Rotating CPS Spacetime Insights

• Rotating CPS spacetimes are stationary, axisymmetric geometries with a core where all photon orbits are closed and share a fixed impact parameter.
• They feature segmented photon rings with constant angular bounds, producing distinct shadow and imaging patterns compared to traditional Kerr models.
• The unique geodesic dynamics in these spacetimes imply robust photon trapping, with observable 'frozen' ring structures detectable by high-resolution VLBI.

A rotating CPS (Continuous Photon Sphere) spacetime is a general relativistic solution engineered so that within a certain core region, all null geodesics at fixed radius are closed, constituting a family of continuous photon spheres. These spacetimes generalize the concept of the photon sphere in Schwarzschild geometry to settings with axial symmetry and rotation. They feature a core where every photon trajectory is a closed orbit, and in the rotating case possess distinctly non-Kerr-like photon regions with unique observational signatures, particularly in high-precision imaging scenarios.

1. Definition and Metric Structure of Rotating CPS Spacetimes

A rotating CPS spacetime is a stationary, axisymmetric geometry containing a core in which every unstable photon orbit satisfies the photon sphere condition. The prototypical metric ansatz for the rotating CPS core is

$$ds^2 = -\left[1 - \frac{r^2 + a^2 - \Delta(r)}{\rho^2}\right]dt^2 + \frac{\rho^2}{\Delta(r)}dr^2 - \frac{2a\sin^2\theta}{\rho^2}(r^2 + a^2 - \Delta(r))dtd\phi + \rho^2 d\theta^2 + \frac{\Sigma\sin^2\theta}{\rho^2}d\phi^2$$

where

$$\rho^2 = r^2 + a^2\cos^2\theta, \quad \Sigma = (r^2 + a^2)^2 - a^2\Delta(r)\sin^2\theta.$$

Within the core region (typically $r < r_+$ for some edge $r_+$), the metric function $\Delta(r)$ is highly constrained by the requirement that every null geodesic with constant $r$ is closed. Explicitly,

$\Delta(r) = \frac{(r^2 + a^2 - B)^2}{k}$

where the constants $B$ and $k$ are determined by the global structure and continuity conditions (Li et al., 16 Aug 2025).

All photon orbits within the CPS core share the same impact parameter $b = B/a$ and a fixed reduced Carter constant $K_E = k$, in stark contrast to the Kerr spacetime where these quantities vary continuously across the photon region.

2. Photon Regions and Dynamical Distinction from Kerr

In a standard Kerr black hole, the photon region in a constant-$\phi$ cross section is $(r, \theta)$-dependent and forms a smooth crescent that changes with radius. For a rotating CPS spacetime, the photon region in a constant-$\phi$ cross section appears as either a single continuous angular sector (between symmetric latitudinal bounds $(\theta_0, \pi-\theta_0)$) or two disconnected angular bands, but crucially, these angular limits are independent of radius within the CPS core.

The permitted polar motion for photons is governed by the geodesic equation

$$\rho^4\sin^2\theta\, \left(\frac{d\theta}{d\lambda}\right)^2 = -a^2\sin^4\theta + (k + 2B)\sin^2\theta - b^2,$$

with the right-hand side in the form of a quadratic in $x = \sin^2\theta$. The solutions yield polar “bands” (e.g., for $x_1 < x < x_2$ or $0 < x < x_*$) that are constant with respect to $r$, which is not possible in the Kerr photon region. This structural rigidity directly affects photon trapping and the visibility of the photon region.

3. Imaging Signatures and Observable Features

Rotating CPS spacetimes with a CPS core have several novel observable properties:

• Uniform Impact Parameter and Photon Region: All unstable photon orbits within the core have the same $b$ and $K_E$, resulting in a photon region that is uniform across $r$ and $\theta$ sectors rather than showing the smooth, varying structure of Kerr.
• Shadow and Photon Ring Features: The fixed impact parameter and angular segmentation can produce shadow images and photon rings with “frozen” or multi-ring substructure—distinct from the continuous broadening and blurring expected in the Kerr case, especially under perturbations.
• Photon Dynamics: A perturbation to a photon orbit in the CPS core leads to a “sliding” of the photon between neighboring unstable orbits, rather than rapid escape or plunging as in Kerr. This may make the photon ring region more robust and segmented in high-resolution observations.

These features can be detected or constrained by current or future very long baseline interferometry (VLBI), including the Event Horizon Telescope (EHT). Future missions with even finer resolution or time variability (“reverberation mapping”) may be able to resolve the angular sector structure directly.

4. Accretion Flow and Universality of Image Profiles

The universality of the image profile in a spherically symmetric, static spacetime with a luminous CPS core is governed by the specific form of the observed intensity,

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

where $b_c$ is the critical impact parameter that defines the outermost photon trajectory. This universality holds regardless of the details of $g_{rr}$, the emissivity profile $j(r)$, or the gravity theory, so long as the emitting region is static and spherically symmetric.

However, this universal profile is strongly sensitive to the nature of the accretion flow:

• Static Thin Disk or Core: The image is universal up to normalization, and the shadow closely resembles that of the Schwarzschild black hole, even if the photon dynamics differ significantly inside the core (Li et al., 16 Aug 2025).
• Infalling Accretion: When emission arises from matter falling inward (e.g., Bondi flow), Doppler shifts and non-static emission break the simple factorization, and $I_{\mathrm{obs}}$ becomes explicitly dependent on direction, flow velocity, and the geometry, producing new image features sensitive to the metric and dynamics.

5. Distinction from Kerr and Experimental Prospects

The most notable technical distinctions between rotating CPS and Kerr geometries:

Feature	Rotating CPS Core	Kerr Spacetime
Photon region profile	Uniform in $r$, segmented in $\theta$	Smooth, continuous crescent varying in $r,\theta$
Impact parameter ($b$)	Fixed for all unstable orbits	Varies across orbits
Carter constant ($K_E$)	Fixed within core	Varies with radius
Shadow shape	Segmented/“frozen” ring	Blurred crescent

The unique structure of the photon region and its impact on both direct imaging (shadow, photon rings) and strong-lensing observables suggests that rotating CPS spacetimes can, in principle, be distinguished from Kerr by high-resolution, polarimetrically sensitive, or time-resolved observations of black hole candidates.

6. Theoretical and Astrophysical Implications

Rotating CPS spacetimes challenge the conventional uniqueness of the Kerr solution as the endpoint for realistic rotating compact objects:

• The universality of image profiles in static accretion scenarios implies that shadow measurements alone may not suffice to distinguish exotic spacetimes from black holes unless dynamic or high-precision angular signatures are resolvable.
• The rigidity of the photon region could result in observable “frozen” photon ring substructure not possible in Kerr (Li et al., 16 Aug 2025).
• The construction of rotating CPS solutions relies on highly tuned metric functions (e.g., $\Delta(r)$), and their formation or persistence as astrophysical objects remains a subject for further research.

A plausible implication is that the future detection of segmented photon ring images or violations of the expected Kerr shadow morphology in VLBI data could signal the presence of physics beyond the Kerr paradigm, specifically pointing toward rotating CPS structures or similar non-Kerr compact objects.

7. Outlook and Open Directions

Identifying and interpreting distinctive features due to the CPS photon region is a promising strategy for future observational campaigns. The robustness of the segmented photon region, its resilience to perturbations, and the detailed time-variability of ringdown signatures remain important targets for numerical simulations. Further studies are warranted to:

• Model realistic accretion flows and emission mechanisms in the context of rotating CPS spacetimes.
• Quantify the impact of finite core size, external spacetime matching, and metric perturbations.
• Develop data analysis pipelines capable of testing for segmented photon region structures in VLBI images.

Ongoing advances in black hole imaging, time-domain astronomy, and polarimetric measurements are likely to play a critical role in constraining or revealing rotating CPS geometries as viable models for ultra-compact rotating astrophysical objects.