Hawking Surfaces: Concepts and Applications

• Hawking surfaces are variably defined surfaces in gravitational physics—from quantum extremal surfaces controlling entropy to apparent/trapping horizons driving flux—each tailored to distinct phenomena.
• They play a critical role in semiclassical gravity and holography, helping to model the Page curve of black hole evaporation and verify entropy prescriptions via area-constrained extremization.
• Their study, via variational methods and quasi-local invariants, offers insights into positivity, rigidity, and small-sphere asymptotics, bridging abstract theory with measurable gravitational effects.

Searching arXiv for recent and foundational papers relevant to “Hawking surfaces,” including Hawking energy, Hawking mass, time-flat surfaces, and quantum extremal/island surfaces. “Hawking surfaces” is not a standard term with a single accepted definition in the arXiv literature. Instead, it designates several distinct surface notions that arise in different subfields of gravitational theory, semiclassical black-hole physics, analogue gravity, and quasi-local geometry. In recent work, the expression can refer to quantum extremal or island-boundary surfaces governing the fine-grained entropy of Hawking radiation; local or quasi-local horizon structures such as apparent or trapping horizons relevant to Hawking flux; characteristic null hypersurfaces used to formulate Hadamard states and semiclassical observables; or special closed spacelike $2$-surfaces on which Hawking mass or Hawking energy becomes geometrically well behaved, especially area-constrained critical surfaces of the Hawking functional, time flat surfaces, photon surfaces, and stable CMC spheres (Almheiri et al., 2020, Visser, 2014, Diaz, 22 Jul 2025).

1. Terminological status and major usages

The phrase “Hawking surfaces” functions as a contextual label rather than a fixed term of art. Several of the relevant papers state explicitly that the term is not used in their formal vocabulary and then identify the technically correct replacements.

Usage domain	Closest formal surface notion	Representative papers
Fine-grained entropy of Hawking radiation	quantum extremal surfaces, island boundaries, RT/HRT surfaces, boundary RT surfaces	(Almheiri et al., 2020, Almheiri et al., 2019, Chou et al., 2021)
Hawking flux and horizon kinematics	apparent/trapping horizons, characteristic null cones, analogue horizons	(Visser, 2014, Janssen et al., 2022, Morresi et al., 2019)
Quasi-local mass and energy	area-constrained critical surfaces of the Hawking functional, time flat surfaces, photon surfaces, stable CMC spheres	(Diaz, 22 Jul 2025, Bray et al., 2013, Bengtsson, 2020, Sun, 2017)

In the entropy literature, the closest meaning is the codimension-$2$ surface $X$ that extremizes generalized entropy and, in evaporation problems, becomes the boundary of an island. In spacetime-horizon discussions, the relevant surfaces are the quasi-local horizon structures that generate Hawking radiation kinematically, often without requiring a global event horizon. In quasi-local geometry, the phrase is most naturally attached to closed surfaces singled out by Hawking mass or Hawking energy, especially when those functionals admit positivity, monotonicity, foliation, or rigidity statements only on distinguished surface classes.

2. Quantum-extremal and island interpretations

In semiclassical gravity and holography, the best technical replacement for “Hawking surfaces” is the quantum extremal surface. The review “The entropy of Hawking radiation” states that the term itself is not used there; the correct language is “generalized-entropy extremizing surfaces,” “extremal surfaces,” “quantum extremal surfaces,” “island boundaries,” and, in replica derivations, surfaces emerging from replica-wormhole saddles (Almheiri et al., 2020). The governing prescription is

$$S= \min_X \left\{ \operatorname{ext}_X \left[ \frac{\operatorname{Area}(X)}{4G_N} + S_{\rm semi\text{-}cl}(\Sigma_X) \right] \right\},$$

with generalized entropy

$$S_{\rm gen}(X)=\frac{\operatorname{Area}(X)}{4G_N}+S_{\rm semi\text{-}cl}(\Sigma_X),$$

and extremality condition

$\delta_X S_{\rm gen}(X)=0.$

For Hawking radiation specifically, the island formula becomes

$$S_{\mathrm{Rad}}=\min_X\Bigg\{\operatorname{ext}_X\left[\frac{\operatorname{Area}(X)}{4G_N}+S_{\rm semi\text{-}cl}(\Sigma_{\rm Rad}\cup\Sigma_{\rm Island})\right]\Bigg\},$$

where $X=\partial(\Sigma_{\rm Island})$ is the island boundary. In this usage, a “Hawking surface” is best understood as the QES or island boundary that corrects Hawking’s monotonic semiclassical entropy and yields the Page curve (Almheiri et al., 2020).

The holographic reformulation sharpens this identification. “The Page curve of Hawking radiation from semiclassical geometry” states that the term is again not explicit, and the closest notions are the lower-dimensional QES and the higher-dimensional RT/HRT surface (Almheiri et al., 2019). In that setting, the lower-dimensional generalized entropy

$$S_{\mathrm{gen}}(y)=\frac{\phi(y)}{4G_N^{(2)}}+S_{\mathrm{Bulk}\text{-}2d}[{\cal I}_y]$$

is equivalent, to leading order, to a higher-dimensional RT/HRT extremization problem, and after the Page time the relevant surface lies just behind the horizon and bounds an island in the black-hole interior. The central geometric consequence is that the interior enters the entanglement wedge of the radiation (Almheiri et al., 2019).

A further refinement appears in “Page Curve of Effective Hawking Radiation,” which introduces an ETW brane representing the front of the Hawking radiation that has actually reached the bath and, with it, a third extremal surface besides the Hartman–Maldacena and island surfaces: the boundary RT surface (BRT) (Chou et al., 2021). In that paper the BRT surface is anchored on the radiation cutoff and the ETW brane and computes the entropy of a finite, time-dependent effective Hawking radiation region. The resulting Page-curve story is therefore a competition among three surfaces,

$$\mathcal S=\min(\mathcal S_{\rm HM},\mathcal S_{\rm BRT},\mathcal S_{\rm IRT}),$$

with the physically realized sequence depending on temperature (Chou et al., 2021). In this branch of the literature, “Hawking surfaces” are therefore not event horizons but entropy-controlling extremal surfaces.

3. Horizon, characteristic, and analogue-surface meanings

A second meaning of “Hawking surfaces” concerns the surfaces that are relevant to producing Hawking radiation kinematically. “Thermality of the Hawking flux” argues that Hawking radiation is associated with the apparent or trapping horizon and “couldn’t care less about the event horizon,” whereas unitarity issues, if present, are tied to the event horizon (Visser, 2014). In that framework the decisive local structure is the horizon-like surface responsible for the exponential peeling or redshift of null rays, not a teleological global boundary. This distinction underlies the paper’s broader claim that the Hawking flux is only approximately Planckian over the bounded interval

$2$0

with greybody suppression

$2$1

and that exact thermality or exact absence of correlations requires extra assumptions about event horizons (Visser, 2014). In this sense, the relevant “surface” is the local horizon structure generating the Hawking effect.

A related but technically distinct usage appears in the characteristic-initial-data formulation of semiclassical gravity. “Hadamard states on spherically symmetric characteristic surfaces, the semi-classical Einstein equations and the Hawking effect” studies quasi-free Hadamard states defined from null cones

$2$2

and the family $2$3 in spherically symmetric spacetimes (Janssen et al., 2022). These outgoing null cones are the relevant operational surfaces because the bulk state is reconstructed from characteristic data via the boundary map

$2$4

and renormalized observables are computed from the corresponding null-boundary two-point functions. In the collapse application, null surfaces adapted to a collapsing shell are used to analyze vacuum polarization associated with Hawking radiation near the collapsing body (Janssen et al., 2022). Here the phrase points neither to the event horizon nor to a codimension-$2$5 QES, but to characteristic null hypersurfaces on which the state and observables are encoded.

Analogue-gravity work supplies yet another interpretation. “Exploring Event Horizons and Hawking Radiation through Deformed Graphene Membranes” proposes Beltrami’s pseudosphere as a material surface whose low-energy graphene quasiparticles experience an effective curved spacetime with an analogue horizon (Morresi et al., 2019). The Gaussian curvature is

$2$6

the Beltrami metric is conformal to a Rindler metric,

$2$7

and the effective Hawking/Unruh temperature is

$2$8

at the horizon (Morresi et al., 2019). The associated LDOS is predicted to take a thermal form near the horizon. However, “Note on Hawking-Unruh effects in graphene” argues that realistic STM/STS measurements on such curved graphene surfaces are likely to be dominated by strain-induced pseudomagnetic fields and near-field plasmon/polariton physics, so the experimentally relevant interpretation is closer to the membrane paradigm than to a clean Hawking-radiation analogue (Chen et al., 2012). Thus even within analogue gravity, “Hawking surfaces” may mean either horizon-bearing effective curved surfaces or membrane-like electromagnetic surfaces, depending on the experimental regime.

4. Hawking mass, Hawking energy, and distinguished geometric surfaces

In differential geometry and mathematical relativity, the term is most naturally tied to closed spacelike $2$9-surfaces on which Hawking mass or Hawking energy is evaluated. A central spacetime definition is the Hawking mass

$X$0

for a closed spacelike $X$1-surface $X$2 with spacelike mean-curvature vector $X$3 (Bray et al., 2013). In initial-data-set form, the Hawking energy is written as

$X$4

and the associated Hawking functional is

$X$5

(Diaz, 22 Jul 2025). In this setting, Hawking surfaces are defined explicitly as the area-constrained critical surfaces of $X$6, equivalently of $X$7, and they satisfy the Euler–Lagrange equation

$X$8

When $X$9, these reduce exactly to area-constrained Willmore surfaces (Diaz, 22 Jul 2025).

A different distinguished class is that of time flat surfaces. For an admissible spacelike $$S= \min_X \left\{ \operatorname{ext}_X \left[ \frac{\operatorname{Area}(X)}{4G_N} + S_{\rm semi\text{-}cl}(\Sigma_X) \right] \right\},$$0-surface with spacelike mean curvature vector, Bray and Jauregui define the normal connection $$S= \min_X \left\{ \operatorname{ext}_X \left[ \frac{\operatorname{Area}(X)}{4G_N} + S_{\rm semi\text{-}cl}(\Sigma_X) \right] \right\},$$1-form

$$S= \min_X \left\{ \operatorname{ext}_X \left[ \frac{\operatorname{Area}(X)}{4G_N} + S_{\rm semi\text{-}cl}(\Sigma_X) \right] \right\},$$2

with $$S= \min_X \left\{ \operatorname{ext}_X \left[ \frac{\operatorname{Area}(X)}{4G_N} + S_{\rm semi\text{-}cl}(\Sigma_X) \right] \right\},$$3, and call the surface time flat when

$$S= \min_X \left\{ \operatorname{ext}_X \left[ \frac{\operatorname{Area}(X)}{4G_N} + S_{\rm semi\text{-}cl}(\Sigma_X) \right] \right\},$$4

(Bray et al., 2013). This condition removes the sign-indefinite divergence term in the spacetime Hawking-mass variation formula and yields monotonicity under outward-spacelike uniformly area expanding flows satisfying the dominant energy condition (Bray et al., 2013). The sequel paper recasts the same idea as control of the obstruction generated by “timelike squiggles” and supplies further divergence-type sufficient conditions for monotonicity (Bray et al., 2014).

Another special surface class arises from timelike totally umbilic hypersurfaces. “The Hawking energy on photon surfaces” shows that timelike totally umbilic hypersurfaces are photon surfaces and that, on such constant-scalar-curvature hypersurfaces in Einstein spacetimes, Hawking energy is monotone under inverse mean curvature flow: increasing in the spacelike case and decreasing in the timelike case (Bengtsson, 2020). In particular, Schwarzschild photon surfaces furnish timelike examples on which Hawking energy has a controlled monotone evolution (Bengtsson, 2020).

Stable CMC spheres, nearly round surfaces, isoperimetric surfaces, and small perturbed geodesic spheres provide further important instances in which Hawking mass is nonnegative, zero Hawking mass is rigid, or the sign of Hawking mass detects the local geometry. These are not all “Hawking surfaces” in the variational sense of the Hawking functional, but they are central to the quasi-local mass branch of the literature because they determine when the Hawking mass behaves as a meaningful geometric invariant.

5. Foliations, monotonicity, rigidity, and small-sphere asymptotics

A substantial modern development is the construction of canonical families of Hawking-energy-critical surfaces. “Local foliations by critical surfaces of the Hawking energy and small sphere limit” proves that near a nondegenerate critical point of

$$S= \min_X \left\{ \operatorname{ext}_X \left[ \frac{\operatorname{Area}(X)}{4G_N} + S_{\rm semi\text{-}cl}(\Sigma_X) \right] \right\},$$5

one can construct a unique local foliation by area-constrained critical spheres of the Hawking functional in a general initial data set $$S= \min_X \left\{ \operatorname{ext}_X \left[ \frac{\operatorname{Area}(X)}{4G_N} + S_{\rm semi\text{-}cl}(\Sigma_X) \right] \right\},$$6 (Diaz, 2022). In the time-symmetric case this reduces to the earlier Willmore foliation theory. The same paper shows that these critical surfaces are perturbations of small geodesic spheres and therefore satisfy the small-sphere expansion

$$S= \min_X \left\{ \operatorname{ext}_X \left[ \frac{\operatorname{Area}(X)}{4G_N} + S_{\rm semi\text{-}cl}(\Sigma_X) \right] \right\},$$7

rather than the Horowitz–Schmidt null-cone small-sphere formula (Diaz, 2022).

The asymptotic counterpart at infinity is developed in “Foliations by critical surfaces of the Hawking energy in asymptotically flat initial data sets,” which constructs large-scale foliations near infinity by Hawking surfaces in asymptotically Schwarzschild data sets (Diaz, 19 Aug 2025). Under the dominant energy condition, the paper proves positivity of the Hawking energy along the leaves, convergence to the ADM energy in the large-sphere limit, and monotonicity along the foliation subject to an explicit integral sign condition. It also studies the coordinate center of the foliation and shows that the resulting center generally differs from the STCMC center because the Hawking foliation remains sensitive to asymptotic $$S= \min_X \left\{ \operatorname{ext}_X \left[ \frac{\operatorname{Area}(X)}{4G_N} + S_{\rm semi\text{-}cl}(\Sigma_X) \right] \right\},$$8-data through the $$S= \min_X \left\{ \operatorname{ext}_X \left[ \frac{\operatorname{Area}(X)}{4G_N} + S_{\rm semi\text{-}cl}(\Sigma_X) \right] \right\},$$9-term (Diaz, 19 Aug 2025).

Positivity and rigidity are pushed further in “Rigidity and positivity of Hawking quasi-local energy on area-constrained critical surfaces,” which proves that the Hawking energy is nonnegative on its natural area-constrained critical surfaces under the dominant energy condition, and that vanishing energy forces the enclosed region to be flat: Euclidean in the time-symmetric case and Minkowskian as a spacelike hypersurface in the fully dynamical case, subject to the paper’s explicit integral conditions (Diaz, 22 Jul 2025). The time-symmetric section extends the discussion to electric charge, nonzero cosmological constant, and higher dimensions (Diaz, 22 Jul 2025).

Parallel rigidity theories exist for Hawking mass in purely Riemannian settings. “Rigidity of Hawking mass for surfaces in three manifolds” proves partial rigidity for nearly round stable CMC spheres with zero Hawking mass in asymptotically flat and asymptotically hyperbolic manifolds, and global flatness from small or large zero-Hawking-mass isoperimetric surfaces (Sun, 2017). “On the Hawking mass for CMC surfaces in positive curved 3-manifolds” proves that zero Hawking mass for stable CMC spheres in $$S_{\rm gen}(X)=\frac{\operatorname{Area}(X)}{4G_N}+S_{\rm semi\text{-}cl}(\Sigma_X),$$0 is rigid under approximate roundness or even symmetry, forcing the surface to be round and, under $$S_{\rm gen}(X)=\frac{\operatorname{Area}(X)}{4G_N}+S_{\rm semi\text{-}cl}(\Sigma_X),$$1, the enclosed region to be a geodesic ball in $$S_{\rm gen}(X)=\frac{\operatorname{Area}(X)}{4G_N}+S_{\rm semi\text{-}cl}(\Sigma_X),$$2 (Melo, 2023). “Some rigidity results for the Hawking mass and a lower bound for the Bartnik capacity” uses optimally perturbed geodesic spheres

$$S_{\rm gen}(X)=\frac{\operatorname{Area}(X)}{4G_N}+S_{\rm semi\text{-}cl}(\Sigma_X),$$3

and the expansion

$$S_{\rm gen}(X)=\frac{\operatorname{Area}(X)}{4G_N}+S_{\rm semi\text{-}cl}(\Sigma_X),$$4

to show that local nonpositivity of Hawking mass characterizes local flatness or constant-curvature geometry and motivates the sup-Hawking mass $$S_{\rm gen}(X)=\frac{\operatorname{Area}(X)}{4G_N}+S_{\rm semi\text{-}cl}(\Sigma_X),$$5 (Mondino et al., 2021).

Variational small-surface analysis provides a unifying framework for these local constructions. “Concentration of Small Hawking Type Surfaces” treats Hawking energy as a Hawking type functional

$$S_{\rm gen}(X)=\frac{\operatorname{Area}(X)}{4G_N}+S_{\rm semi\text{-}cl}(\Sigma_X),$$6

with the Hawking-energy case corresponding to $$S_{\rm gen}(X)=\frac{\operatorname{Area}(X)}{4G_N}+S_{\rm semi\text{-}cl}(\Sigma_X),$$7, proves existence of area-constrained minimizers in the compactified class of haunted bubble trees, and shows that sufficiently small minimizers are smooth embedded spheres (Friedrich, 2019). It also derives the concentration law

$$S_{\rm gen}(X)=\frac{\operatorname{Area}(X)}{4G_N}+S_{\rm semi\text{-}cl}(\Sigma_X),$$8

for small area-constrained critical Hawking surfaces and the corresponding small-surface expansion of Hawking energy (Friedrich, 2019).

Numerical evidence complements the analytic results. “Evolution of Hawking mass under hypersurface-restricted expanding flows” studies small spherical-harmonic perturbations of round spheres in Minkowski spacetime and finds that monotonicity of Hawking mass persists robustly for several single-mode nonspherical perturbations under an in-slice inverse-mean-curvature-type flow, while mixed-mode data reveal numerical instabilities rather than a clean geometric breakdown (Williams, 17 Jan 2026).

6. Conceptual distinctions and recurring controversies

The first recurring source of confusion is terminological. “Hawking surfaces” may refer to QES or island boundaries, apparent or trapping horizons, null characteristic hypersurfaces, graphene pseudospheres, or variationally distinguished closed $$S_{\rm gen}(X)=\frac{\operatorname{Area}(X)}{4G_N}+S_{\rm semi\text{-}cl}(\Sigma_X),$$9-surfaces for Hawking mass or energy. The term therefore acquires meaning only from context.

A second distinction concerns horizon concepts. In the semiclassical-flux literature, the local or quasi-local surface relevant for Hawking radiation is the apparent or trapping horizon, whereas event horizons control the global information-loss question (Visser, 2014). In the fine-grained-entropy literature, neither of these is the decisive surface: the entropy is controlled instead by a QES or island boundary (Almheiri et al., 2020). Conflating these two uses obscures the difference between flux generation and entropy bookkeeping.

A third controversy concerns small-sphere limits. The null-cone small-sphere limit of Horowitz–Schmidt yields the matter-density combination

$\delta_X S_{\rm gen}(X)=0.$0

whereas geodesic-sphere-based Hawking critical surfaces and Hawking type minimizers yield

$\delta_X S_{\rm gen}(X)=0.$1

(Diaz, 2022, Friedrich, 2019). The discrepancy is not a formal contradiction: it arises because the two calculations use different shrinking surface families, namely null light-cone cuts versus perturbations of spacelike geodesic spheres (Diaz, 2022).

A fourth issue is positivity. Hawking energy or Hawking mass is not nonnegative on arbitrary surfaces; this is one of the persistent criticisms of the functional. The recent variational program responds by restricting attention to area-constrained critical surfaces of the Hawking functional, where positivity and rigidity can be proved under the dominant energy condition (Diaz, 22 Jul 2025). Even here, the current dynamical theorems require explicit integral hypotheses involving $\delta_X S_{\rm gen}(X)=0.$2 or $\delta_X S_{\rm gen}(X)=0.$3, and the paper gives Minkowski examples of Hawking surfaces with zero Hawking energy that fail those sufficient conditions, indicating that the conditions are not optimal (Diaz, 22 Jul 2025).

A fifth distinction arises in analogue gravity. Beltrami-shaped graphene provides a formally attractive horizon-bearing surface for Dirac quasiparticles, but near-field experiments may probe strain-induced gauge fields, plasmonic surface modes, and membrane-paradigm analogies rather than a clean Hawking-Unruh spectrum (Morresi et al., 2019, Chen et al., 2012). In this branch of the literature, “surface” refers simultaneously to an effective curved spacetime for quasiparticles and to an actual condensed-matter interface with its own dominant near-field physics.

Taken together, these literatures show that “Hawking surfaces” is best understood as a family resemblance term for surfaces singled out by Hawking phenomena: entropy-controlling extremal surfaces, horizon-generating or horizon-mimicking structures, and quasi-local mass or energy probes on which Hawking functionals become mathematically or physically natural.