Gravitational waveforms and accretion characteristics in a quantum-corrected black hole without Cauchy horizons

Abstract: The use of physical phenomena in the strong-field regime has become a primarily methodology for probing quantum-corrected gravity. This paper investigates periodic orbits, gravitational waves, and accretion disk radiation for a quantum-corrected black hole without Cauchy horizons. First, by analyzing the trajectory equations of massive particles in the equatorial plane, we study the influence of the quantum parameter $ζ$ on the stability of circular orbits. The results show that an increase in $ζ$ leads to an outward migration of both the innermost stable circular orbit and the marginally bound orbit, accompanied by an increase in the required specific angular momentum for particle motion on these two orbits. Then, we further investigate the periodic orbit characteristics of particles and compute the associated gravitational waveforms for extreme mass-ratio inspirals. It is demonstrated that quantum corrections induce a cumulative phase shift in the gravitational wave signal, leading to significant dephasing compared to the classical Schwarzschild case. Furthermore, based on the Novikov-Thorne thin accretion disk model, we evaluate the radiation characteristics of the accretion disk around this quantum-corrected black hole. The results indicate that the introduction of the quantum parameter suppresses the radiant energy flux, effective temperature, and overall radiative efficiency of the disk. These distinctive dynamical and radiative deviations provide potential phenomenological support for distinguishing quantum-corrected geometries from classical black holes using multiple observational means in the future.

• The paper shows that increasing the quantum parameter ζ moves the ISCO and marginally bound orbit outward, raises critical orbital energies and angular momenta, and increasingly alters zoom-whirl trajectories.
• The paper finds cumulative dephasing in extreme-mass-ratio inspiral waveforms that grows with ζ, suggesting LISA-class detectors could probe quantum corrections, although numerical-kludge modeling limits precise forecasts.
• The paper predicts weaker thin-disk emission as ζ increases, with radiative efficiency declining from 5.72% for Schwarzschild to 5.49% near ζ = 3.9M and high-frequency spectra most affected.

Background and motivation

Classical general relativity predicts spacetime singularities inside black holes, and resolving them is generally expected to require a quantum theory of gravity. Loop quantum gravity (LQG) offers a non-perturbative quantization of geometry, but constructing effective black hole solutions that are simultaneously singularity-free and manifestly covariant has been a persistent difficulty in the Hamiltonian formulation. Recent work by Zhang, Lewandowski, Ma, and Yang addressed this by imposing diffeomorphism invariance at the effective level, yielding a family of quantum-corrected black hole (QCBH) solutions that are either regular black holes or traversable wormholes depending on parameters, and — crucially — lack Cauchy horizons, which may avoid the mass inflation instability afflicting many regular black hole models (Zhang et al., 2024, Zhang et al., 2024).

The paper under review connects this covariant QCBH geometry to observables. The authors analyze timelike geodesics, periodic ("zoom-whirl") orbits, gravitational waveforms from extreme mass-ratio inspirals (EMRIs), and Novikov-Thorne thin-disk radiation, all as functions of the single quantum parameter $\zeta$. The aim is to establish whether $\zeta$ leaves measurable signatures in both gravitational-wave and electromagnetic channels.

The spacetime and geodesic structure

The metric is spherically symmetric with

$$g_{tt} = 1 - \frac{r^2\zeta^2\arcsin\!\left(\frac{2M\zeta^2}{r^3}\right)}{2M}, \qquad g_{rr} = \frac{1 - 4M^2\zeta^4/r^6}{g_{tt}},$$

for the asymptotically flat branch ($n=0$). For $\zeta < \sqrt{\pi^3/2}\,M \approx 3.937\,M$ the solution is a regular black hole whose interior contains a wormhole throat at $r_{\rm m} = (2M\zeta^2)^{1/3}$; above this bound the horizon disappears and the geometry becomes a traversable wormhole. The analysis is accordingly restricted to $\zeta < 3.937\,M$, and $\zeta \to 0$ recovers Schwarzschild.

The radial equation for equatorial timelike motion takes the form $$\dot{r}^2 = (1 - 4M^2\zeta^4/r^6)[\tilde{E}^2 - V_{\rm eff}(r)]$$ with $V_{\rm eff}(r) = g_{tt}(1 + \tilde{L}^2/r^2)$; the prefactor reflects the throat structure and vanishes at $\zeta$0. Bound orbits exist only when $\zeta$1 lies between the minimum and maximum of $\zeta$2, and the allowed energy width grows with $\zeta$3.

Circular orbits, ISCO, MBO, and periodic orbits

At fixed radius, particles on circular orbits around the QCBH have larger specific energy, angular momentum, and angular velocity than in Schwarzschild. Solving $\zeta$4 conditions yields closed-form expressions for $\zeta$5 and $\zeta$6 in terms of $\zeta$7 and its derivatives. The central quantitative result is that increasing $\zeta$8 pushes both critical radii outward: for the near-extremal case $\zeta$9, the marginally bound orbit sits at $$g_{tt} = 1 - \frac{r^2\zeta^2\arcsin\!\left(\frac{2M\zeta^2}{r^3}\right)}{2M}, \qquad g_{rr} = \frac{1 - 4M^2\zeta^4/r^6}{g_{tt}},$$0 with $$g_{tt} = 1 - \frac{r^2\zeta^2\arcsin\!\left(\frac{2M\zeta^2}{r^3}\right)}{2M}, \qquad g_{rr} = \frac{1 - 4M^2\zeta^4/r^6}{g_{tt}},$$1, compared to the Schwarzschild values of $$g_{tt} = 1 - \frac{r^2\zeta^2\arcsin\!\left(\frac{2M\zeta^2}{r^3}\right)}{2M}, \qquad g_{rr} = \frac{1 - 4M^2\zeta^4/r^6}{g_{tt}},$$2 and $$g_{tt} = 1 - \frac{r^2\zeta^2\arcsin\!\left(\frac{2M\zeta^2}{r^3}\right)}{2M}, \qquad g_{rr} = \frac{1 - 4M^2\zeta^4/r^6}{g_{tt}},$$3. Both $$g_{tt} = 1 - \frac{r^2\zeta^2\arcsin\!\left(\frac{2M\zeta^2}{r^3}\right)}{2M}, \qquad g_{rr} = \frac{1 - 4M^2\zeta^4/r^6}{g_{tt}},$$4, $$g_{tt} = 1 - \frac{r^2\zeta^2\arcsin\!\left(\frac{2M\zeta^2}{r^3}\right)}{2M}, \qquad g_{rr} = \frac{1 - 4M^2\zeta^4/r^6}{g_{tt}},$$5, $$g_{tt} = 1 - \frac{r^2\zeta^2\arcsin\!\left(\frac{2M\zeta^2}{r^3}\right)}{2M}, \qquad g_{rr} = \frac{1 - 4M^2\zeta^4/r^6}{g_{tt}},$$6, $$g_{tt} = 1 - \frac{r^2\zeta^2\arcsin\!\left(\frac{2M\zeta^2}{r^3}\right)}{2M}, \qquad g_{rr} = \frac{1 - 4M^2\zeta^4/r^6}{g_{tt}},$$7, and $$g_{tt} = 1 - \frac{r^2\zeta^2\arcsin\!\left(\frac{2M\zeta^2}{r^3}\right)}{2M}, \qquad g_{rr} = \frac{1 - 4M^2\zeta^4/r^6}{g_{tt}},$$8 increase monotonically with $$g_{tt} = 1 - \frac{r^2\zeta^2\arcsin\!\left(\frac{2M\zeta^2}{r^3}\right)}{2M}, \qquad g_{rr} = \frac{1 - 4M^2\zeta^4/r^6}{g_{tt}},$$9. This outward migration directly implies that any observable tied to the ISCO — disk inner edge, radiative efficiency, plunge dynamics — shifts systematically with the quantum parameter.

Periodic orbits are classified via the Levin–Perez-Giz scheme: the rational number $n=0$0 labels each orbit by integers $n=0$1 counting zooms, whirls, and vertices. Computing $n=0$2 and $n=0$3 shows that $n=0$4 diverges at the boundaries of the allowed parameter space, and that the boundary values themselves grow with $n=0$5. Tabulated values of $n=0$6 (at fixed $n=0$7 midway between ISCO and MBO) and $n=0$8 (at fixed $n=0$9) are consistently larger than their Schwarzschild counterparts, with deviations growing monotonically in $\zeta < \sqrt{\pi^3/2}\,M \approx 3.937\,M$0 — e.g., $\zeta < \sqrt{\pi^3/2}\,M \approx 3.937\,M$1 rises from 0.965425 (Schwarzschild) to 0.968421 ($\zeta < \sqrt{\pi^3/2}\,M \approx 3.937\,M$2). Numerically integrated trajectories confirm that orbital morphology deviates increasingly from Schwarzschild as $\zeta < \sqrt{\pi^3/2}\,M \approx 3.937\,M$3 grows.

Gravitational waveforms from EMRIs

Using a numerical kludge waveform construction with quadrupole-order polarizations, the authors generate signals for an EMRI with $\zeta < \sqrt{\pi^3/2}\,M \approx 3.937\,M$4, $\zeta < \sqrt{\pi^3/2}\,M \approx 3.937\,M$5, luminosity distance 200 Mpc, and inclination/pericenter angles of $\zeta < \sqrt{\pi^3/2}\,M \approx 3.937\,M$6, following the $\zeta < \sqrt{\pi^3/2}\,M \approx 3.937\,M$7 periodic orbit. The key finding is a cumulative phase dephasing: waveforms initially coincide with the Schwarzschild prediction but progressively drift out of phase over time, with the dephasing growing with $\zeta < \sqrt{\pi^3/2}\,M \approx 3.937\,M$8. This behavior is precisely what makes EMRIs powerful probes — space-based detectors such as LISA, Taiji, and TianQin track long inspiral signals where accumulated phase errors translate into parameter constraints on $\zeta < \sqrt{\pi^3/2}\,M \approx 3.937\,M$9. It should be noted, however, that the calculation uses a kludge scheme rather than full Teukolsky-based self-force evolution, so quantitative dephasing estimates carry the accuracy limitations inherent to that approximation.

Thin accretion disk signatures

Within the Novikov–Thorne model (geometrically thin, optically thick, steady-state Keplerian disk truncated at the ISCO), with $r_{\rm m} = (2M\zeta^2)^{1/3}$0 and $r_{\rm m} = (2M\zeta^2)^{1/3}$1, the authors compute:

• Energy flux $r_{\rm m} = (2M\zeta^2)^{1/3}$2: the profile peaks and declines as usual, but its amplitude decreases monotonically with $r_{\rm m} = (2M\zeta^2)^{1/3}$3.
• Observed temperature $r_{\rm m} = (2M\zeta^2)^{1/3}$4: likewise suppressed by increasing $r_{\rm m} = (2M\zeta^2)^{1/3}$5, via the Stefan–Boltzmann relation and redshift factor.
• Spectral energy distribution: for $r_{\rm m} = (2M\zeta^2)^{1/3}$6 Hz the spectra are nearly indistinguishable from Schwarzschild; quantum corrections suppress the luminosity at higher frequencies. The result is insensitive to the outer disk radius.
• Radiative efficiency $r_{\rm m} = (2M\zeta^2)^{1/3}$7:

Spacetime	$r_{\rm m} = (2M\zeta^2)^{1/3}$8
Schwarzschild	0.05719
QCBH ($r_{\rm m} = (2M\zeta^2)^{1/3}$9)	0.05712
QCBH ($\zeta < 3.937\,M$0)	0.05623
QCBH ($\zeta < 3.937\,M$1)	0.05490

The efficiency drops to 5.49% at maximal $\zeta < 3.937\,M$2, below the Schwarzschild value of $\zeta < 3.937\,M$3. The direction of this effect is notable: unlike some modified-gravity scenarios where corrections enhance radiative output, here the quantum parameter consistently suppresses flux, temperature, and efficiency across the entire parameter range. The suppression is modest — roughly 0.2 percentage points even near extremality — which implies that distinguishing the effect electromagnetically requires high-precision spectroscopy or well-calibrated efficiency measurements.

Limitations and open questions

Several caveats bear on the results. First, the analysis is restricted to the $\zeta < 3.937\,M$4 asymptotically flat branch and to $\zeta < 3.937\,M$5; the wormhole regime beyond this bound is excluded, so no statement is made about horizonless configurations. Second, the waveform computation employs the numerical kludge approximation rather than self-consistent adiabatic or self-force evolution, leaving the precise detectability thresholds for $\zeta < 3.937\,M$6 undetermined. Third, the disk model neglects light bending in the redshift factor and assumes zero torque at the ISCO, standard but simplifying assumptions. Fourth, only the $\zeta < 3.937\,M$7 orbit is used for waveform illustration; a systematic survey over the periodic-orbit taxonomy and matched-filtering studies quantifying measurability remain open. Finally, whether the claimed stability advantage of Cauchy-horizon-free geometries translates into observable differences under realistic perturbations is not addressed here.

Conclusion

This work provides a systematic phenomenological characterization of the covariant, Cauchy-horizon-free QCBH of Zhang et al. through two complementary observational channels. The consistent qualitative picture is that the quantum parameter $\zeta < 3.937\,M$8 acts as a "softening" of the strong-field region: it migrates the ISCO and MBO outward, raises the conserved quantities on critical and periodic orbits, accumulates phase dephasing in EMRI waveforms, and suppresses disk flux, temperature, and radiative efficiency (down to 5.49% versus 5.72% for Schwarzschild). These trends supply concrete templates against which future LISA-class gravitational-wave observations and high-precision electromagnetic measurements could jointly constrain $\zeta < 3.937\,M$9, though quantitative forecasts await full waveform modeling and parameter-estimation studies.