Cloud of Strings Parameter in Black Hole Physics

• Cloud of strings parameter is a measure quantifying the integrated tension and energy density of one-dimensional defects that modify spacetime geometry and black hole horizons.
• Its formulation via the Nambu-Goto action and derived energy-momentum tensor enables modeling of anisotropic pressures and altered thermodynamic properties in various gravitational settings.
• Observational effects on shadow sizes, accretion disk dynamics, and quasinormal modes provide practical constraints using EHT data and gravitational wave measurements.

A cloud of strings parameterizes the spacetime effect of a continuous distribution of one-dimensional string defects ("string cloud") on black hole solutions, wormholes, and related gravitational configurations. In the Letelier model and its generalizations, this parameter—appearing variously as $a$, $\alpha$, $g_s^2$, $b$, $k$, $a_0$, etc.—quantifies the integrated tension or energy density of the string ensemble, which crucially alters geometric, optical, and thermodynamic properties. Recent research has extended this paradigm by introducing additional “magnetic-like” components, nontrivial equations of state, and coupling to matter such as quintessence and dark matter, leading to novel metric structures and observational features. In all cases, the cloud of strings parameter acts as an independent “hair” beyond the standard mass, charge, and spin, and is tightly constrained by contemporary observations.

1. Mathematical Definition and Physical Interpretation

The cloud of strings arises from integrating the Nambu-Goto action over a statistically isotropic population of world-sheets in the ambient spacetime. The energy-momentum tensor has the generic form

$$T^{\mu}{}_{\nu} = \frac{\alpha}{8\pi r^2} \, \mathrm{diag}(-1, 1, 0, 0)$$

for the spherically symmetric case (Ahmed et al., 17 Oct 2025). The key parameter ($\alpha$, $a$, $g_s^2$, $b$, etc.) measures the aggregate string tension per unit length threading through a radius $r$.

Recent generalizations extend the structure to include bivector components $\Sigma_{01}$ (“electric-like”) and $\Sigma_{23}$ (“magnetic-like”), with corresponding parameters $a$ and $c_0$ or $g_s^2$ and $\ell_s$ (Alencar et al., 11 Jan 2025, Silva et al., 26 Nov 2025). The energy-momentum tensor becomes unique and may include anisotropic pressures or nontrivial equations of state, e.g.,

$$T^{\mu}{}_{\nu} = \mathrm{diag}(\rho, -\rho, p, p), \qquad p(r) = - \biggl[\frac{c_0^4}{c_0^4 + r^4}\biggr]\rho(r)$$

with $\rho(r)$ and $p(r)$ functions of $a$ and $c_0$ (Alencar et al., 11 Jan 2025).

In rotating, charged, or AdS backgrounds, the string-cloud parameter typically enters the metric via a constant or $r^n$ term, e.g.,

$$f(r) = 1 - a - \dots, \quad\quad \Delta(r) = (1-k)r^2 + \dots$$

(Cai et al., 28 Jun 2025, Vishvakarma et al., 2023). In all cases, the physical effect is an effective reduction in the solid angle at infinity—a conical deficit proportional to the parameter.

2. Impact on Black Hole Geometry: Horizons, Photon Spheres, and Orbits

The presence of a cloud of strings modifies the lapse function directly: $$f(r) = 1 - \alpha - \frac{2M}{r}, \qquad r_{ph} = \frac{3M}{1 - \alpha}, \qquad r_{ISCO} = \frac{6M}{1 - \alpha}$$ (Ahmed et al., 17 Oct 2025). As $\alpha$ ($a$, $b$, etc.) increases, the horizon radius, ISCO, and photon sphere shift outward; at extremal values, the solution may become horizonless (naked singularity) (Vishvakarma et al., 2023). For metrics with “magnetic-like” components, corrections appear as hypergeometric functions: $$f(r) = 1 - \frac{2M}{r} + \frac{g_s^2 \ell_s^2}{r^2} \, {}_2F_1\left(-\frac{1}{2}, -\frac{1}{4}; \frac{3}{4}; -\frac{r^4}{\ell_s^4}\right)$$ (Silva et al., 26 Nov 2025, Alencar et al., 11 Jan 2025).

In rotating black holes, the parameter alters the ergoregion, static limit surfaces, and horizon positions; e.g., in the Kerr-Newman-de Sitter case,

$\Delta_r = (1 - b_c) r^2 + \dots$

and the shadow radius is

$R_{sh} = \sqrt{1 - b_c} \, r_{cp}$

(Cao et al., 2023). Increasing the parameter generically enlarges the photon sphere and shadow for all classes of black holes, including Bardeen, Kerr, Hayward, and ABG solutions.

3. Observational Signatures: Shadows, Accretion Disks, and Gravitational Lensing

The cloud parameter leaves characteristic imprints on black hole shadows, photon rings, and the brightness of accretion structures:

• The shadow radius $R_{sh}(a)$ or $b_{ph}(a)$ increases monotonically with $a$ ($\alpha$, $b$, $g_s^2$) (He et al., 2021, Vishvakarma et al., 9 Aug 2024).
• The angular size of EHT-observed shadows places bounds:

$$0 \leq a \lesssim 0.1,\quad 0 \leq b_c \lesssim 0.15$$

for M87* and Sgr A* (Cai et al., 28 Jun 2025, Cao et al., 2023, Vishvakarma et al., 9 Aug 2024).

• Thin-disk models reveal that increasing $a$ pushes the ISCO outwards, dims and cools the disk, and softens the spectral luminosity, offering constraints via disk observations (Cai et al., 28 Jun 2025).
• In strong lensing, the critical impact and Einstein-ring radii grow with $b$; the strong-deflection limit coefficients $\bar a$, $\bar b$ likewise adjust, modifying relativistic image magnifications and separations (Vishvakarma et al., 9 Aug 2024).

The presence of a substantial string cloud is tightly constrained by high-resolution interferometry; allowed values must be subdominant ($\lesssim 10\%$) for observed SMBHs.

4. Dynamical and Thermodynamic Consequences

The impact on dynamical and thermodynamic quantities is explicit:

• The Hawking temperature at the horizon decreases with increasing parameter:

$T_H = \frac{1 - \alpha}{4 \pi r_h}$

(Ahmed et al., 17 Oct 2025, Cai et al., 2019). For more complex backgrounds,

$$T_H(r_h, a, \beta) = \frac{\hbar}{4 \pi r_h}\Big[1 + a(1-2\beta)r_h^{4\beta/(1-2\beta)} / (4\beta-1)\Big]$$

(Cai et al., 2019).

• The entropy and Gibbs free energy similarly acquire deficits proportional to the cloud parameter (Ahmed et al., 17 Oct 2025).
• The heat capacity $C_P$ and location of phase transitions shift; for Bardeen and AdS-like solutions, the critical exponents and phase structure map directly onto the Van der Waals universality class (Rodrigues et al., 2022).
• The area spectrum remains equally spaced ($\Delta A = 8\pi\hbar$) and independent of $a$, but entropy spacing may depend on auxiliary parameters such as the Rastall $\beta$ (Cai et al., 2019).

For regular black hole models (e.g., Hayward), the cloud increases horizon and photon sphere radii, expands ISCO, and prolongs the ringdown phase (Liang et al., 4 Nov 2025).

5. Quasinormal Modes and Scalar/Electromagnetic Perturbations

Quasinormal mode frequencies are sensitive to the cloud parameter:

• Both the real and (absolute) imaginary parts of the QNM frequencies $\omega$ decrease nearly linearly with increasing $a$, leading to slower, longer-lived ringdown waves (Silva et al., 26 Nov 2025, Gogoi et al., 2021, Liang et al., 4 Nov 2025).
• Effective potential for scalar (and electromagnetic) perturbations:

$$V_\ell(r) = f(r)\left[\frac{\ell(\ell+1)}{r^2} + \frac{f'(r)}{r}\right]$$

(Liang et al., 4 Nov 2025).

• In higher-dimensional and wormhole models, the cloud parameter also controls dynamical stability and the position of spectral peaks (Gogoi et al., 2022).
• Eikonal (geometric optics) limit: the Lyapunov exponent and QNM decay rate both decrease with $a$, as exhibited in the eikonal formula

$$\omega_{l\to\infty} \simeq l\Omega_c - i(n+1/2)|\lambda_L|$$

with $\Omega_c$ and $\lambda_L$ functions of $f(r; a)$ (Cai et al., 2019).

6. Generalizations, Couplings, and Physical Constraints

Extensions include anisotropic fluids of strings (Santos, 20 Feb 2025), embedding in AdS or quintessence backgrounds (Deglmann et al., 7 Feb 2025, Ahmed et al., 10 Aug 2025, Ahmed et al., 13 Aug 2025), regularization via magnetic-like components (Alencar et al., 11 Jan 2025, Silva et al., 26 Nov 2025), and couplings to dark matter halos (Ahmed et al., 17 Oct 2025), $f(R)$ gravity (Gogoi et al., 2022), or Rastall theories (Li et al., 2020, Sun et al., 13 Jan 2024).

Key properties:

• Two “independent hairs” in generalized models give rise to modified stress-energy tensors and equations of state (Alencar et al., 11 Jan 2025).
• In all backgrounds, constraints on the parameter are set numerically by requiring existence of horizons, regularity, and compatibility with observational data (EHT, LIGO, lensing, and disk spectra).
• Asymptotic flatness and avoidance of singularities generally require $a,\alpha,b<1$; extremal values eliminate horizons or produce naked geometries (Vishvakarma et al., 2023, Silva et al., 26 Nov 2025).
• Future GW and horizon-scale observations may further constrain the allowed fraction of string clouds in realistic systems.

7. Summary Table: Appearance, Physical Role, and Observational Constraints

Parameter	Appearance in Metric	Principal Physical Effect	Typical Allowed Range
$a$, $\alpha$, $g_s^2$, $b$, $k$, $a_0$	$f(r) = 1 - a - 2M/r$, $1 - \alpha - 2M/r$, ...	String tension/energy density, conical deficit, “hair”	$0 < a, \alpha, b, k < 0.1 \text{ to } 0.8$
$c_0$, $\ell_s$	Hypergeometric corrections (e.g., $c_0^4$)	Radial scale of magnetic component	Tuned to horizon structure
$\alpha(r)$, variable	Metric through integrals (fluid of strings)	Transverse pressure, anisotropy	Model-dependent

The cloud of strings parameter, in both its simplest and most generalized forms, fundamentally enriches the "hair" structure, dynamical stability, and observable properties of black holes and wormholes. Its ongoing paper links classical and quantum gravitational theory to astrophysical phenomenology, with its value tightly bounded by the latest observational results (Alencar et al., 11 Jan 2025, Cai et al., 28 Jun 2025, Cao et al., 2023, Vishvakarma et al., 9 Aug 2024, Silva et al., 26 Nov 2025, Liang et al., 4 Nov 2025, He et al., 2021, Rodrigues et al., 2022).