chi'(2a)=-2\chi(2a)/a$.

3. Classify three black holes

Classify $r_+/L=0.4$, $0.8$, and $1.2$ by local stability and classical dominance. Does any classification prove that the saddle contributes?

Solution

Because $1/\sqrt3\simeq0.577$, the $0.4$ black hole has one TT negative mode and is locally unstable. Both $0.4$ and $0.8$ lie below the Hawking–Page value $1$, so their actions exceed that of thermal AdS; the $0.8$ solution is nevertheless locally stable. The $1.2$ solution is locally stable and has lower classical action than thermal AdS. None is guaranteed to contribute until the parent cycle gives $n_s\ne0$ and fixes the oriented phase.

4. Separate a gauge orbit from a boundary excitation

Let $h_{\mu\nu}=2\nabla_{(\mu}\xi_{\nu)}$. What changes when $\xi$ vanishes at the boundary versus when it changes the boundary metric?

Solution

If $\xi$ preserves all boundary data, $h$ is tangent to a gauge orbit and is excluded from the physical Morse index by the quotient and ghost treatment. If $\xi$ changes a prescribed boundary source, it is not an allowed gauge redundancy of that variational problem. Depending on the new problem, it may be excluded altogether or become a physical boundary excitation.

5. Assemble a saddle with one negative and two zero modes

Suppose $\mu_{\mathrm{phys}}=1$, there are two genuine moduli, and the saddle has $n_s=0$ or $1$. State the one-loop treatment in each case.

Solution

Remove the two zero eigenvalues from the nonzero-mode determinant and replace them by

$$J_{\mathrm{zm}}\int_{\mathcal M_s}d^2a.$$

Rotate the negative coordinate onto the oriented descent contour and retain its phase $\pm i$. If $n_s=0$, the entire saddle contribution vanishes despite these local factors. If $n_s=1$, the saddle contributes with the phase inherited from the parent cycle, but a decay rate still requires the appropriate persistence observable and analytic continuation.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Allen, Bruce. “Euclidean Schwarzschild Negative Mode.” Physical Review D 30 (1984): 1153–1157. DOI.
• Barbón, José L. F., and Eliezer Velasco-Aja. “A Note on Black Hole Entropy and Wormhole Instabilities.” Journal of High Energy Physics 08 (2025): 103. DOI. Open preprint.
• Dafermos, Mihalis, Gustav Holzegel, and Igor Rodnianski. “The Linear Stability of the Schwarzschild Solution to Gravitational Perturbations.” Acta Mathematica 222 (2019): 1–214. DOI. Open preprint.
• Gregory, Ruth, and Simon F. Ross. “Stability and the Negative Mode for Schwarzschild in a Finite Cavity.” Physical Review D 64 (2001): 124006. DOI. Open preprint.
• Gross, David J., Malcolm J. Perry, and Laurence G. Yaffe. “Instability of Flat Space at Finite Temperature.” Physical Review D 25 (1982): 330–355. DOI.
• Hawking, Stephen W., and Don N. Page. “Thermodynamics of Black Holes in Anti-de Sitter Space.” Communications in Mathematical Physics 87 (1983): 577–588. DOI.
• Kol, Barak. “Gauge-Eliminated Derivation of the Black-Hole Negative Mode.” Physical Review D 77 (2008): 044039. DOI. Open preprint.
• Liu, Xiaoyi, Donald Marolf, and Jorge E. Santos. “Stability of Saddles and Choices of Contour in the Euclidean Path Integral for Linearized Gravity: Dependence on the DeWitt Parameter.” Journal of High Energy Physics 05 (2024): 087. DOI. Open preprint.
• Liu, Xiaoyi, Jorge E. Santos, and Toby Wiseman. “New Well-Posed Boundary Conditions for Semi-Classical Euclidean Gravity.” Journal of High Energy Physics 06 (2024): 044. DOI. Open preprint.
• Marolf, Donald, and Jorge E. Santos. “The Canonical Ensemble Reloaded: The Complex-Stability of Euclidean Quantum Gravity for Black Holes in a Box.” Journal of High Energy Physics 08 (2022): 215. DOI. Open preprint.
• Marolf, Donald, and Jorge E. Santos. “Stability of the Microcanonical Ensemble in Euclidean Quantum Gravity.” Journal of High Energy Physics 11 (2022): 046. DOI. Open preprint.
• Prestidge, Tim. “Dynamic and Thermodynamic Stability and Negative Modes in Schwarzschild–Anti-de Sitter.” Physical Review D 61 (2000): 084002. DOI. Open preprint.
• Reall, Harvey S. “Classical and Thermodynamic Stability of Black Branes.” Physical Review D 64 (2001): 044005. DOI. Open preprint.
• Shi, Xiaoyi, and Gustavo J. Turiaci. “The Phase of the Gravitational Path Integral.” Journal of High Energy Physics 07 (2025): 047. DOI. Open preprint.
• Shi, Xiaoyi, Gustavo J. Turiaci, and Chih-Hung Wu. “The Fate of Nucleated Black Holes in de Sitter Quantum Gravity.” arXiv:2605.03015 (2026). Open preprint.
• York, James W., Jr. “Black-Hole Thermodynamics and the Euclidean Einstein Action.” Physical Review D 33 (1986): 2092–2099. DOI.