Boulware-Like State in Curved Spacetime

• Boulware-like state is a quantum vacuum defined to be empty at both past and future null infinity, serving as a reference for static observers in curved spacetimes.
• It features a renormalized stress-energy tensor that vanishes at infinity but diverges near event horizons, highlighting key semiclassical effects.
• Extensions to rotating, charged, and higher-dimensional black holes reveal its role in quantum backreaction, holography, and instability analysis.

A Boulware-like state is a quantum state of a field in curved spacetime defined by the requirement that the field be empty, i.e., devoid of particles, at both past and future null infinity. Historically, the canonical Boulware vacuum is constructed for static, spherically symmetric black hole spacetimes (such as Schwarzschild) and is the unique static vacuum that reduces to the Minkowski vacuum at spatial infinity. However, the general concept of a Boulware-like state has now been extended to more general settings, including rotating (Kerr) black holes, Reissner–Nordström (charged) black holes, higher-dimensional black holes, and even to settings of massive gravity and semiclassical backreaction, often with major technical and physical implications for quantum stress tensors and semiclassical geometry. In quantum gravity and quantum field theory in curved spacetime, Boulware-like states serve as a reference vacuum for static observers and play a central role in the paper of quantum effects near horizons, the structure of the stress-energy tensor, and the viability of candidate semiclassical black hole geometries.

1. Definition and Construction of Boulware-like States

The Boulware-like state is fixed by the choice of positive frequency with respect to the static (Killing) time coordinate for field modes, both at past and future infinity. For the prototypical example, consider a massless scalar field in Schwarzschild spacetime. The field expansion

$$\phi = \sum_{\ell,m} \int_0^\infty d\omega \big[a_{\omega\ell m}^{\rm in} \phi_{\omega\ell m}^{\rm in} + a_{\omega\ell m}^{{\rm up}\,\dagger} \phi_{\omega\ell m}^{{\rm up}}\big]$$

defines "in" and "up" modes as positive frequency with respect to the Schwarzschild time at spatial infinity and on the past horizon, respectively. The Boulware state |B⟩ is annihilated by all $a_{\omega\ell m}^{\rm in}$ and $a_{\omega\ell m}^{\rm up}$. This ensures that, for a static exterior star or black hole spacetime, the stress-energy tensor vanishes at infinity and that no outgoing or incoming particles are detected by static observers at large r.

On rotating backgrounds (Kerr), the construction is complicated by the existence of superradiant modes (for bosons) and the shifted natural frequency near the horizon. The candidate Boulware-like state for fermions is defined by using the positive norm of all fermion modes to construct an analog, empty at both past and future null infinity, and invariant under time–reversal (t–ϕ) (Casals et al., 2012).

For charged fields on charged (Reissner–Nordström) black holes, the construction must take into account the shifted frequency due to the gauge potential, leading to analogous Boulware-like and Hartle–Hawking–like states (Balakumar et al., 2023, Balakumar et al., 2022).

Boulware-like states can also be formulated in higher-curvature or modified gravity backgrounds, such as in higher dimensional Einstein–Gauss–Bonnet black holes or in massive gravity, where the vacuum structure interacts nontrivially with extra degrees of freedom and their constraint algebra (López et al., 2015, Masood, 2 Jul 2024, Mukohyama, 2013).

2. Physical Properties and Stress-Energy Tensor

A distinctive property of the Boulware-like state is the behavior of its renormalized stress–energy tensor (RSET) $\langle T_{\mu\nu}\rangle$. For Schwarzschild and Reissner–Nordström spacetimes, the Boulware vacuum is unique in yielding a regular (Minkowski) RSET at spatial infinity, but the RSET becomes singular as one approaches the event horizon: $$\langle B| T_{\mu\nu} |B \rangle \longrightarrow \infty \quad {\rm as}\ r\to r_+$$ for the horizon at $r_+$. (This divergence persists for non-extremal black holes if one ignores quantum backreaction, but the situation is subtler at extremality, as discussed below.)

On asymptotically AdS backgrounds, especially in the context of holography, the dual of the Boulware-like state is the zero-temperature black droplet—a bulk solution that describes a CFT on a black hole background with zero local and global temperature. The holographic stress tensor calculated from such bulk solutions is consistently regular on the entire Schwarzschild horizon at leading order in $N_c$, in contrast with the singular free-theory result. This regularity is enforced by the interacting dynamics of the strongly coupled field theory and the precise implementation of boundary conditions in the Ricci–DeTurck flow approach (Figueras et al., 2011, Fischetti et al., 2016).

For rotating and charged black holes, the RSET in the candidate Boulware-like state often fails to be globally regular due to the presence of superradiant amplification (for bosons) or due to the lack of an equilibrium state simultaneously regular at both the horizon and infinity (as per the Kay–Wald theorem) (Balakumar et al., 2023, Balakumar et al., 2022).

3. Extensions and Generalizations

a) Semiclassical Backreaction and Geometry

Taking into account quantum back-reaction leads to novel modifications of black hole geometry. In two-dimensional dilaton gravity (CGHS/RST models), the back-reaction of the Boulware state generically eliminates the classical horizon, replacing it with a throat with an extremely small $-g_{tt}$ bounded by the inverse of the black hole entropy. On the other side of the throat, the spacetime ends at a null or naked singularity (Potaux et al., 2021, Potaux et al., 2023). For a suitable dominance of ghost (non-physical) fields in the Boulware state, the resulting geometry can be a geodesically complete, horizonless spacetime mimicking black hole properties and exhibiting a type II wormhole structure, with outgoing flux at infinity and invariably a Page-curve behavior in the entropy.

b) Casimir and Atom Probes

The Boulware vacuum can be operationally probed by considering detectors or atoms near a black hole enclosed by a Casimir mirror. Observables such as the Casimir–Polder–like force or the excitation probability of an Unruh--DeWitt detector reflect the curvature-induced, position-dependent modification of the vacuum fluctuations (Zhang et al., 2011, Masood, 2 Jul 2024). For example, the force on an atom at radius r in the Boulware vacuum is attractive near the horizon and repulsive at large r (falling off as $1/r^3$), and the effect of higher-curvature corrections can be tracked through changes in the excitation rate of the detector, with the Gauss–Bonnet coupling parameter $\alpha$ enhancing or suppressing acceleration radiation.

c) Entanglement and Holography

In the context of AdS/CFT, the Boulware-like state for the boundary theory is dual to a particular bulk geometry (e.g., a black droplet). Holographic calculations of the entanglement entropy for "Boulware" states in Rindler or de Sitter spacetime reveal that, while the classical Ryu–Takayanagi area formula yields a nonzero bridge entropy, the inclusion of quantum fluctuations in the effective two-dimensional throat region leads to a vanishing of the modular entanglement entropy in the $T\rightarrow 0$ (Boulware) limit (Emparan et al., 2023). This resolves a longstanding puzzle about the mismatch between semiclassical and quantum expectations for such entanglement.

4. Regularity, Vacuum Choice, and the Anomaly-Induced Effective Action

A crucial consideration is whether the Boulware-like state is physically viable, specifically whether it produces a regular quantum stress tensor everywhere. In two dimensions, any static, horizonless, regular metric (being conformally flat) allows a Boulware vacuum with a regular renormalized stress tensor that vanishes at infinity, compatible with regularity at the center (Numajiri et al., 19 Nov 2024). In contrast, in four dimensions, the anomaly-induced effective action contains additional (non-conformal) homogeneous modes. Imposing regularity at the center of a static, horizonless spacetime forces the homogeneous part of the auxiliary field (that determines the quantum state) to be nontrivial at infinity. As a result, the regular static vacuum in four-dimensional horizonless spacetimes is not the Boulware vacuum, but a distinct, nontrivial vacuum characterized by a different RSET profile and power-law decay at infinity (Numajiri et al., 19 Nov 2024). This result clarifies that vacuum selection in semiclassical gravity is dictated both by asymptotic behavior and global regularity constraints.

5. Boulware-like States Beyond Standard QFT: Massive Gravity and Ghost Modes

In the context of massive gravity, the "Boulware-Deser" ghost is a distinct concept, referring to an additional (usually ghostly) degree of freedom that arises generically in non-linear extensions of massive gravity unless extra constraints are imposed. In this literature, a Boulware-like state denotes a configuration where the Hamiltonian analysis reveals no primary constraint to remove this mode, leading to potential instability (or, in some backgrounds, "benign" behavior) (Mukohyama, 2013, Bañados et al., 2013, Mukohyama et al., 2018). The relevant technical criterion is the invertibility of the Hessian of the Hamiltonian with respect to lapse and shift variables. In three-dimensional bigravity (Zwei-Dreibein), additional constraints can kill the Boulware–Deser mode only in restricted sectors of phase space. In four dimensions, the Ogievetsky–Polubarinov model displays a Boulware–Deser mode that is ghostlike around flat backgrounds but which can be stable ("benign") in cosmological settings, such as de Sitter or Milne universes, if specific conditions on the expansion rate and theory parameters are met.

6. Summary Table: Characteristic Features of the Boulware-like State

Setting	Vacuum Condition	Stress-Energy Tensor	Regularity on Horizon	Other Features
Schwarzschild (boson, 4D)	Empty at $\mathcal{I}^{\pm}$	Divergent at $r_+$	No	Minkowski at infinity
Kerr (boson, 4D)	Empty at $\mathcal{I}^{\pm}$	Not globally defined	No	Obstructed by superradiance
Kerr (fermion, 4D)	Empty at $\mathcal{I}^{\pm}$	Regular outside SLS	Partial: No in ergosphere	Unique to Fermi fields; mode flexibility
2D Dilaton gravity	Empty at $x^{\pm}\to \infty$	Regular if ghost fluid	Yes/No (depends on total central charge)	Throat or wormhole replaces horizon
4D, horizonless static spacetime	Vanishing at infinity	Not Boulware; new state	Yes	Nontrivial homogeneous auxiliary field
AdS/CFT dual	$T_{\rm field\ theory}=0$	Regular at horizon	Yes	Dual to black droplet geometry

7. Physical and Conceptual Significance

The Boulware-like state, as the canonical "minimal energy" vacuum for a given spacetime, provides an operational reference for defining and interpreting quantum effects near black holes and horizonless compact objects. Its divergent (or, in some cases, regularized) stress tensor structure is central to understanding phenomena such as quantum backreaction, Casimir/acceleration-induced forces, and semiclassical modifications of event horizons. The realization that the interacting strongly coupled CFT Boulware–like state is regular on horizons (whereas the free theory is not) highlights nontrivial nonperturbative effects in holographic duals (Figueras et al., 2011, Fischetti et al., 2016). Conversely, the fact that four-dimensional, horizonless regular geometries generically do not admit the Boulware vacuum simultaneously regular everywhere (Numajiri et al., 19 Nov 2024) illustrates that global spacetime properties and the effective action's structure have a decisive influence on quantum state selection.

In semiclassical gravity, the Boulware-like state illuminates both the strengths and limitations of naive vacuum constructions, demonstrates the subtlety of vacuum regularity and boundary conditions, and remains fundamental to ongoing developments in quantum black hole physics, information loss, and the structure of viable semiclassical geometries.