Charged and Rotating C-Metrics

Charge splits the black-hole root and changes both axis conicities; rotation replaces the static horizon generator by a combination of the stationary and axial Killing fields. A convenient rotating C-metric makes these changes explicit and removes the spurious axis rotation present in less carefully normalized forms. The admissible spacetime still depends on root ordering, $P(\theta)>0$, conical identification, and the absence of closed timelike curves.

Required background. Killing Fields and Isometry Groups supplies stationary, axial, and horizon generators.

Helpful background. The Einstein Equation supplies Einstein–Maxwell equations; The Riemann Curvature Tensor supplies invariant checks; Exact Solutions of Einstein’s Equations places the family; Global Structure develops horizon extensions; and Frames, Coframes, and Anholonomy supports electrovac tetrads.

The charged nonrotating family

Fix the geometrized Einstein–Maxwell normalization

$$S=\frac{1}{16\pi G}\int d^4x\sqrt{-g}\, \left(R-F_{ab}F^{ab}\right).$$

Thus $\nabla_aF^{ab}=0$ and $G_{ab}=2(F_{ac}F_b{}^c-\tfrac14g_{ab}F^2)$. Let $q^2=e^2+g^2$, where $e$ and $g$ are length-valued electric and magnetic metric parameters. Their relation to physical charges is fixed only after the flux normalization and the azimuthal period have been specified. The spherical-type line element retains

$$ds^2=\frac1{\Omega^2} \left[-Qdt^2+\frac{dr^2}{Q} +r^2\left(\frac{d\theta^2}{P} +\frac{P\sin^2\theta}{K^2}d\varphi^2\right)\right],$$

but now

$$\begin{aligned} \Omega&=1+Ar\cos\theta,\\ P&=1+2mA\cos\theta+A^2q^2\cos^2\theta,\\ Q&=(1-A^2r^2) \left(1-\frac{2m}{r}+\frac{q^2}{r^2}\right). \end{aligned}$$

A local Maxwell potential can be chosen as

$$\mathcal A=-\frac{e}{r}dt+\frac{g\cos\theta}{K}d\varphi.$$

The magnetic term requires different gauge patches if the axis is included. Its field strength is regular away from the central and conical singularities. Because the angular range is normalized by $K$, globally measured charges can carry factors of $K^{-1}$; $e$ and $g$ should be called metric parameters until the flux integral over a specified horizon or boundary is performed.

For $m^2\ge q^2$, the Reissner–Nordström factors have roots

$$r_\pm=m\pm\sqrt{m^2-q^2}.$$

A standard static exterior requires $r_+<A^{-1}$, $Q>0$, $P>0$, and $\Omega>0$ throughout its chosen domain. The two axis values are

$$P_\pm=1\pm2mA+A^2q^2,$$

so charge shifts both conicities equally but does not remove their difference $P_+-P_-=4mA$. A single $K$ still cannot regularize both axes when $mA\ne0$.

Adding rotation

The accelerating Kerr–Newman form can be written

$$\begin{aligned} ds^2=\frac1{\Omega^2}\Bigg[{} &-\frac{\Delta_r}{\Sigma} \left(dt-\frac{a\sin^2\theta}{K}d\varphi\right)^2 +\frac{\Sigma}{\Delta_r}dr^2+\frac{\Sigma}{\Delta_\theta}d\theta^2\\ &+\frac{\Delta_\theta\sin^2\theta}{\Sigma} \left(a\,dt-\frac{r^2+a^2}{K}d\varphi\right)^2 \Bigg], \end{aligned}$$

with

$$\begin{aligned} \Sigma&=r^2+a^2\cos^2\theta, &\Omega&=1+Ar\cos\theta,\\ \Delta_r&=(1-A^2r^2)(r^2-2mr+a^2+q^2), &\Delta_\theta&=1+2mA\cos\theta +A^2(a^2+q^2)\cos^2\theta. \end{aligned}$$

A compatible dyonic potential is

$$\mathcal A=-\frac{er}{\Sigma} \left(dt-\frac{a\sin^2\theta}{K}d\varphi\right) -\frac{g\cos\theta}{\Sigma} \left(a\,dt-\frac{r^2+a^2}{K}d\varphi\right).$$

Substitution verifies the Einstein–Maxwell equations with zero cosmological constant. The limits $A\to0$, $a\to0$, and $q\to0$ give Kerr–Newman, the charged C-metric, and the rotating vacuum C-metric respectively. These three independent limits are more discriminating than any one of them alone.

The black-hole radii are now

$$r_\pm=m\pm\sqrt{m^2-a^2-q^2},$$

so subextremality requires $m^2>a^2+q^2$. Acceleration horizons remain at $r=\pm A^{-1}$. A usual exterior needs the ordered positive roots $r_+<A^{-1}$ and positive $\Delta_\theta$ over $0\le\theta\le\pi$.

Horizon rotation and torsion checks

At a root $r=r_h$ of $\Delta_r$, the horizon generator is

$$\chi=\partial_t+\Omega_h\partial_\varphi, \qquad \Omega_h=\frac{aK}{r_h^2+a^2},$$

for the displayed coordinate normalization. This follows because the one-form $a\,dt-(r_h^2+a^2)d\varphi/K$ annihilates $\chi$. Physical angular velocity may require subtraction of a rotating boundary frame, especially with $\Lambda\ne0$.

On an axis, the degenerating Killing orbit must be generated by the same periodically identified axial field. In the displayed form $g_{t\varphi}$ vanishes with $\sin^2\theta$ on both axes, so there is no relative time shift built into the two axial identifications. In a more general stationary polynomial form, different axis angular velocities indicate a torsion singularity; imposing periodicity can then generate closed timelike curves. One must remove that relative rotation before interpreting the solution as an ordinary accelerating black hole. Hong and Teo’s factorized rotating form was constructed to meet this condition Hong and Teo 2005, §§ 2–4.

Finally, check $g_{\varphi\varphi}>0$ off the axes in the proposed domain. Correct horizon roots do not by themselves exclude an azimuthal closed-timelike-curve region.

The electromagnetic sector provides an additional invariant check. In the limit $A\to0$, the potential must reduce to the standard dyonic Kerr–Newman gauge field, while $a\to0$ gives a radial Coulomb electric field plus a monopole magnetic flux. Integrating $F=d\mathcal A$ and its dual over a closed two-surface fixes the relation between $(e,g)$ and physical charges for the chosen angular period. A sign error in the magnetic term can survive the metric limits but fails this flux test; an omitted factor of $K$ changes the global charge even though the local Maxwell equations still hold.

Exercises

Derive the angular velocity $\Omega_h$ by requiring the surviving square at $\Delta_r=0$ to annihilate the horizon generator.

Solution

At $r=r_h$, the term proportional to $\Delta_r$ vanishes. The remaining Killing-sector one-form is

$$\omega=a\,dt-\frac{r_h^2+a^2}{K}d\varphi.$$

For $\chi=\partial_t+\Omega_h\partial_\varphi$ to be null along the horizon generator, require

$$\omega(\chi)=a-\frac{r_h^2+a^2}{K}\Omega_h=0.$$

Therefore $\Omega_h=aK/(r_h^2+a^2)$. When $A\to0$ and $K\to1$, this is the familiar Kerr–Newman result.

References

• Kenneth Hong and Edward Teo, “A New Form of the Rotating C-Metric,” Classical and Quantum Gravity 22, 109–118 (2005). DOI.