Accelerating and C-Metric Families

The C-metric describes black holes accelerated by axis defects, electromagnetic fields in broader families, or cosmological boundary conditions. Its local line element is compact, but its interpretation is unusually dependent on polynomial roots, coordinate rectangles, azimuthal normalization, and limiting procedures. This chapter develops those structures before assigning names such as “black-hole horizon,” “acceleration horizon,” or “infinity.”

Helpful background. The Einstein Equation supplies the vacuum and electrovac equations; The Riemann Curvature Tensor distinguishes curvature singularities; Killing Fields and Isometry Groups supplies the stationary and axial symmetries; Exact Solutions of Einstein’s Equations places the family; Global Structure develops horizons and conformal boundaries; and Frames, Coframes, and Anholonomy supports invariant frame checks.

Enter the C-metric

A useful factorized vacuum form is

$$ds^2=\frac{1}{A^2(x-y)^2} \left[ -F(y)d\tau^2+\frac{dy^2}{F(y)} +\frac{dx^2}{G(x)}+G(x)d\phi^2 \right],$$ $$G(x)=(1-x^2)(1+2mAx), \qquad F(y)=-(1-y^2)(1+2mAy).$$

Here $A>0$ is an acceleration parameter and $m$ becomes the Schwarzschild mass parameter in a controlled $A\to0$ limit. With $G(x)>0$, either sign of nonzero $F(y)$ gives Lorentzian signature: $F>0$ makes $\tau$ timelike and selects a static region, while $F<0$ makes $y$ timelike. For $0<2mA<1$, a standard angular interval is $-1<x<1$, bounded by the two pieces of the symmetry axis. Relevant $y$-roots include $y=-1$ and $y=-1/(2mA)$, but which root is an acceleration or black-hole horizon depends on the chosen connected domain. The compactifying factor $A(x-y)$ vanishes at $x=y$, while the displayed physical-metric prefactor diverges there; this locates conformal infinity when that locus lies on the domain boundary.

The factorized form makes roots visible but does not by itself fix global identifications. The period of $\phi$ determines an axis deficit. One can usually regularize one axis segment, not both, leaving a string or strut whose tension supplies the acceleration. Kinnersley and Walker established the accelerating-black-hole interpretation and its global properties in Kinnersley and Walker 1970, pp. 1359–1370; the factorized coordinates used here follow the simplification in Hong and Teo 2003, pp. 3269–3277.

From roots to a spacetime

The construction proceeds in a strict order:

1. Factor the structure functions and order all real roots for the stated parameters.
2. Begin with a connected static rectangle in the $(x,y)$ plane on which $F>0$, $G>0$, and $x\ne y$; after crossing a regular horizon, $F<0$ cells remain Lorentzian but are not static in this chart.
3. Identify axes from zeros of $G$, Killing horizons from zeros of $F$, and conformal infinity from $x=y$.
4. Fix the period or normalization of $\phi$, compute both axis conicities, and state the remaining defect.
5. Inspect curvature and Maxwell invariants at every endpoint before deciding whether it is a horizon, axis, infinity, or singularity.
6. Only then translate to spherical-type, Weyl, or boost-rotation coordinates and take physical limits.

This order prevents a common failure: a polynomial root can change role or leave the coordinate domain when parameters cross a discriminant. A label inherited from a neighboring regime is not evidence that the same hypersurface still belongs to the spacetime.

Six questions developed here

C-Metric Coordinate Forms and Parameters relates the factorized chart to spherical-type coordinates and explains which parameter statements survive coordinate changes. Coordinate Domains and Horizon Structure turns root ordering into signature, axis, horizon, and boundary data. Conical Defects and Acceleration Sources derives the deficit angles and tensions rather than inserting a string interpretation by analogy.

Charged and Rotating C-Metrics tracks electromagnetic roots, angular velocity, and the extra regularity problem associated with rotation. C-Metrics with Cosmological Constant compares the acceleration scale $A^{-1}$ with the curvature scale $\lvert\Lambda\rvert^{-1/2}$ and distinguishes slow and fast acceleration. Extensions and Controlled Limits of the C-Metric shows why the Schwarzschild and flat limits require simultaneous coordinate rescalings.

These pages classify exact geometries and their domains. Thermodynamic first laws, radiation extraction, pair-creation rates, and nonlinear evolution are separate problems even when they use the same metrics.

Checks that travel across the family

Three checks are especially discriminating. In the zero-acceleration limit, a spherical chart must recover Schwarzschild, Reissner–Nordström, or Kerr–Newman with the advertised normalization. In the zero-mass limit, any remaining curvature must be explained by charge, rotation, or $\Lambda$; the vacuum uncharged result is flat spacetime in accelerated coordinates, possibly with a conical identification. On either axis, the ratio of circumference to proper radius must reproduce the claimed deficit and string tension.

Curvature invariants add a fourth check. In the vacuum type-D family the Weyl curvature scales with the mass parameter and a power of the conformal factor; it diverges at the black-hole singularity but remains finite at a regular Killing horizon. A coordinate pole in $F^{-1}$ is therefore insufficient to diagnose a singularity.

Review the chapter

For the factorized vacuum functions, show that $G(x)>0$ on $-1<x<1$ when $0<2mA<1$.

Solution

Inside the interval, $1-x^2>0$. Moreover $x>-1$, so

$$1+2mAx>1-2mA>0.$$

Both factors are positive, hence $G(x)>0$. The endpoints $x=\pm1$ are the two axis segments. This argument also shows why the parameter restriction matters: if $2mA$ crosses unity, the third root $x=-1/(2mA)$ reaches the angular interval and changes its admissible domains.

References

• Kenneth Hong and Edward Teo, “A New Form of the C-Metric,” Classical and Quantum Gravity 20, 3269–3277 (2003). DOI.
• William Kinnersley and Martin Walker, “Uniformly Accelerating Charged Mass in General Relativity,” Physical Review D 2, 1359–1370 (1970). DOI.