Magnetized Kerr–Taub–NUT Spacetime

• The magnetized Kerr–Taub–NUT solution is an exact Einstein–Maxwell spacetime generated via the Harrison–Ernst transformation applied to a Kerr–Taub–NUT seed.
• It features a dual interpretation, where the external magnetic parameter induces a genuine electromagnetic field while the NUT charge accounts for intrinsic gravitomagnetic effects.
• The analysis covers horizon structure, quasi-local thermodynamics, Kerr/CFT correspondence, and lensing effects that differentiate electromagnetic magnetization from NUT-induced deflection.

Searching arXiv for recent and foundational papers on magnetized Kerr–Taub–NUT and related constructions. The magnetized Kerr–Taub–NUT solution is an exact Einstein–Maxwell spacetime obtained by applying the Harrison–Ernst magnetization procedure to the Kerr–Taub–NUT seed geometry, thereby placing a rotating NUT spacetime in a Melvin-type magnetic universe (Siahaan, 2021). In the recent literature, however, the word “magnetized” is not always used in the same sense: in work on light propagation in Kerr–Taub–NUT geometry, “magnetism” denotes the NUT parameter’s gravitomagnetic monopole effect rather than a physical electromagnetic field (Chakrabortya et al., 2015). For that reason, the topic has two closely related but conceptually distinct layers: an exact magnetized electrovac solution with external magnetic parameter $b$, and a gravitomagnetic interpretation in which the NUT charge itself modifies null geodesics and lensing.

1. Terminological scope and conceptual distinction

In Kerr–Taub–NUT geometry, the NUT parameter is a gravitomagnetic quantity. The lensing analysis of Chakraborty and Sen states explicitly that the paper’s “magnetism” refers to the gravitomagnetic effect encoded by the NUT parameter $n$, not to a physical electromagnetic field; no electromagnetic $4$-potential $A_\mu$ nor field tensor $F_{\mu\nu}$ is introduced, and light propagation is governed by null geodesics of the Kerr–Taub–NUT vacuum spacetime (Chakrabortya et al., 2015). In that usage, the spacetime is “magnetic” only in the sense of NUT-induced gravitomagnetism.

The exact-solution literature uses a different construction. Ghezelbash and Siahaan, and independently Siahaan in the Kerr/CFT context, construct a genuine magnetized Kerr–Taub–NUT spacetime in Einstein–Maxwell theory by Harrison magnetization of a Kerr–Taub–NUT seed, with external magnetic field controlled by a parameter $b$ and physical field strength $B = 2b$ (Ghezelbash et al., 2021). In this second usage, the spacetime carries an induced Maxwell field and is not asymptotically flat.

A common misconception is therefore to identify the NUT parameter itself with an electromagnetic magnetic charge. The papers do not do this. In the exact magnetized solution, the seed Kerr–Taub–NUT geometry is vacuum and the electromagnetic field is generated by the Harrison transform; in the lensing paper, the “magnetism” is the NUT gravitomagnetic monopole effect rather than an electromagnetic sector (Siahaan, 2021).

2. Seed Kerr–Taub–NUT geometry

The seed spacetime used for magnetization is the Kerr–Taub–NUT metric in Boyer–Lindquist–like coordinates $(t,r,x,\phi)$, with $x=\cos\theta$, mass parameter $m$, rotation parameter $n$0, and NUT charge $n$1 ($n$2 in alternate notation). In the form used for magnetization, the metric is written as

$n$3

with

$n$4

The same Kerr–Taub–NUT structure also appears as the $n$5 specialization of the Kerr–Newman–Taub–NUT seed used in the more general magnetized Kerr–Newman–Taub–NUT construction, where

$n$6

before the neutral specialization is taken (Siahaan, 2021).

The horizon radii of the neutral seed are determined by the roots of $n$7,

$n$8

and the extremal condition is

$n$9

The lensing paper emphasizes the analogous condition $4$0 as the parameter domain in which horizons exist, regarding $4$1 as a naked-singularity regime forbidden by cosmic censorship (Chakrabortya et al., 2015).

For magnetization, the seed metric is recast into Lewis–Papapetrou–Weyl form,

$4$2

with $4$3. In the seed Kerr–Taub–NUT case,

$4$4

$4$5

$4$6

3. Harrison–Ernst magnetization and the exact solution

The exact magnetized Kerr–Taub–NUT solution is produced by the Harrison transformation acting on Ernst potentials. In the general stationary-axisymmetric setting,

$4$7

and the Harrison transformation is

$4$8

with

$4$9

For the Kerr–Taub–NUT vacuum seed, $A_\mu$0, so the transformation simplifies to

$A_\mu$1

In the small-$A_\mu$2 regime, the exact-solution analysis states that the Maxwell field is $A_\mu$3 and the metric deformation is $A_\mu$4; more explicitly,

$A_\mu$5

while

$A_\mu$6

The shift function $A_\mu$7 differs from $A_\mu$8 by $A_\mu$9 terms (Ghezelbash et al., 2021).

The magnetized metric is presented in Lewis–Papapetrou–Weyl form as

$F_{\mu\nu}$0

with $F_{\mu\nu}$1 given as a rational function of $F_{\mu\nu}$2,

$F_{\mu\nu}$3

and with the electromagnetic Ernst potential written as

$F_{\mu\nu}$4

The induced vector potential is reconstructed from $F_{\mu\nu}$5, with

$F_{\mu\nu}$6

and $F_{\mu\nu}$7 determined by

$F_{\mu\nu}$8

The paper notes that $F_{\mu\nu}$9 built from $b$0 solves the Einstein–Maxwell equations (Siahaan, 2021).

A broader exact family is obtained by starting from Kerr–Newman–Taub–NUT and then setting $b$1. In that construction, the full magnetized Kerr–Newman–Taub–NUT metric takes the form

$b$2

with

$b$3

and the neutral magnetized Kerr–Taub–NUT solution is the $b$4 specialization of this exact family (Ghezelbash et al., 2021).

4. Asymptotics, regularity, horizons, and quasi-local thermodynamics

The magnetized Kerr–Taub–NUT spacetime is not asymptotically flat. Its large-$b$5 behavior is Melvin-like. In the general magnetized Kerr–Newman–Taub–NUT analysis, the asymptotic Melvin metric is written as

$b$6

while the NUT parameter produces an asymptotic off-diagonal correction

$b$7

The Kerr/CFT treatment describes the same large-distance structure as a Melvin-type magnetic universe with external field $b$8, with $b$9 and $B = 2b$0 at leading order (Ghezelbash et al., 2021).

Global structure remains influenced by the NUT sector. The papers recall that the NUT parameter introduces Misner strings along the axis and the possibility of closed timelike curves; imposing periodicity on $B = 2b$1 can remove Misner string singularities but typically induces closed timelike curves in the bulk. The external magnetic field alters the asymptotics but does not eliminate Misner strings or closed timelike curves (Siahaan, 2021).

At the level of curvature invariants, the exact-solution analysis finds that the Ricci scalar vanishes and that $B = 2b$2 is regular everywhere, including $B = 2b$3. The Kretschmann scalar is written schematically as

$B = 2b$4

and the analysis concludes that the magnetized Kerr–Newman–Taub–NUT spacetime is regular at $B = 2b$5; the same regularity conclusion is stated to apply when $B = 2b$6, i.e. for magnetized Kerr–Taub–NUT (Ghezelbash et al., 2021). The Kerr/CFT paper likewise states that the Kretschmann scalar is regular at $B = 2b$7 when $B = 2b$8, reflecting the known absence of a Kerr ring singularity in Kerr–Taub–NUT geometry (Siahaan, 2021).

Horizons are still controlled by the seed function $B = 2b$9. For magnetized Kerr–Taub–NUT,

$(t,r,x,\phi)$0

and the external magnetic field does not move these roots. The Kerr/CFT paper gives the horizon area and entropy as

$(t,r,x,\phi)$1

together with

$(t,r,x,\phi)$2

The magnetized Kerr–Newman–Taub–NUT analysis writes the area in quasi-local form as

$(t,r,x,\phi)$3

and states that this reflects the angular rescaling required to remove conical defects in magnetized spacetimes (Siahaan, 2021).

Because the spacetime is not asymptotically flat, conserved quantities are formulated quasi-locally. In the magnetized Kerr–Newman–Taub–NUT family, the generalized Smarr relation is

$(t,r,x,\phi)$4

with $(t,r,x,\phi)$5, and the first law is

$(t,r,x,\phi)$6

The NUT-tube potentials are

$(t,r,x,\phi)$7

These formulas reduce to the $(t,r,x,\phi)$8 magnetized Kerr–Taub–NUT case and make explicit that Misner-string sectors contribute to the thermodynamics (Ghezelbash et al., 2021).

5. Extremal limit and Kerr/CFT correspondence

The extremal magnetized Kerr–Taub–NUT solution satisfies

$(t,r,x,\phi)$9

and the near-horizon limit is obtained by the coordinate scaling

$x=\cos\theta$0

with $x=\cos\theta$1 and $x=\cos\theta$2. The resulting near-horizon geometry is

$x=\cos\theta$3

with $x=\cos\theta$4, $x=\cos\theta$5, and explicit functions $x=\cos\theta$6, $x=\cos\theta$7, and $x=\cos\theta$8 given in the paper. The near-horizon Maxwell field is

$x=\cos\theta$9

with $m$0 likewise given explicitly (Siahaan, 2021).

The near-horizon isometry group is $m$1, generated by

$m$2

together with $m$3. Within the Kerr/CFT correspondence, the central charge is

$m$4

which evaluates to

$m$5

The Frolov–Thorne temperature is

$m$6

equivalently,

$m$7

Using the Cardy formula,

$m$8

the extremal entropy is

$m$9

with

$n$00

The paper emphasizes that both $n$01 and $n$02 enter the central charge and the Frolov–Thorne temperature, but their product reproduces the Bekenstein–Hawking entropy of the extremal horizon (Siahaan, 2021).

6. Null geodesics, weak-field bending, and the gravitomagnetic interpretation

In the null-geodesic analysis of Kerr–Taub–NUT spacetime, the metric is expressed in Boyer–Lindquist-type coordinates $n$03 with

$n$04

and line element

$n$05

On the equatorial plane, $n$06, one has $n$07. Using Hamilton–Jacobi separation with constants $n$08, $n$09, and $n$10, the equatorial null geodesics are obtained by setting $n$11, and the bending angle is defined by

$n$12

The weak-deflection expansion introduces

$n$13

together with

$n$14

The authors expand the equatorial deflection angle in a Taylor series in $n$15 and $n$16 up to fourth order, keeping mixed terms $n$17 with $n$18 (Chakrabortya et al., 2015).

Several limiting cases are singled out. Setting $n$19 reduces the bending angle to the Kerr weak-field equatorial result. Setting $n$20 yields the Taub–NUT limit, and the further limit $n$21 recovers the Schwarzschild weak-field expansion. Most notably, the paper gives a massless-NUT limit in which $n$22, $n$23, $n$24, and

$n$25

so the bending angle is nonzero even for a hypothetical massless body with nonzero NUT charge. The same analysis states that increasing $n$26 increases $n$27, that prograde/retrograde asymmetry persists through $n$28, and that the NUT parameter tends to increase the deflection relative to pure Kerr at the same mass and spin (Chakrabortya et al., 2015).

This suggests a useful division of physical content. In the exact magnetized Einstein–Maxwell solution, the parameter $n$29 embeds Kerr–Taub–NUT into a Melvin magnetic universe and generates an external electromagnetic field. In the lensing problem, the NUT parameter $n$30 already acts as a gravitomagnetic monopole and produces additional deflection even in the absence of an electromagnetic sector. A plausible implication is that precise lensing analyses must distinguish electromagnetic magnetization effects from NUT-induced gravitomagnetic corrections rather than treating them as interchangeable descriptions.