Misner Strings in NUT-Charged Spacetimes

• Misner strings are semi-infinite line singularities in spacetimes with nonzero NUT charge, acting as gravitational analogues to Dirac strings with topological significance.
• They notably affect black hole thermodynamics by partitioning conserved charges, such as mass and angular momentum, between horizons and the strings.
• Their study employs geometric, holographic, and rod-structure formalisms to address topological flux, entropy contributions, and gauge singularities.

A Misner string is a semi-infinite line singularity—the gravitational analogue of a Dirac string—arising in spacetimes with nonzero NUT charge, most notably in Taub–NUT and its generalizations. In such solutions, the off-diagonal components of the metric, typically of the form $dt+2n\cos\theta\,d\phi$ or generalizations thereof, are not globally well-defined and exhibit string-like singularities along certain polar axes. These structures play central roles in both the geometric and thermodynamic properties of the underlying spacetime, and feature as essential carriers of topological flux, conserved charges, and gauge singularities. In black hole thermodynamics and AdS/CFT contexts, the physicality and contributions of Misner strings are crucial to the definition of conserved charges, the first law, and entropy.

1. Geometric Structure and Occurrence

Misner strings appear in exact solutions of Einstein’s equations possessing a NUT parameter $n$ (or $\ell$), which encodes a topological twist in the $t$–$\phi$ fibration. The canonical example is the Taub–NUT metric: $$ds^2 = -f(r)\left(dt + 2n\cos\theta\,d\phi\right)^2 + f(r)^{-1}dr^2 + (r^2 + n^2)\left(d\theta^2 + \sin^2\theta\,d\phi^2\right)$$ with $f(r) = \frac{r^2 - 2mr - n^2}{r^2+n^2}$, and the one-form $dt + 2n\cos\theta\,d\phi$ developing coordinate singularities at $\theta=0,\pi$ (Ciambelli et al., 2020, Gera et al., 2019, Gal'tsov et al., 1 Feb 2026). These same structures are present in numerous relatives: Taub–NUT–AdS, Taub–Bolt–AdS, charged and rotating NUTty black holes, as well as in extended theories such as Einstein–Maxwell–Dilaton–Axion (EMDA).

The string’s location and “strength” can be parameterized by constants that redistribute the singularities between the axes, e.g., a shift $s$ in $dt + 2(n\cos\theta+s)d\phi$ for asymmetric configurations. The essential property is that, unless $t$ is artificially identified periodically (the Misner prescription), the coordinate singularities are not removable and correspond to genuine, physical line defects—Misner strings.

2. Physical Interpretation and Topological Aspects

Misner strings carry topological flux analogous to magnetic flux in the Dirac monopole, with the NUT charge acting as a “magnetic mass.” The string introduces a twist in the spacetime bundle structure—for Taub–NUT, a nontrivial $U(1)$ fibration over $S^2$. Singularities in the metric’s off-diagonal terms mirror the Dirac string's role in electrodynamics: they source delta-function curvature contributions localized on the axes (Gera et al., 2019, Gal'tsov et al., 1 Feb 2026).

In the first-order (Hilbert–Palatini) formulation, the NUT parameter is realized as a topological torsion charge, codified in the Nieh–Yan class, entirely bypassing the need for distributional sources or periodic identification. A degenerate-metric extension can eliminate both the Misner string and associated closed timelike curves while preserving the topological character of the NUT charge as a torsional flux (Gera et al., 2019).

3. Thermodynamics, Conserved Charges, and First Laws

The presence of Misner strings has profound implications for black hole thermodynamics. Unlike horizon-localized charges, total mass, NUT charge, angular momentum, and electromagnetic charges become partitioned between the horizon and the strings themselves (Awad et al., 2022, Bordo et al., 2019, Bordo et al., 2019, Gal'tsov et al., 2021).

Let $S_\infty$ be a two-sphere at infinity and $S_H$ the horizon: $Q_\infty - Q_H = \text{flux through strings}$ For mass and NUT charge,

$$M_\infty = m,\qquad M_H = m - 2n\phi_n,\qquad \phi_n\equiv -\frac{n}{2r_+}$$

The difference is carried by the strings, necessitating the explicit inclusion of string charge densities for a consistent first law. Analogous flux partitioning holds for electric and magnetic charges in dyonic solutions.

To account for these localized sources, new conjugate pairs $(\psi_\pm, N_\pm)$ are introduced, where $\psi_\pm$ are the “string potentials,” identified with the surface gravities of the Misner string horizons, and $N_\pm$ are “Misner charges” extracted via generalized Komar integrals over tubes surrounding the string defects (Bordo et al., 2019, Bordo et al., 2019). The generalized first law reads: $$\delta M = T\,\delta S + \Omega\,\delta J + \psi_+\,\delta N_+ + \psi_-\,\delta N_- + V\,\delta P + \dots$$ where $J$ and $V$ also receive explicit Misner-string contributions, and $N_\pm$ can be varied independently if the distribution of string strengths is asymmetric.

4. Entropy and Topological Renormalization

Misner strings contribute nontrivially to gravitational entropy, particularly in asymptotically locally AdS spaces. Standard Noether–Wald entropy computations yield divergent string-associated terms that are rendered finite by adding a topological term—specifically the Gauss–Bonnet invariant with coupling $\alpha = \ell^2/4$ (“topological renormalization”) to the bulk action (Ciambelli et al., 2020). The total entropy for a Taub–Bolt–AdS solution splits: $S = S_{\text{bolt}} + S_{\text{string}}$ with $S_{\text{string}}$ capturing the finite contribution of the Misner strings. Closed-form expressions for entropy, mass, and other thermodynamic potentials are in full agreement with AdS/CFT-motivated counterterm methods.

When a parity-odd Chern–Pontryagin term is added, and the self-duality condition is imposed, the Taub–NUT solution becomes the zero-entropy reference, shifting all physical charges and entropy to differences between bolt and nut solutions (Ciambelli et al., 2020).

5. Distributional Charges and Electromagnetic Hair

In charged (Einstein–Maxwell–NUT) spacetimes, Misner strings support delta-like electromagnetic field components and encode effective electric and magnetic line-charge densities: $$\rho_e(r) = \frac{n[p(r^2 - n^2) + 2rnq]}{(r^2 + n^2)^2}, \quad \rho_m(r) = -\frac{n[q(r^2 - n^2) - 2rnp]}{(r^2 + n^2)^2}$$ These string-localized charges create a complex “hair zone” around the black hole, with field lines connecting string segments, the horizon, or even forming closed loops (when rotation is present) (Gal'tsov et al., 1 Feb 2026). The zone extends only over a few units of the NUT parameter, and mass/angular momentum integrals acquire further string contributions from these distributional sources. Such structures are not smeared out by coordinate choices and have physical consequences for both local and asymptotic observables.

6. Rod-Structure Formalism and Generalizations

In stationary, axisymmetric solutions (including Kerr–NUT, Kerr–Sen–NUT, and multiple-hole systems), Misner strings manifest naturally as spacelike rods along the symmetry axis in the Weyl–Papapetrou formalism. The Tomimatsu “rod decomposition” demonstrates that the entire mass, angular momentum, and charge content of a spacetime can be assigned to individual rods—horizons, Misner strings, cosmic strings, or struts: $$M_\infty = \sum_n M_n, \quad J_\infty = \sum_n J_n, \quad Q_\infty = \sum_n Q_n$$ Each rod is characterized by a length $L_n$, surface gravity $\kappa_n$, and potentials, with explicit Smarr-type relations: $$M_n = \textstyle\frac{1}{2}L_n + 2\Omega_n J_n + \Phi_n Q_n$$ For half-infinite Misner rods, the thermodynamic length is typically divergent and must be regularized, but their tension and contributions to the first law are essential for the full accounting of physical quantities (Gal'tsov et al., 2021).

7. Removal of Misner Strings and Alternative Formulations

Standard removals of Misner strings by periodic identification of $t$ (Misner’s prescription) introduce closed timelike curves and relate $n$ to coordinate periodicity. In contrast, first-order (tetrad–connection) approaches admit smooth extensions where the Taub–NUT phase is glued to a degenerate-metric region with nontrivial torsion, eradicating both string singularities and closed timelike curves—without periodic identification or causality violation. The NUT charge remains as a topological invariant (Nieh–Yan index) of the degenerate region, providing a purely geometric interpretation (Gera et al., 2019).

8. Holography, Parity Structure, and Boundary Stress Tensors

In AdS/CFT and fluid/gravity duality, Misner strings encode nontrivial topological features in the bulk that map to even- or odd-parity structures in the dual boundary theory. Inclusion of the Chern–Pontryagin invariant in the bulk action introduces a parity-odd Cotton tensor component in the holographic stress tensor: $$T^{\pm}_{ab} = T_{ab} \mp \frac{\ell^2}{8\pi G} C_{ab}$$ On self-dual (anti-self-dual) backgrounds, the total holographic stress tensor vanishes—consistent with the redefined zero-energy ground state structure of the theory (Ciambelli et al., 2020). The interplay between bulk topological invariants and boundary stress tensors links the physics of Misner strings to broader questions in quantum gravity and holography.

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Misner strings are topological and geometric structures essential to the full characterization of NUT-charged spacetimes in general relativity and related theories. Their physicality, thermodynamic roles, and holographic signatures underlie much of the modern understanding of black hole mechanics, conserved charges, and topological effects in gravitational systems (Ciambelli et al., 2020, Bordo et al., 2019, Gal'tsov et al., 1 Feb 2026, Gal'tsov et al., 2021, Awad et al., 2022, Gera et al., 2019, Bordo et al., 2019).