Hartle–Hawking Vacuum

• Hartle–Hawking vacuum is a quantum state in curved spacetime that embodies thermal equilibrium and regularity on both future and past horizons via analytic continuation.
• Its Euclidean construction employs Wick rotation and periodicity in imaginary time, directly linking surface gravity with the Hawking temperature.
• The state underpins black hole thermodynamics and quantum cosmology, influencing renormalized stress-tensor calculation and no-boundary proposals.

The Hartle–Hawking vacuum is a distinguished state in quantum field theory on curved spacetime, defined specifically for black hole backgrounds and more generally for manifolds with bifurcate Killing horizons. It arises from imposing regularity conditions on the analytically continued (Euclidean) section of the spacetime, ensuring that the quantum state is thermal at the Hawking temperature and regular on both future and past horizons. The Hartle–Hawking state plays a central role in black hole thermodynamics and in the formulation of quantum cosmology, where it inspires the eponymous no-boundary proposal for the wave function of the universe.

1. Euclidean Construction and Regularity Conditions

The essential construction of the Hartle–Hawking vacuum proceeds via analytic continuation (Wick rotation) of the black hole metric, $t \rightarrow i\tau$. In the resulting Euclidean geometry, smoothness at the horizon enforces periodicity in imaginary time: $\beta = \frac{2\pi}{\kappa}$ where $\kappa$ is the surface gravity of the black hole. This periodicity immediately yields the Hawking temperature: $T = \frac{\kappa}{2\pi}$ The requirement of absence of conical singularities at the horizon uniquely fixes the quantum state to be a thermal (KMS) state with respect to the natural time translation symmetry. The Green’s functions constructed in the Euclidean section (by solving

$$[-\nabla^2_{g_E} + m^2]\, G_E(x, x') = \delta(x, x')$$

and imposing the corresponding periodicity in $\tau$) are then analytically continued back to Lorentzian signature to yield correlation functions in the physical (Lorentzian) spacetime.

This construction guarantees that the Hartle–Hawking state is regular on both the future and past event horizons and is manifestly invariant under the Killing flow—a property that is essential for its thermodynamic interpretation. The state is also of Hadamard form, ensuring that its ultraviolet singularities match those of the Minkowski vacuum, a property necessary for local renormalizability (Sanders, 2013).

2. Stress–Energy Tensor and Hadamard Renormalization

To paper the physical properties of the Hartle–Hawking state, particularly the expectation value of the local stress–energy tensor $\langle T_{\mu\nu} \rangle$, one must handle divergences arising in the two-point function. The Hadamard renormalization procedure is employed, based on subtraction of the singular part of the Green’s function, which has the universal form: $$G_{\text{sing}}(x, x') = \frac{1}{8\pi^2} \left[\frac{\Delta^{1/2}}{\sigma} + V(x,x')\ln(\sigma/\lambda^2) + W(x,x')\right]$$ where $\sigma$ is half the square of the geodesic distance and $\Delta$ the Van Vleck–Morette determinant. The renormalized stress tensor is then computed from

$$\langle T_{\mu\nu} \rangle_{\text{ren}} = \lim_{x' \to x} \left[ \mathcal{D}_{\mu\nu} \left( G(x, x') - G_{\text{sing}}(x, x') \right) \right]$$

where $\mathcal{D}_{\mu\nu}$ is a specific differential operator encoding the geometric structure and spin of the field (Breen et al., 2011). The regularity of the Hartle–Hawking state on both the event and cosmological (for multi-horizon spacetimes) horizons can be rigorously established by verifying that combinations of stress-tensor components, such as $$(T^r_r)_{\text{ren}} - (T^\theta_\theta)_{\text{ren}}$$, vanish appropriately at the horizon, ensuring smoothness in a freely falling frame.

3. Thermal Properties and KMS Condition

The Hartle–Hawking vacuum is a Kubo–Martin–Schwinger (KMS) state at temperature $T = \kappa/(2\pi)$. In the algebraic approach, the two-point function $w_2(x, y)$ satisfies the KMS condition with respect to the Killing flow: $$w_2(f, \alpha_{t + i\beta}(g)) = w_2(\alpha_t(g), f)$$ for all $t$ and $\beta = 2\pi/\kappa$, where $\alpha_t$ is the automorphism associated with the Killing flow. The state is thus stationary and displays the correlation structure of a thermal equilibrium. In exterior (static) regions, restriction of the state yields a thermal density matrix, while globally it is pure and regular across the (bifurcate) horizon (Sanders, 2013, Higuchi et al., 2021).

This property underlies the Hawking effect: observers at infinity perceive a thermal flux of radiation at the Hawking temperature, while freely falling observers on the horizon encounter a regular (smooth) vacuum.

4. Generalizations and Stability in Various Settings

The Hartle–Hawking state has been extended to a range of circumstances:

• Spherically symmetric, multi-horizon spacetimes: In, for example, lukewarm Reissner–Nordström–de Sitter black holes (where black hole and cosmological horizons share the same temperature), a global Euclidean regularity can be maintained, and the Hartle–Hawking state is regular on all horizons if suitable conditions on the renormalized stress tensor are satisfied (Breen et al., 2011).
• Quantum collapse in AdS: For black holes forming via collapse in asymptotically AdS spacetimes, initial Boulware-like states (no flux at infinity) generically evolve—under rapid collapse and appropriate boundary conditions—into states characterized by thermal equilibrium at the Hawking temperature, with both ingoing and outgoing thermal flux, analogous to the Hartle–Hawking state (Abel et al., 2015).
• Loop Quantum Gravity: Attempts to define an initial “Hartle–Hawking state” in the context of background-independent approaches such as LQG demonstrate subtleties. A naive transcription using the Euclidean action does not solve the Lorentzian Hamiltonian constraint; instead, using the Lorentzian action provides a formal candidate state that, in symmetry-reduced sectors, can approach the Ashtekar–Lewandowski vacuum (Dhandhukiya et al., 2016).
• 3D Quantum Gravity: In three-dimensional AdS gravity, the Hartle–Hawking state admits a nonperturbative expression in terms of Liouville conformal field theory boundary states, with the thermal properties and factorization properties encoded in overlaps of “ZZ” boundary states (Chua et al., 2023).

5. Role in Quantum Cosmology and No-Boundary Proposal

The Euclidean construction of the Hartle–Hawking vacuum directly inspired the "no-boundary" wave function of the universe, positing that the quantum state of the cosmos is obtained by summing over regular Euclidean four-geometries interpolating between “nothing” (no boundary) and the observed universe. In cosmological applications, the Hartle–Hawking wave function is defined by a Euclidean path integral weighted by $\exp(-S_E)$, where $S_E$ is the Euclidean action. This construction—supported by analysis of the Wheeler–DeWitt equation and instanton solutions—leads to boundary conditions that determine the quantum state of the universe and strongly influence predictions for the primordial fluctuation spectrum, cosmic microwave background, and implications for inflation (Yeom, 2017, Kang et al., 2022).

The no-boundary proposal, and by extension the Hartle–Hawking vacuum, thus provide a unifying approach to initial conditions in quantum cosmology and to the thermodynamics of horizons in gravitational physics (Horowitz et al., 16 Sep 2025).

6. Limitations and Quantum Gravity Corrections

While the Hartle–Hawking vacuum possesses a unique set of regularity, thermal, and symmetry properties in semiclassical gravity, quantum gravity corrections can substantially modify its structure:

• String theory corrections: Non-perturbative $\alpha'$ effects in models such as $SL(2,\mathbb{R})_k / U(1)$ (the Euclidean two-dimensional black hole) induce an energy-dependent phase shift in string scattering, leading to divergences in the horizon energy density and “firewall”-like singularities, even in the eternal black hole geometry. This result indicates a failure of the classical Hartle–Hawking picture at the non-perturbative level (Ben-Israel et al., 2015).
• Vacuum dependence of decay rates: In catalysis of false vacuum decay by black holes, the Hartle–Hawking vacuum (being fully thermal) can lead to unsuppressed decay rates at high temperatures, in contrast to the Unruh vacuum which retains exponential suppression due to the absence of ingoing flux and greybody factors. This distinction is crucial in semiclassical investigations of vacuum stability and early universe scenario-building (Shkerin et al., 2021).
• Lattice simulations: Numerical investigations of the Hartle–Hawking (or Hawking–Hartle) vacuum using staggered fermion discretizations and Hadamard renormalization confirm its regularity across horizons and provide a platform for exploring quantum field theory in time-dependent or dynamical geometries (Lewis et al., 2019).

7. Mathematical Formulations and Representative Equations

The construction of the Hartle–Hawking vacuum involves several key mathematical structures:

Aspect	Mathematical Expression	Reference
Euclidean time periodicity	$\beta = 2\pi/\kappa$	(Horowitz et al., 16 Sep 2025)
Hawking temperature	$T = \kappa/(2\pi)$	(Horowitz et al., 16 Sep 2025)
Renormalized stress tensor	$$\langle T_{\mu\nu} \rangle_{\text{ren}} = \lim_{x' \to x} \{ \mathcal{D}_{\mu\nu} [G(x, x')-G_{\text{sing}}(x, x')] \}$$	(Breen et al., 2011)
KMS condition	$w_2(f, \alpha_{i\beta}(g)) = w_2(g, f)$	(Sanders, 2013)
KMS/thermal Wightman function	$$\langle \phi(x)\phi(x')\rangle = \int_0^\infty \frac{d\omega}{2\omega}\sum_\sigma \varphi_{\omega\sigma}(x)\varphi_{\omega\sigma}(x')\left[\frac{e^{-i\omega (t-t')}}{1-e^{-\omega\beta}}+\frac{e^{i\omega (t-t')}}{e^{\omega\beta}-1}\right]$$	(Higuchi et al., 2021)

These formulas encapsulate the analytic and algebraic aspects of the Hartle–Hawking state, and underlie its thermal, regularity, and symmetry properties in stationary black hole spacetimes and more general quantum gravitational systems.

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In summary, the Hartle–Hawking vacuum is defined by its analytic construction via Euclidean methods, its KMS thermal equilibrium condition at the Hawking temperature, and its regularity across Killing horizons. It serves as a canonical global state for matter fields on black hole spacetimes, grounds the thermodynamic interpretation of black holes, and motivates the no-boundary approach in quantum cosmology. Its structure remains robust in semiclassical gravity, but string theoretic and further quantum gravity corrections may introduce novel phenomena that challenge its traditional interpretation (Ben-Israel et al., 2015, Horowitz et al., 16 Sep 2025).