Generalized Letelier–Alencar Solution

• The Generalized Letelier–Alencar solution is a family of static, spherically symmetric black-hole metrics that combine a Schwarzschild mass, a cloud of strings, and cosmological fields like quintessence and dark matter.
• It derives a parametric metric function via nonhomogeneous Euler-type differential equations, allowing detailed studies of geodesics, photon spheres, tidal forces, and thermodynamics.
• Extensions include AdS corrections and regularized cores using the Dagum distribution, enabling practical predictions for black-hole shadows, phase transitions, and observational tests.

The Generalized Letelier–Alencar solution is a family of static, spherically symmetric black-hole metrics describing the mutual gravitational influence of a Schwarzschild mass, a radial cloud of strings, and cosmological fields such as quintessence or perfect-fluid dark matter. It generalizes the original Letelier angular-deficit geometry, embedding it in backgrounds relevant for observational cosmology and astrophysical modeling. The solution is characterized by a parameter-rich metric function and admits various extensions including AdS, regularized cores, and phase-transition structures. Its versatility supports studies of geodesics, shadows, tidal forces, and thermodynamics under well-defined physical and mathematical constraints.

1. Metric Structure, Matter Content, and Parameters

Letelier–Alencar solutions proceed from the spherical metric ansatz in Schwarzschild-like coordinates: $$ds^2 = -f(r)dt^2 + \frac{dr^2}{f(r)} + r^2(d\theta^2 + \sin^2\theta\,d\varphi^2)$$ The classical solution involves two noninteracting sources:

• Cloud of Strings (Letelier tensor):

$$T^t{}_t = T^r{}_r = a\,r^{-2}, \quad T^\theta{}_\theta = T^\varphi{}_\varphi = 0$$

with $a>0$ controlling string density, yielding a solid angle deficit $4\pi a$.

• Quintessence Fluid (Kiselev tensor):

$$T^t{}_t = T^r{}_r = \rho_q(r),\qquad T^\theta{}_\theta = T^\varphi{}_\varphi = -\frac{1}{2}(3\omega_q+1)\,\rho_q(r)$$

where

$$\rho_q(r) = -\alpha \frac{3\omega_q}{2\,r^{3(\omega_q+1)}}$$

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The total energy-momentum tensor is a direct sum: $$T^\mu_\nu = T^\mu_\nu\text{(string)} + T^\mu_\nu\text{(quintessence)}$$

Including perfect-fluid dark matter (PFDM) and a cosmological constant, the most general solution extends the metric function to: $$\boxed{ f(r) = 1 -\alpha -\frac{2M}{r} + \frac{\lambda}{r}\ln\frac{r}{|\lambda|} -\frac{N}{r^{3w+1}} + \frac{r^2}{\ell_p^2} }$$ where terms denote respectively flat background, string deficit, Schwarzschild mass, PFDM, quintessence, and AdS curvature (Ahmed et al., 24 Oct 2025, Sood et al., 25 Mar 2024).

2. Field Equations, Solution Derivation, and Limiting Cases

The Einstein equations $$R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R = 8\pi T_{\mu\nu}$$ are solved under the above sources, yielding a nonhomogeneous Euler-type ordinary differential equation for $f(r)$ (Costa et al., 2018): $$r^2 f'' + 3(\omega_q+1) r f' + (3\omega_q+1)\left[ f + a \right]= 0$$ Its general solution is

$$f(r) = 1 - a - \frac{2M}{r} - \frac{\alpha}{r^{3\omega_q+1}}$$

Special limits reproduce classical results:

• $\alpha=0$: Letelier string cloud
• $a=0$: Kiselev quintessence
• $a=\alpha=0$: Schwarzschild

Further, the inclusion of PFDM ($\lambda$), cosmological constant ($\ell_p$), or regularization (see Section 6) modifies the solution but preserves its analytic structure (Sood et al., 25 Mar 2024).

3. Physical Interpretation, Horizons, and Energy Conditions

Physical parameter roles:

• $M$: ADM mass
• $a,\;\alpha$: string density and angular deficit (dimensionless, $a \in [0,1)$)
• $\omega_q$: quintessence state parameter ($-1<\omega_q<-1/3$ for cosmic acceleration)
• $\lambda$: PFDM density scaling ($r^{-3}$, barotropic relations)
• $N$: quintessence normalization
• $\ell_p$: AdS radius

Horizons are positive real roots of $f(r)=0$; their number and location depend sensitively on parameter values. For example, at $\omega_q = -2/3$ the horizon equation becomes quadratic: $\alpha r^2 - (1-a) r + 2M = 0$ with discriminant $\Delta=(1-a)^2-8M\alpha$.

Energy conditions:

• Weak: $a/r^2 + \rho_q \geq 0$; $c_s \geq 0$
• Null and strong: detailed verification via stress tensor components
• Admissible ranges: $$M>0,\;\omega_q\in(-1,-1/3),\;a,\alpha,\lambda,N\geq0$$ must ensure at least one real positive root (event horizon) (Simão et al., 28 Apr 2025).

4. Geodesic Structure, Photon Spheres, and Tidal Forces

The effective potentials for null and timelike geodesics are: $$V_{\text{eff}}(r) = f(r) \left( \frac{L^2}{r^2} + \delta \right)$$ with $\delta = 0$ (null) or $1$ (timelike).

Photon sphere radius is defined by (Silva et al., 26 Nov 2025): $r_{\rm ph} f'(r_{\rm ph}) - 2 f(r_{\rm ph}) = 0$ Numerical studies show $r_{\rm ph}$ increases with $g_s$ (cloud strength) and decreases with $\ell_s$ (spread), with divergence at extremal limits.

ISCO (innermost stable circular orbit) is found from: $$2 r_I f(r_I) f'(r_I) - 4 r_I (f'(r_I))^2 + 6 f(r_I) f'(r_I) = 0$$ String cloud parameters shift both photon sphere and ISCO outward, modifying orbital frequencies and stability maps (Silva et al., 26 Nov 2025).

Radial infall and circular motion experience tidal forces characterized by directional compression/stretching, with the Kretschmann scalar diverging as $r^{-8}$ in generalized solutions versus $r^{-6}$ in Schwarzschild/Letelier: $$K(r) \approx \frac{56 g_s^4 \ell_s^4}{r^8} - \frac{96 g_s^2 \ell_s^2 M}{r^7} + \frac{48 M^2}{r^6} + \mathcal{O}(r^{-5})$$ A plausible implication is enhanced singular behavior near $r\to0$, with parametric control of tidal anisotropy.

5. Thermodynamics, Shadows, and Phase Transitions

Thermodynamic properties are derived from horizon quantities:

• Hawking temperature: $T_H = \frac{f'(r_h)}{4\pi}$
• Entropy: $S = \pi r_h^2$ or, in regularized models, functions of core scale $r_0$, independent of string parameter $a$, e.g.

$$S = \pi[r_h^2 + 8 r_h r_0 - \frac{8 r_0^3}{r_h} - \frac{r_0^4}{r_h^2} + 6 r_0^2 \ln\left(r_h^2/r_0^2\right)]$$

• Heat capacity: sign change marks a second-order phase transition (Muniz et al., 14 Nov 2025).

In Letelier–Alencar–AdS–PFDM models, the first law generalizes: $dM = T dS + V dP + \Pi d\beta$ where $\beta$ parameterizes PFDM.

Phase transition phenomena appear in the behavior of the photon sphere radius and impact parameter, where abrupt changes act as order parameters; critical exponents are found to be $1/2$, matching expectations in ordinary thermal systems (Sood et al., 25 Mar 2024).

Black-hole shadows and photon spheres are actively modeled in these backgrounds for constraints from Event Horizon Telescope observations (e.g., Sgr~A*, M87*) (Muniz et al., 14 Nov 2025).

6. Regularization: Dagum Distribution and Core Physics

The central singularity of classical Letelier–Alencar solutions can be regularized by "smearing" the mass and string source terms with a rational cutoff (Dagum distribution): $D(r) = \left(1 + \frac{r_0}{r}\right)^{-4}$ yielding a regularized metric function: $$f(r) = 1 - \left[\frac{2M}{r} - \frac{|a| r_0^2}{r^2}{}_2F_1\left(-\frac12, -\frac14; \frac34; -\frac{r^4}{r_0^4}\right)\right]\left(1 + \frac{r_0}{r}\right)^{-4}$$ This construction ensures curvature invariants remain finite: $K(0) = \frac{24 |a|^2}{r_0^4}$ The innermost region behaves as anti–de Sitter with

$f(r\to0) \simeq 1 + \frac{r^2}{\ell^2}$

with $\ell = r_0/\sqrt{|a|}$.

Energy-condition analysis reveals multiple regimes: NEC/SEC hold at the core, WEC/DEC are violated in a thin intermediate band, and all conditions are satisfied at large $r$. This structure is required for singularity resolution per the negative Tolman mass theorem (Muniz et al., 14 Nov 2025).

A plausible implication is that such regularized black holes admit stable thermodynamic branches (Rényi non-extensive entropy), loss of classical phase transitions, and externally constrained shadow radii compatible with current EHT bounds.

7. Summary and Applications

The generalized Letelier–Alencar solution encompasses, in a unified analytic framework, the black hole interactions with cosmic fields: string clouds, quintessence, perfect-fluid dark matter, and AdS terms. The parametric freedom enables tailored studies of geodesic motion, observational signatures (shadows, lensing), phase transitions, and regularization of singularities. The formalism reduces seamlessly to classical Schwarzschild, Letelier, and Kiselev solutions under appropriate limits, and admits extensions to regular black-hole interiors and AdS cores.

Active applications include modeling astrophysical observables, testing thermodynamic universality, and constructing singularity-resolving geometries consistent with theoretical and experimental constraints. The solution space is modulated by mass, string density/profile, dark-energy fluid parameters, and regularization scale, admitting rich mathematical and physical phenomena relevant for contemporary black hole research (Costa et al., 2018, Ahmed et al., 24 Oct 2025, Sood et al., 25 Mar 2024, Silva et al., 26 Nov 2025, Simão et al., 28 Apr 2025, Muniz et al., 14 Nov 2025).