New Cloud of Strings in Gravitational Physics

• New Cloud of Strings (NCS) is a generalized string-cloud model in gravitational physics that extends Letelier’s construction by incorporating both electric-like and magnetic-like bivector components.
• The two-parameter model (a and c₀) produces a unique anisotropic stress tensor with two independent black-hole hairs, modifying horizons, geodesics, and thermodynamic behavior.
• Additional approaches using variable string-fluid equations of state and MGD deformations demonstrate diverse extensions that impact observable phenomena like shadows and phase transitions.

Searching arXiv for papers explicitly using or clarifying “New Cloud of Strings” and closely related generalized cloud/fluid-of-strings models. New Cloud of Strings (NCS) denotes a family of generalized string-cloud constructions in gravitational physics that extend, reinterpret, or embed Letelier’s cloud of strings in broader settings. In its strictest recent usage, NCS refers to a two-parameter generalization of Letelier’s model obtained by allowing both an electric-like bivector component $\Sigma_{01}$ and a magnetic-like component $\Sigma_{23}$, yielding a uniquely fixed anisotropic stress tensor of the form $\mathrm{diag}(\rho,-\rho,p,p)$ and a black-hole metric with two independent hairs (Alencar et al., 11 Jan 2025). In a broader literature, the label also covers variable-equation-of-state string fluids, MGD-deformed string-cloud geometries, and composite configurations in which an otherwise standard Letelier cloud is placed in a new gravitational, electromagnetic, or thermodynamic environment (Santos, 20 Feb 2025, Panotopoulos et al., 2018, Gogoi et al., 2021). The common thread is that NCS departs from the original constant-deficit Letelier sector by introducing either additional string degrees of freedom, new radial structure, or new couplings that materially alter horizons, geodesics, thermodynamics, and, in some cases, regularity.

1. Letelier cloud of strings and the meaning of “new”

Letelier’s original cloud of strings treats a spacetime as sourced by a continuous distribution of Nambu–Goto strings. The basic object is the antisymmetric bivector

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

with induced world-sheet metric $h_{AB}$ or $\gamma_{AB}$ depending on notation, and an effective stress tensor built from $\Sigma^{\mu\nu}$ and a string density. In the standard spherically symmetric case, only the electric-like component $\Sigma_{01}$ is retained, which produces the familiar Letelier stress tensor with radial tension and vanishing tangential pressures, and leads to a Schwarzschild-like lapse $f(r)=1-|a|-2M/r$ or equivalent notation $f(r)=1-a-2M/r$ (Alencar et al., 11 Jan 2025, Santos, 20 Feb 2025).

Recent work uses “new” in more than one sense. The most literal usage appears in “A New Cloud of Strings” (Alencar et al., 11 Jan 2025), where the cloud sector itself is generalized by introducing $\Sigma_{23}$0 in addition to $\Sigma_{23}$1. A broader but still intrinsic generalization appears in “Revisiting black holes surrounded by cloud and fluid of strings in general relativity” (Santos, 20 Feb 2025), where the equation-of-state parameter of a fluid of strings is promoted from a constant to a radial function $\Sigma_{23}$2. By contrast, some papers employ “new” only for the total background, not for the string sector itself. The Rastall-gravity quasinormal-mode study explicitly states that there is no new type of cloud-of-strings model introduced; the cloud remains the standard Letelier cloud, and the novelty lies in the combined configuration of Rastall gravity, non-linear electrodynamics, and the surrounding cloud (Gogoi et al., 2021).

This distinction is central. In a strict encyclopedia sense, NCS is best understood as an umbrella term whose most precise realizations are either: a generalized string bivector sector with two independent string hairs (Alencar et al., 11 Jan 2025), or a generalized string fluid with variable $\Sigma_{23}$3 (Santos, 20 Feb 2025). Other uses are best read as phenomenological extensions of the environment in which the standard Letelier cloud is embedded.

2. Two-parameter NCS from electric-like and magnetic-like string bivectors

The clearest formal realization of NCS generalizes Letelier’s cloud by allowing a magnetic-like bivector component $\Sigma_{23}$4 alongside the electric-like component $\Sigma_{23}$5. In a static, spherically symmetric metric

$\Sigma_{23}$6

the conservation equations imply

$\Sigma_{23}$7

with

$\Sigma_{23}$8

introducing a second constant $\Sigma_{23}$9 in addition to $\mathrm{diag}(\rho,-\rho,p,p)$0 (Alencar et al., 11 Jan 2025).

Substituting this into the coarse-grained string stress tensor yields a unique anisotropic form,

$\mathrm{diag}(\rho,-\rho,p,p)$1

In Schwarzschild gauge $\mathrm{diag}(\rho,-\rho,p,p)$2,

$\mathrm{diag}(\rho,-\rho,p,p)$3

The model is therefore fully specified by two parameters, $\mathrm{diag}(\rho,-\rho,p,p)$4, and realizes the diagonal structure $\mathrm{diag}(\rho,-\rho,p,p)$5 emphasized in the abstract (Alencar et al., 11 Jan 2025).

The resulting black-hole solution is

$\mathrm{diag}(\rho,-\rho,p,p)$6

with

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

This geometry carries two independent hairs, $\mathrm{diag}(\rho,-\rho,p,p)$8 and $\mathrm{diag}(\rho,-\rho,p,p)$9, both sourced by the string cloud rather than by gauge fields. In the limit $$\Sigma^{\mu\nu}=\epsilon^{AB}\frac{\partial x^{\mu}}{\partial \xi^A}\frac{\partial x^{\nu}}{\partial \xi^B},$$0, the magnetic-like sector vanishes and the model reduces to Letelier’s cloud,

$$\Sigma^{\mu\nu}=\epsilon^{AB}\frac{\partial x^{\mu}}{\partial \xi^A}\frac{\partial x^{\nu}}{\partial \xi^B},$$1

while $$\Sigma^{\mu\nu}=\epsilon^{AB}\frac{\partial x^{\mu}}{\partial \xi^A}\frac{\partial x^{\nu}}{\partial \xi^B},$$2 gives Schwarzschild (Alencar et al., 11 Jan 2025).

The same paper identifies a thermodynamically relevant parameter range $$\Sigma^{\mu\nu}=\epsilon^{AB}\frac{\partial x^{\mu}}{\partial \xi^A}\frac{\partial x^{\nu}}{\partial \xi^B},$$3. In this regime the Hawking temperature

$$\Sigma^{\mu\nu}=\epsilon^{AB}\frac{\partial x^{\mu}}{\partial \xi^A}\frac{\partial x^{\nu}}{\partial \xi^B},$$4

vanishes at a finite remnant radius

$$\Sigma^{\mu\nu}=\epsilon^{AB}\frac{\partial x^{\mu}}{\partial \xi^A}\frac{\partial x^{\nu}}{\partial \xi^B},$$5

with remnant entropy

$$\Sigma^{\mu\nu}=\epsilon^{AB}\frac{\partial x^{\mu}}{\partial \xi^A}\frac{\partial x^{\nu}}{\partial \xi^B},$$6

The paper further states that the specific heat changes sign around $$\Sigma^{\mu\nu}=\epsilon^{AB}\frac{\partial x^{\mu}}{\partial \xi^A}\frac{\partial x^{\nu}}{\partial \xi^B},$$7, indicating a second-order phase transition and a stable remnant endpoint (Alencar et al., 11 Jan 2025).

3. Variable-$$\Sigma^{\mu\nu}=\epsilon^{AB}\frac{\partial x^{\mu}}{\partial \xi^A}\frac{\partial x^{\nu}}{\partial \xi^B},$$8 string fluids and regular NCS geometries

A second major line of generalization starts from the distinction between a cloud of strings and a fluid of strings. For static spherical symmetry, the generalized fluid-of-strings stress tensor is written as

$$\Sigma^{\mu\nu}=\epsilon^{AB}\frac{\partial x^{\mu}}{\partial \xi^A}\frac{\partial x^{\nu}}{\partial \xi^B},$$9

where the ratio $h_{AB}$0 is allowed to vary with radius (Santos, 20 Feb 2025). This promotion of $h_{AB}$1 from a constant to a function is the core new ingredient in that construction.

With

$h_{AB}$2

Einstein’s equations reduce to a master equation for $h_{AB}$3, and after introducing $h_{AB}$4 one integration yields

$h_{AB}$5

so that the general solution for arbitrary $h_{AB}$6 is

$h_{AB}$7

Constant $h_{AB}$8 reproduces the standard Soleng fluid of strings, $h_{AB}$9 gives the Letelier cloud limit $\gamma_{AB}$0, and $\gamma_{AB}$1 maps the string fluid to Kiselev’s anisotropic fluid (Santos, 20 Feb 2025).

The same work constructs a novel regular black hole,

$\gamma_{AB}$2

which tends at large radius to

$\gamma_{AB}$3

exactly the Schwarzschild solution with a cloud of strings. Near the origin,

$\gamma_{AB}$4

so the geometry has a de Sitter-like core with finite invariants

$\gamma_{AB}$5

The paper gives an explicit $\gamma_{AB}$6 that reproduces this solution via the general formula, with $\gamma_{AB}$7 and $\gamma_{AB}$8 (Santos, 20 Feb 2025).

This line of work therefore defines NCS not by a new bivector content, but by a generalized anisotropic string medium with variable tangential pressure. It also establishes a decomposition of the resulting anisotropic fluid into a perfect fluid, an electromagnetic field, and a minimally coupled scalar field through the relations

$\gamma_{AB}$9

together with inverse formulas for $\Sigma^{\mu\nu}$0, and $\Sigma^{\mu\nu}$1 (Santos, 20 Feb 2025).

4. Alternative NCS constructions: MGD deformations, AdS embeddings, and composite environments

A distinct construction appears in three-dimensional gravity, where Minimal Geometric Deformation (MGD) is applied to the known $\Sigma^{\mu\nu}$2-dimensional cloud-of-strings black hole. There the seed cloud is

$\Sigma^{\mu\nu}$3

with seed solution

$\Sigma^{\mu\nu}$4

The MGD prescription keeps $\Sigma^{\mu\nu}$5 fixed and deforms only the radial sector,

$\Sigma^{\mu\nu}$6

with $\Sigma^{\mu\nu}$7 determined from a linear constraint on the added source $\Sigma^{\mu\nu}$8 (Panotopoulos et al., 2018).

Two explicit NCS families are obtained. For $\Sigma^{\mu\nu}$9, the deformation is

$\Sigma_{01}$0

and the metric becomes

$\Sigma_{01}$1

For a traceless extra source, $\Sigma_{01}$2, one finds

$\Sigma_{01}$3

leading to

$\Sigma_{01}$4

In both cases, the original horizon $\Sigma_{01}$5 is preserved, while the radial geometry and effective anisotropy are altered (Panotopoulos et al., 2018).

Other papers extend string clouds primarily through the surrounding sector. In extended black-hole thermodynamics, a Schwarzschild–AdS black hole minimally coupled to a cloud of strings has

$\Sigma_{01}$6

and in five dimensions the equation of state becomes

$\Sigma_{01}$7

with critical quantities

$\Sigma_{01}$8

This realizes a Van der Waals–like phase transition absent in pure Schwarzschild–AdS and introduces topological charge as an additional thermodynamic variable (Ghaffarnejad et al., 2018).

In other settings, the phrase NCS is best interpreted cautiously. The Rastall-gravity paper states explicitly that the cloud sector remains Letelier’s model with

$\Sigma_{01}$9

and that the novelty lies in the coupled system rather than in a new cloud-of-strings energy–momentum tensor (Gogoi et al., 2021).

5. Geodesics, photon spheres, shadows, and tidal structure

Across the NCS literature, one of the clearest signatures of generalized string clouds is the systematic modification of null and timelike effective potentials. For static spherical metrics,

$f(r)=1-|a|-2M/r$0

or $f(r)=1-|a|-2M/r$1 for timelike geodesics in the usual sign convention (Ahmed et al., 29 Sep 2025, Santos, 20 Feb 2025).

In the hypergeometric NCS plus dark matter halo model, the photon sphere radius $f(r)=1-|a|-2M/r$2 is determined numerically from the generalized circular-null-orbit condition, and both $f(r)=1-|a|-2M/r$3 and the shadow radius

$f(r)=1-|a|-2M/r$4

increase with the NCS intensity parameter $f(r)=1-|a|-2M/r$5, while increasing $f(r)=1-|a|-2M/r$6 decreases them slightly (Ahmed et al., 29 Sep 2025). The same paper finds that the ISCO is obtained from

$f(r)=1-|a|-2M/r$7

and that $f(r)=1-|a|-2M/r$8 also increases strongly with $f(r)=1-|a|-2M/r$9 and decreases mildly with $f(r)=1-a-2M/r$0 (Ahmed et al., 29 Sep 2025).

The regular fluid-of-strings construction likewise modifies photon spheres and shadow sizes. For the regular metric

$f(r)=1-a-2M/r$1

the photon sphere approximately satisfies

$f(r)=1-a-2M/r$2

for $f(r)=1-a-2M/r$3, and the shadow radius depends on the corresponding critical impact parameter $f(r)=1-a-2M/r$4 (Santos, 20 Feb 2025).

A complementary perspective comes from the Letelier–Alencar generalized cloud, where the stress tensor

$f(r)=1-a-2M/r$5

produces a hypergeometric metric similar in structure to the two-hair NCS. There, the radii of the photon sphere and ISCO increase with the cloud parameter $f(r)=1-a-2M/r$6 and decrease with the scale $f(r)=1-a-2M/r$7, while circular orbits cease to exist in parts of parameter space (Silva et al., 26 Nov 2025).

The same generalized model alters tidal forces in ways absent in the original Letelier spacetime. For radial infall, the tidal eigenvalues are

$f(r)=1-a-2M/r$8

and the paper shows that a sign inversion between stretching and compression can occur, although typically inside the event horizon. For circular motion, the cloud modifies the effective Keplerian frequency even at large radii and produces no sign change of the tidal components in the stable-orbit region (Silva et al., 26 Nov 2025).

6. Thermodynamics, criticality, and phase transitions

NCS sectors also change black-hole thermodynamics in systematic ways. In the strict two-hair NCS model, the horizon equation determines a nontrivial $f(r)=1-a-2M/r$9, the Hawking temperature vanishes at a finite $\Sigma_{23}$00, and the entropy remains the standard area law

$\Sigma_{23}$01

with a stable remnant of finite entropy $\Sigma_{23}$02 for $\Sigma_{23}$03 (Alencar et al., 11 Jan 2025).

In regular fluid-of-strings models, the Hawking temperature for reduced constant-$\Sigma_{23}$04 solutions is

$\Sigma_{23}$05

for $\Sigma_{23}$06, while the regular black hole exhibits two horizons for typical parameters and allows horizon merging at critical $\Sigma_{23}$07, producing a horizonless regular geometry beyond that point (Santos, 20 Feb 2025).

Thermodynamic phase structure becomes especially rich in AdS backgrounds. In the Bardeen solution with a cloud of strings,

$\Sigma_{23}$08

the cloud parameter makes the spacetime singular at the origin but preserves the same event-horizon characteristics as the Bardeen solution. The enthalpy is

$\Sigma_{23}$09

and the heat capacity

$\Sigma_{23}$10

has two divergences, dividing the system into three phases: small stable, intermediate unstable, and large stable. The Gibbs free energy exhibits a swallow-tail structure, and the critical exponents are

$\Sigma_{23}$11

the same as in Van der Waals theory (Rodrigues et al., 2022).

In the AdS black hole with hypergeometric NCS and dark matter halo, the Hawking temperature, equation of state, Gibbs free energy, and heat capacity all depend explicitly on the NCS hypergeometric sector. The paper shows that increasing $\Sigma_{23}$12 or the halo scale $\Sigma_{23}$13 lowers the temperature at small $\Sigma_{23}$14, shifts the heat-capacity divergences, and moves the Hawking–Page transition and critical point. For $\Sigma_{23}$15, $\Sigma_{23}$16 develops a swallow-tail corresponding to a first-order small/large black-hole transition; at $\Sigma_{23}$17 this shrinks to a cusp (Ahmed et al., 29 Sep 2025).

The broader lesson is consistent across the literature: NCS parameters usually do not abolish the standard black-hole chemistry framework, but rather supply new thermodynamic work terms, shift coexistence curves, and in several cases generate remnants or modify critical points without changing the mean-field universality class.

7. Conceptual scope, ambiguities, and current status

NCS is not a single universally fixed model. The literature supports at least three technically distinct meanings.

First, in the narrowest and most literal sense, NCS is the two-parameter string-cloud sector with both $\Sigma_{23}$18 and $\Sigma_{23}$19 nonzero, characterized by $\Sigma_{23}$20 and $\Sigma_{23}$21, unique anisotropic equations of state, two independent hairs, and a hypergeometric black-hole solution (Alencar et al., 11 Jan 2025).

Second, NCS can denote generalized cloud/fluid-of-strings media in which the algebraic structure of Letelier’s matter is retained but its effective equation of state is promoted, most notably through $\Sigma_{23}$22 in the fluid-of-strings framework. In that reading, NCS is a broader family that includes regular black holes, Kiselev-like subfamilies, and effective perfect-fluid/electromagnetic/scalar decompositions (Santos, 20 Feb 2025).

Third, some works use language suggestive of novelty for systems where the cloud sector itself is not new. The Rastall-gravity quasinormal-mode paper is explicit that the cloud is the standard Letelier cloud generalized only through Rastall conservation, and that “new” properly describes the full configuration rather than a new cloud-of-strings stress tensor (Gogoi et al., 2021). Similar caution applies to many thermodynamic or astrophysical embeddings.

This suggests that NCS is best treated as a research category rather than a single equation of state. Its defining feature is the attempt to go beyond the original constant-deficit Letelier cloud, either by enlarging the string bivector content, introducing a radial equation-of-state function, deforming the geometry through MGD, or embedding the cloud in richer matter or gravity sectors that produce observationally relevant modifications. Within that category, the most concrete recent advances are the two-hair hypergeometric NCS black hole (Alencar et al., 11 Jan 2025), the variable-$\Sigma_{23}$23 regular string-fluid black hole (Santos, 20 Feb 2025), and the AdS hypergeometric NCS with dark matter halo, which ties NCS directly to shadows, ISCOs, and phase transitions (Ahmed et al., 29 Sep 2025).