RG-Improved Kerr Black Hole

• RG-improved Kerr black holes are rotating solutions enhanced by quantum corrections via a running Newton’s constant, fundamentally altering their geometry and horizon structure.
• The improvement replaces the fixed G₀ with a position-dependent G(r), derived from RG equations, leading to measurable shifts in ergospheres, geodesics, and black hole shadows.
• Quantum modifications also adjust thermodynamic laws and regularize classical singularities, providing testable predictions for gravitational wave observations and accretion phenomena.

A renormalization group improved Kerr black hole is a rotating black hole solution whose classical geometry is modified by quantum corrections arising from the scale dependence of the gravitational coupling. Implemented within the framework of Quantum Einstein Gravity (QEG) and the broader asymptotic safety scenario, this approach systematically incorporates nonperturbative renormalization group (RG) effects by promoting Newton’s constant to a running coupling, G(k), and replacing it with a position-dependent function in the metric. These quantum modifications impact the global structure, observables, thermodynamics, and astrophysical signatures of Kerr black holes, with measurable consequences for horizon locations, geodesic motion, gravitational waveforms, thermodynamics, and the black hole shadow.

1. Renormalization Group Improvement: Formalism and Metric Construction

The essential principle is the “RG improvement” of the classical Kerr metric by replacing the fixed Newton’s constant $G_0$ with a scale-dependent coupling $G(k)$, where the scale $k$ is dynamically identified with local geometric invariants or proper distances. In the context of QEG, the running is determined by the RG equation

$G(k) = \frac{G_0}{1 + \omega G_0 k^2} \ ,$

where $\omega$ is a fixed-point parameter set by the UV completion of gravity (1009.3528).

For rotating spacetimes, the RG scale $k$ is typically taken as a function of the radial coordinate and possibly the polar angle. A practical identification is

$$G_0 \rightarrow G(r) = G\left(k = \xi/d(r)\right) \quad \text{with} \quad G(r) = \frac{G_0 r^2}{r^2 + \bar{w} G_0},$$

with $d(r)$ an invariant distance (often $r$), $\xi \sim \mathcal{O}(1)$, and $\bar{w} = w\xi^2$. For Kerr, the improved metric in Boyer–Lindquist coordinates retains its classical structure but with the replacement

$$\Delta \rightarrow \Delta_i(r) = r^2 + a^2 - 2M G(r) r.$$

Thus, every instance of $G_0$ in the Kerr metric is substituted by $G(r)$ (1009.3528).

Alternate proposals employ coordinate-invariant scale identifications—such as a function of the Weyl invariant $\mathcal{I}$,

$$k^4 = |\mathcal{I}| = \frac{48 G_0^2 M^2}{(r^2 + a^2 \cos^2\theta)^3},$$

which makes $G(r,\theta)$ explicitly depend on local curvature, ensuring coordinate independence (Held, 2021).

2. Causal Structure, Horizons, and Ergosphere in the Improved Geometry

A central consequence of the running coupling is that the positions of event horizons and static limit surfaces become $r$-dependent, non-algebraic functions:

• The horizons are found by solving $\Delta_i(r) = 0$, while the outer and inner static limits (ergosphere boundaries) are determined by $g_{tt}=0$ with the running G(r).
• For classical Kerr: $r_\pm = M G_0 \pm \sqrt{M^2 G_0^2 - a^2}$. For the RG-improved case, this generalizes to involve $G(r)$ and must be solved numerically (1009.3528, Chen et al., 2023).

The quantum corrections result in modest shifts for macroscopic (astrophysical) black holes, but for masses near the Planck scale, horizon radii and ergospheres are dramatically altered. Notably, below a critical mass the two horizons merge and disappear, indicating the occurrence of a quantum censorship bound (1009.3528). The positions of the ergosphere and frame dragging angular velocities are also modified, affecting energy extraction processes such as the Penrose process and superradiance. Frame-dragging is encoded via

$$\omega(r, \theta) = -\frac{g_{t\phi}}{g_{\phi\phi}} = \frac{2MG(r)ra}{\Sigma_i \sin^2\theta} \ .$$

3. Mass, Angular Momentum “Dressing," and Komar Integrals

The spacetime remains asymptotically flat ($r \to \infty$, $G(r)\to G_0$), allowing standard ADM mass and angular momentum definitions. The RG-improved geometry, however, induces a split between the “intrinsic” (bare) and “dressed” mass/rotation, captured by Komar integrals: $$M = M_H + (\text{quantum-induced contribution}), \quad J = J_H + (\text{additional quantum contribution}).$$ The bare quantities $M_H, J_H$ are calculated as horizon Komar integrals, acquiring corrections due to the effective pseudo-matter (arising from $G(r)$ gradients). Importantly, they continue to obey the classical Smarr formula,

$$M_H = 2\Omega_H J_H + \frac{\kappa \mathcal{A}}{4\pi G_0},$$

but with $\kappa$ and $\Omega_H$ modified: $$\Omega_H = \frac{a}{r_+^2 + a^2}, \quad \kappa = \frac{r_+ - MG(r_+) - M r_+ G'(r_+)}{r_+^2 + a^2},$$ where $r_+$ is the quantum-corrected horizon location (1009.3528).

4. Thermodynamic Laws and Entropy Modifications

The RG-improved Kerr spacetime exhibits substantial departures from classical black hole thermodynamics. While the standard identification $T = \kappa/(2\pi)$ and Bekenstein-Hawking area law $S = \mathcal{A}/(4G_0)$ hold for the classical or minimally corrected (Schwarzschild) case, for the full RG-improved rotating solution the temperature–surface gravity relation is violated.

• The “first law” in its classical form, $\delta M - \Omega_H \delta J = T \delta S$, no longer admits an exact state function for entropy, i.e., the differential form $(2\pi/\kappa)(\delta M - \Omega_H \delta J)$ is not closed.
• For small angular momentum ($J \ll M^2$), an expansion yields

$$T(M, J) = T_0(M) + T_2(M) J^2 + \mathcal{O}(J^4), \quad S(M, J) = S_0(M) + S_2(M) J^2 + \mathcal{O}(J^4),$$

with quantum corrections $T_2(M)$ and $S_2(M)$ uniquely determined such that the first law holds up to $\mathcal{O}(J^2)$ (1009.3528).

• Both the temperature and entropy are reduced relative to their Schwarzschild values by the $J^2$ quantum corrections (1009.3528).

For a broader class of field theories and in the presence of non-minimal couplings, the total black hole entropy can be partitioned into a gravitational contribution (running with the RG scale) and a quantum piece related to the entanglement entropy of field modes. For Kerr, a similar split is theoretically plausible, but the subtleties of rotation complicate the explicit function form (Satz et al., 2013, Miqueleto et al., 2020).

5. Regularity, Causality, and Singularity Resolution

Mass-dependent cutoff identifications and coordinate-invariant RG-improvement procedures yield metrics where classical singularities are resolved:

• The improved coupling $G(r) \to 0$ as $r \to 0$ (for mass-dependent identifications), causing the ring singularity at $r=0, \theta=\pi/2$ to be “smoothed out” (Chen et al., 2023).
• The Kretschmann scalar and other curvature invariants remain finite at the would-be singularity, indicating the interior is regularized, often evolving to a de Sitter–like core structure (Platania, 2019).
• Closed timelike curves, ubiquitous in classical Kerr for $r<0$, are partially eliminated in RG-improved metrics, as quantum corrections modify $g_{\phi\phi}$ such that causal anomalies are avoided in a significant fraction of the interior (Chen et al., 2023, Cao et al., 21 Oct 2024).

The approach preserves thermodynamic coherence, i.e., the first law on the horizon and a well-defined entropy, when the scale identification is a suitable function of mass and horizon area (e.g., $G = G(Mr, A)$, $A = 4\pi(r^2 + a^2)$).

6. Dynamical and Observational Implications

Geodesics and Accretion Disks

• The radii of the ISCO, marginally bound orbit (MBO), and event horizon all decrease as the quantum parameters in the running coupling increase, reflecting a “shrinking” of the black hole’s strong-field region (Li et al., 9 Sep 2025, Sánchez, 7 Apr 2024).
• In accretion disk models (Novikov–Page–Thorne), quantum corrections induce inward ISCO shifts, substantial enhancements in energy flux, temperature, and luminosity for prograde disks, and increased accretion efficiency, with corrections amplified for rapidly rotating black holes (Sánchez, 7 Apr 2024).
• These deviations from the classical Kerr predictions are parameterized directly by the strength of the quantum corrections (e.g., via $\xi$ or $\omega$), which can be constrained observationally.

Gravitational Waveforms

• The waveforms emitted from extreme mass-ratio inspirals (EMRIs) in the RG-improved Kerr background exhibit frequency and amplitude shifts, most pronounced in the prograde case.
• The characteristic strain of these waveforms with quantum corrections may lie within the sensitivity bands of future gravitational wave detectors (LISA, Taiji, DECIGO) (Li et al., 9 Sep 2025).

Black Hole Shadow and EHT Observations

• Quantum improvements produce a more compact black hole shadow and, for high spins and quantum parameter values, a cusp-like deformation on the prograde side. However, simulated images for a wide range of quantum parameters and spins remain within the error bars of current EHT measurements for Sgr A* and M87* (Sánchez, 1 Aug 2024, Cao et al., 21 Oct 2024).
• Systematic reductions in the observed intensity (from simulated thin-disk images) are the primary quantum-improved signature, with only subtle modifications to the shadow contour (Cao et al., 21 Oct 2024). The parameter space in which such spacetimes remain regular and free of closed timelike curves while possessing an event horizon can be tightly constrained with high-resolution shadow observables.

7. Conceptual and Methodological Issues

• The coordinate dependence of metric-level RG improvement is a recognized issue; invariant RG improvements (directly on curvature invariants) offer more robust, coordinate-independent corrections (Held, 2021).
• Cutoff identification remains a non-universal aspect: approaches based on proper distances, curvature scalars (Kretschmann, Weyl), or self-consistent iterative procedures yield similar qualitative corrections but may differ in quantitative predictions (Chen et al., 2023, Platania, 2019).
• In the entropy sector, the entanglement entropy of quantum fields, combined with renormalization of gravitational couplings, contributes decisively to the Bekenstein–Hawking entropy, with subleading corrections (including logarithmic and $\sqrt{A}$ terms) in the presence of running couplings and additional charges (Jana et al., 29 Jan 2025, Miqueleto et al., 2020).

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This framework, rooted in asymptotically safe gravity and QEG, allows systematic computation and prediction of quantum gravity effects on rotating black holes. The RG-improved Kerr solution provides a robust, physically motivated, and observationally relevant extension of classical black hole theory, enabling new tests via multi-messenger astrophysics, gravitational wave astronomy, and horizon-scale imaging.