Boulware Vacuum in Curved Spacetime

• Boulware vacuum is a static, zero-flux quantum state defined via positive-frequency mode decomposition with respect to a timelike Killing field.
• It is characterized by vanishing incoming and outgoing flux at infinity while exhibiting divergent stress-energy near non-extremal black hole horizons.
• Recent semiclassical and holographic analyses show that backreaction and extremal configurations can regularize its behavior, prompting renewed interest in its physical implications.

The Boulware vacuum is the static, zero-flux quantum state associated with positive-frequency mode decomposition with respect to a timelike Killing field in an asymptotically static region. In asymptotically flat black-hole exteriors it is the state that looks empty to static observers at infinity, while in Rindler language it is the unentangled product of left and right wedge vacua. Across the literature, it is characterized by vanishing incoming and outgoing flux at infinity, regular vacuum behavior in the asymptotic region, and, in generic non-extremal black-hole geometries, a divergent renormalized stress-energy tensor at the horizon. More recent work has refined this picture in several directions: horizonless compact objects can support Boulware-like states once backreaction is included, extremal horizons exhibit subtler behavior, and holographic constructions reinterpret the Boulware limit as a zero-entanglement modular ground state (Zhou et al., 2012, Emparan et al., 2023, Arrechea et al., 2023, Arrechea et al., 2024).

1. Definition in static black-hole and Rindler geometries

In Schwarzschild spacetime, the Boulware vacuum is defined by requiring that all annihilation operators for modes of positive Killing frequency $\omega>0$ with respect to the static Killing vector $\partial_t$ annihilate the vacuum (Zhou et al., 2012). In the scalar-field treatment on Reissner–Nordström, the state is built from mode functions

$$u_{\omega lm}(x)=\frac{1}{\sqrt{4\pi\,\omega}}\,e^{-i\omega t}\,Y_{lm}(\theta,\phi)\,\Phi_{\omega l}(r),$$

with the vacuum selected by the absence of thermal occupation factors in the two-point function (Arrechea et al., 2023). In the $1+1$-dimensional Schwarzschild reduction, the positive-frequency modes are $e^{-i\omega u}$ and $e^{-i\omega v}$, where $u=t-r_*$ and $v=t+r_*$, and the corresponding Wightman function is

$$W_B(x,x')=-\frac{1}{4\pi}\log\!\left[-\Lambda^2(\Delta u-i\epsilon)(\Delta v-i\epsilon)\right]$$

(Tjoa et al., 2020).

This definition generalizes to asymptotically anti-de Sitter and shell geometries. In the collapsing-shell Schwarzschild–AdS model, the early-time vacuum is defined by static modes

$$\phi_\omega=\frac{1}{\sqrt{4\pi|\omega|}}\left(e^{-i\omega u}-e^{-i\omega v}\right),$$

with $\partial_t$0, $\partial_t$1, and is explicitly identified as a Boulware-like vacuum for the static exterior region (Abel et al., 2015). In the Euclidean thin-shell construction with Minkowski interior and Schwarzschild exterior, the Boulware state is the zero-temperature state with continuous Euclidean frequency spectrum, no periodic identification of $\partial_t$2, and asymptotically Minkowskian static boundary conditions (Arrechea et al., 8 Jul 2026).

A distinct but closely related formulation appears in Rindler space. In $\partial_t$3-dimensional Minkowski spacetime with Rindler coordinates

$\partial_t$4

the Minkowski vacuum can be written as a thermofield double of left and right wedges at modular temperature $\partial_t$5, while the Boulware vacuum is the limiting product state

$\partial_t$6

obtained as $\partial_t$7 in

$\partial_t$8

In that context it is explicitly the unentangled modular ground state, singular on the Rindler horizon and expected to have vanishing entanglement across the wedges (Emparan et al., 2023).

2. Two-point functions, mode sums, and state selection

The Boulware vacuum is operationally specified through its two-point function or Euclidean Green function. In the Reissner–Nordström scalar-field analysis, the Boulware two-point function is

$\partial_t$9

with no thermal occupation factors, in contrast to the Hartle–Hawking and Unruh states (Arrechea et al., 2023). In the direct Euclidean treatment of the Boulware renormalized stress tensor, the Green function is represented as

$$u_{\omega lm}(x)=\frac{1}{\sqrt{4\pi\,\omega}}\,e^{-i\omega t}\,Y_{lm}(\theta,\phi)\,\Phi_{\omega l}(r),$$0

with continuous $$u_{\omega lm}(x)=\frac{1}{\sqrt{4\pi\,\omega}}\,e^{-i\omega t}\,Y_{lm}(\theta,\phi)\,\Phi_{\omega l}(r),$$1, reflecting the zero-temperature character of the state (Arrechea et al., 2024).

In Bertotti–Robinson spacetime, the Boulware vacuum is defined by positive frequency with respect to the black-hole patch time $$u_{\omega lm}(x)=\frac{1}{\sqrt{4\pi\,\omega}}\,e^{-i\omega t}\,Y_{lm}(\theta,\phi)\,\Phi_{\omega l}(r),$$2, using radial modes $$u_{\omega lm}(x)=\frac{1}{\sqrt{4\pi\,\omega}}\,e^{-i\omega t}\,Y_{lm}(\theta,\phi)\,\Phi_{\omega l}(r),$$3. The resulting propagator is obtained in closed form from a new summation formula for associated Legendre functions, leading to

$$u_{\omega lm}(x)=\frac{1}{\sqrt{4\pi\,\omega}}\,e^{-i\omega t}\,Y_{lm}(\theta,\phi)\,\Phi_{\omega l}(r),$$4

(Ottewill et al., 2012). The same work shows that thermal states can be generated from the Boulware propagator by imaginary-time image sums, and the Hartle–Hawking propagator is recovered when the temperature equals the Hawking temperature (Ottewill et al., 2012).

In the false-vacuum decay analysis on a two-dimensional black-hole background, the Boulware Green function is

$$u_{\omega lm}(x)=\frac{1}{\sqrt{4\pi\,\omega}}\,e^{-i\omega t}\,Y_{lm}(\theta,\phi)\,\Phi_{\omega l}(r),$$5

and the corresponding boundary coefficients in the complex-tunneling formalism are

$$u_{\omega lm}(x)=\frac{1}{\sqrt{4\pi\,\omega}}\,e^{-i\omega t}\,Y_{lm}(\theta,\phi)\,\Phi_{\omega l}(r),$$6

This identifies the Boulware state with vacuum/Feynman boundary conditions in the asymptotic past, without thermal mixing of positive and negative frequencies (Shkerin et al., 2021).

These constructions share a common structure: the Boulware vacuum is picked out by static positive-frequency analyticity and by the absence of thermal factors in mode occupation numbers. A plausible implication is that its pathology at horizons is not tied to the two-point function as a distribution, but to local composite observables obtained from its short-distance derivatives.

3. Horizon singularity, asymptotic emptiness, and detector response

The canonical physical interpretation of the Boulware vacuum is that it is empty at infinity and singular at a non-extremal horizon. In the electromagnetic-atom analysis in Schwarzschild spacetime, it is described as the “non-thermal, star-like” vacuum: no incoming or outgoing radiation at infinity, no spontaneous excitation of a ground-state atom, but divergent local behavior at the horizon (Zhou et al., 2012). The total rate of change of the mean atomic energy is

$$u_{\omega lm}(x)=\frac{1}{\sqrt{4\pi\,\omega}}\,e^{-i\omega t}\,Y_{lm}(\theta,\phi)\,\Phi_{\omega l}(r),$$7

so upward transitions cancel exactly and spontaneous excitation does not occur in the Boulware vacuum (Zhou et al., 2012). Near the horizon, however, the proper acceleration

$$u_{\omega lm}(x)=\frac{1}{\sqrt{4\pi\,\omega}}\,e^{-i\omega t}\,Y_{lm}(\theta,\phi)\,\Phi_{\omega l}(r),$$8

diverges, and the spontaneous emission rate acquires a term proportional to $$u_{\omega lm}(x)=\frac{1}{\sqrt{4\pi\,\omega}}\,e^{-i\omega t}\,Y_{lm}(\theta,\phi)\,\Phi_{\omega l}(r),$$9, signaling pathological behavior for static observers (Zhou et al., 2012).

In two-dimensional Schwarzschild, the renormalized stress tensor in the Boulware state is

$1+1$0

$1+1$1

which fall off at large $1+1$2 and have no constant asymptotic flux, but fail the regularity conditions at the horizon (Anderson et al., 18 Oct 2025). In the same model, the renormalized vacuum polarization is

$1+1$3

so it diverges logarithmically at $1+1$4 and at the singularity $1+1$5 (Anderson et al., 18 Oct 2025).

Operationally, the Boulware vacuum still contains nontrivial nonlocal correlations. In Schwarzschild entanglement harvesting with Unruh–DeWitt detectors, the Boulware Wightman function is used as the baseline “cold” state, and the derivative two-point function along static trajectories is

$1+1$6

(Tjoa et al., 2020). Far from the horizon, detector response approaches the Minkowski-vacuum result; near the horizon, both concurrence and mutual information are suppressed, and the horizon inhibits correlations in all three canonical vacua (Tjoa et al., 2020). In $1+1$7-dimensional Schwarzschild, entanglement harvesting from the Boulware vacuum is strongly amplified near null caustics, and pre-existing vacuum entanglement can be harvested even for lightlike separations (Caribé et al., 2023).

4. Semiclassical backreaction and horizonless Boulware geometries

A major development in the modern literature is the claim that the Boulware vacuum need not be discarded once backreaction is treated self-consistently. In the static semiclassical analysis based on a two-dimensional anomaly model for vacuum energy, the vacuum choice is fixed by

$1+1$8

which is explicitly identified as the Boulware vacuum suitable for static configurations (Ho et al., 2017). Solving the semiclassical Einstein equations nonperturbatively shows that $1+1$9 never reaches zero at finite $e^{-i\omega u}$0; instead, the would-be Schwarzschild horizon is replaced by a local minimum of the areal radius, a wormhole-like neck (Ho et al., 2017). Near the neck $e^{-i\omega u}$1,

$e^{-i\omega u}$2

with

$e^{-i\omega u}$3

and the geometry is smooth there (Ho et al., 2017).

This framework leads to several concrete claims. First, Buchdahl’s inequality can be violated without divergence in pressure, even if the stellar surface is below the classical Schwarzschild radius (Ho et al., 2017). Second, the proper distance from the neck to the surface or interior core is of order $e^{-i\omega u}$4, so the high-redshift region is thin in proper distance even when the asymptotic Schwarzschild radius is macroscopic (Ho et al., 2017). Third, the results imply that “in principle the Boulware vacuum can be physical for black holes,” provided the geometry is horizonless rather than exactly Schwarzschild (Ho et al., 2017).

A related anomaly-induced analysis of Schwarzschild backreaction reaches a similar qualitative conclusion. Using the Riegert–Mottola–Vaulin stress tensor in the Boulware state, an order-reduced and conserved treatment yields horizonless geometries with a regular ultracompact throat slightly outside $e^{-i\omega u}$5, whereas a non-conserved order-reduced treatment tends to produce naked singularities (Carballo-Rubio et al., 11 Dec 2025). This suggests that the fate of the would-be horizon is sensitive to how higher-derivative semiclassical dynamics is reduced, but the horizon-avoidance tendency itself is robust within Boulware-like negative-energy backreaction (Carballo-Rubio et al., 11 Dec 2025).

The thin-shell study sharpens this picture for material sources. It computes $e^{-i\omega u}$6 and $e^{-i\omega u}$7 for massless scalar fields in the Boulware state in a spacetime with Minkowski interior and Schwarzschild exterior, and finds that outside a highly compact shell these quantities rapidly approach the Schwarzschild Boulware values (Arrechea et al., 8 Jul 2026). This supports a possible universality of the exterior Boulware vacuum for highly compact horizonless objects (Arrechea et al., 8 Jul 2026).

5. Extremal horizons, near-horizon limits, and regularity questions

Although the Boulware vacuum is generically singular on non-extremal horizons, the extremal case is more subtle. In the direct mode-sum study of the renormalized stress tensor in the Boulware state on Reissner–Nordström spacetimes, the sub-extremal case reproduces the standard picture: every renormalized stress-energy tensor component in the Boulware state diverges at the horizon as $e^{-i\omega u}$8 for all the couplings considered (Arrechea et al., 2023). By contrast, in extremal Reissner–Nordström the later direct Euclidean calculation finds numerical evidence that the Boulware-state renormalized stress tensor is finite on the extremal horizon regardless of the field mass and its coupling (Arrechea et al., 2024).

For a massless conformally coupled scalar at the extremal horizon, the result is

$e^{-i\omega u}$9

coinciding with the Bertotti–Robinson value (Arrechea et al., 2024). For massive fields, the analytic and numeric pieces each diverge logarithmically as $e^{-i\omega v}$0, but the divergences cancel in the full renormalized tensor, leaving a finite result (Arrechea et al., 2024). This does not make the Boulware vacuum universally regular at all horizons; it isolates extremality as a special limit in which the usual non-extremal pathology can fail.

The Bertotti–Robinson analysis provides the corresponding near-horizon field theory. In that $e^{-i\omega v}$1 geometry, the Boulware and Schwarzschild vacua are equivalent, while the Poincaré, Global, and Hartle–Hawking vacua are equivalent (Ottewill et al., 2012). The renormalized stress tensor in the Boulware vacuum takes the form

$e^{-i\omega v}$2

with analogous expressions for the other diagonal components, so the horizon divergence is explicit in the near-horizon $e^{-i\omega v}$3 black-hole patch (Ottewill et al., 2012). This is consistent with the sub-extremal picture and helps explain why the extremal regularity result is nontrivial rather than automatic.

A plausible synthesis is that extremal Boulware regularity is a statement about the full renormalized stress tensor in the exact extremal geometry, whereas Bertotti–Robinson Boulware singularity reflects the static patch state adapted to the $e^{-i\omega v}$4 black-hole time. The data support this distinction, but do not collapse it into a single universal rule.

6. Holographic, modular, and cosmological generalizations

Recent work extends the notion of the Boulware vacuum beyond black-hole exteriors into modular Hamiltonian and holographic settings. In Rindler space, the Boulware vacuum is the unentangled product state $e^{-i\omega v}$5, obtained as the $e^{-i\omega v}$6 limit of the thermofield-double family

$e^{-i\omega v}$7

(Emparan et al., 2023). For a generic local quantum field theory, if the modular vacuum is nondegenerate, the Rindler entanglement entropy must vanish as $e^{-i\omega v}$8 (Emparan et al., 2023).

At strong coupling, the naive holographic dual of these low-temperature states is a family of hyperbolic AdS black holes with metric

$e^{-i\omega v}$9

whose extremal limit has nonzero area entropy density

$u=t-r_*$0

(Emparan et al., 2023). This creates the holographic Boulware puzzle: field theory expects zero entanglement in the Boulware limit, while the classical extremal bridge has finite area (Emparan et al., 2023). The resolution is an effective $u=t-r_*$1 throat description governed by JT gravity plus Schwarzian boundary modes. The exact low-temperature partition function is

$u=t-r_*$2

and the corresponding density of states is

$u=t-r_*$3

which vanishes as $u=t-r_*$4 (Emparan et al., 2023). The interpretation given is that quantum fluctuations of the long $u=t-r_*$5 throat remove the would-be extremal entropy as a measure of entanglement, restoring the expectation that the Boulware vacuum has no macroscopic entanglement (Emparan et al., 2023).

The same bulk family, after a Weyl transformation, yields a de Sitter boundary metric

$u=t-r_*$6

and the $u=t-r_*$7 limit is interpreted as a Boulware–de Sitter state with minimal entanglement across the cosmological horizon (Emparan et al., 2023). This suggests that “Boulware vacuum” is best viewed not only as a black-hole state but as a broader modular concept: the static, minimal-entanglement ground state associated with a horizon-generating Killing flow.

The concept also becomes nontrivial in horizonless regular geometries. In the anomaly-induced effective-action analysis of Bardeen-type spacetimes, the preferred regular vacuum in four dimensions is not the Boulware vacuum; central regularity forces a nonzero auxiliary-field charge $u=t-r_*$8, so the resulting static state is asymptotically vacuum-like but not Boulware in the strict Schwarzschild sense (Numajiri et al., 2024). This indicates that “empty at infinity” and “globally regular” need not select the same state once horizons are absent and the Weyl tensor is nonzero (Numajiri et al., 2024).

Taken together, these developments show that the Boulware vacuum remains a central organizing concept in quantum field theory on curved backgrounds: it is the static zero-flux Killing vacuum in asymptotically stationary exteriors, the product modular ground state in Rindler space, the zero-entanglement limit in holographic modular thermodynamics, and a stringent diagnostic for how semiclassical backreaction, extremality, and global regularity reshape the meaning of vacuum in curved spacetime (Emparan et al., 2023, Arrechea et al., 2024, Numajiri et al., 2024).