Schwarzschild MOG Black Holes

• Schwarzschild MOG black holes are static, spherically symmetric solutions in Modified Gravity featuring an enhanced gravitational coupling and a gravitational charge linked to mass.
• They exhibit modified horizon structures, including dual horizons and enlarged shadows, which alter lensing and wave signatures compared to classical Schwarzschild solutions.
• Extensions of this model reveal regular interiors, traversable wormholes, and modified thermodynamic properties, offering rich avenues for theoretical and observational tests.

A Schwarzschild MOG black hole is the static, spherically symmetric solution of Scalar-Tensor-Vector Gravity (STVG), also known as Modified Gravity (MOG), characterized by an enhanced gravitational coupling $G=G_N(1+\alpha)$ and an additional gravitational “charge” $Q=\sqrt{\alpha G_N}M$ tied to the black hole mass. This solution generalizes the Schwarzschild metric, introduces new horizon and causal structure features, allows for regular (singularity-free) interiors under certain nonlinear extensions, and exhibits distinctive observable phenomena such as enlarged shadows, modified lensing, altered wave signatures, and exotic traversable wormhole solutions.

1. Field Equations, Metric, and Horizon Structure

The Schwarzschild–MOG black hole emerges from the STVG field equations

$R_{\mu\nu} = -8\pi G\, T_{\phi\, \mu\nu},$

where $T_{\phi\, \mu\nu}$ is the energy-momentum tensor of the vector field $\phi_\mu$. The vector field’s contribution enters the metric function as a gravitational analog to charge, with

$$B_{\mu\nu} = \partial_\mu \phi_\nu - \partial_\nu \phi_\mu,$$

and

$$T_{\phi\, \mu\nu} = -\frac{1}{4\pi}\left[ B_{\mu}{}^{\alpha}B_{\nu\alpha} - \frac{1}{4}g_{\mu\nu} B^{\alpha\beta}B_{\alpha\beta} \right].$$

Assuming $\phi_0 \neq 0$ (static potential), and substituting $G=G_N(1+\alpha)$ and $Q=\sqrt{\alpha G_N}M$, the Schwarzschild–MOG metric is

$$ds^2 = \left[ 1 - \frac{2GM}{r} + \frac{\alpha G_N G M^2}{r^2} \right] dt^2 - \left[ 1 - \frac{2GM}{r} + \frac{\alpha G_N G M^2}{r^2} \right]^{-1} dr^2 - r^2 d\Omega^2.$$

This reduces to Schwarzschild as $\alpha\to 0$.

The horizon structure is governed by solving

$$1 - \frac{2GM}{r} + \frac{\alpha G_N G M^2}{r^2} = 0,$$

yielding two real, positive horizons for $\alpha > 0$:

$$r_\pm = G_N M \left[ (1+\alpha) \pm \sqrt{1+\alpha} \right].$$

$r_+$ is the outer event horizon and $r_-$ is the inner Cauchy horizon. For certain nonlinear vector field dynamics, regular (nonsingular) solutions exist, resulting in curvature invariants remaining finite at $r\to 0$. In such regular interiors, the spacetime may approach an (anti-)de Sitter geometry.

2. Enhanced Gravitational Coupling and Physical Parameters

A key feature of Schwarzschild–MOG black holes is the effective gravitational constant,

$G = G_N (1+\alpha),$

where the dimensionless parameter $\alpha$ quantifies the departure from general relativity. The gravitational “charge” appears as

$Q = \sqrt{\alpha G_N}M,$

entering the metric similarly to the electric charge in Reissner–Nordström but arising from the gravitational vector field, not electromagnetism. This “charge” scales with the mass, so extremality (merging of the horizons) can only occur for negative $\alpha\to -1$.

3. Kruskal–Szekeres Extension and Global Structure

The maximal analytic extension of the Schwarzschild–MOG spacetime proceeds via a Kruskal–Szekeres–type transformation, writing the metric:

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

with $\gamma(r) = 1 - (2GM)/r + (\alpha G_N G M^2)/r^2$. By switching to coordinates $(u, v)$, all coordinate singularities at $r_\pm$ are removed, allowing construction of the full Penrose diagram. The global causal structure matches that of Reissner–Nordström: two horizons (event and Cauchy), with the possibility of naked regular cores in special cases.

4. Particle Orbits and Motion

The motion of test particles incorporates the modified geometry and vector field through the equation

$$\frac{d^2 x^\mu}{ds^2} + \Gamma^\mu_{\alpha\beta} \frac{dx^\alpha}{ds} \frac{dx^\beta}{ds} = \kappa B_\sigma{}^\mu \frac{dx^\sigma}{ds},$$

with $\kappa = \sqrt{\alpha G_N}$, so the “gravitational charge to mass” ratio $q/m = \kappa$. For $m=0$ (photons), standard geodesic motion holds; omnipresent energy and angular momentum conservation permit a radial effective potential analysis:

$$\left(\frac{dr}{ds}\right)^2 + V_{\rm eff}(r) = \frac{r^4(C^2 - 1)}{J^2}.$$

The effective potential contains a positive-definite repulsive component $\sim\alpha G_N G M^2/r^2$, which prevents access to $r=0$ and stabilizes orbits.

Stable circular orbits are shifted outwards compared to Schwarzschild, and the ISCO increases with $\alpha$. For the photon sphere, the radius is

$$r_{\rm ps} = \frac{3}{2} G_N(1+\alpha)M \left[ 1 + \sqrt{1 - \frac{8\alpha}{9(1+\alpha)}} \right ].$$

5. Shadows, Lensing, and Observational Signatures

The shadow cast by the Schwarzschild–MOG black hole is determined by the closed photon orbits near the photosphere. The shadow size grows with $\alpha$ according to analytic formulae:

$$r_{\rm shad} \approx (2.59 + 2\alpha) r_s,\quad r_s=2G_N M,$$

and the photosphere/critical impact parameter expands with increasing $\alpha$. For events like Sgr A* and M87*, observed shadow diameters measured by EHT can be compared to these predictions, potentially constraining $\alpha$ (Moffat, 2015, Moffat et al., 2019, Izmailov et al., 2019).

Gravitational lensing observables, including image positions, time delays, and Einstein ring radii, acquire corrections proportional to $\alpha$, especially pronounced in the strong field regime. The MOG metric reproduces Schwarzschild lensing for $\alpha=0$, but with positive $\alpha$ the bending angle and related observables increase, while for $-1<\alpha<0$ (interpreted as brane-world tidal charge) the effects are reversed and even singular as $\alpha \to -1$. Subtle deviations in the weak field can be reached by high-precision astrometric or VLBI experiments (Rehman et al., 10 Feb 2025, Izmailov et al., 2019).

6. Thermodynamics, Regularity, and Quantum Effects

Thermodynamics of Schwarzschild–MOG black holes significantly deviate from general relativity:

• The Hawking temperature is

$$T = \frac{1}{2\pi G_N M} \frac{1}{(1+\sqrt{1+\alpha})(1+\alpha+\sqrt{1+\alpha})},$$

always lower than in Schwarzschild for $M$ fixed.

• Entropy is modified:

$S_A = \pi G_N^2 (1+\sqrt{1+\alpha})^2,$

with additional $\alpha$-dependent corrections if the first law is integrated directly.

In non-linear extensions, a critical value $\alpha_{\rm crit} \simeq 0.673$ marks the transition to “gray holes”: configurations with no horizon but substantial gravitational redshift. These prevent total information loss, offering a possible resolution to the information loss paradox (Mureika et al., 2015, Saghafi et al., 2019).

Quantum corrections to the entropy, arising from thermal fluctuations, lead to the standard logarithmic terms $S = S_0 - \frac{1}{2}\ln(S_0 T^2)$, again with $\alpha$-dependent coefficients.

In a noncommutative geometry variant, mass smearing results in a remnant at the end of evaporation, characterized by zero temperature and vanishing entropy, with the emission spectrum displaying nonthermal correlations between different modes, allowing for possible information retrieval (Saghafi et al., 2019).

7. Extensions: Accretion, Scattering, Wormholes, and Gravitational Waves

Accretion: The mass accretion rate for polytropic and isothermal fluids onto Schwarzschild–MOG black holes is increased relative to Schwarzschild. For a fluid with pressure $p = k e$, analysis shows that the accretion rate $\dot{M}$ increases monotonically with both radial distance and the MOG parameter $\alpha$. The specific form of $\dot{M}$ incorporates both the modified geometry and the fluid’s equation of state, yielding direct $\alpha$-dependence (John, 2016, Ahmed et al., 6 Sep 2025).

Quantum Scattering: The differential cross-section for fermions scattering off Schwarzschild–MOG black holes increases with $\alpha$, displaying enhanced glory (backscattering peak) and spiral/orbiting features in the angular profile. These arise from the deeper gravitational potential induced by a larger $\alpha$, substantially modifying scattering vis-à-vis Schwarzschild (Sporea, 2018).

Wormholes: By balancing gravitational attraction with the repulsive component from the vector field (the $Q^2$ term), Schwarzschild–MOG admits traversable wormhole solutions whose throats are stabilized without violating the weak energy condition (i.e., without exotic matter). The construction is enabled by matching two regularized spherical regions with a nonzero “charge,” leading to a geometry where travelers pass from one asymptotically flat region to another via a stable throat (Moffat, 2014).

Gravitational Waves, Quasinormal Modes: The quasinormal mode (QNM) spectrum is sensitive to $\alpha$. For the same mass, increasing $\alpha$ decreases both the real (oscillation) and imaginary (damping) parts of QNM frequencies. When the ADM mass is rescaled appropriately ($M \to M/(1+\alpha)$), the ringdown frequencies show larger real parts (oscillation frequency) while damping remains close to Schwarzschild values. Detailed studies indicate that deviations in the QNM spectrum—identifiable in the ringdown phase of mergers—are a strong discriminant for MOG versus GR (Manfredi et al., 2017, Liu et al., 2023, Al-Badawi, 2023). Greybody factors, governing Hawking emission, are also enhanced with increasing $\alpha$.

Summary Table: Core Schwarzschild–MOG Black Hole Properties

Property	Dependence/Formula	Effect of $\alpha$
Gravitational “charge”	$Q = \sqrt{\alpha G_N} M$	Induces $Q^2$ term in metric
Metric function	$$f(r) = 1 - \frac{2GM}{r} + \frac{\alpha G_N G M^2}{r^2}$$	Reduces to Schwarzschild if $\alpha=0$; adds repulsive core
Horizons	$r_\pm = G_N M ((1+\alpha)\pm\sqrt{1+\alpha})$	Two horizons for $\alpha>0$
Photon sphere/shadow	$r_{\rm ps} = \ldots$ (see above), $r_{\rm shad}\sim (2.59 + 2\alpha)r_s$	Enlarged for $\alpha>0$
Hawking temperature	$$T = 1/(2\pi G_N M [(1+\sqrt{1+\alpha})(1+\alpha+\sqrt{1+\alpha})])$$	Decreases with $\alpha$
Entropy (area law)	$S_A = \pi G_N^2 (1+\sqrt{1+\alpha})^2$	Explicit $\alpha$ correction
QNM frequencies	$\omega$ depends on $(1+\alpha)$ rescaling	Lowered for fixed $M$, raised for ADM mass rescaling
Mass accretion rate	$\dot{M} \propto (1+\alpha)^2$ (exact form depends on EoS)	Larger for $\alpha>0$
Regularity	For suitable nonlinearities, curvature invariants are finite at $r=0$	Allows singularity-free solutions

Significance and Observational Implications

The Schwarzschild–MOG solution offers a phenomenologically viable framework for testing strong-field gravity. Its distinctive features—such as an enlarged shadow, modifiable horizon structure, altered ringdown signals, the possibility of regular or horizonless compact objects (“gray holes”), and traversable wormholes stabilized without exotic matter—can in principle be probed via current and upcoming observations, notably shadow imaging (EHT), gravitational wave spectroscopy (LIGO/Virgo/LISA), and high-precision lensing measurements. These properties enable stringent constraints on $\alpha$ and the viability of STVG/MOG as an alternative or extension to general relativity (Moffat, 2014, Moffat, 2015, Moffat et al., 2019, Manfredi et al., 2017).