Euclidean Schwarzschild Black Hole

Updated 27 December 2025

Euclidean Schwarzschild black hole is a Riemannian manifold obtained by Wick rotating the Lorentzian metric, ensuring the elimination of conical singularities via periodic Euclidean time.
The formulation facilitates a gravitational path integral approach that reveals semiclassical thermodynamics, deriving key quantities like Hawking temperature and the Bekenstein–Hawking entropy.
Quantum corrections, through functional determinants and limiting curvature principles, integrate matter effects and stabilize the thermodynamic description in line with the generalized second law.

The Euclidean Schwarzschild black hole is defined by the analytic continuation of the Lorentzian Schwarzschild solution to Euclidean signature, yielding a Riemannian manifold pivotal in quantum gravity and black hole thermodynamics. The Euclidean formulation provides a geometric route to deriving black hole entropy and temperature, underpins the semiclassical gravitational path integral, and serves as the foundational instanton geometry in studies of thermodynamic stability, quantum corrections, and entropy laws.

1. Euclidean Section of the Schwarzschild Geometry

The Euclidean Schwarzschild metric arises from the Wick rotation $t \to -i\tau$ applied to the Lorentzian Schwarzschild line element: $ds^2_\mathcal{L} = -\left(1 - \frac{2GM}{r}\right)dt^2 + \left(1 - \frac{2GM}{r}\right)^{-1}dr^2 + r^2 d\Omega^2.$ After analytic continuation, the Euclidean metric is

$ds^2_E = \left(1 - \frac{2GM}{r}\right)d\tau^2 + \left(1 - \frac{2GM}{r}\right)^{-1}dr^2 + r^2 d\Omega^2.$

Analysis near the horizon $r=2GM$ shows that regularity of the $(\tau, r)$ submanifold requires the Euclidean time $\tau$ to be periodic with period $\beta = 8\pi GM$ , eliminating a conical singularity at the horizon. This fixes the Hawking temperature to

$T_H = \frac{1}{8\pi GM}.$

The period $\beta$ encodes the thermality of the Schwarzschild black hole in the Euclidean formalism and is essential for consistent gravitational thermodynamics (Heymans et al., 15 Apr 2024, El-Menoufi, 2017, Hirayama, 2010, Boos, 2023).

2. Thermodynamics and Path Integral Formulation

The Euclidean Schwarzschild space provides a saddle point for the gravitational path integral, with its semiclassical thermodynamics extracted from the partition function $Z(\beta) \sim e^{-I_{\text{tot}}}$ , where $I_{\text{tot}}$ consists of the Einstein–Hilbert (bulk) action and the Gibbons–Hawking–York (GHY) boundary term: $I_{\text{tot}} = I_{EH} + I_{GHY}.$ For Schwarzschild geometry, $R=0$ everywhere outside the central singularity, so the bulk term vanishes, and the entire entropy originates from the GHY term evaluated at large radius. Explicit evaluation yields the Bekenstein–Hawking area law for entropy,

$S = \frac{A}{4G},\quad A=4\pi (2GM)^2,$

where $A$ is the horizon area. Perturbations localized in the core do not affect the entropy provided the limiting curvature condition (core curvature independent of $M$ ) is respected, ensuring the dominance of the boundary contribution and the robustness of the area law (Boos, 2023).

3. Functional Determinants and Generalized Entropy

The inclusion of quantum matter, external fields, and horizon-localized degrees of freedom modifies the Euclidean partition function and black hole entropy. Considering a real scalar field $\phi$ coupled to a quenched Gaussian disorder $\chi$ , the effective Euclidean action is

$S_E[\phi,\chi] = \frac12 \int d^4x\,\sqrt{g}\left(g^{\mu\nu}\partial_\mu\phi\partial_\nu\phi + m_0^2\phi^2 + \frac{\lambda_0}{12}\phi^4\right) + \int d^4x\,\sqrt{g}\, \chi(x)\phi(x),$

with $\chi$ characterized by a variance $V(r)$ . The free energy, averaged over disorder, is computed using the distributional zeta-function method, resulting in a series representation: $\langle \ln Z \rangle = \sum_{k=1}^\infty c_k \langle Z^k \rangle_\chi$ where each moment $\langle Z^k \rangle_\chi$ involves functional determinants of generalized Schrödinger operators,

$L_k = -\Delta_s + m_0^2 - kV(r).$

These determinants are regularized via zeta function or heat kernel methods. The generalized entropy density,

$s_{\text{gen}}(r) = s_{\text{geom}} + s_{\text{matter}},$

combines the geometric Bekenstein–Hawking term with positive corrections from matter and disorder, ensuring a locally smooth and positive entropy density except for mild divergences near the horizon in the absence of a cutoff (Heymans et al., 15 Apr 2024).

4. Quantum Corrections and Thermodynamic Stability

Quantum gravitational and matter sector corrections to the Schwarzschild free energy are systematically included within the Euclidean approach using effective field theory and the nonlocal heat kernel formalism. At one-loop, logarithmic nonlocal terms of the form $R_{\mu\nu\rho\sigma}\ln(\Box/\mu^2)R^{\mu\nu\rho\sigma}$ emerge, leading to corrections of the free energy: $F(\beta) = \frac{\beta}{16\pi G} - \frac{128\pi^2\Xi}{\beta}\ln\left(\frac{\beta}{\beta_{QG}}\right),$ where $\Xi$ encapsulates the field content. A large number of gauge fields ( $\Xi<0$ ) stabilizes the hole above a critical mass $M_c\sim\sqrt{N}M_P$ , making the specific heat positive and removing the thermodynamic instability otherwise inherent to Schwarzschild black holes. If $\Xi>0$ , the instability persists, but the entropy at high curvature may be suppressed or vanish (El-Menoufi, 2017).

5. Negative Modes and the Spectrum of Fluctuations

The path integral formulation in Euclidean signature permits explicit analysis of small metric fluctuations via the Lichnerowicz operator $\Delta_L$ . One finds a single normalizable non-conformal negative mode when the Schwarzschild heat capacity is negative, signaling a 'tachyonic' instability at the quadratic level. The existence of this negative mode is intimately linked to thermodynamic instability and is essential for interpreting the nucleation and decay processes in the semiclassical ensemble. This correspondence holds in asymptotically flat and, more generally, Schwarzschild–AdS backgrounds, where the disappearance of the negative mode coincides with the transition to thermodynamic stability for large AdS black holes (Hirayama, 2010).

6. Core Structure and the Limiting Curvature Principle

Attempts to regularize the Schwarzschild singularity with a finite core of radius $\ell$ introduce new interior geometric contributions. However, in the Euclidean formulation, corrections to the entropy scale with $(\ell/2M)^{1-a}$ , where $a$ parametrizes the scaling of $\ell$ with the black hole mass. Imposing both exact agreement with the area law and the first law of thermodynamics at all orders enforces $a=1$ , the "limiting curvature" hypothesis: the maximal interior curvature is independent of $M$ and set by a universal bound. Thus, no modifications to the entropy arise from the core provided this condition is met, supporting the insensitivity of the Euclidean entropy to UV bulk structure (Boos, 2023).

7. Generalized Second Law in Euclidean Functional Methods

Within the Euclidean framework, the generalized entropy

$S_{\text{total}} = S_{\text{geom}} + S_{\text{matter}}$

is shown to obey the generalized second law: $\Delta S_{\text{total}} \geq 0$ under arbitrary small exchanges of energy or matter across the horizon. The geometric entropy grows with any accretion, while the matter/disorder contribution is positive-definite and monotonic under physical processes. The functional determinants appearing in the generalized entropy encode horizon and near-horizon effects, including those from disorder or emergent quantum gravitational degrees of freedom, and ensure compatibility with the generalized second law as derived in Euclidean thermodynamics (Heymans et al., 15 Apr 2024).