Schwarzschild-Like Black Hole Models

• Schwarzschild-like black holes are generalized solutions of Einstein's equations that modify the classic Schwarzschild metric via additional matter fields or altered gravitational actions.
• They incorporate dark energy, dark matter halos, and regular de Sitter cores to address singularity issues and emulate effects from extended gravity theories.
• These models serve as theoretical laboratories to study black hole thermodynamics, geodesic completeness, and observational signatures while probing energy condition implications.

A Schwarzschild-like black hole is a generalization of the Schwarzschild solution to Einstein’s field equations, in which the standard, spherically symmetric, asymptotically flat vacuum spacetime is modified via additional matter content, nontrivial asymptotics, altered gravitational actions, or additional external or environmental effects. This class of solutions provides a framework for investigating the dynamical, thermodynamic, and geometrical properties of black holes in settings that depart from the vacuum general relativity scenario, including the presence of dark energy, dark matter halos, deviations motivated by quantum gravity, or spontaneous Lorentz symmetry breaking.

1. General Definition and Metric Forms

A Schwarzschild-like black hole maintains the static, spherically symmetric line element of the form

$$ds^2 = -F(r)\, dt^2 + F(r)^{-1}\, dr^2 + r^2(d\theta^2 + \sin^2\theta\, d\phi^2),$$

where the lapse function $F(r)$ is modified from the Schwarzschild $F_{\rm Sch}(r) = 1 - 2M/r$ to a form reflecting new physics. Typical modifications of $F(r)$ incorporate:

• Dark Energy or Cosmological Constant: $F(r) = 1 - 2M/r - \Lambda r^2/3$ (Schwarzschild–de Sitter)
• Deformed/Parametrized Metrics: $F(r)$ with deformation parameters, e.g., $F(r) = 1 - 2M/r + \epsilon (M/r)^k$
• Environmental Effects (Dark Matter Halos): $F(r)$ includes a term $\propto$ mass profile or density, e.g., logarithmic or power-law corrections (Al-Badawi et al., 2 Nov 2024, Uktamov et al., 26 May 2025)
• Regularized (Singularity-free) Cores: $F(r)$ transitions to de Sitter-like behavior at $r \to 0$ (Ghosh et al., 27 Mar 2025)
• Modified Gravity Theories: $F(r)$ results as a solution to $f(R)$ gravity or other extended theories (Ghosh et al., 27 Mar 2025)

Embedded or axisymmetric generalizations, e.g., the Schwarzschild–Levi-Civita black hole, further relax asymptotic properties and symmetry structures (Mazharimousavi, 4 Mar 2024).

2. Modifications due to Matter and Extended Gravity

Dark Energy and Matter Embeddings

The Schwarzschild–dark energy black hole is constructed by embedding the Schwarzschild mass into a non-vacuum, repulsive background. The metric has

$$ds^2 = \left[1 - \frac{2(M + m r^2)}{r}\right] du^2 + 2 du\, dr - r^2(d\theta^2 + \sin^2\theta\, d\phi^2),$$

where $m$ encodes the dark energy density. The stress-energy satisfies weak and dominant energy conditions but violates the strong energy condition (SEC), due to negative pressure and $w = -1/2$ (Ishwarchandra et al., 2014). The violation of SEC implies a repulsive matter component and an expanding, accelerating observer congruence.

Dark Matter Halo Spacetimes

Schwarzschild-like solutions in dark matter backgrounds replace the vacuum with a realistic mass profile, such as Dehnen-type

$$\rho(r) = \rho_s \left(\frac{r}{r_s}\right)^{-\gamma} \left[\left(\frac{r}{r_s}\right)^\alpha + 1\right]^{(\gamma-\beta)/\alpha}$$

or other composite forms. The metric function $F(r)$ acquires a correction determined by the cumulative mass: $$F(r) = 1 - \frac{2M}{r} - 8\pi\rho_sr_s^2 \frac{\log(1 + r_s/r)}{1 + r_s/r}$$ (Uktamov et al., 26 May 2025), or

$$F(r) = 1 - \frac{2M}{r} - 32\pi \rho_sr_s^3\sqrt{\frac{r + r_s}{r r_s^2}}$$

(Al-Badawi et al., 2 Nov 2024). Curvature invariants diverge at $r=0$ but vanish at infinity (asymptotic flatness). All major energy conditions (WEC, NEC, DEC, SEC) are satisfied for the DM parameters considered.

3. Regular and Geodesically Complete Models

To eliminate the central singularity, $F(r)$ can be engineered to interpolate to a de Sitter limit at the origin, e.g.: $$F(r) = 1 - \frac{2mr^2}{(r + l)^3}, \quad \text{with}~ l = \frac{8}{27} m$$ (Ghosh et al., 27 Mar 2025). Near $r \to 0$, $F(r) \to 1 - (1/3)\Lambda r^2$ with $\Lambda = 6m/l^3$. For $r \to \infty$, $F(r) \to 1 - 2m/r$, recovering Schwarzschild asymptotics. The event horizon is at $r_h = 2l$ (single horizon). All principal energy conditions (except SEC near $r=0$) are satisfied, and all invariants are manifestly finite everywhere—confirming geodesic completeness and the absence of curvature singularities. The underlying gravitational action is constructed via an $f(R)$ model, determined by integrating the modified field equations numerically and approximated by a Padé form in $R$.

4. Energy Conditions and Physical Acceptability

Schwarzschild-like solutions must respect the principal energy conditions for physical viability, particularly:

Condition	Mathematical Form	Validity in Schwarzschild-like Models
Weak (WEC)	$T_{ab} t^a t^b \geq 0$	Satisfied for all $r$ in regular & DM halo models
Null (NEC)	$T_{ab} n^a n^b \geq 0$	Satisfied everywhere (often saturated)
Dominant (DEC)	$T_{ab} t^a t^b \geq 0$ and $T^{ab} t_b$ is causal	Satisfied in all examined cases
Strong (SEC)	$R_{ab} t^a t^b \geq 0$	Holds for $r \geq r_h/2$; violated near $r=0$ in regular models

Regular black holes, by Zaslavskii's criterion, can violate SEC inside the horizon (allowing a de Sitter core), but must satisfy SEC outside $r_h/2$ to ensure gravitational attraction and the avoidance of caustics in timelike congruences.

5. Geometric, Thermodynamic, and Observational Features

Horizons and Geodesic Motion

• The horizon location is determined by the largest positive root of $F(r)$.
• The existence of a photon sphere or null circular orbit is not guaranteed in all cases; for example, the Schwarzschild–Levi-Civita black hole lacks such a structure outside the horizon (Mazharimousavi, 4 Mar 2024).
• Timelike geodesics in DM–modified metrics shift the innermost stable circular orbit (ISCO) and bound state regions outward, with the effective potential barrier and allowed orbit structure being DM–density-dependent.

Thermodynamics and Quantum Gravity Corrections

• In non-minimally coupled models (e.g., bumblebee gravity), black hole thermodynamic variables are generically modified, with entropy and energy gaining dependence on the Lorentz symmetry breaking parameter (e.g., $E = \sqrt{1+l} M$, $S = \pi r_h^2 (1+l)$) (An, 27 Jan 2024).
• Models incorporating quantum gravity corrections (e.g., loop quantum gravity) exhibit corrections to both horizon structure and phase behavior, potentially with new critical points and altered critical ratios (e.g., $7/18$ instead of $3/8$ as in the Van der Waals fluid) (Wang et al., 13 May 2024).

Observational Signatures and Astrophysics

• The presence of DM halos or nontrivial core structure may impact observable quantities such as black hole shadows, lensing angles, accretion rates, and quasi-periodic oscillation frequencies.
• Schwarzschild-like metrics with de Sitter-like centers are singularity-free and exhibit geodesic completeness, potentially providing preferred endpoints for gravitational collapse in generalized gravity.

Modification Physics	Metric Function $F(r)$ Modification	Observational/Physical Feature
Dark Energy	$m r^2$ term, quadratic in $r$	Violates SEC, repulsive gravity
Dark Matter Halo	Logarithmic or power-law correction (via mass profile)	Shifts ISCO, affects lensing/shadow
Regularization (f(R))	$r^2$ (de Sitter core); $F(r)\sim 1 - (1/3)\Lambda r^2$	No singularity at $r=0$, unique horizon
Deformed Gravity/Bumblebee	$(1+l)$ (radial component), other coefficients	Modified temperature, entropy, evaporation time

6. Theoretical and No-Hair Properties

Regular Schwarzschild-like black holes constructed within f(R) gravity frameworks can adhere strictly to the no-hair theorem—i.e., the entire geometry is uniquely determined by the mass, with no free functional parameters other than those fixed by matching asymptotic and regularity conditions (Ghosh et al., 27 Mar 2025). Generalized spacetimes with fixed deformation functions or environmental parameters can still retain no-hair behavior if non-mass parameters are functionally related or determined by consistency requirements.

7. Significance and Outlook

Schwarzschild-like black holes serve as crucial theoretical laboratories for understanding:

• The fate of singularities and cosmic censorship in extended gravity
• The impact of environmental matter fields on black hole structure and dynamics
• The interface between semiclassical gravity, energy conditions, and astrophysical signatures
• The potential observational distinction between vacuum black holes and those immersed in dark matter or other non-vacuum surroundings

By satisfying (or tacitly violating for controlled, physical reasons) canonical energy conditions, and by presenting geodesically complete, curvature-regular models, Schwarzschild-like black holes provide compelling alternatives and generalizations of the canonical black hole paradigm for a wide class of gravitational, astrophysical, and cosmological phenomena (Ishwarchandra et al., 2014, Al-Badawi et al., 2 Nov 2024, Uktamov et al., 26 May 2025, Ghosh et al., 27 Mar 2025).