Euclidean Kerr Metric and Conformal Kähler Structure

• Euclidean Kerr metric is the Riemannian analytic continuation of Kerr black-hole geometry, featuring Wick rotation and a positive-definite metric.
• It exhibits a global conformally Kähler structure through an exact Lee form and self-dual Weyl tensor, situating it within the ambitoric, ambi–Kähler class.
• The derivation integrates complex geometry, T² isometry, and regularity conditions including bolt analysis to ensure analytical and geometrical consistency.

Searching arXiv for the cited papers to ground the article in current metadata. The Euclidean Kerr metric is the Riemannian, positive-definite analytic continuation of the Kerr black-hole geometry. In Boyer–Lindquist–type coordinates it is obtained by Wick rotation, admits a $T^2$ isometry generated by Euclidean time translation and axial rotation, and occupies a distinguished position in four-dimensional complex geometry because it is not merely almost Hermitian: it is globally conformally Kähler, with an explicit conformal factor determined either from the Lee form or from the self-dual Weyl tensor (Kelekçi, 2022). A complementary derivation places it inside the ambitoric, ambi–Kähler class characterized by two commuting complex structures and two commuting Killing fields, from which the Euclidean Kerr line element emerges as a special two-parameter vacuum subfamily (Krasnov et al., 2024).

1. Definition and explicit line element

After Wick rotations $t \to i\,t$, $a \to i\,a$, the Euclidean Kerr metric can be written as

$$g_E = \frac{\Sigma}{\Delta}\,dr^2 +\Sigma\,d\theta^2 +\frac{\sin^2\theta}{\Sigma}\,\bigl((r^2-a^2)\,d\phi-a\,dt\bigr)^2 +\frac{\Delta}{\Sigma}\,\bigl(dt+a\sin^2\theta\,d\phi\bigr)^2,$$

with

$$\Delta(r)=r^2-2Mr-a^2, \qquad \Sigma(r,\theta)=r^2-a^2\cos^2\theta,$$

where $M>0$ is the mass parameter and $a$ is the rotation parameter (Kelekçi, 2022).

An equivalent presentation uses $\rho^2=r^2-a^2\cos^2\theta$ and writes

$$ds^2 = \frac{\Delta(r)}{\rho^2}\,\bigl(dt-a\sin^2\theta\,d\phi\bigr)^2 +\frac{\rho^2}{\Delta(r)}\,dr^2 +\rho^2\,d\theta^2 +\frac{\sin^2\theta}{\rho^2}\,\bigl((r^2-a^2)\,d\phi+a\,dt\bigr)^2,$$

with the same structure function $\Delta(r)=r^2-2Mr-a^2$ (Krasnov et al., 2024).

In component form,

$t \to i\,t$0

$t \to i\,t$1

with all other $t \to i\,t$2 (Krasnov et al., 2024).

The same source describes the coordinate ranges as

$t \to i\,t$3

with angular coordinates

$t \to i\,t$4

capturing the $t \to i\,t$5 isometry (Krasnov et al., 2024).

2. Euclidean continuation and geometric setting

The Euclidean metric is obtained from the Lorentzian Kerr geometry in Boyer–Lindquist form by the substitutions

$t \to i\,t$6

which transform

$t \to i\,t$7

and

$t \to i\,t$8

yielding a real, positive-definite metric (Krasnov et al., 2024).

Krasnov and Shaw formulate the Euclidean Kerr geometry within a broader four-dimensional vacuum type-D framework. Their starting point is that, after analytic continuation to Euclidean signature, each half of the Weyl tensor is type D, and by a theorem of Derdziński each half defines an integrable almost-complex structure of opposite orientation. Two opposite-orientation complex structures in four dimensions commute. Together with stationarity and axisymmetry, this places Euclidean Kerr in the class of ambitoric, ambi–Kähler geometries: Riemannian four-manifolds with two commuting integrable complex structures $t \to i\,t$9 of opposite orientation and a torus $a \to i\,a$0 of isometries that is holomorphic for both $a \to i\,a$1 and $a \to i\,a$2 (Krasnov et al., 2024).

In their adapted ansatz, one introduces commuting vector fields

$a \to i\,a$3

with dual one-forms $a \to i\,a$4, chooses coordinates

$a \to i\,a$5

and then writes

$a \to i\,a$6

Orthogonality under the product structure $a \to i\,a$7 forces a diagonal metric form, while the ambi–Kähler conditions constrain the metric functions to separable structures. Imposing the vacuum Einstein equation $a \to i\,a$8 makes the remaining functions polynomial, and the Euclidean Kerr metric is recovered by restricting to zero NUT charge, zero magnetic mass, zero cosmological constant, and asymptotic flatness, with

$a \to i\,a$9

(Krasnov et al., 2024).

This construction is significant because it shows that the Euclidean Kerr line element is not an isolated coordinate ansatz but a rigid consequence of compatible complex-geometric and symmetry data.

3. Orthonormal coframe and almost-Hermitian structure

A convenient orthonormal vierbein is

$$g_E = \frac{\Sigma}{\Delta}\,dr^2 +\Sigma\,d\theta^2 +\frac{\sin^2\theta}{\Sigma}\,\bigl((r^2-a^2)\,d\phi-a\,dt\bigr)^2 +\frac{\Delta}{\Sigma}\,\bigl(dt+a\sin^2\theta\,d\phi\bigr)^2,$$0

$$g_E = \frac{\Sigma}{\Delta}\,dr^2 +\Sigma\,d\theta^2 +\frac{\sin^2\theta}{\Sigma}\,\bigl((r^2-a^2)\,d\phi-a\,dt\bigr)^2 +\frac{\Delta}{\Sigma}\,\bigl(dt+a\sin^2\theta\,d\phi\bigr)^2,$$1

for which

$$g_E = \frac{\Sigma}{\Delta}\,dr^2 +\Sigma\,d\theta^2 +\frac{\sin^2\theta}{\Sigma}\,\bigl((r^2-a^2)\,d\phi-a\,dt\bigr)^2 +\frac{\Delta}{\Sigma}\,\bigl(dt+a\sin^2\theta\,d\phi\bigr)^2,$$2

(Kelekçi, 2022).

With respect to this coframe, the almost-Hermitian structure is defined by

$$g_E = \frac{\Sigma}{\Delta}\,dr^2 +\Sigma\,d\theta^2 +\frac{\sin^2\theta}{\Sigma}\,\bigl((r^2-a^2)\,d\phi-a\,dt\bigr)^2 +\frac{\Delta}{\Sigma}\,\bigl(dt+a\sin^2\theta\,d\phi\bigr)^2,$$3

together with

$$g_E = \frac{\Sigma}{\Delta}\,dr^2 +\Sigma\,d\theta^2 +\frac{\sin^2\theta}{\Sigma}\,\bigl((r^2-a^2)\,d\phi-a\,dt\bigr)^2 +\frac{\Delta}{\Sigma}\,\bigl(dt+a\sin^2\theta\,d\phi\bigr)^2,$$4

The associated differential identity is

$$g_E = \frac{\Sigma}{\Delta}\,dr^2 +\Sigma\,d\theta^2 +\frac{\sin^2\theta}{\Sigma}\,\bigl((r^2-a^2)\,d\phi-a\,dt\bigr)^2 +\frac{\Delta}{\Sigma}\,\bigl(dt+a\sin^2\theta\,d\phi\bigr)^2,$$5

(Kelekçi, 2022).

The same geometry can be described in the self-dual/anti-self-dual language used by Krasnov and Shaw. In an orthonormal SD/ASD basis the Weyl halves take the diagonal type-D form

$$g_E = \frac{\Sigma}{\Delta}\,dr^2 +\Sigma\,d\theta^2 +\frac{\sin^2\theta}{\Sigma}\,\bigl((r^2-a^2)\,d\phi-a\,dt\bigr)^2 +\frac{\Delta}{\Sigma}\,\bigl(dt+a\sin^2\theta\,d\phi\bigr)^2,$$6

with

$$g_E = \frac{\Sigma}{\Delta}\,dr^2 +\Sigma\,d\theta^2 +\frac{\sin^2\theta}{\Sigma}\,\bigl((r^2-a^2)\,d\phi-a\,dt\bigr)^2 +\frac{\Delta}{\Sigma}\,\bigl(dt+a\sin^2\theta\,d\phi\bigr)^2,$$7

and each half is trace-free and divergence-free (Krasnov et al., 2024).

These two descriptions emphasize different aspects of the same Riemannian geometry. The coframe presentation isolates the almost-Hermitian structure directly, while the Weyl-tensor description links Euclidean Kerr to the complex geometry of type-D Einstein metrics.

4. Lee form and global conformal Kähler structure

From

$$g_E = \frac{\Sigma}{\Delta}\,dr^2 +\Sigma\,d\theta^2 +\frac{\sin^2\theta}{\Sigma}\,\bigl((r^2-a^2)\,d\phi-a\,dt\bigr)^2 +\frac{\Delta}{\Sigma}\,\bigl(dt+a\sin^2\theta\,d\phi\bigr)^2,$$8

the Lee form is

$$g_E = \frac{\Sigma}{\Delta}\,dr^2 +\Sigma\,d\theta^2 +\frac{\sin^2\theta}{\Sigma}\,\bigl((r^2-a^2)\,d\phi-a\,dt\bigr)^2 +\frac{\Delta}{\Sigma}\,\bigl(dt+a\sin^2\theta\,d\phi\bigr)^2,$$9

Since the Lee form is exact, the locally conformally Kähler criterion implies that the Euclidean Kerr metric is globally conformally Kähler (Kelekçi, 2022).

A convenient global conformal factor is

$$\Delta(r)=r^2-2Mr-a^2, \qquad \Sigma(r,\theta)=r^2-a^2\cos^2\theta,$$0

Defining

$$\Delta(r)=r^2-2Mr-a^2, \qquad \Sigma(r,\theta)=r^2-a^2\cos^2\theta,$$1

one finds

$$\Delta(r)=r^2-2Mr-a^2, \qquad \Sigma(r,\theta)=r^2-a^2\cos^2\theta,$$2

so $$\Delta(r)=r^2-2Mr-a^2, \qquad \Sigma(r,\theta)=r^2-a^2\cos^2\theta,$$3 is a genuine Kähler structure (Kelekçi, 2022).

A common point of confusion is the status of the unscaled Euclidean Kerr metric itself. The precise statement is not that $$\Delta(r)=r^2-2Mr-a^2, \qquad \Sigma(r,\theta)=r^2-a^2\cos^2\theta,$$4 is Kähler, but that $$\Delta(r)=r^2-2Mr-a^2, \qquad \Sigma(r,\theta)=r^2-a^2\cos^2\theta,$$5 is globally conformally Kähler and that the conformally related metric $$\Delta(r)=r^2-2Mr-a^2, \qquad \Sigma(r,\theta)=r^2-a^2\cos^2\theta,$$6 is Kähler. This distinction is explicit in the calculation of the Lee form and in the closure of $$\Delta(r)=r^2-2Mr-a^2, \qquad \Sigma(r,\theta)=r^2-a^2\cos^2\theta,$$7 rather than $$\Delta(r)=r^2-2Mr-a^2, \qquad \Sigma(r,\theta)=r^2-a^2\cos^2\theta,$$8 (Kelekçi, 2022).

5. Weyl-tensor characterization and Derdziński’s theorem

The conformal factor can also be derived from curvature. Because the Euclidean Kerr metric is Ricci-flat and of Petrov type D, the self-dual Weyl endomorphism

$$\Delta(r)=r^2-2Mr-a^2, \qquad \Sigma(r,\theta)=r^2-a^2\cos^2\theta,$$9

is diagonal in the self-dual basis $M>0$0: $M>0$1 Its norm satisfies

$M>0$2

(Kelekçi, 2022).

By Derdziński’s theorem, this yields the same conformal scaling, up to an overall constant, and therefore confirms that

$M>0$3

is Kähler (Kelekçi, 2022).

This curvature-based derivation is important because it makes the conformal Kähler structure intrinsic. The factor $M>0$4 is not merely a coordinate artifact arising from an adapted almost-Hermitian ansatz; it is encoded in the Weyl tensor itself. A plausible implication is that the Euclidean Kerr metric exemplifies the tight relation between type-D curvature algebra and complex geometry in four-dimensional Riemannian Einstein spaces.

6. Regularity, bolts, and related Euclidean instantons

In the Euclidean section, the bolt sits at the largest real root $M>0$5 of

$M>0$6

Near $M>0$7, Krasnov and Shaw write

$M>0$8

Absence of a conical tip forces the Euclidean time $M>0$9 to be periodic, while $a$0 has period $a$1; they further state that no misidentification arises on the axis $a$2 (Krasnov et al., 2024).

Related Euclidean black-hole instantons extend the same logic. For the Euclidean Kerr–Newman metric, the near-horizon $a$3 plane takes the form

$a$4

so regularity requires Euclidean time period

$a$5

and the angular potential enters through the identification

$a$6

(Raha, 2019).

For Euclidean Kerr–Newman–de Sitter, smooth compact non-singular instantons arise only when the bolt equations and Dirac quantisation conditions are solved by a discrete set of parameters labeled by integer pairs $a$7 and $a$8. In that setting the total Euclidean action

$a$9

is finite and bounded below, and the minimal value occurs at $\rho^2=r^2-a^2\cos^2\theta$0 (Chruściel et al., 2015).

These extensions show that the Euclidean Kerr metric belongs to a broader instanton framework in which regularity, periodicity, and global field definitions are central. Within that framework, the pure Euclidean Kerr geometry is distinguished by the explicit and complete verification that its Riemannian metric is globally conformally Kähler, with the conformal factor determined independently by the Lee form and by the self-dual Weyl tensor (Kelekçi, 2022).