Schwarzschild-Like Black Hole

• Schwarzschild-like black holes are static, spherically symmetric modifications of the Schwarzschild solution that incorporate extra physical effects via additional fields and deformation parameters.
• They modify key observables by altering spacetime structure, thermodynamic relations, quasinormal mode spectra, and gravitational lensing signatures compared to classical black holes.
• These models offer practical insights into Lorentz violation, dark matter impacts, and accretion dynamics, enabling multi-messenger tests of strong gravity in astrophysical settings.

A Schwarzschild-like black hole is a static, spherically symmetric solution to modified gravity theories or GR with extensions, which preserves the coordinate structure of the Schwarzschild metric but incorporates additional physical effects via new fields, non-linear terms, or surrounding matter distributions. Such metrics arise in contexts ranging from Lorentz-violating bumblebee models, modified gravity frameworks (MOG, bumblebee, Starobinsky-Bel-Robinson), quantum-corrected scenarios, and as effective black holes embedded in dark matter halos. These solutions are distinguished by parameterized deformations, altered thermodynamic and dynamical properties, and modified observational signatures relative to the classical Schwarzschild spacetime.

1. Spacetime Structure and Key Metrics

Schwarzschild-like metrics generally retain the canonical form

$$ds^2 = -f(r)\,dt^2 + \frac{dr^2}{f(r)} + r^2 (d\theta^2 + \sin^2\theta\, d\phi^2)$$

with the lapse function $f(r)$ admitting model-specific corrections. In the bumblebee gravity scenario, $f(r) = 1 - 2M/r$, but the $dr^2$ term is rescaled by a Lorentz-breaking parameter, yielding $g_{rr} \propto (1+l)/(1-2M/r)$, where $l$ quantifies Lorentz violation (An, 27 Jan 2024). For dark-matter-immersed black holes, $f(r)$ contains analytic halo-induced terms: $$f(r) = 1 - \frac{2M}{r} - 8\pi\rho_s r_s^2 (1 + r_s/r)^2\ln(1 + r_s/r)$$ with Dehnen-profile halo parameters ($\rho_s$, $r_s$) (Pathrikar, 4 Nov 2025, Uktamov et al., 26 May 2025, Al-Badawi et al., 2 Nov 2024).

Event horizon radii are modified by the non-Schwarzschild terms but commonly remain near $r_h = 2M$ unless the corrections are large. In models like Starobinsky-Bel-Robinson gravity ($\beta$ parameter), the lapse function includes rapid-decay $O(r^{-10})$ terms that are negligible at large $r$ but modify the near-horizon structure (Arora et al., 2023). Metrics with explicit Lorentz symmetry breaking (bumblebee, bumblebee-metric-affine) introduce further modifications in $A(r), B(r)$ but preserve $r_h$.

2. Thermodynamics, Entropy, and First Law Deviations

Thermodynamic analysis reveals that Schwarzschild-like black holes frequently exhibit shifted relationships among entropy $S$, temperature $T$, and mass $M$, dictated by the deformation parameters.

In bumblebee gravity,

$$S = \pi r_h^2(1 + l), \quad T = \frac{1}{8\pi M \sqrt{1 + l}}, \quad E = \sqrt{1 + l}\, M, \quad dE = T\, dS$$

The non-minimal coupling $B^\mu B^\nu R_{\mu\nu}$ alters the identification of thermodynamic energy, leading to discrepancies between ADM mass and thermodynamic mass (An, 27 Jan 2024). The Iyer–Wald phase-space formalism is essential for correct entropy computation, due to horizon-divergent fields analogous to Horndeski gravity (An, 27 Jan 2024). Quantum corrections (e.g. loop quantum gravity, Barbero–Immirzi parameter $\gamma$) induce critical points, zeroth-order phase transitions, and nontrivial Joule-Thomson behavior; the modified first law often requires correction functions, restoring area entropy (Wang et al., 13 May 2024).

For deformed metrics (running $G$, Hayward regular BH, Tsallis–Rényi entropy), the Hawking temperature and entropy depend explicitly on the deformation parameters, shifting the possible thermodynamic Ricci curvature scalars and admitting offshell thermodynamic geometries (Wen, 2016).

3. Quasinormal Modes and Spectral Stability

Ringdown spectra of Schwarzschild-like black holes exhibit parameter-dependent shifts and instabilities. The master wave equation takes the Regge–Wheeler form with $f(r)$ modified,

$$\frac{d^2\psi}{dr_*^2} + [\omega^2 - V(r)] \psi = 0$$

where $V(r)$ is model-dependent (Siqueira et al., 23 Jan 2025, Pathrikar, 4 Nov 2025, Liu et al., 30 Sep 2024). WKB+Padé analysis shows QNM frequencies $\omega_n$ decrease ($\mathrm{Re}\,\omega$ lower) and damping ($|\mathrm{Im}\,\omega|$) is reduced for denser, more extended dark matter halos. Physically meaningful deformations (e.g. Rezzolla–Zhidenko parametrization) induce persistent pseudospectral instability—QNM overtone spectra become fragile, and distinguishing physical origins of spectral shifts is challenging (Siqueira et al., 23 Jan 2025).

In the presence of dark halos, the connection between the eikonal QNM limit and the photon sphere-shadow radius ($R_{sh}$) remains robust, allowing ringdown observations to probe halo properties (Liu et al., 30 Sep 2024).

4. Particle Dynamics, Accretion, and Magnetic Effects

Steady-state accretion onto Schwarzschild-like backgrounds is consistently suppressed by deformation parameters. For bumblebee gravity, the steady Bondi accretion rate and Vlasov current are reduced by $(1+l)^{-1/2}$, though radial flow profiles below the sonic point mirror the Schwarzschild case (Cai et al., 2022, Yang et al., 2018). Polytropic gas accretion obeys modified critical point conditions,

$u_c^2 = \frac{M}{2(1+l)r_c}$

with mass accretion rate scaling as $(1+l)^{-1/2}$ times the Schwarzschild value (Yang et al., 2018).

Schwarzschild-like black holes with external magnetic fields exhibit distinct ISCO characteristics: charged-particle ISCOs uniformly shrink relative to neutral ones and, in strong fields, can approach the event horizon arbitrarily closely (Mannobova et al., 29 Jun 2025). Collision energies between charged ISCO and infalling neutral particles can diverge near the horizon, providing an efficient astrophysical high-energy accelerator mechanism, with implications for X-ray QPOs and relativistic precession models (Mannobova et al., 29 Jun 2025, Boboqambarova et al., 2021).

5. Photon Sphere, Shadow, and Gravitational Lensing

The photon sphere radius $r_{ph}$ and derived shadow size are sensitive diagnostics of Schwarzschild-like deformations. For standard models, $r_{ph}=3M$, $b_c=3\sqrt{3}M$, but halo and parameter-induced corrections shift $r_{ph}$:

• Dehnen-type halo: $r_{ph}$ and shadow radius $R_{sh}$ decrease with increasing halo mass or scale radius, so EHT shadow observations constrain halo parameters $\rho_s$, $r_s$ (Pathrikar, 4 Nov 2025, Al-Badawi et al., 2 Nov 2024, Uktamov et al., 26 May 2025).
• Bumblebee gravity: $r_{ph}$ unaffected, but the critical impact parameter $b_c$ and lensing deflection receive corrections proportional to Lorentz violation parameter $\ell$ (Güllü et al., 2020, Gao, 19 Sep 2024).
• Starobinsky-Bel-Robinson: Stringy corrections monotonically reduce the shadow radius; current EHT error bars envelop such deviations for observationally allowed $\beta$ (Arora et al., 2023).

The weak-field light-deflection angle generally takes the form

$$\hat\alpha(b) = \frac{4M}{b} + \mathrm{deformation\ terms}$$

with analytic coefficients depending on physical parameters of the extension (e.g. Lorentz-violating terms, dark-matter halo, global monopole) (Gao, 19 Sep 2024, Güllü et al., 2020).

6. Hawking Radiation and Greybody Factors

Greybody transmission factors $|T_\ell|^2$ for Hawking emission are enhanced at low frequency in Schwarzschild-like metrics with flattened potential barriers. Halo parameters ($\rho_s$, $r_s$) lower the barrier height in the effective potential, shifting transmission curves to lower frequencies and softening the spectrum (Pathrikar, 4 Nov 2025). While not directly observable with current instrumentation, this effect constitutes a theoretical signature distinguishing Schwarzschild-like black holes from the vacuum case.

7. Physical and Observational Implications

Schwarzschild-like black holes provide a unified formalism for studying the impact of physical deformations (Lorentz or CPT violation, quantum corrections, exotic matter, dark-matter environment) on black hole observables:

• Ringdown frequencies and shadow sizes serve as precision probes of strong-field environment and deformation parameters.
• Accretion rate modifications due to Lorentz symmetry breaking are subject to constraint by SMBH inflow observations, though deviations are extremely small for realistic parameter values ($l \lesssim 10^{-13}$) (Cai et al., 2022).
• Magnetic field effects shift ISCOs inward for charged particles, possibly implicating high-energy emissions and resonant QPO frequencies (Mannobova et al., 29 Jun 2025, Boboqambarova et al., 2021).
• Gravitational lensing angle and Einstein ring observations yield upper bounds on deformation parameters ($\ell \lesssim 10^{-6}$ for bumblebee models) (Gao, 19 Sep 2024).
• Quantum and strong-gravity inspired horizonless configurations, such as the 2-2-hole, exhibit distinctive time-delay echo signatures and gravitational trapping, with implications for gravitational wave observations (Holdom et al., 2016).

Astrophysical measurements—Event Horizon Telescope imaging, ringdown mode spectroscopy, lensing surveys—continue to place dynamic bounds on allowed parameter space for Schwarzschild-like extensions, but current limits show that deviations from GR remain tightly constrained at horizon scales. Careful consideration of multi-messenger observations could reveal subtle but robust imprints of non-vacuum structure in the near-horizon geometry.