--- title: New Cloud of Strings in Gravitational Physics url: https://www.emergentmind.com/topics/new-cloud-of-strings-ncs type: topic --- # New Cloud of Strings in Gravitational Physics Searching arXiv for recent 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 [2501.07609]. 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 [2502.15846], [1810.08830], [2111.00854]. 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 \(f(r)=1-a-2M/r\) [2501.07609], [2502.15846]. Recent work uses “new” in more than one sense. The most literal usage appears in “A New Cloud of Strings” [2501.07609], where the cloud sector itself is generalized by introducing \(\Sigma_{23}\) in addition to \(\Sigma_{01}\). A broader but still intrinsic generalization appears in “Revisiting black holes surrounded by cloud and fluid of strings in general relativity” [2502.15846], where the equation-of-state parameter of a fluid of strings is promoted from a constant to a radial function \(\alpha(r)\). 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 [2111.00854]. 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 [2501.07609], or a generalized string fluid with variable \(\alpha(r)\) [2502.15846]. 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 \(\Sigma_{23}\) alongside the electric-like component \(\Sigma_{01}\). In a static, spherically symmetric metric \[ ds^2=-A(r)dt^2+B(r)dr^2+C^2(r)(d\theta^2+\sin^2\theta\,d\phi^2), \] the conservation equations imply \[ \rho\Sigma_{01}=\frac{a\sqrt{AB}}{C^2},\qquad \rho\Sigma_{23}=b(r)\sin\theta, \] with \[ b(r)=\frac{a c_0^2}{\sqrt{c_0^4+C^4}}, \] introducing a second constant \(c_0\) in addition to \(a\) [2501.07609]. Substituting this into the coarse-grained string stress tensor yields a unique anisotropic form, \[ T^{0}_{\ 0}=T^{1}_{\ 1}=-\frac{|a|}{C^4}\sqrt{c_0^4+C^4},\qquad T^{2}_{\ 2}=T^{3}_{\ 3}=\frac{|a|c_0^4}{C^4\sqrt{c_0^4+C^4}}. \] In Schwarzschild gauge \(C(r)=r\), \[ \rho=\frac{|a|}{r^4}\sqrt{c_0^4+r^4},\qquad p_r=-\rho,\qquad p_t=\rho\,\frac{c_0^4}{c_0^4+r^4}. \] The model is therefore fully specified by two parameters, \((a,c_0)\), and realizes the diagonal structure \(\mathrm{diag}(\rho,-\rho,p,p)\) emphasized in the abstract [2501.07609]. The resulting black-hole solution is \[ ds^2=-f(r)dt^2+\frac{dr^2}{f(r)}+r^2d\Omega^2, \] with \[ f(r)=1-\frac{2M}{r}+\frac{|a|c_0^2}{r^2}\,{}_2F_1\!\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};-\frac{r^4}{c_0^4}\right). \] This geometry carries two independent hairs, \(a\) and \(c_0\), both sourced by the string cloud rather than by gauge fields. In the limit \(c_0\to0\), the magnetic-like sector vanishes and the model reduces to Letelier’s cloud, \[ f(r)\to 1-|a|-\frac{2M}{r}, \] while \(a\to0\) gives Schwarzschild [2501.07609]. The same paper identifies a thermodynamically relevant parameter range \(0<|a|<1\). In this regime the Hawking temperature \[ T_H=\frac{1}{4\pi r_h}\left(1-\frac{|a|}{r_h^2}\sqrt{c_0^4+r_h^4}\right) \] vanishes at a finite remnant radius \[ r_0=\frac{|c_0||a|^{1/2}}{(1-a^2)^{1/4}}, \] with remnant entropy \[ S_{\rm rem}=\pi\frac{|c_0|^2|a|}{\sqrt{1-a^2}}. \] The paper further states that the specific heat changes sign around \(r_0\), indicating a second-order phase transition and a stable remnant endpoint [2501.07609]. ## 3. Variable-\(\alpha(r)\) 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 \[ T^{\mu}{}_{\nu}=\bigl[-\rho(r),-\rho(r),\rho(r)/\alpha(r),\rho(r)/\alpha(r)\bigr], \] where the ratio \(\rho/p=\alpha(r)\) is allowed to vary with radius [2502.15846]. This promotion of \(\alpha\) from a constant to a function is the core new ingredient in that construction. With \[ ds^2=-f(r)\,dt^2+f(r)^{-1}dr^2+r^2d\Omega^2, \] Einstein’s equations reduce to a master equation for \(f(r)\), and after introducing \(X(r)=rf(r)\) one integration yields \[ \frac{d}{dr}(r f(r)) - 1 = c_1 \exp\!\left( \int \frac{-2}{\alpha(r) r}\, dr \right), \] so that the general solution for arbitrary \(\alpha(r)\neq2\) is \[ f(r)=1+\frac{c_2}{r}+\frac{c_1}{r}\int \exp\!\left(\int \frac{-2}{\alpha(r) r}\,dr \right)\,dr. \] Constant \(\alpha\) reproduces the standard Soleng fluid of strings, \(\alpha\to\infty\) gives the Letelier cloud limit \(f(r)=1-2M/r-\epsilon\), and \(\alpha=2/(3w+1)\) maps the string fluid to Kiselev’s anisotropic fluid [2502.15846]. The same work constructs a novel regular black hole, \[ f(r)=1-\left(\frac{r_g}{r}+\epsilon\right)\left(1-e^{-r^3/r_0^3}\right), \] which tends at large radius to \[ f(r)\xrightarrow[]{r\gg r_0}1-\frac{r_g}{r}-\epsilon =1-\frac{2M}{r}-\epsilon, \] exactly the Schwarzschild solution with a cloud of strings. Near the origin, \[ f(r)\approx 1-\frac{r_g}{r_0^3}r^2-\epsilon\frac{r^3}{r_0^3}, \] so the geometry has a de Sitter-like core with finite invariants \[ R=-\frac{12r_g}{r_0^3},\qquad R_{\mu\nu}R^{\mu\nu}=\frac{36r_g^2}{r_0^6},\qquad K=\frac{24r_g^2}{r_0^6}. \] The paper gives an explicit \(\alpha(r)\) that reproduces this solution via the general formula, with \(c_1=-1/r_0^3\) and \(c_2=-r_g\) [2502.15846]. 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 \[ \rho=\rho_f+\frac12E^2+\frac12(\nabla\phi)^2,\quad p_r=p_f-\frac12E^2+\frac12(\nabla\phi)^2,\quad p_t=p_f+\frac12E^2-\frac12(\nabla\phi)^2, \] together with inverse formulas for \(p_f,\rho_f,E^2\), and \((\nabla\phi)^2\) [2502.15846]. ## 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 \(2+1\)-dimensional cloud-of-strings black hole. There the seed cloud is \[ M^{\mu}{}_{\nu}=\frac{\xi}{r}\,\mathrm{diag}(1,1,0), \] with seed solution \[ e^{\nu(r)}=\mu(r)=-M+2\xi r,\qquad r_H=\frac{M}{2\xi}. \] The MGD prescription keeps \(e^\nu\) fixed and deforms only the radial sector, \[ e^{-\lambda(r)}=\mu(r)+\alpha h(r), \] with \(h(r)\) determined from a linear constraint on the added source \(\theta^\mu{}_\nu\) [1810.08830]. Two explicit NCS families are obtained. For \(\theta^1{}_1=\theta^2{}_2\), the deformation is \[ h(r)=c_1(\xi r)^2(-M+2\xi r), \] and the metric becomes \[ ds^2=(-M+2\xi r)\,dt^2-\frac{dr^2}{[1+\alpha c_1(\xi r)^2](2\xi r-M)}-r^2d\phi^2. \] For a traceless extra source, \(\theta^\mu{}_\mu=0\), one finds \[ h(r)=c_1\frac{-M+2\xi r}{(-M+3\xi r)^{4/3}}, \] leading to \[ ds^2=(-M+2\xi r)\,dt^2-\frac{dr^2}{[1+\alpha c_1(3\xi r-M)^{-4/3}](2\xi r-M)}-r^2d\phi^2. \] In both cases, the original horizon \(r_H=M/(2\xi)\) is preserved, while the radial geometry and effective anisotropy are altered [1810.08830]. 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 \[ f(r)=k-\frac{2m}{r^{\,n-2}}+\frac{r^2}{l^2}-\frac{2a}{(n-1)\,r^{\,n-3}}, \] and in five dimensions the equation of state becomes \[ P=\frac{1}{27}\frac{27\pi T v^2-18kv+8a}{\pi v^3}, \] with critical quantities \[ v_c=\frac{4}{3}\frac{a}{k},\qquad P_c=\frac{1}{8\pi}\frac{k^3}{a^2},\qquad T_c=\frac{1}{2\pi}\frac{k^2}{a}. \] This realizes a Van der Waals–like phase transition absent in pure Schwarzschild–AdS and introduces topological charge as an additional thermodynamic variable [1806.06688]. 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 \[ \mathcal{T}^{\mu}{}_{\nu}=\mathrm{diag}(\rho_c,\rho_c,0,0),\qquad \rho_c(r)=b\,r^{-\frac{2(4\beta-1)}{2\beta-1}}, \] and that the novelty lies in the coupled system rather than in a new cloud-of-strings energy–momentum tensor [2111.00854]. ## 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, \[ V_{\rm eff}(r)=\left(-\epsilon+\frac{L^2}{r^2}\right)f(r), \] or \(V_{\rm eff}=f(r)\left(1+L^2/r^2\right)\) for timelike geodesics in the usual sign convention [2509.24833], [2502.15846]. In the hypergeometric NCS plus dark matter halo model, the photon sphere radius \(r_{\rm ph}\) is determined numerically from the generalized circular-null-orbit condition, and both \(r_{\rm ph}\) and the shadow radius \[ R_s=\frac{r_{\rm ph}}{\sqrt{f(r_{\rm ph})}} \] increase with the NCS intensity parameter \(\alpha\), while increasing \(b\) decreases them slightly [2509.24833]. The same paper finds that the ISCO is obtained from \[ 3f(r)f'(r)+rf''(r)f(r)-2r(f'(r))^2=0, \] and that \(r_{\rm ISCO}\) also increases strongly with \(\alpha\) and decreases mildly with \(b\) [2509.24833]. The regular fluid-of-strings construction likewise modifies photon spheres and shadow sizes. For the regular metric \[ f(r)=1-\left(\frac{r_g}{r}+\epsilon\right)\left(1-e^{-r^3/r_0^3}\right), \] the photon sphere approximately satisfies \[ r_m\approx\frac{3r_g}{2(1-\epsilon)}=\frac{3M}{1-\epsilon}, \] for \(r_m\gg r_0\), and the shadow radius depends on the corresponding critical impact parameter \(b_c\) [2502.15846]. A complementary perspective comes from the Letelier–Alencar generalized cloud, where the stress tensor \[ T^t{}_t=T^r{}_r=-\frac{g_s^2}{r^4}\sqrt{l_s^4+r^4},\qquad T^\theta{}_\theta=T^\varphi{}_\varphi=\frac{g_s^2l_s^4}{r^4\sqrt{l_s^4+r^4}} \] 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 \(g_s\) and decrease with the scale \(l_s\), while circular orbits cease to exist in parts of parameter space [2511.21604]. The same generalized model alters tidal forces in ways absent in the original Letelier spacetime. For radial infall, the tidal eigenvalues are \[ k_1=-\frac{f''(r)}{2},\qquad k_2=-\frac{f'(r)}{2r}, \] 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 [2511.21604]. ## 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 \(M(r_h)\), the Hawking temperature vanishes at a finite \(r_0\), and the entropy remains the standard area law \[ S=\pi r_h^2, \] with a stable remnant of finite entropy \(S_{\rm rem}\) for \(0<|a|<1\) [2501.07609]. In regular fluid-of-strings models, the Hawking temperature for reduced constant-\(\alpha\) solutions is \[ T_H(\alpha,\epsilon)=\frac{\hbar \epsilon}{2\pi l (\alpha - 2)} \left( \frac{\alpha - 2}{\alpha \epsilon} \right)^{\frac{\alpha}{2}+1}, \] for \(\alpha\neq2\), while the regular black hole exhibits two horizons for typical parameters and allows horizon merging at critical \(r_0\), producing a horizonless regular geometry beyond that point [2502.15846]. Thermodynamic phase structure becomes especially rich in AdS backgrounds. In the Bardeen solution with a cloud of strings, \[ f(r)=1-a-\frac{2Mr^2}{(q^2+r^2)^{3/2}}-\frac{\lambda r^2}{3}, \] the cloud parameter makes the spacetime singular at the origin but preserves the same event-horizon characteristics as the Bardeen solution. The enthalpy is \[ M(S,q,a,P)=\frac{(\pi q^2+S)^{3/2}(-3a+8PS+3)}{6\sqrt{\pi}S}, \] and the heat capacity \[ C_P=\frac{2S(\pi q^2+S)\left(S(-a+8PS+1)+2\pi(a-1)q^2\right)} { S^2(a+8PS-1)-4\pi(a-1)q^2S-8\pi^2(a-1)q^4} \] 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 \[ \alpha=0,\qquad \beta=\frac12,\qquad \gamma=1,\qquad \delta=3, \] the same as in Van der Waals theory [2210.06531]. 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 \(\alpha\) or the halo scale \(r_s\) lowers the temperature at small \(r_h\), shifts the heat-capacity divergences, and moves the Hawking–Page transition and critical point. For \(P