Improved Schwarzschild-like Black Hole Models

• Improved Schwarzschild-like black holes are static, spherically symmetric spacetimes that extend the Schwarzschild solution with quantum corrections and environmental effects.
• Modifications include running Newton couplings, higher-curvature terms, and dark matter halo embeddings that adjust horizon structure, photon spheres, and geodesic behavior.
• These models facilitate studies of quasinormal modes, thermodynamic phase transitions, and regularization of singularities to probe quantum gravitational impacts.

An improved Schwarzschild-like black hole is a static, spherically symmetric geometry that reduces to the Schwarzschild solution in an appropriate limit while incorporating additional structure such as a running Newton coupling, higher-curvature or regularizing corrections, or an external matter sector such as a dark matter halo. In the recent literature, the label is used for several closely related constructions rather than for a single universal metric. These constructions share the goal of modifying the Schwarzschild exterior or interior in a controlled way and are typically analyzed through their horizon structure, photon sphere and shadow, quasinormal modes, geodesics, scattering and absorption, and thermodynamic or information-theoretic properties (Rincon et al., 2020, Alencar et al., 5 Mar 2026, Liu et al., 2024).

1. Terminology and canonical geometric form

Most improved Schwarzschild-like models preserve the standard static, spherically symmetric ansatz

$ds^2=-f(r)\,dt^2+f(r)^{-1}dr^2+r^2 d\Omega^2,$

so that all deviations from Schwarzschild are encoded in the lapse $f(r)$. This is the form used in renormalization-group improved geometries, in dark-matter halo embeddings, in regularized one-horizon models, and in several perturbative analyses of shadows, ringdown, and absorption (Rincon et al., 2020, Pedraza et al., 27 Mar 2026, Ghosh et al., 27 Mar 2025).

There are also non-minimal Schwarzschild-like extensions in which the radial and angular sectors are modified separately. In Einstein-Hilbert-Bumblebee gravity with a global monopole, the metric is

$$ds^{2} = -\left(1-\frac{2M}{r}\right)dt^{2} + (1+\ell)\left(1-\frac{2M}{r}\right)^{-1}dr^{2} + g^2 r^2 \left(d\theta^2 + \sin^2\theta\, d\phi^2\right),$$

with $\ell=\xi b^2$ and $g^2=1-\bar\mu$, so that the deformation is not exhausted by a single lapse function (Güllü et al., 2020). Observationally calibrated halo models can also separate the temporal and radial functions, as in the NGC 4649 construction

$$ds^2=-F(r)\,dt^2+\frac{dr^2}{1-\frac{2M(r)}{r}}+r^2 d\Omega^2,$$

with $M(r)=m_{\rm BH}+\frac{r^3V_c^2}{a^2+r^2}$ and $F(r)=\frac{(r-\lambda)^\xi}{r}$ (Lobo et al., 7 May 2025).

This suggests that “improved Schwarzschild-like black hole” is best understood as a family label for non-rotating, spherically symmetric spacetimes that retain the Schwarzschild limit while altering either the ultraviolet core, the near-horizon geometry, or the exterior environment.

2. Quantum-improved and asymptotically safe constructions

A central line of work constructs the improved Schwarzschild black hole by promoting Newton’s constant to a running coupling. In the asymptotic-safety framework summarized in “Quasinormal modes of an improved Schwarzschild black hole” (Rincon et al., 2020), the running coupling is

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

with $\tilde{\omega}=118/(15\pi)$, and the scale identification is taken as

$f(r)$0

This yields

$f(r)$1

and, in geometrized units,

$f(r)$2

At large distances the classical Schwarzschild form is recovered, while deviations become relevant at short distances (Rincon et al., 2020).

A closely related functional-renormalization-group construction uses

$f(r)$3

and produces two horizons,

$f(r)$4

In that formulation, the improved geometry introduces an inner Cauchy horizon absent in Schwarzschild and provides the background for island and Page-curve analyses (Saha et al., 25 Mar 2025).

A newer renormalization-group scheme replaces the simpler running by an exact “Scheme B” coupling together with a proper-distance interpolating function,

$f(r)$5

so that

$f(r)$6

Its large-$f(r)$7 expansion recovers Schwarzschild, while the short-distance behavior is de Sitter-like (Alencar et al., 5 Mar 2026). The same lapse is used in the later study of shadow, ringdown, and strong cosmic censorship, where the geometry acquires an outer event horizon and an inner Cauchy horizon for an allowed region of $f(r)$8 (Al-Badawi et al., 26 Apr 2026).

Not all quantum improvements are equivalent. In the action-improvement approach based on the curvature invariant $f(r)$9, the coupling is promoted directly in the action,

$$ds^{2} = -\left(1-\frac{2M}{r}\right)dt^{2} + (1+\ell)\left(1-\frac{2M}{r}\right)^{-1}dr^{2} + g^2 r^2 \left(d\theta^2 + \sin^2\theta\, d\phi^2\right),$$0

with

$$ds^{2} = -\left(1-\frac{2M}{r}\right)dt^{2} + (1+\ell)\left(1-\frac{2M}{r}\right)^{-1}dr^{2} + g^2 r^2 \left(d\theta^2 + \sin^2\theta\, d\phi^2\right),$$1

That construction preserves general covariance, but the resulting singularity is not regularized (Moti et al., 2018).

3. Environmental and modified-gravity Schwarzschild-like geometries

A second major class embeds a black hole in an external medium. For a pseudo-isothermal dark matter halo, the lapse is

$$ds^{2} = -\left(1-\frac{2M}{r}\right)dt^{2} + (1+\ell)\left(1-\frac{2M}{r}\right)^{-1}dr^{2} + g^2 r^2 \left(d\theta^2 + \sin^2\theta\, d\phi^2\right),$$2

where $$ds^{2} = -\left(1-\frac{2M}{r}\right)dt^{2} + (1+\ell)\left(1-\frac{2M}{r}\right)^{-1}dr^{2} + g^2 r^2 \left(d\theta^2 + \sin^2\theta\, d\phi^2\right),$$3 is the central halo density and $$ds^{2} = -\left(1-\frac{2M}{r}\right)dt^{2} + (1+\ell)\left(1-\frac{2M}{r}\right)^{-1}dr^{2} + g^2 r^2 \left(d\theta^2 + \sin^2\theta\, d\phi^2\right),$$4 the halo core radius; $$ds^{2} = -\left(1-\frac{2M}{r}\right)dt^{2} + (1+\ell)\left(1-\frac{2M}{r}\right)^{-1}dr^{2} + g^2 r^2 \left(d\theta^2 + \sin^2\theta\, d\phi^2\right),$$5 gives Schwarzschild (Liu et al., 2024).

For a Dehnen-type $$ds^{2} = -\left(1-\frac{2M}{r}\right)dt^{2} + (1+\ell)\left(1-\frac{2M}{r}\right)^{-1}dr^{2} + g^2 r^2 \left(d\theta^2 + \sin^2\theta\, d\phi^2\right),$$6 halo, the density profile is

$$ds^{2} = -\left(1-\frac{2M}{r}\right)dt^{2} + (1+\ell)\left(1-\frac{2M}{r}\right)^{-1}dr^{2} + g^2 r^2 \left(d\theta^2 + \sin^2\theta\, d\phi^2\right),$$7

with mass profile

$$ds^{2} = -\left(1-\frac{2M}{r}\right)dt^{2} + (1+\ell)\left(1-\frac{2M}{r}\right)^{-1}dr^{2} + g^2 r^2 \left(d\theta^2 + \sin^2\theta\, d\phi^2\right),$$8

and the combined black-hole-plus-halo metric may be written as

$$ds^{2} = -\left(1-\frac{2M}{r}\right)dt^{2} + (1+\ell)\left(1-\frac{2M}{r}\right)^{-1}dr^{2} + g^2 r^2 \left(d\theta^2 + \sin^2\theta\, d\phi^2\right),$$9

This spacetime is asymptotically flat and reduces to Schwarzschild for $\ell=\xi b^2$0 (Uktamov et al., 26 May 2025, Pathrikar, 4 Nov 2025).

A different Dehnen-type halo with $\ell=\xi b^2$1 gives

$\ell=\xi b^2$2

again with the Schwarzschild limit recovered as $\ell=\xi b^2$3 (Al-Badawi et al., 2024). In an empirically calibrated galactic environment, the NGC 4649 model uses

$\ell=\xi b^2$4

so that the metric depends on $\ell=\xi b^2$5, $\ell=\xi b^2$6, and $\ell=\xi b^2$7 and smoothly reduces to Schwarzschild as $\ell=\xi b^2$8 and $\ell=\xi b^2$9 (Lobo et al., 7 May 2025).

Modified-gravity examples include the bumblebee-global-monopole solution quoted above (Güllü et al., 2020), the Starobinsky-Bel-Robinson metric

$g^2=1-\bar\mu$0

which reduces to Schwarzschild for $g^2=1-\bar\mu$1 (Arora et al., 2023), and the Johannsen-Psaltis deformation

$g^2=1-\bar\mu$2

which parameterizes deviations from general relativity while keeping the horizon at $g^2=1-\bar\mu$3 for the allowed range $g^2=1-\bar\mu$4 (Magalhães et al., 2020).

4. Horizons, regular cores, singularities, and energy conditions

The horizon and core structure is strongly model dependent. In the asymptotic-safety inspired metric $g^2=1-\bar\mu$5, the number and position of horizons depend sensitively on $g^2=1-\bar\mu$6, and for Planck-scale masses quantum effects are most important (Pedraza et al., 27 Mar 2026). In the FRG-improved geometry with $g^2=1-\bar\mu$7, the improvement explicitly generates an inner Cauchy horizon in addition to the outer horizon (Saha et al., 25 Mar 2025). In the later two-parameter RG model, horizons satisfy

$g^2=1-\bar\mu$8

and the quartic can admit both $g^2=1-\bar\mu$9 and $$ds^2=-F(r)\,dt^2+\frac{dr^2}{1-\frac{2M(r)}{r}}+r^2 d\Omega^2,$$0; a critical curve $$ds^2=-F(r)\,dt^2+\frac{dr^2}{1-\frac{2M(r)}{r}}+r^2 d\Omega^2,$$1 marks the merger of the two horizons (Al-Badawi et al., 26 Apr 2026).

By contrast, some improved metrics are designed to preserve a single-horizon structure. “Geodesically Complete Regularized Schwarzschild Black Holes” proposes

$$ds^2=-F(r)\,dt^2+\frac{dr^2}{1-\frac{2M(r)}{r}}+r^2 d\Omega^2,$$2

for which the spacetime has a single event horizon, is asymptotically Schwarzschild, and develops a de Sitter core near $$ds^2=-F(r)\,dt^2+\frac{dr^2}{1-\frac{2M(r)}{r}}+r^2 d\Omega^2,$$3 through

$$ds^2=-F(r)\,dt^2+\frac{dr^2}{1-\frac{2M(r)}{r}}+r^2 d\Omega^2,$$4

The associated Ricci and Kretschmann scalars,

$$ds^2=-F(r)\,dt^2+\frac{dr^2}{1-\frac{2M(r)}{r}}+r^2 d\Omega^2,$$5

remain finite at the origin (Ghosh et al., 27 Mar 2025). The new RG-improved solution of (Alencar et al., 5 Mar 2026) likewise replaces the central singularity by a regular de Sitter-like core and yields finite curvature invariants.

Regularization is therefore not a generic consequence of “improvement.” In the Ricci-tensor-squared action-improvement program, the singularity at $$ds^2=-F(r)\,dt^2+\frac{dr^2}{1-\frac{2M(r)}{r}}+r^2 d\Omega^2,$$6 survives and can diverge at least as fast as in the classical case; that approach may produce only one event horizon or even a naked singularity, depending on parameters (Moti et al., 2018). Conversely, in dark-matter-halo solutions the central singularity generally persists. The Dehnen-halo letter reports that the Ricci scalar becomes nonzero, the Kretschmann scalar acquires halo-dependent corrections, and all curvature invariants still diverge as $$ds^2=-F(r)\,dt^2+\frac{dr^2}{1-\frac{2M(r)}{r}}+r^2 d\Omega^2,$$7, while vanishing at spatial infinity (Uktamov et al., 26 May 2025). The $$ds^2=-F(r)\,dt^2+\frac{dr^2}{1-\frac{2M(r)}{r}}+r^2 d\Omega^2,$$8 Dehnen model likewise remains singular at the center and asymptotically flat at infinity (Al-Badawi et al., 2024).

Energy conditions also vary by construction. The regularized one-horizon model satisfies DEC, WEC, and NEC throughout spacetime, and satisfies SEC for $$ds^2=-F(r)\,dt^2+\frac{dr^2}{1-\frac{2M(r)}{r}}+r^2 d\Omega^2,$$9, in accordance with the quoted Zaslavskii regularity criterion (Ghosh et al., 27 Mar 2025). The Dehnen-halo solutions report that NEC, WEC, DEC, and SEC are satisfied everywhere outside the central singularity (Uktamov et al., 26 May 2025, Al-Badawi et al., 2024).

5. Perturbations, ringdown, photon sphere, and shadow

Perturbative analyses usually reduce field dynamics to a Regge-Wheeler-type equation,

$M(r)=m_{\rm BH}+\frac{r^3V_c^2}{a^2+r^2}$0

with $M(r)=m_{\rm BH}+\frac{r^3V_c^2}{a^2+r^2}$1. For scalar perturbations of the asymptotically safe improved Schwarzschild geometry,

$M(r)=m_{\rm BH}+\frac{r^3V_c^2}{a^2+r^2}$2

while for electromagnetic perturbations

$M(r)=m_{\rm BH}+\frac{r^3V_c^2}{a^2+r^2}$3

The sixth-order WKB approximation is used with the standard ingoing-at-the-horizon and outgoing-at-infinity boundary conditions (Rincon et al., 2020).

In the asymptotically safe model, all computed scalar and electromagnetic modes are stable. As $M(r)=m_{\rm BH}+\frac{r^3V_c^2}{a^2+r^2}$4 increases, both the real part and the absolute imaginary part of $M(r)=m_{\rm BH}+\frac{r^3V_c^2}{a^2+r^2}$5 decrease. For Planck-scale objects, the difference between improved and classical Schwarzschild quasinormal frequencies is of order a few per cent, whereas for astrophysical black holes it becomes negligible, below $M(r)=m_{\rm BH}+\frac{r^3V_c^2}{a^2+r^2}$6 (Rincon et al., 2020).

Dark-matter environments alter the ringdown in model-dependent ways. For the pseudo-isothermal halo, scalar perturbations are governed by

$M(r)=m_{\rm BH}+\frac{r^3V_c^2}{a^2+r^2}$7

and the sixth-order WKB study reports that increasing $M(r)=m_{\rm BH}+\frac{r^3V_c^2}{a^2+r^2}$8 increases the real part of $M(r)=m_{\rm BH}+\frac{r^3V_c^2}{a^2+r^2}$9 and decreases the magnitude of the imaginary part, with increasing $F(r)=\frac{(r-\lambda)^\xi}{r}$0 producing a smaller but similar effect. The same work connects the eikonal regime to the photon sphere through

$F(r)=\frac{(r-\lambda)^\xi}{r}$1

and

$F(r)=\frac{(r-\lambda)^\xi}{r}$2

with a larger $F(r)=\frac{(r-\lambda)^\xi}{r}$3 or $F(r)=\frac{(r-\lambda)^\xi}{r}$4 shrinking the shadow seen by a distant observer (Liu et al., 2024).

For the Dehnen $F(r)=\frac{(r-\lambda)^\xi}{r}$5 halo, scalar, electromagnetic, and axial gravitational perturbations are computed with WKB plus Padé approximants. Increasing $F(r)=\frac{(r-\lambda)^\xi}{r}$6 and $F(r)=\frac{(r-\lambda)^\xi}{r}$7 lowers the peak of the effective potential and shifts it outward, decreases both $F(r)=\frac{(r-\lambda)^\xi}{r}$8 and $F(r)=\frac{(r-\lambda)^\xi}{r}$9, moves the photon sphere outward, enlarges the shadow, and enhances greybody factors by lowering the effective barrier (Pathrikar, 4 Nov 2025).

The two-parameter RG-improved black hole of (Al-Badawi et al., 26 Apr 2026) unifies scalar, electromagnetic, and Dirac perturbations and computes fundamental and overtone modes with sixth-order WKB cross-checked against time-domain ringdown. Increasing $G(k)=\frac{G_0}{1+\tilde{\omega}G_0 k^2},$0 and, more mildly, $G(k)=\frac{G_0}{1+\tilde{\omega}G_0 k^2},$1, raises $G(k)=\frac{G_0}{1+\tilde{\omega}G_0 k^2},$2 and lowers $G(k)=\frac{G_0}{1+\tilde{\omega}G_0 k^2},$3. The study further analyzes strong cosmic censorship at the inner Cauchy horizon through

$G(k)=\frac{G_0}{1+\tilde{\omega}G_0 k^2},$4

finding $G(k)=\frac{G_0}{1+\tilde{\omega}G_0 k^2},$5 to be multipole-independent at the $G(k)=\frac{G_0}{1+\tilde{\omega}G_0 k^2},$6 level and approximately geometric,

$G(k)=\frac{G_0}{1+\tilde{\omega}G_0 k^2},$7

with a thin crescent near the extremal boundary where Christodoulou-SCC marginally fails (Al-Badawi et al., 26 Apr 2026).

6. Geodesics, scattering, absorption, and orbital phenomenology

Geodesic structure is another major diagnostic. In “Geodesic Motions near an improved Schwarzschild black hole,” the lapse is written as

$G(k)=\frac{G_0}{1+\tilde{\omega}G_0 k^2},$8

and, after expansion,

$G(k)=\frac{G_0}{1+\tilde{\omega}G_0 k^2},$9

For timelike and null geodesics,

$\tilde{\omega}=118/(15\pi)$0

with

$\tilde{\omega}=118/(15\pi)$1

That study reports that only one event horizon is possible for physical parameter choices, that the effective potential is always negative, that all physically allowed massive-particle orbits are bound, and that timelike phase trajectories are ellipses while null ones are straight lines through the origin in the $\tilde{\omega}=118/(15\pi)$2 plane (Mandal, 2022).

Wave scattering and absorption provide a complementary probe of quantum corrections. For the improved Schwarzschild black hole analyzed in (Pedraza et al., 27 Mar 2026), the classical differential scattering section is computed from

$\tilde{\omega}=118/(15\pi)$3

the glory approximation is

$\tilde{\omega}=118/(15\pi)$4

and the partial-wave absorption section is

$\tilde{\omega}=118/(15\pi)$5

The main result is that classical scattering differs only slightly from Schwarzschild, whereas semi-classical and full partial-wave treatments reveal altered interference patterns and amplitudes, especially in the low-frequency and small-mass regime; the low-frequency absorption approaches the corrected horizon area, while the high-frequency behavior is well reproduced by the sinc approximation (Pedraza et al., 27 Mar 2026).

External fields and additional deformations further enrich the phenomenology. In the singularity-free Schwarzschild-like metric

$\tilde{\omega}=118/(15\pi)$6

immersed in an external magnetic field with

$\tilde{\omega}=118/(15\pi)$7

the effective potential for charged equatorial motion is

$\tilde{\omega}=118/(15\pi)$8

Increasing the magnetic parameter shrinks the charged-particle ISCO toward the event horizon, allows unbounded center-of-mass energies in collisions between a charged ISCO particle and a neutral infaller, and shifts relativistic-precession-model frequencies and the $\tilde{\omega}=118/(15\pi)$9 QPO resonance radius inward (Mannobova et al., 29 Jun 2025).

In the bumblebee-global-monopole geometry, the horizon remains at $f(r)$00, the photon sphere at $f(r)$01, the shadow radius becomes $f(r)$02, the Hawking temperature is

$f(r)$03

and the weak-field deflection angle is

$f(r)$04

The global monopole parameter increases both the shadow radius and the deflection angle, while the Lorentz-symmetry-breaking parameter lowers the temperature and increases the deflection angle (Güllü et al., 2020).

7. Thermodynamics, thermodynamic geometry, and information

Thermodynamic analysis has become one of the principal criteria for judging whether an improved Schwarzschild-like geometry behaves as a plausible quantum-corrected black hole. In the new RG-improved model, the Hawking temperature

$f(r)$05

departs from the Schwarzschild $f(r)$06 law, reaches a maximum, and then vanishes at a remnant radius. The entropy obtained from the first law acquires the logarithmic correction

$f(r)$07

and the heat capacity diverges at the temperature maximum, signaling a phase transition. A thermodynamic topological analysis based on a generalized free energy preserves the global topological number $f(r)$08 of Schwarzschild while shifting the critical point (Alencar et al., 5 Mar 2026).

The subsequent study of the same geometry emphasizes a Davies-type phase transition, a bell-curve temperature profile peaking at $f(r)$09, a nontrivial Weinhold-Ruppeiner geometry on the $f(r)$10 slice, and a small-black-hole branch with positive heat capacity. The sparsity of Hawking radiation and the energy-emission rate are written in terms of a single auxiliary function $f(r)$11,

$f(r)$12

$f(r)$13

making the quantum-correction dependence explicit (Al-Badawi et al., 26 Apr 2026).

Thermodynamic geometry can also be constructed off shell by enlarging the state space. “Offshell thermodynamic metrics of the Schwarzschild black hole” introduces an extra deformation variable such as a running Newton constant, a cutoff scale $f(r)$14, a noncommutative parameter $f(r)$15, or a Tsallis-Rényi parameter $f(r)$16, thereby converting the degenerate one-parameter Schwarzschild thermodynamics into a nontrivial two-variable thermodynamic geometry. The on-shell Schwarzschild metric then appears as a gauge-fixed submanifold (Wen, 2016).

Information-theoretic extensions of the improved Schwarzschild geometry use the island formula

$f(r)$17

with quantum-corrected horizon data. In the FRG-improved Schwarzschild black hole, the island boundary receives corrections, the late-time fine-grained entropy becomes

$f(r)$18

and the scrambling time is

$f(r)$19

That analysis reports $f(r)$20, a reduced Page time relative to Schwarzschild, and a mutual-information criterion $f(r)$21 for fixing the island contribution (Saha et al., 25 Mar 2025).

Taken together, these studies show that the improved Schwarzschild-like black hole is not a single metric but a structured research program. Depending on the improvement scheme, one obtains few-per-cent Planck-scale shifts of quasinormal spectra, regular de Sitter cores or unresolved singularities, one- or two-horizon configurations, dark-matter-induced changes in shadows and geodesics, modified scattering and absorption, and thermodynamic phase structures ranging from off-shell curvature diagnostics to remnants, Davies points, and island-based Page-curve corrections.