Boulware-like Quantum State Overview

• Boulware-like quantum states are defined via positive-frequency modes with respect to a timelike Killing vector at infinity, ensuring an asymptotic vacuum with no thermal flux.
• In static black hole geometries, the Boulware state exhibits singular stress-energy behavior at the horizon while maintaining a zero particle flux at spatial infinity.
• Modified constructions in rotating, horizonless, and holographic settings adjust near-horizon regularity and flux properties, leading to observable variations in semiclassical backreaction and entanglement.

Searching arXiv for recent and foundational papers on Boulware-like quantum states. arXiv search query: all:"Boulware-like quantum state" OR all:"Boulware state" black hole quantum state arXiv search query: ti:"Boulware" OR abs:"Boulware" AND (black hole OR Kerr OR Reissner-Nordstrom) A Boulware-like quantum state is a stationary, asymptotically vacuum state defined by positive-frequency modes with respect to a preferred timelike Killing generator at infinity, or by closely analogous no-flux asymptotic boundary data in settings where the standard Schwarzschild construction is generalized. In static black hole spacetimes, the Boulware vacuum is the state with no particles for static observers at infinity and no thermal occupation; in non-extremal cases it is typically singular at the horizon, whereas in rotating, horizonless, semiclassically back-reacted, and holographic settings the label “Boulware-like” is used for related constructions that preserve the asymptotic vacuum character while modifying regularity, flux, or state-selection properties (Arrechea et al., 2024).

1. Definition and state-theoretic status

In a static, spherically symmetric black hole spacetime with timelike Killing vector $\partial_t$, the Boulware state is the vacuum defined by positive-frequency modes with respect to the Killing time $t$. One expands the field in modes $\phi \propto e^{-i\omega t}$, associates annihilation operators with $\omega>0$, and thereby obtains a state that is vacuum at infinity, with no thermal occupation and no Hawking flux (Arrechea et al., 2024). In the Reissner–Nordström case the same definition is stated in Lorentzian language as positive frequency with respect to the static Killing time $t$ together with the “no-particles/no-thermal-flux” condition at spatial infinity (Arrechea et al., 2023).

This state is conventionally contrasted with the Hartle–Hawking and Unruh vacua. The Hartle–Hawking state is a thermal equilibrium state at the Hawking temperature and is regular on a non-extremal horizon, while the Unruh state describes a black hole formed by collapse, is regular on the future horizon, and carries outgoing Hawking radiation at infinity (Arrechea et al., 2024). In the anomaly-induced effective-action formulation, “Boulware-like” is also used for the static, asymptotically Minkowski auxiliary-field choice with vanishing fields at infinity and no flux, analogous to the Schwarzschild Boulware vacuum (Numajiri et al., 2024).

The phrase acquires further variants in nonstatic or nonstandard constructions. In the Kerr fermion problem, a candidate Boulware-like state is defined to be empty at both past and future null infinity (Casals et al., 2012). In the accreting Boulware scenario for evaporation, the relevant state is a running Boulware ground state generated by gravitational vacuum polarization and supplying negative energy to the hole while maintaining a positive energy outflux to infinity (Israel, 2015). These variants preserve the central asymptotic idea—vacuum with respect to a stationary notion of time at infinity—even when horizon behavior or interior interpretation differs.

2. Canonical construction on static black-hole backgrounds

For static Reissner–Nordström geometries, the background metric is

$$ds^2 = -f(r)\,dt^2 + f(r)^{-1}\,dr^2 + r^2(d\theta^2 + \sin^2\theta\,d\phi^2), \qquad f(r)=1-\frac{2M}{r}+\frac{Q^2}{r^2},$$

with horizon radii

$$r_\pm = M \pm \sqrt{M^2-Q^2}, \qquad \kappa=\frac12 f'(r_+).$$

A scalar field with action

$$S=\int d^4x\,\sqrt{-g}\left[-\tfrac12 g^{ab}\nabla_a\phi\nabla_b\phi-\tfrac12 m^2\phi^2-\tfrac12 \xi R\phi^2\right]$$

is decomposed as

$$\phi_{\omega\ell m}(t,r,\theta,\phi) = e^{-i\omega t}Y_{\ell m}(\theta,\phi)\frac{u_{\omega\ell}(r)}{r},$$

with the radial equation written in Schrödinger form using the tortoise coordinate $dr_*/dr=f^{-1}$ (Arrechea et al., 2024).

In Lorentzian quantization on Reissner–Nordström, one introduces “in” and “up” modes. The Boulware two-point function has no thermal population in either sector: $t$0 where $t$1 (Arrechea et al., 2023). This formulation makes explicit that the state has neither thermal occupation nor Hawking flux.

The Euclidean renormalization of the Boulware state is subtler because the zero-temperature limit removes the periodic identification of Euclidean time. A direct mode-sum prescription generalizing the extended coordinate method treats the Euclidean Green function with continuous $t$2 and no periodic identification of $t$3,

$t$4

so that the Euclideanized geometry carries a conical singularity at a non-extremal horizon. This conical singularity manifests in divergent expectation values at non-extremal horizons, consistent with the known divergence of the Boulware renormalized stress-energy tensor there (Arrechea et al., 2024).

3. Rotating, horizonless, and modified Boulware-like constructions

For a massless quantized spin-$t$5 field on non-extremal Kerr, the construction differs qualitatively from the bosonic case because all fermion modes have positive norm. In Boyer–Lindquist coordinates, the relevant horizon generator is

$t$6

and near the horizon the separated radial modes behave as

$t$7

Using the orthonormal “in”$t$8“up” basis, the candidate Boulware-like vacuum $t$9 is defined by

$\phi \propto e^{-i\omega t}$0

Because the fermionic “out/down”–“in/up” relations do not involve complex conjugation, this state is empty at both $\phi \propto e^{-i\omega t}$1 and $\phi \propto e^{-i\omega t}$2; the bosonic obstruction from superradiant negative-norm modes is absent (Casals et al., 2012).

In anomaly-induced descriptions of horizonless regular spacetimes, the meaning of “Boulware-like” shifts from mode occupation to auxiliary-field boundary data. In two dimensions, the static solution

$\phi \propto e^{-i\omega t}$3

shows that the Boulware-like choice $\phi \propto e^{-i\omega t}$4 is regular and preferred in horizonless regular static spacetimes, because regularity at the center requires $\phi \propto e^{-i\omega t}$5 (Numajiri et al., 2024). In four-dimensional Bardeen-type geometries, however, the boundary-value problem for the anomaly-induced auxiliary fields generically requires a nonzero $\phi \propto e^{-i\omega t}$6, so the preferred static regular state is not the Boulware state even though the physical renormalized stress-energy tensor still decays at infinity (Numajiri et al., 2024).

A further modification appears in the Casimir-like probe of four-dimensional Einstein–Gauss–Bonnet gravity. There, “Boulware-like” means that the field is prepared in a state whose positive-frequency content is defined with respect to $\phi \propto e^{-i\omega t}$7 at infinity, together with a mirror-imposed near-horizon boundary condition. For an $\phi \propto e^{-i\omega t}$8-wave scalar mode with a Dirichlet mirror at $\phi \propto e^{-i\omega t}$9,

$\omega>0$0

and the associated Wightman function is built from these standing-wave modes rather than from the unmodified horizon basis (Masood, 2024). This suggests that the asymptotic Boulware condition can be retained while the near-horizon mode structure is deliberately altered.

4. Stress tensor, regularity, and quantum inequalities

The defining physical feature of the standard Boulware state on a non-extremal black hole is the singular behavior of the renormalized stress-energy tensor at the horizon. In the direct Reissner–Nordström computation, the leading divergence in the sub-extremal case scales like $\omega>0$1 and is captured entirely by the analytic part of the renormalized stress-energy tensor, while the numeric part contributes a subleading $\omega>0$2 term (Arrechea et al., 2024). As $\omega>0$3, all components decay to zero, consistent with the zero-temperature, no-flux character of the state (Arrechea et al., 2024).

The scalar Reissner–Nordström analysis reaches the same conclusion from a different renormalization route. In the Boulware state all diagonal components diverge as $\omega>0$4, with

$\omega>0$5

approaching nonzero finite limits, while in the Unruh state $\omega>0$6 and $\omega>0$7 diverge only as $\omega>0$8 and the Hartle–Hawking state remains regular on the outer horizon (Arrechea et al., 2023). For minimal coupling, the Boulware state typically has negative energy density near the horizon and violates the null energy condition for $\omega>0$9, while for $t$0 it satisfies the null energy condition everywhere outside the horizon (Arrechea et al., 2023).

Extremality changes the picture. For extremal Reissner–Nordström with $t$1, direct Boulware-state renormalization yields strong numerical evidence that the renormalized stress-energy tensor is regular at the extremal horizon for all field masses $t$2 and couplings $t$3. In the massless case,

$t$4

so the horizon value is finite and isotropic. In the massive case, analytic and numeric contributions separately develop logarithmic divergences near $t$5, but those divergences cancel in the total (Arrechea et al., 2024).

The accreting Boulware scenario introduces negative energy in a controlled semiclassical way. In the underlying $t$6-dimensional spherical reduction, the stress tensor obeys the anomaly and conservation equations

$t$7

and the negative energy density of the static Boulware state is stated to marginally satisfy Flanagan’s quantum energy inequality when the sampling function is set to unity (Israel, 2015). The same work characterizes these negative energies as “of the innocuous kind, like those of the Casimir effect,” thereby locating them within accepted semiclassical constraints rather than outside them (Israel, 2015).

5. Backreaction, throat formation, and horizonless mimickers

Exact semiclassical backreaction in two-dimensional dilaton gravity shows that a Boulware-like state can qualitatively change the causal structure. In the Russo–Susskind–Thorlacius model, the Boulware state is defined by vanishing stress-energy at infinity, which fixes

$t$8

On a fixed classical background the corresponding quantum stress tensor diverges negatively near the classical horizon, but the fully back-reacted geometry does not preserve that horizon. Instead, the classical horizon is replaced by a throat in which the $t$9 component of the metric is extremely small but nonzero, with

$$ds^2 = -f(r)\,dt^2 + f(r)^{-1}\,dr^2 + r^2(d\theta^2 + \sin^2\theta\,d\phi^2), \qquad f(r)=1-\frac{2M}{r}+\frac{Q^2}{r^2},$$0

and beyond the throat the spacetime ends at a null singularity (Potaux et al., 2021). By contrast, the Hartle–Hawking state yields a smooth horizon with regular curvature (Potaux et al., 2021).

Hybrid constructions extend this mechanism. In the two-dimensional RST model with physical fields in the Hartle–Hawking or Unruh state and wrong-sign fields in the Boulware state, a wide domain with dominating non-physical fields yields a geodesically complete, asymptotically flat causal diamond free of horizon or curvature singularity (Potaux et al., 2023). In the static Hartle–Hawking–Boulware hybrid with $$ds^2 = -f(r)\,dt^2 + f(r)^{-1}\,dr^2 + r^2(d\theta^2 + \sin^2\theta\,d\phi^2), \qquad f(r)=1-\frac{2M}{r}+\frac{Q^2}{r^2},$$1, the geometry is horizonless and asymptotically flat at both ends, yet still supports thermal radiation from the physical sector at infinity (Potaux et al., 2023). This suggests that Boulware-like components can act as a mechanism for producing black-hole mimickers rather than singular horizon states when semiclassical backreaction is solved exactly.

A qualitatively different strong-coupling realization appears in AdS/CFT. For a large-$$ds^2 = -f(r)\,dt^2 + f(r)^{-1}\,dr^2 + r^2(d\theta^2 + \sin^2\theta\,d\phi^2), \qquad f(r)=1-\frac{2M}{r}+\frac{Q^2}{r^2},$$2 CFT on a Schwarzschild background, the leading $$ds^2 = -f(r)\,dt^2 + f(r)^{-1}\,dr^2 + r^2(d\theta^2 + \sin^2\theta\,d\phi^2), \qquad f(r)=1-\frac{2M}{r}+\frac{Q^2}{r^2},$$3 stress tensor extracted from a smooth classical AdS$$ds^2 = -f(r)\,dt^2 + f(r)^{-1}\,dr^2 + r^2(d\theta^2 + \sin^2\theta\,d\phi^2), \qquad f(r)=1-\frac{2M}{r}+\frac{Q^2}{r^2},$$4 bulk dual is static, has no flux at infinity, and is regular on both the future and past Schwarzschild horizons—even in the Boulware-like state (Figueras et al., 2011). In that construction the distinction between Unruh and Boulware vacua is deferred to the subleading $$ds^2 = -f(r)\,dt^2 + f(r)^{-1}\,dr^2 + r^2(d\theta^2 + \sin^2\theta\,d\phi^2), \qquad f(r)=1-\frac{2M}{r}+\frac{Q^2}{r^2},$$5 sector, where bulk one-loop effects are expected to introduce flux for Unruh and horizon singularities for Boulware (Figueras et al., 2011). A plausible implication is that the singularity of the free-field Boulware stress tensor is not universal across coupling regimes.

6. Entanglement, evaporation, and observational probes

The Boulware-like idea also appears in attempts to reformulate Hawking evaporation. In the accreting Boulware scenario, the gravitational field of a collapsing compact object polarizes the surrounding vacuum and produces a nett negative energy density. As the object contracts, negative Boulware energy drains inward while polarization forces expel an equal amount of positive energy outward. The result is that a positive flux $$ds^2 = -f(r)\,dt^2 + f(r)^{-1}\,dr^2 + r^2(d\theta^2 + \sin^2\theta\,d\phi^2), \qquad f(r)=1-\frac{2M}{r}+\frac{Q^2}{r^2},$$6 remains an outflux at infinity but is interpreted near the horizon as an influx of negative energy (Israel, 2015). The outgoing radiation remains nearly thermal for observers at infinity, yet pair creation does not appear in the semiclassical description, so the usual entanglement-breaking problem of pair production is avoided (Israel, 2015). The same work states that, for external observers, the pair-creation and accreting-Boulware scenarios are indistinguishable in terms of flux and spectrum, but only the latter does not run foul of unitarity violation (Israel, 2015).

In holographic entanglement, the Boulware limit is the $$ds^2 = -f(r)\,dt^2 + f(r)^{-1}\,dr^2 + r^2(d\theta^2 + \sin^2\theta\,d\phi^2), \qquad f(r)=1-\frac{2M}{r}+\frac{Q^2}{r^2},$$7 limit of an entangled Rindler thermofield double. Classically, hyperbolic AdS black holes retain a finite bridge width as $$ds^2 = -f(r)\,dt^2 + f(r)^{-1}\,dr^2 + r^2(d\theta^2 + \sin^2\theta\,d\phi^2), \qquad f(r)=1-\frac{2M}{r}+\frac{Q^2}{r^2},$$8, producing a nonzero entropy density

$$ds^2 = -f(r)\,dt^2 + f(r)^{-1}\,dr^2 + r^2(d\theta^2 + \sin^2\theta\,d\phi^2), \qquad f(r)=1-\frac{2M}{r}+\frac{Q^2}{r^2},$$9

which appears to contradict the expectation that the Boulware vacuum should be unentangled (Emparan et al., 2023). The resolution is supplied by quantum fluctuations of the AdS$$r_\pm = M \pm \sqrt{M^2-Q^2}, \qquad \kappa=\frac12 f'(r_+).$$0 throat, governed by a Schwarzian effective theory with exact one-loop partition function

$$r_\pm = M \pm \sqrt{M^2-Q^2}, \qquad \kappa=\frac12 f'(r_+).$$1

leading to a density of states

$$r_\pm = M \pm \sqrt{M^2-Q^2}, \qquad \kappa=\frac12 f'(r_+).$$2

which vanishes as $$r_\pm = M \pm \sqrt{M^2-Q^2}, \qquad \kappa=\frac12 f'(r_+).$$3 (Emparan et al., 2023). The same mechanism yields unentangled Boulware states in de Sitter space after a second Weyl transformation (Emparan et al., 2023).

Observable consequences of Boulware-like states also arise in detector and atom probes. For a static two-level atom outside Schwarzschild in the Boulware vacuum, the renormalized position-dependent Lamb shift produces a Casimir–Polder-like force

$$r_\pm = M \pm \sqrt{M^2-Q^2}, \qquad \kappa=\frac12 f'(r_+).$$4

which is attractive near the horizon and repulsive far away with $$r_\pm = M \pm \sqrt{M^2-Q^2}, \qquad \kappa=\frac12 f'(r_+).$$5 behavior (Zhang et al., 2011). In the Einstein–Gauss–Bonnet Casimir setup, a freely falling Unruh–DeWitt detector interacting with Boulware-like mirror-modified modes exhibits acceleration radiation whose intensity is enhanced for negative $$r_\pm = M \pm \sqrt{M^2-Q^2}, \qquad \kappa=\frac12 f'(r_+).$$6 and suppressed for positive $$r_\pm = M \pm \sqrt{M^2-Q^2}, \qquad \kappa=\frac12 f'(r_+).$$7 (Masood, 2024). These examples indicate that Boulware-like states are not merely formal vacuum choices but can imprint measurable signatures on response functions and effective forces.

The modern use of the term therefore covers a family of states sharing asymptotic vacuum character but differing sharply in horizon behavior, regularity, and dynamical role. In free-field static black-hole settings the Boulware state is vacuum at infinity and singular on non-extremal horizons; in extremal, fermionic Kerr, anomaly-induced horizonless, back-reacted dilaton, detector-modified, and holographic constructions, Boulware-like states can instead be regular, horizonless, entanglement-suppressing, or thermodynamically nontrivial, while still being anchored by the absence of thermal population at infinity or its closest available analogue (Arrechea et al., 2024).