Continuous Photon Spheres in Compact Objects

• Continuous Photon Spheres (CPS) are defined as extended spacetime regions where all circular photon orbits share the same critical impact parameter, forming a continuous shell rather than isolated radii.
• Their unique imaging profiles under static emission reveal a universal intensity distribution that mimics Schwarzschild black holes despite differing core photon dynamics.
• In rotating CPS spacetimes, photon orbits form fixed angular sectors with invariant impact parameters, leading to distinctive lensing features and potential time-dependent echoes.

A Continuous Photon Sphere (CPS) is a spacetime region, typically arising in self-gravitating, isotropic, spherically symmetric or certain rotating metrics, in which all circular photon orbits exist at the same critical impact parameter across an extended core volume rather than at a single, isolated radius. This structure leads to universal features in gravitational lensing and imaging, and often induces distinctive photon region geometries in rotating solutions. CPSs have been proposed in various theoretical models, including horizonless naked singularities, self-gravitating compact objects without event horizons, and as mathematically idealized cores of ultracompact objects. Their observational and theoretical properties offer both a window into non-black-hole spacetimes and a challenge for interpretation of high-resolution astronomical images.

1. Definition and Geometric Structure of Continuous Photon Spheres

A CPS is defined by the property that, within a finite spatial region (the "CPS region"), circular photon orbits exist for every radius in that region, all sharing the same critical impact parameter $b_c$. The canonical example is a static self-gravitating, spherically symmetric spacetime in which the time-time metric component inside the CPS core is

$g_{tt} = -C r^2$

for constant $C$ and $r$ within the CPS region (Li et al., 16 Aug 2025). For such metrics, the effective potential for massless particles (photons) allows a continuous family of circular null geodesics for all radii where this form holds, not just at isolated points as in Schwarzschild (where the photon sphere is at $r=3M$).

The critical impact parameter for these orbits is

$b_c = \frac{1}{\sqrt{C}}$

and all photon orbits inside the CPS core satisfy $b=b_c$, rendering the set of circular orbits non-isolated and forming a "shell" or extended core of photon trapping (Li et al., 16 Aug 2025). In rotating generalizations, subject to metric constraints, all unstable photon orbits in the continuous region share the same $b$ and reduced Carter constant $k$; unlike Kerr, the angular latitude range of photon region orbits is frozen across the CPS core.

This differs fundamentally from black hole photon spheres, which exist only at discrete radii where the effective potential for null geodesics has a stationary point.

2. Imaging, Lensing, and Universality for Static Luminous Cores

Central to the observational signatures of CPS spacetimes is the formation of images—particularly under thin disk illumination or static, spherically symmetric emission models. For any static, spherically symmetric CPS spacetime with a luminous core, the observed intensity profile as a function of impact parameter $b$ for rays that reach the observer ($b < b_c$) takes the universal form

$$I_{\textrm{obs}} \propto \frac{1}{\sqrt{1 - b^2/b_c^2}}$$

regardless of the detailed metric component $g_{rr}$, the emission profile $j(r)$, or specific energy-momentum content of the spacetime (Li et al., 16 Aug 2025). This result arises because, in such CPS spacetimes, photons from different radii inside the core all emerge with the same impact parameter and redshift structure, leading to an invariant functional shape for the image.

As a result, images of thin disks or static spherically symmetric emission from CPS spacetimes are practically indistinguishable from those of Schwarzschild black holes for $b > b_c$, even though the photon dynamics inside the core differ drastically. This makes the distinction between a true black hole and a horizonless CPS core fundamentally ambiguous based solely on these profiles.

The table below summarizes universal image features in various CPS emission models:

Emission Scenario	Image Profile ($I_{\rm obs}$ vs $b$)	Universality Valid?
Thin disk, static emission	$I_{\textrm{obs}} \propto 1/\sqrt{1-b^2/b_c^2}$	Yes
Spherical infall, non-static	Depends on $b$, $g_i$ (Doppler shifts)	No
Rotating thin disk	$b$-dependence may be affected by rotation	No/Model-dependent

For spherically symmetric infalling accretion flows, the Doppler shifts and non-static emission introduce $b$-dependent modulations to the observed intensity, breaking the universal profile and resulting in image features sensitive to both the accretion dynamics and spatial metric (Li et al., 16 Aug 2025).

3. Photon Regions and Structure in Rotating CPS Spacetimes

In rotating CPS spacetimes, the photon region geometry differs qualitatively from that in Kerr. The metric ansatz for rotation adopts the generic form: $$ds^2 = -\left[1 - \frac{r^2 + a^2 - \Delta}{\rho^2}\right] dt^2 + \frac{\rho^2}{\Delta} dr^2 - \frac{2a \sin^2 \theta}{\rho^2} (r^2 + a^2 - \Delta) dt d\phi + \rho^2 d\theta^2 + \frac{\Sigma \sin^2 \theta}{\rho^2} d\phi^2$$ with suitably chosen $\Delta(r)$ to enforce $R(r) = 0$ and $R'(r)=0$ for all $r$ inside the CPS core, so all circular photon orbits share the same impact parameter $b$ and Carter constant $k$ (Li et al., 16 Aug 2025). The latitudinal motion equation in terms of $x = \sin^2\theta$ gives allowed angular bands where photon orbits can exist: $A(x) = -a^2 x^2 + (k + 2B)x - b^2$ Depending on $a^2$, $b$, $k$ and $B$, the photon region in a constant-$\phi$ cross-section can appear as one or two angular sectors—a distinctive difference from the crescent-shaped and $r$-dependent photon region of Kerr.

These properties imply that CPS photon regions, in principle, should manifest observational signatures such as shadow boundaries or brightness distributions that are robustly distinct from the Kerr scenario—e.g., "frozen" angular photon region sectors and subring features that do not shift with $r$ (Li et al., 16 Aug 2025).

4. Theoretical Implications and Microscopic Photon Dynamics

The existence of a CPS core requires, at minimum, a spacetime region where $g_{tt}\propto -r^2$. This manifests in certain horizonless naked singularity models or engineered matter distributions but does not typically arise in conventional black hole solutions constrained by regularity and energy conditions.

The photon dynamics inside such a CPS region are markedly different from black holes. While all photon orbits with $b=b_c$ are unstable, perturbations merely shift photons onto neighboring circular orbits rather than causing rapid infall or escape, as occurs for isolated photon spheres (Li et al., 16 Aug 2025). This behavior can, in principle, induce slow time-dependent features or "echoes" persistent in lensed images, distinct from black hole expectations.

In addition, the presence of a continuous (as opposed to discrete) photon region creates a different topology for photon trapping, potentially impacting quasinormal mode spectra, the time profile of ringdown signals, and even the propagation of waves through the core.

5. Observational Signatures and Distinguishability from Black Holes

Although the direct imaging of thin disks around a CPS core mimics that of a Schwarzschild black hole (universal $b$-profile for static emission), any deviation from the static, spherically symmetric emission—such as bowl-shaped or off-axis accretion, significant infall velocities, or rotation—can break the universality and reintroduce metric- and dynamics-specific image features.

Key signatures that could distinguish CPS spacetimes from black holes include:

• Non-universal, $b$-dependent substructure in images under non-static emission, potentially resolvable with next-generation very-long-baseline interferometry (VLBI).
• Frozen angular sector photon regions in rotating spacetimes, leading to shadow and photon ring features that are inconsistent with Kerr predictions if sufficiently resolved.
• Possible persistence of photons within the CPS region (delayed escapes, time-domain features) if perturbed, manifesting as distinct temporal decay profiles or echo phenomena.

In all cases, observational diagnosis requires going beyond the image envelope and analyzing the detailed image substructure, velocity-dependent photon fluxes, or time-resolved imaging.

6. Context and Significance in Ultractompact Object Studies

The paper of CPS spacetimes is motivated by the desire to characterize alternatives to black holes—especially horizonless ultracompact objects that may mimic black holes in astrophysical images yet differ in their core photon dynamics and gravitational lensing properties. Universal image features under thin disk illumination challenge the community to develop more refined observational diagnostics to distinguish such possibilities.

Moreover, CPS models provide mathematically explicit test cases for the limits of black hole uniqueness and for probing strong-field regimes where standard photon sphere theory requires extension.

7. Summary Table: CPS vs. Schwarzschild/Kerr Photon Regions

Feature	Standard Black Hole (Schwarzschild/Kerr)	Continuous Photon Sphere (CPS)
Photon sphere location(s)	Discrete radius (e.g., $r=3M$), impact parameter depends on $r$	All $r$ inside CPS core, $b=b_c$ $\forall$ $r$
Shadow/bright ring image	Sharp ring, profile set by $b_{ph}$, off-core photon dynamics	Same universal profile under static disk
Photon region in rotation	Crescent, $r$-dependent sector in $(\theta,\phi)$ cross-section	One/two angular sectors, independent of $r$
Dynamics of perturbed photons	Rapid infall/escape near photon sphere	Shift to neighboring circular orbits, possible photon trapping
Universality of image profile	Only for Schwarzschild under static disk	Universal for all static luminous CPS cores
Subtlety of metric diagnosis	Possible with time-resolved or velocity-sensitive imaging	Nontrivial—requires departures from thin disk imaging

Continuous photon spheres thus stand at the intersection of geometric analysis, gravitational lensing, and observational astrophysics—a domain where the fine structure of photon trapping and emission opens a window into the deepest properties of compact spacetimes (Li et al., 16 Aug 2025).