Orbital, Shadow, and Thin-Disk Signatures of a Regular Black Hole with Gravitational Self-Energy

Abstract: We investigate geodesic motion, shadow observables and thin-disk accretion for the regular black hole generated by a non-local gravitational self-energy contribution. The geometry is controlled by a zero-point length and can be followed smoothly from the Schwarzschild limit to a cold extremal remnant. We compute the photon ring, critical impact parameter, apparent shadow radius, null Lyapunov exponent, innermost stable circular orbit, orbital frequencies and Novikov-Thorne flux profile. As the self-energy scale grows, both the photon ring and the ISCO move outward, while the photon-ring frequency and instability exponent decrease. The horizon-normalized shadow area increases by about a factor of 2.7 for the near-extremal benchmark, although the same shadow radius decreases when normalized by the ADM mass. The ISCO binding efficiency grows modestly, whereas the zero-torque thin-disk flux peak moves outward and falls to about 61% of the Schwarzschild peak as the solution approaches the remnant regime. These trends identify a coherent set of optical, orbital and accretion signatures of the gravitational-self-energy regularization.

• The paper develops geodesic and accretion diagnostics for a gravitational-self-energy regular black hole, finding that the photon ring shifts outward, its instability weakens, and the ISCO moves from 3 to 4.359 horizon radii as the zero-point-length parameter approaches its extremal value.
• The paper shows that the shadow area grows by a factor of 2.695 at fixed horizon radius but the shadow radius decreases from 5.196 to 4.192 when normalized by ADM mass, demonstrating that observational conclusions depend critically on the reference scale.
• The paper finds that thin-disk emission shifts outward, with the flux peak moving from 4.776 to 6.993 horizon radii and declining to 61.1% of the Schwarzschild peak near the remnant branch, while ISCO efficiency rises modestly from 5.72% to 7.62%.

Geometry and parametrization

The paper analyzes a static, spherically symmetric regular black hole whose lapse function is modified by a non-local gravitational self-energy contribution, following the solution of Jusufi and Singleton. The metric retains the standard form $ds^2 = -f(r)dt^2 + dr^2/f(r) + r^2 d\Omega_2^2$, with the deformation controlled by a zero-point length $l_0$ that smears the central source. At large radius the geometry is asymptotically Schwarzschild with an ADM mass $M_{\rm ADM} = m(1 + 3\pi m/32l_0)$, so the self-energy correction renormalizes the asymptotic mass upward relative to the bare parameter $m$.

The analysis is performed in horizon-normalized variables $x = r/r_+$ and $\lambda = l_0/r_+$. The condition $f(1)=0$ fixes $\bar m = m/r_+$ on the branch continuously connected to Schwarzschild ($\bar m(0)=1/2$). The branch terminates at an extremal, zero-temperature remnant at $\lambda_{\rm ext} = 0.6526091837\ldots$, determined by simultaneous degeneracy of the horizon and vanishing surface gravity. The benchmark sequence $l_0$0 spans nearly the full black-hole branch; at $l_0$1 the inner horizon sits at $l_0$2 and $l_0$3. This fixed-horizon normalization is a deliberate choice: it isolates near-horizon optical structure but must be distinguished from ADM-mass normalization when comparing to observations.

Photon ring and null instability

Null circular orbits satisfy $l_0$4. As $l_0$5 increases, the photon orbit moves outward from $l_0$6 (Schwarzschild) to $l_0$7 at $l_0$8, and the critical impact parameter grows from $l_0$9 to $M_{\rm ADM} = m(1 + 3\pi m/32l_0)$0. Two dynamical quantities decrease monotonically: the photon-ring angular frequency falls from $M_{\rm ADM} = m(1 + 3\pi m/32l_0)$1 to $M_{\rm ADM} = m(1 + 3\pi m/32l_0)$2, and the null Lyapunov exponent drops faster still, so that the ratio $M_{\rm ADM} = m(1 + 3\pi m/32l_0)$3 decreases from unity to $M_{\rm ADM} = m(1 + 3\pi m/32l_0)$4. In horizon units, the self-energy regularization therefore produces a larger photon ring that is less rapidly unstable.

The author connects these quantities to quasinormal modes through the eikonal correspondence $M_{\rm ADM} = m(1 + 3\pi m/32l_0)$5, $M_{\rm ADM} = m(1 + 3\pi m/32l_0)$6, which would suggest lower real frequencies and weaker damping at large multipole. This inference is stated with appropriate caution: the correspondence is not a theorem for all backgrounds or perturbation sectors, and known counterexamples are cited. A direct perturbative calculation for this geometry remains undone.

Shadow radius and its scale dependence

For a static observer at spatial infinity, $M_{\rm ADM} = m(1 + 3\pi m/32l_0)$7. At fixed event-horizon radius the shadow grows monotonically, and the apparent area increases by a factor of 2.695 at $M_{\rm ADM} = m(1 + 3\pi m/32l_0)$8 relative to Schwarzschild. However, normalized by the ADM mass — which itself grows along the branch from $M_{\rm ADM} = m(1 + 3\pi m/32l_0)$9 to $m$0 — the same shadow radius decreases, from $m$1 to $m$2. This is the paper's most consequential observational point: a model-independent angular measurement constrains only $m$3, and converting it into a test of the metric requires an independent mass calibration. Horizon-normalized and mass-normalized comparisons yield opposite conclusions about whether the shadow is enlarged or shrunk, so the choice of held-fixed scale is not innocuous.

Since $m$4, the growth of the horizon-normalized shadow follows directly from the outward displacement and frequency reduction of the photon ring.

Timelike circular orbits and the ISCO

The ISCO moves outward from $m$5 to $m$6 over the benchmark sequence, with specific angular momentum rising from $m$7 to $m$8 and specific energy falling from $m$9 to $x = r/r_+$0. The binding efficiency $x = r/r_+$1 consequently rises from 5.72% to 7.62%, a modest increase that reflects deeper geodesic binding, not enhanced Hawking luminosity. Despite this deepening, the orbital frequency measured at infinity falls to $x = r/r_+$2 at $x = r/r_+$3, because the outward displacement of the ISCO dominates over the increased binding energy.

Novikov–Thorne flux

Using the circular-orbit data, the paper computes the zero-torque Novikov–Thorne flux for a geometrically thin, optically thick, radiatively efficient disk with inner edge at the ISCO. The idealizations are stated explicitly: no disk self-gravity, finite-thickness or pressure corrections, magnetic stresses, returning radiation, or spectral hardening; the result is used as a geometric comparison between backgrounds rather than a radiative-transfer prediction.

The flux peak moves outward monotonically for all nonzero $x = r/r_+$4, from $x = r/r_+$5 to $x = r/r_+$6. Its height is non-monotonic: it slightly exceeds the Schwarzschild value at $x = r/r_+$7 (ratio 1.086), is essentially unchanged at $x = r/r_+$8 (1.003), then falls to 0.611 of the Schwarzschild peak at $x = r/r_+$9. The dominant accretion signature is thus the outward displacement of the bright disk region, with substantial flux suppression appearing only near the remnant branch.

Limitations and open questions

Several caveats bound the results. All quantitative trends are quoted in horizon-normalized units on a single benchmark sequence; behavior at fixed ADM mass is discussed only for the shadow radius, and other observables could differ under that normalization. The eikonal quasinormal-mode interpretation is asserted only schematically and is explicitly not a substitute for computing the perturbation spectrum of this metric. The thin-disk analysis inherits the full set of Novikov–Thorne idealizations, so the 61% peak-flux reduction should not be read as a direct spectral prediction. Finally, the analysis covers only the neutral, non-rotating sector; whether the signatures persist under rotation, and how they compare against EHT-grade imaging or X-ray spectroscopy data, are questions the paper leaves open.

Conclusion

The paper establishes a coherent set of geodesic and accretion diagnostics for a gravitational-self-energy regular black hole: an outward-moving photon ring and ISCO, a growing horizon-normalized shadow (factor 2.7 in area) that shrinks under ADM normalization, decreasing photon-ring frequency and Lyapunov exponent, modestly increased binding efficiency (5.72% → 7.62%), and a displaced, suppressed thin-disk flux peak. The central methodological lesson is the strong dependence of observational interpretation on the chosen reference scale, and the natural next steps are direct perturbation spectra, ray-traced images, and dynamical accretion modeling for this geometry.