Cloud of Strings (CoS) in Black Hole Physics

• Cloud of Strings (CoS) is an effective string-fluid model constructed from a continuum of Nambu–Goto strings that deform black hole metrics through deficit-like shifts or hypergeometric corrections.
• Generalizations retaining both electric-like and magnetic-like bivector components produce anisotropic stress-energy profiles, influencing photon spheres, lensing, and potential observational signatures.
• The CoS framework impacts astrophysical observables by altering black hole shadows, ISCO, orbital dynamics, and thermodynamic behavior, with implications for EMRI analyses and gravitational-wave signals.

Cloud of Strings (CoS) denotes an effective matter source built from a continuum or macroscopic average of one-dimensional Nambu–Goto strings. In the Letelier construction, the source is encoded by a string-worldsheet bivector $\Sigma^{\mu\nu}$ and commonly deforms static, spherically symmetric metrics through a deficit-like shift in the lapse function, such as $f(r)=1-\frac{2M}{r}-\alpha$ or $f(r)=1-a-\frac{2M}{r}$. More recent generalizations retain both the electric-like bivector component $\Sigma_{01}$ and a magnetic-like component $\Sigma_{23}$, producing anisotropic sources of the form $\mathrm{diag}(\rho,-\rho,p,p)$ and hypergeometric corrections to the geometry. Across recent work, CoS has been used to study black-hole shadows, strong lensing, quasinormal modes, extended thermodynamics, EMRI waveforms, holography, and string cosmology (Alencar et al., 11 Jan 2025, Alloqulov et al., 14 Dec 2025).

1. Source model and stress-energy structure

The common starting point is the Nambu–Goto action

$$S_{NG}=\int \sqrt{-\gamma}\,\mathcal{M}\, d\lambda^0 d\lambda^1,$$

with induced worldsheet metric $$\gamma_{AB}=g_{\mu\nu}\frac{\partial x^\mu}{\partial \lambda^A}\frac{\partial x^\nu}{\partial \lambda^B}$$, bivector

$$\Sigma^{\mu\nu}=\epsilon^{AB}\frac{\partial x^\mu}{\partial \lambda^A}\frac{\partial x^\nu}{\partial \lambda^B},$$

and $\gamma=\frac12 \Sigma^{\mu\nu}\Sigma_{\mu\nu}$. In this framework the CoS stress tensor is written as

$f(r)=1-\frac{2M}{r}-\alpha$0

or in equivalent mixed-index forms used in recent black-hole papers (Sadeghi et al., 2019, Alencar et al., 11 Jan 2025).

In the original Letelier setup, spherical symmetry keeps only the electric-like component $f(r)=1-\frac{2M}{r}-\alpha$1. This yields the familiar anisotropic source

$f(r)=1-\frac{2M}{r}-\alpha$2

and the corresponding lapse $f(r)=1-\frac{2M}{r}-\alpha$3. In that sense, the CoS parameter $f(r)=1-\frac{2M}{r}-\alpha$4 acts as a string-cloud density or deficit-angle parameter, and $f(r)=1-\frac{2M}{r}-\alpha$5 returns ordinary Schwarzschild (Silva et al., 26 Nov 2025, Alloqulov et al., 14 Dec 2025).

A major recent extension keeps both $f(r)=1-\frac{2M}{r}-\alpha$6 and $f(r)=1-\frac{2M}{r}-\alpha$7. In that model,

$f(r)=1-\frac{2M}{r}-\alpha$8

with

$f(r)=1-\frac{2M}{r}-\alpha$9

leading to the unique anisotropic source

$f(r)=1-a-\frac{2M}{r}$0

where, in the standard gauge $f(r)=1-a-\frac{2M}{r}$1,

$f(r)=1-a-\frac{2M}{r}$2

and the equation of state is

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

This generalized CoS is parameterized by the string-cloud strength $f(r)=1-a-\frac{2M}{r}$4 and the new scale $f(r)=1-a-\frac{2M}{r}$5, and it reduces to Letelier’s original cloud when $f(r)=1-a-\frac{2M}{r}$6 (Alencar et al., 11 Jan 2025).

The same matter language also appears in cosmology. In cylindrically symmetric inhomogeneous string-fluid models, the source is written as

$f(r)=1-a-\frac{2M}{r}$7

with total density $f(r)=1-a-\frac{2M}{r}$8, so the cloud of strings is interpreted as a mixture of string tension density $f(r)=1-a-\frac{2M}{r}$9 and attached massive particles of density $\Sigma_{01}$0 (Yadav et al., 2011).

2. Metric deformations and parameterizations

A recurring structural feature is that the CoS enters the geometry either as a constant subtraction from the vacuum lapse or as a hypergeometric deformation generated by the generalized two-component source. Representative realizations are listed below.

Model class	Representative metric insertion	Reported role of CoS
Letelier-type static black hole	$\Sigma_{01}$1	deficit-like shift
Bardeen black hole in CoS	$\Sigma_{01}$2	enlarges horizon and photon scales with $\Sigma_{01}$3
Generalized two-component CoS	$\Sigma_{01}$4	introduces second hair $\Sigma_{01}$5

In the Bardeen case, the CoS parameter $\Sigma_{01}$6 appears through the replacement $\Sigma_{01}$7, so the cloud changes the asymptotic redshift structure and the effective gravitational potential felt by photons. In that model horizons exist only for $\Sigma_{01}$8, increasing $\Sigma_{01}$9 enlarges the event horizon, and increasing the magnetic charge parameter $\Sigma_{23}$0 tends to shrink it (Vishvakarma et al., 2024).

The same constant-shift mechanism reappears in rotating geometries. In the rotating Letelier–Bardeen–Kerr spacetime,

$\Sigma_{23}$1

so the CoS modifies the event-horizon equation, the static limit surface, the ergoregion, photon orbits, and the shadow (Vishvakarma et al., 2023).

By contrast, the generalized Letelier–Alencar construction replaces the constant subtraction by a hypergeometric term,

$\Sigma_{23}$2

with effective coupling $\Sigma_{23}$3 and length scale $\Sigma_{23}$4. This model can admit two horizons, an extremal case, or no horizon, and the horizon structure is no longer analytically solvable (Silva et al., 26 Nov 2025).

Collectively, these constructions suggest two broad CoS regimes. In one regime, the cloud acts as a global deficit contribution to the lapse. In the other, it generates a genuinely new radial profile with an additional scale and extra hair. The literature treats both as cloud-of-strings geometries, but they need not be observationally equivalent (Alencar et al., 11 Jan 2025, Silva et al., 26 Nov 2025).

3. Photon dynamics, shadows, and strong lensing

CoS effects are especially visible in null geodesics, photon spheres, shadows, and strong-deflection lensing. In the Bardeen black hole in CoS, the null effective potential is

$\Sigma_{23}$5

and the paper reports that increasing $\Sigma_{23}$6 increases the photon-sphere radius and the critical impact parameter $\Sigma_{23}$7, while increasing $\Sigma_{23}$8 decreases them. Sample values at $\Sigma_{23}$9 are

$\mathrm{diag}(\rho,-\rho,p,p)$0

for $\mathrm{diag}(\rho,-\rho,p,p)$1, respectively (Vishvakarma et al., 2024).

In the same strong-deflection analysis, CoS enhances strong-field lensing. The deflection angle becomes larger when $\mathrm{diag}(\rho,-\rho,p,p)$2 is increased, magnification diverges as the source approaches the optic axis, and for fixed source position the magnification is larger when $\mathrm{diag}(\rho,-\rho,p,p)$3 is turned on. The Einstein-ring position

$\mathrm{diag}(\rho,-\rho,p,p)$4

increases with $\mathrm{diag}(\rho,-\rho,p,p)$5, decreases with $\mathrm{diag}(\rho,-\rho,p,p)$6, and becomes smaller for higher-order rings. Using EHT shadow measurements, the paper quotes the bounds

$\mathrm{diag}(\rho,-\rho,p,p)$7

with corresponding bounds on $\mathrm{diag}(\rho,-\rho,p,p)$8 reported for fixed $\mathrm{diag}(\rho,-\rho,p,p)$9 (Vishvakarma et al., 2024).

Rotation adds a further CoS signature. In the rotating Bardeen–Kerr case, the shadow size increases with the CoS parameter $$S_{NG}=\int \sqrt{-\gamma}\,\mathcal{M}\, d\lambda^0 d\lambda^1,$$0, while the distortion parameter $$S_{NG}=\int \sqrt{-\gamma}\,\mathcal{M}\, d\lambda^0 d\lambda^1,$$1 decreases with $$S_{NG}=\int \sqrt{-\gamma}\,\mathcal{M}\, d\lambda^0 d\lambda^1,$$2. The same study reports that increasing $$S_{NG}=\int \sqrt{-\gamma}\,\mathcal{M}\, d\lambda^0 d\lambda^1,$$3 makes the ergoregion larger, even though the static-limit-surface thickness can decrease in the tabulated examples (Vishvakarma et al., 2023).

Several AdS constructions display the same outward displacement of optical scales. In the generalized CoS plus quintessence model, the photon-sphere condition is $$S_{NG}=\int \sqrt{-\gamma}\,\mathcal{M}\, d\lambda^0 d\lambda^1,$$4, and the numerical tables show that increasing $$S_{NG}=\int \sqrt{-\gamma}\,\mathcal{M}\, d\lambda^0 d\lambda^1,$$5 generally pushes the photon sphere outward; with $$S_{NG}=\int \sqrt{-\gamma}\,\mathcal{M}\, d\lambda^0 d\lambda^1,$$6 and $$S_{NG}=\int \sqrt{-\gamma}\,\mathcal{M}\, d\lambda^0 d\lambda^1,$$7, the reported values are $$S_{NG}=\int \sqrt{-\gamma}\,\mathcal{M}\, d\lambda^0 d\lambda^1,$$8 at $$S_{NG}=\int \sqrt{-\gamma}\,\mathcal{M}\, d\lambda^0 d\lambda^1,$$9, $$\gamma_{AB}=g_{\mu\nu}\frac{\partial x^\mu}{\partial \lambda^A}\frac{\partial x^\nu}{\partial \lambda^B}$$0 at $$\gamma_{AB}=g_{\mu\nu}\frac{\partial x^\mu}{\partial \lambda^A}\frac{\partial x^\nu}{\partial \lambda^B}$$1, and $$\gamma_{AB}=g_{\mu\nu}\frac{\partial x^\mu}{\partial \lambda^A}\frac{\partial x^\nu}{\partial \lambda^B}$$2 at $$\gamma_{AB}=g_{\mu\nu}\frac{\partial x^\mu}{\partial \lambda^A}\frac{\partial x^\nu}{\partial \lambda^B}$$3 (Ahmed et al., 10 Aug 2025). In the AdS black hole with a dark-matter halo, the photon sphere satisfies

$$\gamma_{AB}=g_{\mu\nu}\frac{\partial x^\mu}{\partial \lambda^A}\frac{\partial x^\nu}{\partial \lambda^B}$$4

and both $$\gamma_{AB}=g_{\mu\nu}\frac{\partial x^\mu}{\partial \lambda^A}\frac{\partial x^\nu}{\partial \lambda^B}$$5 and the shadow radius grow with $$\gamma_{AB}=g_{\mu\nu}\frac{\partial x^\mu}{\partial \lambda^A}\frac{\partial x^\nu}{\partial \lambda^B}$$6 (Ahmed et al., 13 Sep 2025).

An important nuance is that the physical interpretation is not uniform across papers. Several studies state that the CoS weakens effective gravitational binding because it lowers the lapse or the effective potential. The Bardeen strong-lensing paper instead describes increasing $$\gamma_{AB}=g_{\mu\nu}\frac{\partial x^\mu}{\partial \lambda^A}\frac{\partial x^\nu}{\partial \lambda^B}$$7 as making the gravitational field effectively “stronger” in the sense of producing larger photon capture scales, larger impact parameters, larger photon-sphere radii, and stronger lensing effects. This suggests that CoS interpretations depend on which observable—local binding, capture scale, or optical size—is being emphasized (Vishvakarma et al., 2024, Ahmed et al., 12 Sep 2025).

4. Massive orbits, tidal response, and EMRIs

For timelike motion, the CoS modifies circular-orbit structure, marginal stability, inspiral phasing, and tidal response. In the Schwarzschild-plus-CoS model used for EMRIs,

$$\gamma_{AB}=g_{\mu\nu}\frac{\partial x^\mu}{\partial \lambda^A}\frac{\partial x^\nu}{\partial \lambda^B}$$8

and the effective potential is

$$\gamma_{AB}=g_{\mu\nu}\frac{\partial x^\mu}{\partial \lambda^A}\frac{\partial x^\nu}{\partial \lambda^B}$$9

The paper reports that the effective potential decreases as $$\Sigma^{\mu\nu}=\epsilon^{AB}\frac{\partial x^\mu}{\partial \lambda^A}\frac{\partial x^\nu}{\partial \lambda^B},$$0 increases, while larger $$\Sigma^{\mu\nu}=\epsilon^{AB}\frac{\partial x^\mu}{\partial \lambda^A}\frac{\partial x^\nu}{\partial \lambda^B},$$1 raises the potential barrier. Both the marginally bound orbit and the ISCO radius increase with $$\Sigma^{\mu\nu}=\epsilon^{AB}\frac{\partial x^\mu}{\partial \lambda^A}\frac{\partial x^\nu}{\partial \lambda^B},$$2, and the ISCO energy decreases (Alloqulov et al., 14 Dec 2025).

Closed-form Letelier limits are available in the Schwarzschild black hole immersed in a King dark matter halo. In the halo-free case,

$$\Sigma^{\mu\nu}=\epsilon^{AB}\frac{\partial x^\mu}{\partial \lambda^A}\frac{\partial x^\nu}{\partial \lambda^B},$$3

so the string cloud pushes both the photon sphere and the ISCO outward relative to Schwarzschild. The same paper states that increasing $$\Sigma^{\mu\nu}=\epsilon^{AB}\frac{\partial x^\mu}{\partial \lambda^A}\frac{\partial x^\nu}{\partial \lambda^B},$$4 lowers the effective potential and weakens the effective radial force, while raising the specific energy and angular momentum required for stable circular motion (Ahmed et al., 17 Oct 2025).

The generalized Letelier–Alencar geometry adds a second scale. There, the light-ring radius and ISCO both increase with $$\Sigma^{\mu\nu}=\epsilon^{AB}\frac{\partial x^\mu}{\partial \lambda^A}\frac{\partial x^\nu}{\partial \lambda^B},$$5 and decrease with $$\Sigma^{\mu\nu}=\epsilon^{AB}\frac{\partial x^\mu}{\partial \lambda^A}\frac{\partial x^\nu}{\partial \lambda^B},$$6, and circular orbits cease to exist in certain regions of parameter space. In the Letelier limit $$\Sigma^{\mu\nu}=\epsilon^{AB}\frac{\partial x^\mu}{\partial \lambda^A}\frac{\partial x^\nu}{\partial \lambda^B},$$7,

$$\Sigma^{\mu\nu}=\epsilon^{AB}\frac{\partial x^\mu}{\partial \lambda^A}\frac{\partial x^\nu}{\partial \lambda^B},$$8

again showing outward migration of the ISCO with increasing cloud strength (Silva et al., 26 Nov 2025).

EMRIs provide a waveform-based probe of these orbital changes. In the Schwarzschild-plus-CoS EMRI study, the large-$$\Sigma^{\mu\nu}=\epsilon^{AB}\frac{\partial x^\mu}{\partial \lambda^A}\frac{\partial x^\nu}{\partial \lambda^B},$$9 expansions show that the leading radial frequency $\gamma=\frac12 \Sigma^{\mu\nu}\Sigma_{\mu\nu}$0 is explicitly sensitive to $\gamma=\frac12 \Sigma^{\mu\nu}\Sigma_{\mu\nu}$1, whereas $\gamma=\frac12 \Sigma^{\mu\nu}\Sigma_{\mu\nu}$2 enters the angular frequency $\gamma=\frac12 \Sigma^{\mu\nu}\Sigma_{\mu\nu}$3 only at subleading order beyond the usual $\gamma=\frac12 \Sigma^{\mu\nu}\Sigma_{\mu\nu}$4 term. Under the adiabatic approximation, the CoS changes the inspiral trajectory in $\gamma=\frac12 \Sigma^{\mu\nu}\Sigma_{\mu\nu}$5 and $\gamma=\frac12 \Sigma^{\mu\nu}\Sigma_{\mu\nu}$6, and the waveform difference becomes prominent after about one year because of secular dephasing. The mismatch analysis uses the detectability threshold $\gamma=\frac12 \Sigma^{\mu\nu}\Sigma_{\mu\nu}$7 and concludes that LISA could distinguish the CoS-modified spacetime from ordinary Schwarzschild if

$\gamma=\frac12 \Sigma^{\mu\nu}\Sigma_{\mu\nu}$8

in that setup (Alloqulov et al., 14 Dec 2025).

Tidal-force studies sharpen the distinction between the original Letelier model and its generalization. For radial free fall in the original Letelier spacetime, the tidal components coincide with Schwarzschild because they depend on $\gamma=\frac12 \Sigma^{\mu\nu}\Sigma_{\mu\nu}$9 and $f(r)=1-\frac{2M}{r}-\alpha$00, not on the additive constant shift in $f(r)=1-\frac{2M}{r}-\alpha$01. In the Letelier–Alencar model, by contrast, the tidal divergence near the center is stronger, and the paper reports that an inversion between stretching and compression may occur in radial motion, although this regime is typically hidden inside the event horizon. For observers in circular motion, the cloud modifies the Keplerian frequency and the tidal-force profile even at large distances, and there is no sign change of the tidal components in the stable-orbit region (Silva et al., 26 Nov 2025).

5. Wave propagation and thermodynamic phase structure

CoS deformations also control scalar barriers, quasinormal spectra, greybody factors, and black-hole phase structure. In the black hole immersed in modified Chaplygin-like dark fluid and CoS, the lapse is

$f(r)=1-\frac{2M}{r}-\alpha$02

and the scalar perturbation potential is

$f(r)=1-\frac{2M}{r}-\alpha$03

The paper states that the CoS intensity has the primary influence on shadows, QNMs, and greybody factors: larger $f(r)=1-\frac{2M}{r}-\alpha$04 makes the shadow significantly larger, lowers the barrier height and broadens the barrier, and reduces both the real and imaginary parts of the QNM frequencies. For the fundamental mode $f(r)=1-\frac{2M}{r}-\alpha$05, the reported WKB values shift monotonically as

$f(r)=1-\frac{2M}{r}-\alpha$06

when $f(r)=1-\frac{2M}{r}-\alpha$07 increases from $f(r)=1-\frac{2M}{r}-\alpha$08 to $f(r)=1-\frac{2M}{r}-\alpha$09 to $f(r)=1-\frac{2M}{r}-\alpha$10. In the same model, at $f(r)=1-\frac{2M}{r}-\alpha$11 the 50% greybody transmission point moves from about $f(r)=1-\frac{2M}{r}-\alpha$12 to $f(r)=1-\frac{2M}{r}-\alpha$13 when $f(r)=1-\frac{2M}{r}-\alpha$14 is raised from $f(r)=1-\frac{2M}{r}-\alpha$15 to $f(r)=1-\frac{2M}{r}-\alpha$16 (Yan et al., 26 Nov 2025).

Thermodynamically, CoS can change not only the location of critical points but the very existence of van der Waals-like structure. In five-dimensional Einstein–Gauss–Bonnet AdS black holes, the string-cloud parameter $f(r)=1-\frac{2M}{r}-\alpha$17 enters the metric inside the square root and modifies the mass, temperature, and equation of state. A central result is that when the Gauss–Bonnet coupling is sent to zero but the cloud remains, the Hawking–Page transition disappears and the SBH/LBH transition recovers. The critical exponents remain

$f(r)=1-\frac{2M}{r}-\alpha$18

so the CoS shifts the phase structure without changing mean-field universality (Ghaffarnejad et al., 2018).

The Bardeen solution with a cloud of strings exhibits the same mean-field critical exponents and a three-branch phase structure. In that model the cloud parameter $f(r)=1-\frac{2M}{r}-\alpha$19 shifts the lapse to

$f(r)=1-\frac{2M}{r}-\alpha$20

lowers the temperature, and yields two heat-capacity divergences. The Gibbs free-energy analysis identifies three phases—small black hole, small-large intermediate phase, and large black hole—with the two outer branches stable and the intermediate branch unstable (Rodrigues et al., 2022).

A distinct dynamical characterization appears in Schwarzschild-AdS black holes with generalized CoS parameter $f(r)=1-\frac{2M}{r}-\alpha$21. There the thermodynamic branches SBH, IBH, and LBH are mirrored by multivalued Lyapunov-exponent branches $f(r)=1-\frac{2M}{r}-\alpha$22, $f(r)=1-\frac{2M}{r}-\alpha$23, and $f(r)=1-\frac{2M}{r}-\alpha$24. For both photons and massive particles, the discontinuity obeys a square-root law with universal critical exponent $f(r)=1-\frac{2M}{r}-\alpha$25, so the Lyapunov exponent functions as an order parameter for the phase structure (Kumar et al., 4 Aug 2025).

Not every observable is CoS-sensitive. In an Einstein–AdS black brane with dRGT massive gravity and a string cloud background, the final result for the shear-viscosity ratio is independent of the cloud of strings. The paper concludes that KSS-bound violation is caused by the massive gravity sector, not by the string cloud itself (Sadeghi et al., 2019).

6. Generalizations, singularity issues, and broader settings

A recent development is the explicit construction of a “new” cloud of strings with two independent hairs. The corresponding black-hole metric

$f(r)=1-\frac{2M}{r}-\alpha$26

has two horizons for the physical branch $f(r)=1-\frac{2M}{r}-\alpha$27, and its thermodynamics predicts a remnant at

$f(r)=1-\frac{2M}{r}-\alpha$28

with finite remnant entropy

$f(r)=1-\frac{2M}{r}-\alpha$29

The paper emphasizes that the additional parameters $f(r)=1-\frac{2M}{r}-\alpha$30 and $f(r)=1-\frac{2M}{r}-\alpha$31 amount to two hairs and could influence shadows and gravitational-wave signals (Alencar et al., 11 Jan 2025).

Regularity is a recurrent fault line. In the Bardeen solution with a cloud of strings, the string cloud preserves the basic horizon structure but makes the spacetime singular at $f(r)=1-\frac{2M}{r}-\alpha$32, with $f(r)=1-\frac{2M}{r}-\alpha$33. The same pattern appears in the newly constructed AdS black hole coupled to nonlinear electrodynamics and CoS: the Bardeen-like spacetime is regular when the string cloud is absent, but becomes singular once the cloud is included (Rodrigues et al., 2022, Sudhanshu et al., 2024). The Frolov regular black hole shows an even sharper version of this effect: adding a CoS term shifts the lapse by $f(r)=1-\frac{2M}{r}-\alpha$34, changes the Kretschmann scalar from $f(r)=1-\frac{2M}{r}-\alpha$35 in the original model to $f(r)=1-\frac{2M}{r}-\alpha$36, and thereby destroys the nonsingular core. The same paper, however, reports geodesic completeness in the parameter ranges examined, so the singularity interpretation is presented as subtle rather than settled (Nascimento et al., 13 Jan 2026).

Outside static black holes, CoS sources support radiating and cosmological spacetimes. In higher-dimensional Lovelock gravity with null dust plus a string cloud, the cloud function $f(r)=1-\frac{2M}{r}-\alpha$37 shifts or splits horizons and can eliminate real roots of the horizon equation, producing naked singularities for some parameter ranges (Ghosh et al., 2014). In cylindrically symmetric inhomogeneous universes sourced by a string fluid, the exact solutions fall into three classes: a non-singular expanding model, a singular big-bang expanding model, and an oscillatory model. These cosmologies are generally shearing, non-rotating, and anisotropic, with $f(r)=1-\frac{2M}{r}-\alpha$38, and isotropy is recovered only when a special parameter choice $f(r)=1-\frac{2M}{r}-\alpha$39 is made (Yadav et al., 2011).

Taken together, these results indicate that “Cloud of Strings” is not a single metric ansatz but a family of effective string-fluid sources. In the simplest Letelier limit, it acts as a constant deficit-like deformation. In generalized models, it introduces additional scales, extra hair, hypergeometric structure, remnant thermodynamics, and new strong-field signatures. The literature also shows that CoS can enlarge photon spheres, shadow radii, Einstein rings, and ISCOs; shift QNM spectra and heat-capacity divergences; and, in some regular-black-hole backgrounds, destroy regularity. This suggests that CoS is best understood as a flexible geometric source whose observational imprint depends strongly on how the string bivector sector is modeled and on which observable is being probed.