Spinning Cosmic String Dynamics

• Spinning cosmic strings are line-like spacetime defects carrying mass, angular momentum, and torsion, marked by a conical deficit and off-diagonal frame-dragging terms.
• They are modeled through modified cylindrical metrics that incorporate rotation and screw dislocations, capturing curvature, torsion, and even closed timelike curves.
• Quantum analyses in these backgrounds reveal modified Landau levels, relativistic oscillator spectra, and scattering phase shifts, with analogies in optical and condensed-matter systems.

Searching arXiv for relevant papers on spinning cosmic strings and closely related analyses. arXiv search query: "spinning cosmic string" A spinning cosmic string is a line-like topological defect in spacetime that carries both mass and angular momentum; in broader Einstein–Cartan and defect-theoretic formulations it may also carry torsion through a screw-dislocation term. Its defining geometric features are a conical defect, encoded by an angular-deficit parameter, and an off-diagonal time–angle coupling that represents frame dragging. In the idealized case the spacetime is locally flat outside the core but globally nontrivial, while more general models include finite-width cores, internal vacuum energy, torsion, and coupling to black holes, quantum fields, and optical analogues (Muniz et al., 2014, Boumali, 20 Nov 2025).

1. Geometric structure and standard metrics

A standard ideal spinning cosmic string metric is

$$ds^2 = -\big(c\,dt + a\,d\phi\big)^2 + dp^2 + \alpha^2 p^2 d\phi^2 + dz^2,$$

where $p$ is the radial distance from the string, $\phi$ is the azimuthal angle, $z$ is the coordinate along the string, $\alpha \le 1$ is the angular-deficit parameter, and $a = 4GJ/c^3$ is the rotational parameter. In this form, $\alpha < 1$ describes a conical defect, while the off-diagonal time–angle structure encodes frame dragging (Muniz et al., 2014).

A more general stationary, cylindrically symmetric metric used for a spinning cosmic string endowed with both curvature and torsion is

$$ds^{2} = -\left(dt + 4GJ^{t} d\varphi\right)^{2} + d\rho^{2} + \alpha^{2}\rho^{2} d\varphi^{2} + \left(dz + 4GJ^{z} d\varphi\right)^{2}.$$

Here $\alpha\in(0,1)$ is the conical parameter, $J^{t}$ is the linear density of angular momentum, and $p$0 is a screw-dislocation parameter. In the language of defects, $p$1 encodes conical curvature or disclination, $p$2 encodes rotation or frame dragging, and $p$3 encodes torsion or screw dislocation (Boumali, 20 Nov 2025).

Several limiting cases recur in the literature. Setting $p$4 gives the pure spinning cosmic string, setting $p$5 gives the pure screw dislocation, and setting both to zero gives the static cosmic string. In another common convention, used in relativistic quantum studies, the metric is written as

$p$6

with $p$7 and $p$8, so that the nonzero $p$9 component directly expresses the rotational character of the defect (Cunha et al., 2020).

2. Causal structure, closed timelike curves, and dimensional subtleties

Spinning cosmic strings are closely associated with closed timelike curves. In the ideal spinning-string geometry, the boundary of the CTC region lies at a radius proportional to $\phi$0, and the nonrelativistic Landau-level analysis assumes that the quantum particle remains at a safe distance,

$\phi$1

so that the CTC region does not directly affect the Schrödinger problem (Muniz et al., 2014).

In the curvature-plus-torsion geometry, the azimuthal metric component is

$\phi$2

If $\phi$3, positivity of $\phi$4 requires

$\phi$5

so strong rotation relative to torsion creates a geometric radial cutoff that acts as a hard wall in scattering problems (Boumali, 20 Nov 2025).

The existence and physical relevance of CTCs remain controversial. In a self-gravitating spinning cosmic string with a $\phi$6 scalar gauge field, the causality-breaking boundary $\phi$7 can approach the string-core radius $\phi$8. In that limit the metric components become singular, the proper time on the core stops flowing, and the energy-momentum tensor components diverge; the same analysis expects the angular momentum $\phi$9 to decrease due to the emission of gravitational energy triggered by scalar perturbations, which aligns the discussion with chronology protection rather than with accessible time-travel geometries (Slagter, 2015).

An exact straight spinning cosmic string solution in Brans–Dicke gravity reaches a related conclusion from another direction. It was shown that there is a suitable choice for the integration constants in which closed timelike curves are not allowed, even though the spacetime still describes a straight spinning cosmic string (Santos et al., 2017).

A separate controversy concerns torsion after dimensional reduction. A later comment argued that a “spinning cosmic string with torsion” becomes physically incoherent if it is reduced to $z$0-dimensions with $z$1, because screw dislocations require the $z$2-direction; in that view, a $z$3-dimensional model may retain spin but not torsion (Oliveira, 2024). This suggests that statements about torsionful spinning strings depend sensitively on whether the full $z$4-dimensional geometry is preserved.

3. Quantum spectra in spinning cosmic string backgrounds

The spinning cosmic string is a standard background for bound-state problems. In the nonrelativistic Landau problem for a spinless charged particle in a static and uniform magnetic field parallel to the string, the energy levels become

$z$5

The first bracket is the cosmic-string-modified Landau structure, while the last two terms give rotation-dependent shifts. In the weak-rotation regime the correction is interpreted as an analogue of a quadratic Zeeman effect (Muniz et al., 2014).

For relativistic spin-$z$6 motion in a uniform magnetic field plus an Aharonov–Bohm flux, the radial equation depends on the effective angular momentum

$z$7

so the string’s rotation enters by coupling energy into the angular sector. The corresponding bound-state solutions are given in terms of Kummer functions, and the energy spectrum contains both geometry-independent relativistic Landau branches and branches that depend on $z$8, $z$9, $\alpha \le 1$0, and the Aharonov–Bohm flux (Cunha et al., 2020).

For relativistic spin-0 bosons, the DKP oscillator in a spinning cosmic string background yields a radial equation with the centrifugal term

$\alpha \le 1$1

so the deficit angle rescales the effective angular momentum while the rotation parameter $\alpha \le 1$2 couples to the energy $\alpha \le 1$3. In that model, increasing $\alpha \le 1$4 increases $\alpha \le 1$5, and increasing $\alpha \le 1$6 decreases $\alpha \le 1$7 (Hosseinpour et al., 2018). A generalized DKP oscillator with Cornell potential $\alpha \le 1$8 preserves the same structural dependence on $\alpha \le 1$9 and shows that the spectrum depends strongly on the linear part $a = 4GJ/c^3$0 of the Cornell potential (Yang et al., 2021).

Relativistic oscillator models have also been extended to fully solvable Dirac systems with disclination and dislocation. In the Dirac oscillator for spin-$a = 4GJ/c^3$1 particles, the flat-space Moshinsky spectrum is replaced by energy spectra expressed in terms of effective angular quantum numbers that depend on $a = 4GJ/c^3$2, $a = 4GJ/c^3$3, $a = 4GJ/c^3$4, $a = 4GJ/c^3$5, and $a = 4GJ/c^3$6; temporal torsion produces energy-dependent shifts, spatial torsion produces momentum-dependent shifts, and curvature and torsion lift degeneracies (Boumali, 19 Sep 2025). A supersymmetric treatment of the Dirac problem further transforms the system into a relativistic nonlinear isotonic oscillator, with rational extensions expressed in terms of exceptional orthogonal Laguerre $a = 4GJ/c^3$7 polynomials (Yesiltas et al., 2024).

4. Scattering, optical geometry, and effective-medium formulations

Scattering theory reveals the same topological structure from a different angle. For spin-$a = 4GJ/c^3$8 particles in the general spinning-string spacetime with curvature and torsion, the relevant quantity is the modified azimuthal index

$a = 4GJ/c^3$9

This quantity controls the centrifugal barrier, the radial Bessel or confluent-hypergeometric solutions, and the partial-wave phase shifts. In the Coulomb problem the result is topology-renormalised Mott/Rutherford scattering, while in the Coulomb-free limit the scattering becomes purely geometric yet still exhibits forward enhancement governed by defect parameters and the cutoff $\alpha < 1$0 (Boumali, 20 Nov 2025).

Optical studies treat the stationary spinning-string metric through a Randers-type optical geometry. In one such treatment the equatorial-plane metric

$\alpha < 1$1

leads to a String-Randers optical metric and a Gauss–Bonnet calculation of weak light deflection with static, spin, and torsion contributions (Jusufi, 2016). As noted above, a later comment challenged the torsion interpretation after reduction to $\alpha < 1$2-dimensions (Oliveira, 2024).

The geometric-optics approximation also permits an exact effective-medium formulation. In the Tamm-medium representation of a spinning cosmic string, numerical ray tracing shows that rays never cross the string’s boundary, that the medium supports evanescent waves in regions of phase space corresponding to those spacetime regions which could support closed timelike curves, and that a spinning string can be slightly visible while a non-spinning string is almost perfectly invisible (Anderson et al., 2010).

In non-local gravity, a slowly spinning cosmic string is regularized by a Poincaré-invariant non-locality $\alpha < 1$3. The angle deficit then becomes a function of the radial distance, the non-local gravitomagnetic field is smooth rather than distributional, and the spacetime is simply connected rather than multiply connected (Boos, 2020).

5. Interior structure, realistic cores, and dynamical string models

Idealized line defects are often replaced by finite-width sources. A spinning cosmic string with internal structure may be modeled on the Gott–Hiscock cosmic string, with a finite core filled with material and vacuum energies. In that setting the effective angular parameter depends on the internal volumetric energy density $\alpha < 1$4 and the vacuum energy $\alpha < 1$5, and the Landau spectrum acquires explicit dependence on $\alpha < 1$6 and $\alpha < 1$7, reducing to the ideal spinning-string result when $\alpha < 1$8 (Muniz et al., 2014).

A more dynamical treatment uses an Abelian Higgs or global-string interior. In the stationary cylindrically symmetric metric

$\alpha < 1$9

the cross term $$ds^{2} = -\left(dt + 4GJ^{t} d\varphi\right)^{2} + d\rho^{2} + \alpha^{2}\rho^{2} d\varphi^{2} + \left(dz + 4GJ^{z} d\varphi\right)^{2}.$$0 represents the intrinsic spin of the string, while matching to an exterior conical metric fixes the asymptotic angle deficit. Numerical interior solutions show a moving causality-breaking boundary and a direct coupling between the angular-momentum parameter and the interior fields (Slagter, 2015).

The same analysis argues that the physically dangerous limit is $$ds^{2} = -\left(dt + 4GJ^{t} d\varphi\right)^{2} + d\rho^{2} + \alpha^{2}\rho^{2} d\varphi^{2} + \left(dz + 4GJ^{z} d\varphi\right)^{2}.$$1. In that regime the metric becomes singular, the proper time required to make a complete loop becomes infinite, and the energy-momentum tensor components diverge. The expectation that angular momentum is radiated away by scalar-triggered gravitational emission implies that accessible CTCs are exceedingly unlikely (Slagter, 2015). This suggests that finite-core and field-theoretic completions soften the interpretation of the formal CTC regions present in ideal stationary metrics.

6. Black holes, analogue systems, and broader significance

Spinning cosmic strings also appear in black-hole spacetimes. A Schwarzschild or Reissner–Nordström black hole pierced by an infinitely long spinning cosmic string and a global monopole is described by a metric in which the string contributes an off-diagonal $$ds^{2} = -\left(dt + 4GJ^{t} d\varphi\right)^{2} + d\rho^{2} + \alpha^{2}\rho^{2} d\varphi^{2} + \left(dz + 4GJ^{z} d\varphi\right)^{2}.$$2 term and an angular rescaling $$ds^{2} = -\left(dt + 4GJ^{t} d\varphi\right)^{2} + d\rho^{2} + \alpha^{2}\rho^{2} d\varphi^{2} + \left(dz + 4GJ^{z} d\varphi\right)^{2}.$$3. In tunneling calculations, the ADM mass, energy, angular momentum, and charge are reduced by defect factors, but the Hawking temperature remains unchanged (Jusufi, 2015).

In a different regime, cosmic strings attached to rapidly spinning black holes can extract significant amounts of rotational energy and angular momentum. For primordial black holes, it was argued that if there are cosmic strings with tension greater than $$ds^{2} = -\left(dt + 4GJ^{t} d\varphi\right)^{2} + d\rho^{2} + \alpha^{2}\rho^{2} d\varphi^{2} + \left(dz + 4GJ^{z} d\varphi\right)^{2}.$$4, the spins of large primordial black holes of mass greater than $$ds^{2} = -\left(dt + 4GJ^{t} d\varphi\right)^{2} + d\rho^{2} + \alpha^{2}\rho^{2} d\varphi^{2} + \left(dz + 4GJ^{z} d\varphi\right)^{2}.$$5 should consequently be observed to be near zero (Ahmed et al., 2024). A complementary analysis of captured cosmic string loops around spinning black holes showed competing effects of loop growth by the superradiant extraction of the black-hole spin energy and loop decay by friction against the horizon, with possible asymptotic states that are strong emitters of gravitational waves (Xing et al., 2020).

The topic also has an extensive analogue literature. Dirac-material realizations were proposed in which disclinations, dislocations, and local rotations mimic $$ds^{2} = -\left(dt + 4GJ^{t} d\varphi\right)^{2} + d\rho^{2} + \alpha^{2}\rho^{2} d\varphi^{2} + \left(dz + 4GJ^{z} d\varphi\right)^{2}.$$6, $$ds^{2} = -\left(dt + 4GJ^{t} d\varphi\right)^{2} + d\rho^{2} + \alpha^{2}\rho^{2} d\varphi^{2} + \left(dz + 4GJ^{z} d\varphi\right)^{2}.$$7, and $$ds^{2} = -\left(dt + 4GJ^{t} d\varphi\right)^{2} + d\rho^{2} + \alpha^{2}\rho^{2} d\varphi^{2} + \left(dz + 4GJ^{z} d\varphi\right)^{2}.$$8, respectively, so that topology-renormalised scattering translates into modified electronic transport in strained or defective graphene (Boumali, 20 Nov 2025). Other work notes analogues in vortices in superfluids, where quantized circulation and topological defects mimic cosmic strings (Muniz et al., 2014). A plausible implication is that spinning cosmic strings occupy a dual role: they remain a gravitational and cosmological construct, but they also provide an organizing geometry for quantum, optical, and condensed-matter systems in which curvature, torsion, frame dragging, and topology are implemented effectively rather than literally.