Hillingar Black Holes: Extended Photon Sphere

• Hillingar black holes are static, spherically symmetric solutions with a Schwarzschild core surrounded by a null-particle ocean, creating an extended photon sphere.
• The model uses a piecewise mass function to delineate three regions, ensuring optical mimicry of a Schwarzschild black hole from external observers.
• Thermodynamic analysis shows that under equilibrium, the HBH replicates the temperature, entropy, and free energy of a Schwarzschild black hole, though wave dynamics may differ.

Hillingar black holes are static, spherically symmetric solutions of Einstein’s equations consisting of a central Schwarzschild black hole of mass $m$ surrounded by a thick shell, or “ocean,” of orbiting massless particles, with total ADM mass $M$. The surrounding matter is a null Einstein cluster, or null cluster (NC), with zero radial pressure, and its backreaction converts the usual single Schwarzschild photon sphere into an extended photon sphere of finite depth. In asymptotically flat space the resulting object is optically indistinguishable from an ordinary Schwarzschild black hole of mass $M$ for observers at infinity, and, under a formal assumption of thermal equilibrium, it also reproduces the temperature, entropy, and free energy of that Schwarzschild black hole (Riojas et al., 29 Jun 2026, Riojas et al., 29 Jun 2026).

1. Definition and spacetime structure

In four-dimensional asymptotically flat spacetime, the metric is written as

$ds^2=-f(r)\,dt^2+j(r)^{-1}dr^2+r^2d\Omega_2^2,$

with

$$j(r)=1-\frac{2\widehat m(r)}{r}, \qquad f(r)=\left[\frac{\widehat m(r)}{M}\right]^2 j(r).$$

The mass function is piecewise: $\widehat m(r)= \begin{cases} m, & r\in(2m,\,3m),\[2pt] \dfrac{r}{3}, & r\in(3m,\,3M),\[2pt] M, & r\in(3M,\,\infty). \end{cases}$ Thus the geometry has three regions: an inner Schwarzschild core of mass $m$, the null-cluster ocean occupying $3m$M$. The shell begins at the photon sphere of the inner black hole and ends at the photon sphere associated with the total mass, which the authors describe as an aligned extended photon sphere (Riojas et al., 29 Jun 2026).

The HBH has a true event horizon at

$r_h=2m$

in asymptotically flat $M$0. The shell thickness is

$M$1

and for fixed $M$2, $M$3 can be made arbitrarily large, so the ocean can be arbitrarily deep. This depth is not an auxiliary cutoff but part of the exact solution family.

A convenient equivalent notation uses

$M$4

so that in the ocean

$M$5

and therefore

$M$6

This makes explicit the self-similar, conical character of the matter region.

2. Null-cluster matter model and the extended photon sphere

The HBH ocean is modeled by an anisotropic stress tensor

$M$7

with

$M$8

The stress tensor is traceless,

$M$9

so the shell is an ultra-relativistic Einstein cluster made of massless particles on randomly oriented circular null orbits. Using $M$0, the density profile is

$M$1

In conventions retaining $M$2, the same condition is written as

$M$3

throughout the ocean (Riojas et al., 29 Jun 2026).

The key optical function is

$M$4

For an ordinary Schwarzschild geometry, $M$5 has a single maximum at the photon sphere. For the HBH,

$M$6

so the maximum becomes a plateau. Equivalently,

$M$7

throughout the ocean. Every radius in $M$8 is therefore locally a photon sphere for the mass interior to it. This is the precise sense in which the HBH extends the photon sphere into a finite region of arbitrary depth (Riojas et al., 29 Jun 2026).

The HBH is also the luminal limit of a more general family of self-similar Einstein clusters with $M$9. For those solutions,

$ds^2=-f(r)\,dt^2+j(r)^{-1}dr^2+r^2d\Omega_2^2,$0

with

$ds^2=-f(r)\,dt^2+j(r)^{-1}dr^2+r^2d\Omega_2^2,$1

and the constituent speed satisfies

$ds^2=-f(r)\,dt^2+j(r)^{-1}dr^2+r^2d\Omega_2^2,$2

For $ds^2=-f(r)\,dt^2+j(r)^{-1}dr^2+r^2d\Omega_2^2,$3, the particles move on stable timelike circular orbits; as $ds^2=-f(r)\,dt^2+j(r)^{-1}dr^2+r^2d\Omega_2^2,$4, one has $ds^2=-f(r)\,dt^2+j(r)^{-1}dr^2+r^2d\Omega_2^2,$5, $ds^2=-f(r)\,dt^2+j(r)^{-1}dr^2+r^2d\Omega_2^2,$6, the inner edge moves to $ds^2=-f(r)\,dt^2+j(r)^{-1}dr^2+r^2d\Omega_2^2,$7, and the HBH emerges as the marginally stable null limit.

3. Geodesics, shadow, and optical mimicry

Null and timelike geodesics obey

$ds^2=-f(r)\,dt^2+j(r)^{-1}dr^2+r^2d\Omega_2^2,$8

with conserved energy and angular momentum

$ds^2=-f(r)\,dt^2+j(r)^{-1}dr^2+r^2d\Omega_2^2,$9

After the reparametrization $$j(r)=1-\frac{2\widehat m(r)}{r}, \qquad f(r)=\left[\frac{\widehat m(r)}{M}\right]^2 j(r).$$0, the radial equation takes the form

$$j(r)=1-\frac{2\widehat m(r)}{r}, \qquad f(r)=\left[\frac{\widehat m(r)}{M}\right]^2 j(r).$$1

For null geodesics, $$j(r)=1-\frac{2\widehat m(r)}{r}, \qquad f(r)=\left[\frac{\widehat m(r)}{M}\right]^2 j(r).$$2, and in the ocean

$$j(r)=1-\frac{2\widehat m(r)}{r}, \qquad f(r)=\left[\frac{\widehat m(r)}{M}\right]^2 j(r).$$3

so the effective potential is exactly flat. Circular null orbits occur when the impact parameter $$j(r)=1-\frac{2\widehat m(r)}{r}, \qquad f(r)=\left[\frac{\widehat m(r)}{M}\right]^2 j(r).$$4 equals

$$j(r)=1-\frac{2\widehat m(r)}{r}, \qquad f(r)=\left[\frac{\widehat m(r)}{M}\right]^2 j(r).$$5

At $$j(r)=1-\frac{2\widehat m(r)}{r}, \qquad f(r)=\left[\frac{\widehat m(r)}{M}\right]^2 j(r).$$6, every radius in the ocean supports a marginally stable circular null orbit. For $$j(r)=1-\frac{2\widehat m(r)}{r}, \qquad f(r)=\left[\frac{\widehat m(r)}{M}\right]^2 j(r).$$7, the trajectories are spirals across the plateau region (Riojas et al., 29 Jun 2026).

Because $$j(r)=1-\frac{2\widehat m(r)}{r}, \qquad f(r)=\left[\frac{\widehat m(r)}{M}\right]^2 j(r).$$8 is constant in the ocean and decreases away from it in both directions, the plateau at $$j(r)=1-\frac{2\widehat m(r)}{r}, \qquad f(r)=\left[\frac{\widehat m(r)}{M}\right]^2 j(r).$$9 is the only local maximum relevant to exterior optical probes. The outer edge $\widehat m(r)= \begin{cases} m, & r\in(2m,\,3m),\[2pt] \dfrac{r}{3}, & r\in(3m,\,3M),\[2pt] M, & r\in(3M,\,\infty). \end{cases}$0 plays the role of the critical Schwarzschild photon sphere of mass $\widehat m(r)= \begin{cases} m, & r\in(2m,\,3m),\[2pt] \dfrac{r}{3}, & r\in(3m,\,3M),\[2pt] M, & r\in(3M,\,\infty). \end{cases}$1. Since the spacetime is exactly Schwarzschild-$\widehat m(r)= \begin{cases} m, & r\in(2m,\,3m),\[2pt] \dfrac{r}{3}, & r\in(3m,\,3M),\[2pt] M, & r\in(3M,\,\infty). \end{cases}$2 for $\widehat m(r)= \begin{cases} m, & r\in(2m,\,3m),\[2pt] \dfrac{r}{3}, & r\in(3m,\,3M),\[2pt] M, & r\in(3M,\,\infty). \end{cases}$3, no incoming null or timelike geodesic that crosses $\widehat m(r)= \begin{cases} m, & r\in(2m,\,3m),\[2pt] \dfrac{r}{3}, & r\in(3m,\,3M),\[2pt] M, & r\in(3M,\,\infty). \end{cases}$4 can later return to infinity. This is the core reason the HBH is an optical black-hole mimic.

For a static spherical metric, the shadow seen by an observer at $\widehat m(r)= \begin{cases} m, & r\in(2m,\,3m),\[2pt] \dfrac{r}{3}, & r\in(3m,\,3M),\[2pt] M, & r\in(3M,\,\infty). \end{cases}$5 obeys

$\widehat m(r)= \begin{cases} m, & r\in(2m,\,3m),\[2pt] \dfrac{r}{3}, & r\in(3m,\,3M),\[2pt] M, & r\in(3M,\,\infty). \end{cases}$6

For the HBH, with $\widehat m(r)= \begin{cases} m, & r\in(2m,\,3m),\[2pt] \dfrac{r}{3}, & r\in(3m,\,3M),\[2pt] M, & r\in(3M,\,\infty). \end{cases}$7,

$\widehat m(r)= \begin{cases} m, & r\in(2m,\,3m),\[2pt] \dfrac{r}{3}, & r\in(3m,\,3M),\[2pt] M, & r\in(3M,\,\infty). \end{cases}$8

This yields three regimes. For $\widehat m(r)= \begin{cases} m, & r\in(2m,\,3m),\[2pt] \dfrac{r}{3}, & r\in(3m,\,3M),\[2pt] M, & r\in(3M,\,\infty). \end{cases}$9, the shadow is identical to that of a Schwarzschild black hole of mass $m$0. For $m$1, it matches Schwarzschild of mass $m$2. For observers inside the ocean,

$m$3

so the shadow fills exactly half the sky. At large $m$4, Synge’s asymptotic behavior is recovered: $m$5

The optical equivalence is not exact at the level of wave propagation. For bosonic fields of spin $m$6, the radial wave equation involves

$m$7

with

$m$8

In the ocean this becomes

$m$9

The potential has finite jumps at $3mquasinormal modes and greybody factors need not coincide with Schwarzschild even when the shadow and primary photon ring do (Riojas et al., 29 Jun 2026).

4. Thermodynamic mimicry in asymptotically flat space

The thermodynamic analysis assumes, formally, that the horizon and the null-cluster ocean can be placed in thermal equilibrium. Under that assumption, the Hawking temperature computed from the surface gravity at the inner horizon is

$3m

This is independent of the horizon mass $3mRiojas et al., 29 Jun 2026).

The local Tolman temperature is

$3m

Inside the ocean this simplifies to

$3m

The paper derives, from entropy maximization for anisotropic matter, the local Euler relation

$3m

Since the null cluster has $3m

$M$0

Using

$M$1

one obtains the entropy density

$M$2

Integrating over the ocean gives

$M$3

Adding the horizon entropy of the central black hole,

$M$4

yields

$M$5

Under the coarse-grained additivity assumed in the paper, the HBH entropy therefore coincides exactly with that of an ordinary Schwarzschild black hole of mass $M$6 (Riojas et al., 29 Jun 2026).

The same conclusion is reproduced in several independent ways. In the Euclidean-action approach, the Ricci-scalar bulk term vanishes because the NC gas is traceless, the matter action vanishes for the $M$7 Einstein cluster, and no extra wall terms arise because the ocean is aligned smoothly with the inner and outer Schwarzschild regions. Consequently only the Gibbons–Hawking–York term remains, and since the asymptotic geometry is exactly Schwarzschild-$M$8, the free energy is the same as for a black hole of mass $M$9.

A York-style finite-cavity derivation gives

$r_h=2m$0

which is York’s Schwarzschild action with $r_h=2m$1. The quasilocal energy is

$r_h=2m$2

and the entropy inside the sphere is

$r_h=2m$3

Hence the entropy is constant in the inner vacuum, grows only through the ocean, and becomes constant again outside it: $r_h=2m$4

$r_h=2m$5

$r_h=2m$6

5. Photon-sphere thermodynamics, cavities, and AdS generalization

The photon sphere is central not only to optics but also to the thermodynamic structure. In ordinary Schwarzschild cavity thermodynamics, the specific heat is

$r_h=2m$7

which diverges at the photon sphere $r_h=2m$8. For the HBH the same formula holds with $r_h=2m$9. Inside the ocean, where $M$00,

$M$01

Thus, at fixed radius, adding or removing energy does not change the local temperature profile; instead, the ocean extends or shrinks. This is why the ocean behaves thermodynamically like a heat bath (Riojas et al., 29 Jun 2026).

The cavity analysis is especially provocative. For ordinary isotropic radiation in flat space, a black hole in a cavity can fully evaporate into radiation only if the cavity is parametrically large,

$M$02

because ordinary radiation has entropy

$M$03

while the black hole has

$M$04

With NC gas, the conclusion changes. In a cavity with

$M$05

the ocean can fill the box all the way to the wall. In flat space, all HBH configurations of fixed total mass $M$06 have the same entropy independent of $M$07, so a mass-$M$08 Schwarzschild black hole and a cavity-filling NC gas state are entropically degenerate. The authors therefore note that an HBH in a cavity of radius $M$09 can evaporate, potentially posing the information puzzle in a small finite volume (Riojas et al., 29 Jun 2026).

In asymptotically AdS space, exact thermodynamic mimicry fails. The metric becomes

$M$10

with

$M$11

The Hawking temperature is then

$M$12

where $M$13 satisfies

$M$14

The entropy becomes

$M$15

Thus the AdS HBH is not thermodynamically equivalent to a Schwarzschild-AdS black hole of mass $M$16. The authors describe a richer phase structure in both canonical and microcanonical ensembles, including a gas-filled limiting state with minimum temperature

$M$17

6. Higher-dimensional extension, interpretation, and limitations

The classical construction extends to all $M$18 and to any cosmological constant. In $M$19 dimensions,

$M$20

with

$M$21

and

$M$22

Inside the ocean,

$M$23

The central qualitative feature therefore persists: a null-matter shell creates an aligned extended photon sphere between the inner and outer photon spheres (Riojas et al., 29 Jun 2026).

The physical interpretation advanced by the authors is deliberately cautious. Classically, HBHs are exact, highly special black-hole-plus-matter solutions with several distinctive properties: the constituents themselves move on the null circular geodesics that generate the extended photon sphere; the construction is wall-free because $M$24; and, unlike more generic self-similar gases or shell assemblies, the smooth HBH ocean is the unique aligned case that optically mimics Schwarzschild-$M$25 from outside. This suggests a precise separation between exterior optical observables and interior matter structure.

At the same time, the papers stress substantial limitations. The thermal-equilibrium assumption is formal and may fail: the relevant quanta would have wavelengths of order the ocean size, the required entropy density

$M$26

is enormous, and a fluid description may be unreliable. The ocean may leak from its edges at $M$27 and $M$28. Thermal coupling seems in tension with the condition $M$29. Formation mechanisms are unclear, and the analysis is restricted to static, spherical configurations; rotating generalizations are left open. Wave dynamics can differ from Schwarzschild even when geometric optics does not, so “optically indistinguishable” is limited to the geodesic and distant-image sense. The authors therefore treat HBHs as a toy model, potentially suggestive for black-hole evolution and finite-volume information-loss questions, but not yet established as realistic astrophysical objects (Riojas et al., 29 Jun 2026, Riojas et al., 29 Jun 2026).