Conical Defects and Acceleration Sources

In the vacuum C-metric, acceleration is paid for by global axis geometry. The azimuthal period can make either half of the symmetry axis locally regular, but for $mA\ne0$ it cannot regularize both. The remaining deficit represents a cosmic string pulling the black hole; a conical excess represents a strut pushing it. Their tensions follow directly from the circumference-to-radius ratio.

Required background. Killing Fields and Isometry Groups supplies the axial Killing field and its fixed sets.

Helpful background. The Einstein Equation relates distributional curvature to stress; The Riemann Curvature Tensor supplies the local-flatness test; Exact Solutions of Einstein’s Equations places the family; Global Structure develops its extensions; and Frames, Coframes, and Anholonomy supports regular-axis frames.

Axis conicity from the metric

Use the angular sector of the spherical-type C-metric,

$$ds^2_{\mathrm{ang}} =\frac{r^2}{\Omega^2} \left[ \frac{d\theta^2}{P(\theta)} +\frac{P(\theta)\sin^2\theta}{K^2}d\varphi^2 \right],$$

with $\varphi\sim\varphi+2\pi$ and

$$P(\theta)=1+2mA\cos\theta.$$

Near the north axis $\theta=0$, define $P_+=1+2mA$. At fixed $r$, the proper distance from the axis and the orbit circumference behave as

$$\ell=\frac{r}{\Omega_+\sqrt{P_+}}\,\theta+O(\theta^2),$$ $$\mathcal C=\frac{2\pi r\sqrt{P_+}}{\Omega_+K}\,\theta+O(\theta^2).$$

Hence

$$\lim_{\ell\to0}\frac{\mathcal C}{\ell} =2\pi\frac{P_+}{K}, \qquad \delta_+=2\pi\left(1-\frac{P_+}{K}\right).$$

The conformal factor cancels. At the south axis $\theta=\pi$, with $P_-=1-2mA$, the same calculation gives

$$\delta_-=2\pi\left(1-\frac{P_-}{K}\right).$$

These formulas are local and geometric. They do not depend on visualizing the coordinate axis as a Euclidean line.

String, strut, and tension

In units $G=1$, a thin conical defect has tension

$$\mu_\pm=\frac{\delta_\pm}{8\pi} =\frac14\left(1-\frac{P_\pm}{K}\right).$$

Positive $\mu$ is a deficit associated with a tensile string; negative $\mu$ is an excess associated with a compressive strut. If the north axis is regularized by choosing $K=P_+$, then

$$\delta_+=0, \qquad \delta_-=\frac{8\pi mA}{1+2mA}, \qquad \mu_-=\frac{mA}{1+2mA}.$$

The remaining south-axis string supplies the force. If instead $K=P_-$, the south axis is regular and

$$\mu_+=-\frac{mA}{1-2mA},$$

so the north segment has a conical excess and acts as a strut. In the weak-acceleration regime, $mA\ll1$, either magnitude begins as $\lvert\mu\rvert\simeq mA$, consistent with a Newtonian force $mA$.

Regularizing both segments would require $K=P_+=P_-$, hence $mA=0$. This is the precise obstruction: it is not a failure to choose a clever angular coordinate. A single global period cannot equal two different regularity periods.

Where the source resides

Away from the axis, the uncharged metric satisfies the vacuum Einstein equation. At a conical defect, curvature has a distributional part supported on the axis, corresponding to the idealized string or strut stress tensor. Removing the axis from the coordinate manifold hides that distribution locally but does not remove the global conicity or its physical role.

The maximal extension can contain more than one accelerating black-hole region, and the axis segments can attach to different portions of infinity or another black hole. Statements such as “a string pulls one black hole” are therefore domain-dependent summaries of a global geometry. The invariant local data are the deficit, its location on a particular fixed-set component, and the normalization of the axial Killing field.

Other mechanisms can alter this balance. An external electromagnetic field introduced by a Harrison transformation can provide acceleration while allowing both axes to be regular for a tuned field strength. A negative cosmological constant can support slowly accelerating black holes without an acceleration horizon, but a conical defect may still be retained as part of the solution. Those are different boundary-value problems, not vacuum C-metrics with the source silently removed.

The relation between factorized coordinates, axis segments, and the string interpretation is worked out in Griffiths, Krtouš, and Podolský 2006, §§ 3–5.

Common pitfalls

Setting $\varphi$ to period $2\pi$ before fixing its normalization. The physical conicity depends on both the metric coefficient and the chosen axial period. Keep $K$ explicit until the axis to be regularized has been selected.

Calling every axis singularity a curvature singularity. A regular axis is a smooth fixed set in polar-type coordinates. A conical defect has distributional curvature but finite ordinary curvature away from the axis; the black-hole central singularity is different again.

Exercises

Show that the difference between the two axis tensions is fixed by $mA/K$, independent of which axis is regularized.

Solution

Using $P_+-P_-=4mA$,

$$\mu_- -\mu_+ =\frac14\left(1-\frac{P_-}{K} -1+\frac{P_+}{K}\right) =\frac{P_+-P_-}{4K} =\frac{mA}{K}.$$

Changing $K$ changes both conicities because it changes the global axial normalization, but their difference retains the stated form. For $mA\ne0$ it cannot vanish, so both axes cannot simultaneously have zero tension.

References

• Jerry B. Griffiths, Pavel Krtouš, and Jiří Podolský, “Interpreting the C-Metric,” Classical and Quantum Gravity 23, 6745–6766 (2006). DOI.