Euclidean Actions, Boundary Terms, and Free-Energy Comparisons

A Euclidean black-hole free energy is meaningful only after the action, boundary data, counterterms, gauge ensemble, and integration contour have been fixed. Dropping a Gibbons–Hawking term, comparing unmatched cutoffs, or switching from fixed potential to fixed charge changes the variational problem rather than producing a harmless convention change.

Required background. Metric Counterterms and the Boundary Stress Tensor constructs the finite gravitational action. Euclidean Saddles, Thermal States, and Hawking–Page Transitions supplies the saddle comparison.

Helpful background. Euclidean Gravitational Saddles and Boundary Terms develops the gravitational-EFT setting.

First application. Evaluate a renormalized Euclidean AdS black-hole action in the canonical ensemble and recover energy, entropy, and free energy by thermodynamic derivatives.

A variationally complete Euclidean action

For Einstein gravity with a radial boundary, a standard Dirichlet action is

$$I_E =-\frac{1}{2\kappa^2}\int_M d^{d+1}x\sqrt g\,(R-2\Lambda) -\frac{1}{\kappa^2}\int_{\partial M}d^dx\sqrt\gamma\,K +I_{\mathrm{ct}}[\gamma] +I_{\mathrm{matter}}.$$

The sign of $K$ follows the declared outward-normal convention. $I_{\mathrm{ct}}$ is a local functional of induced boundary fields; for asymptotically AdS gravity it replaces background subtraction and allows saddles with different interiors to be compared at the same boundary sources. Corners or null pieces require their own terms when present, though a smooth stationary Euclidean black hole has neither at the regular horizon.

For a Maxwell field, the usual bulk action with fixed boundary $A_\tau$ gives the grand-canonical ensemble. Passing to fixed electric flux requires a boundary Legendre term. The on-shell values therefore compute different thermodynamic potentials:

$$I_E^{(\mu)}=\beta\Omega(T,\mu), \qquad I_E^{(Q)}=\beta F(T,Q) =I_E^{(\mu)}+\beta\mu Q,$$

with the final sign tied to the definition of $Q$ and the Euclidean continuation of $A_\tau$.

Recovering energy, entropy, and free energy

At a smooth saddle with fixed external sources,

$$Z(\beta)\simeq e^{-I_E(\beta)}, \qquad F=\frac{I_E}{\beta}, \qquad E=\left.\frac{\partial I_E}{\partial\beta}\right|_{\text{sources}},$$

and

$$S=\beta(E-F) =\left(\beta\frac{\partial}{\partial\beta}-1\right)I_E.$$

These identities are a stringent check because $E$ must also equal the charge obtained from the renormalized boundary stress tensor, and $S$ must agree with the appropriate horizon entropy. For two-derivative Einstein gravity this is $A_h/(4G)$; for a higher-derivative action it is generally a Noether-charge entropy.

A reliable calculation follows this order:

1. declare induced metric, matter sources, and ensemble;
2. make the Euclidean section smooth, thereby fixing the thermal identification;
3. regulate all saddles on matched induced boundary data;
4. include bulk, Gibbons–Hawking, matter, finite-ensemble, and counterterm contributions;
5. remove the cutoff and differentiate while holding the declared sources fixed;
6. verify the first law and the stress-tensor charges.

The counterterm method of Balasubramanian and Kraus 1999, §§2–4 makes steps 3–5 intrinsic to the boundary geometry.

Local versus global stability

The least Euclidean action among allowed saddles gives global dominance at leading semiclassical order. Local thermodynamic stability instead concerns the Hessian of the relevant potential. In the canonical ensemble, a negative heat capacity is a negative direction in energy fluctuations; in the grand-canonical ensemble, charge and angular-momentum susceptibilities enter as well.

A Euclidean negative mode is related to, but not identical with, thermodynamic instability. Gauge fixing, zero modes, conformal-factor rotation, and the integration cycle determine which fluctuations contribute. A stationary point with lower real action is not automatically on the chosen steepest-descent contour. This page uses the Euclidean saddle expansion conditionally; the nonperturbative contour question remains separate.

Adversarial control: omit one term

Omitting the Gibbons–Hawking term leaves uncancelled normal derivatives in the metric variation, so the claimed Dirichlet saddle is not stationary. Omitting the Maxwell Legendre term while calling the result “fixed charge” differentiates the wrong thermodynamic potential. Omitting a logarithmic counterterm changes the anomaly and leaves scale dependence that cannot be removed by taking the cutoff away.

These failures can be caught without knowing the expected free energy: vary the complete action and inspect boundary terms, then compare $\partial_\beta I_E$ with the independently renormalized energy.

Evidence ceiling

A finite, variationally correct on-shell action and consistent thermodynamic derivatives establish the contribution of a specified Euclidean saddle in a specified ensemble. They do not prove that the gravitational path integral converges, that this saddle lies on the defining contour, that no other topology dominates, or that its entropy has been microscopically counted.

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

• Balasubramanian, Vijay, and Per Kraus. “A Stress Tensor for Anti-de Sitter Gravity.” Communications in Mathematical Physics 208, 413–428 (1999). DOI; arXiv:hep-th/9902121.
• Gibbons, G. W., and Stephen W. Hawking. “Action Integrals and Partition Functions in Quantum Gravity.” Physical Review D 15, 2752–2756 (1977). DOI.
• Hawking, Stephen W., and Don N. Page. “Thermodynamics of Black Holes in Anti-de Sitter Space.” Communications in Mathematical Physics 87, 577–588 (1983). DOI.