Taub–NUT space is a solution of Einstein's field equations in general relativity that generalizes the Taub solution by incorporating a parameter known as the NUT (Newman–Unti–Tamburino) charge. The metric describes a four‑dimensional, stationary, and asymptotically locally flat spacetime that exhibits several unusual geometric and physical properties, such as closed timelike curves and a gravitomagnetic monopole moment.
Historical background
- The original Taub solution was discovered by Abraham Taub in 1951 as a homogeneous, anisotropic cosmological model.
- In 1963, E. Newman, L. Tamburino, and T. Unti introduced the NUT parameter in a separate context, leading to the combined Taub–NUT family of metrics.
- The combined solution was subsequently studied by several researchers, notably Misner (1963) and Hawking (1972), who highlighted its significance for understanding gravitational analogues of magnetic monopoles and for exploring the role of topology in general relativity.
Metric form
In Boyer–Lindquist–type coordinates $(t, r, \theta, \phi)$ the Lorentzian Taub–NUT metric can be written as
$$ ds^{2}= -f(r),[dt+2\ell \cos\theta , d\phi]^{2}+ \frac{dr^{2}}{f(r)}+(r^{2}+\ell^{2})(d\theta^{2}+\sin^{2}!\theta, d\phi^{2}), $$
where
$$ f(r)=\frac{r^{2}-2mr-\ell^{2}}{r^{2}+\ell^{2}}. $$
Here $m$ is the mass parameter (analogous to the Schwarzschild mass) and $\ell$ is the NUT charge, which has dimensions of length and introduces a gravitomagnetic component to the spacetime.
Key properties
| Property | Description |
|---|---|
| Stationarity | The spacetime possesses a timelike Killing vector field outside the horizons, reflecting time‑translation symmetry. |
| Axial symmetry | An additional rotational Killing vector exists, associated with the $\phi$ coordinate. |
| Closed timelike curves (CTCs) | For any non‑zero NUT charge, CTCs appear in regions where the coefficient of $d\phi^{2}$ becomes negative, typically near the axis $\theta =0,\pi$. |
| Gravitomagnetic monopole | The NUT parameter $\ell$ behaves analogously to a magnetic monopole in electromagnetism, yielding a non‑trivial off‑diagonal term $g_{t\phi}$. |
| Misner string | The term $2\ell \cos\theta , d\phi$ introduces a singularity along the polar axis, interpreted as a line singularity (the “Misner string”). It can be removed by imposing a periodic identification of the time coordinate, at the expense of introducing CTCs throughout the spacetime. |
| Topology | The spatial sections have the topology of $\mathbb{R}^{3}$ punctured by the Misner string; globally the manifold can be viewed as a non‑trivial $S^{1}$ bundle over $S^{2}$. |
| Geodesic motion | Test particle trajectories exhibit precession effects reminiscent of magnetic monopole dynamics; the NUT charge contributes an additional “magnetic‑type” force term. |
Physical and theoretical relevance
- Gravitational analogues of monopoles – The NUT charge provides a model for gravitomagnetic monopoles, useful in exploring dualities between electric‑type (mass) and magnetic‑type (NUT) sources in gravity.
- Euclidean quantum gravity – By analytically continuing $t$ to imaginary values, the Euclidean Taub–NUT metric yields self‑dual solutions that serve as instantons in quantum gravity and string theory.
- Topological considerations – The necessity of periodic time identifications to eliminate the Misner string ties the solution to discussions of causality, global identifications, and compactified dimensions.
- Extensions – Generalizations include Taub–NUT–AdS (with a negative cosmological constant), higher‑dimensional NUT‑charged metrics, and supersymmetric versions appearing in supergravity and string theory compactifications.
Limitations and interpretative issues
The presence of closed timelike curves and the requirement of a periodic time coordinate raise questions about the physical realism of the spacetime. Consequently, many authors treat Taub–NUT space primarily as a mathematical laboratory for studying the interplay of geometry, topology, and gravitation rather than as a model of an astrophysical object.
References (selected)
- A. Taub, “Empty Space-Times Admitting a Three‑Parameter Group of Motions,” Annals of Mathematics 53, 472 (1951).
- E. Newman, L. Tamburino, T. Unti, “Empty‑Space Generalization of the Schwarzschild Metric,” Journal of Mathematical Physics 4, 915 (1963).
- C. W. Misner, “The Flatter Regions of Taub‑NUT Space,” Journal of Mathematical Physics 4, 924 (1963).
- S. W. Hawking, “Gravitational Instantons,” Physics Letters A 60, 81 (1977).
- D. N. Page, “Taub‑NUT Instantons,” Phys. Lett. B 78, 249 (1978).
Note: The above summary reflects the consensus in the physics and mathematical literature up to the present date.
