C-Metric Coordinate Forms and Parameters

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The factorized C-metric chart turns horizon and axis locations into polynomial roots; a spherical-type chart turns the same roots into recognizable radii and angles. The transformation between them identifies $r=2m$ as the black-hole Killing horizon, $r=A^{-1}$ as the acceleration horizon, and the conformal boundary as $1+Ar\cos\theta=0$ . It also shows why the zero-acceleration limit must keep rescaled time and radius fixed.

Required background. Killing Fields and Isometry Groups supplies the stationary and axial Killing fields.

Helpful background. The Einstein Equation supplies the vacuum equations; The Riemann Curvature Tensor supplies invariant checks; Exact Solutions of Einstein’s Equations places the family; Global Structure develops horizon and boundary language; and Frames, Coframes, and Anholonomy supports tetrad verification.

Here $m$ is a gravitational-length parameter and $A$ is an inverse length. In the asymptotically flat Schwarzschild limit with the usual time and angular normalization, $m=G M_{\mathrm{phys}}$ ; at nonzero acceleration, a physical mass requires a specified charge prescription.

Factorized roots and spherical variables

Begin with

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).

Use the simultaneous change

x=\cos\theta, \qquad y=-\frac{1}{Ar}, \qquad \tau=At, \qquad \phi=\frac{\varphi}{K}.

Then

ds^2=\frac{1}{\Omega^2} \left[-Q(r)dt^2+\frac{dr^2}{Q(r)} +r^2\left(\frac{d\theta^2}{P(\theta)} +\frac{P(\theta)\sin^2\theta}{K^2}d\varphi^2\right)\right],

where

\Omega=1+Ar\cos\theta, \qquad P=1+2mA\cos\theta, \qquad Q=\left(1-A^2r^2\right)\left(1-\frac{2m}{r}\right).

The map sends

y=-\frac{1}{2mA}\longleftrightarrow r=2m, \qquad y=-1\longleftrightarrow r=\frac1A, \qquad x=y\longleftrightarrow\Omega=0.

Thus the two relevant $y$ -roots become the black-hole and acceleration horizons in the usual static domain $2m<r<A^{-1}$ . The equality $x=y$ is not another root of $Q$ ; it is the conformal boundary. The transformation makes that distinction geometrically visible.

Parameter normalization

The dimensions are

[m]=L, \qquad [A]=L^{-1}, \qquad [mA]=1.

In the $A\to0$ limit with $(t,r,\theta,\varphi)$ fixed,

\Omega\to1, \qquad P\to1, \qquad Q\to1-\frac{2m}{r},

so $m$ is normalized as the Schwarzschild gravitational-length parameter. Holding the original $y$ and $\tau$ fixed would not reach this limit: finite $r$ requires $y\sim-A^{-1}$ and finite physical time requires $\tau\sim A$ . This is why a direct substitution $A=0$ in the polynomial chart is singular even though the spacetime limit is smooth.

The constant $K$ fixes the normalization of the axial Killing field when $\varphi$ has period $2\pi$ . It is not removable after a global angular identification has been chosen. Different $K$ values describe different conical deficits, even though they are locally related by rescaling $\varphi$ .

In asymptotically flat Schwarzschild, $m/G=M_{\mathrm{phys}}$ is the conserved mass. In a C-metric with a string, acceleration horizon, or cosmological boundary, the physical charge associated with $\partial_t$ depends on how that Killing field and the boundary are normalized. It is safest to call $m$ a metric parameter until a specified charge prescription has been applied. The same caution applies to $A$ : it sets the acceleration length and becomes the proper acceleration of selected reference curves only in controlled regimes.

Other canonical forms

Older C-metric coordinates often use cubic or quartic structure functions with coefficients rather than explicit roots. Affine transformations of $x$ and $y$ , accompanied by rescalings of the Killing coordinates, can move three roots to convenient values. The transformation changes coefficient labels, so two symbols called $m$ need not agree until the Schwarzschild limit and axial normalization are compared. The factorized nonrotating form makes these roots explicit and simplifies the Weyl-coordinate transformation Hong and Teo 2003, §§ 2–3. Removing torsion singularities on the axes is a separate issue in the rotating extension.

Weyl coordinates are adapted instead to the two commuting Killing fields. The canonical cylindrical radius is determined invariantly, up to normalization, by the Killing block:

\rho^2=-\det(g_{AB}), \qquad A,B\in\{t,\varphi\}.

Axis and horizon intervals then appear as rods on $\rho=0$ . This representation is powerful for global interval structure and solution generation, whereas the $(x,y)$ chart is usually clearer for root domains and the $(r,\theta)$ chart for limits. A coordinate form should be chosen for the question, not mistaken for a new spacetime.

Checks and limitations

Verify a proposed transformation at the level of the entire line element, including the time and azimuthal rescalings. Matching only the zeros of $F$ and $Q$ misses normalization and conicity. Next test the conformal factor: a region crossing $\Omega=0$ is not covered by a single physical chart with the same conformal orientation. Finally compute an invariant or substitute into Einstein’s equation; root matching alone cannot detect an erroneous overall sign.

The spherical form above covers the nonrotating uncharged vacuum family in a static region. Charge changes $Q$ and $P$ ; rotation introduces a $dt\,d\varphi$ term and makes horizon generators linear combinations of the Killing fields. Those extensions require their own parameter maps.

Exercises

Starting from $y=-1/(Ar)$ , transform the factor $F(y)$ and show that it produces the radial function $Q(r)$ .

Solution

Substitution gives

\begin{aligned} F(y) &=-\left(1-\frac{1}{A^2r^2}\right) \left(1-\frac{2m}{r}\right)\\ &=\frac{1}{A^2r^2} \left(1-A^2r^2\right) \left(1-\frac{2m}{r}\right) =\frac{Q(r)}{A^2r^2}. \end{aligned}

Because $d\tau=A\,dt$ and the conformal prefactor becomes $r^2/\Omega^2$ , the time term is $-Qdt^2/\Omega^2$ . Also $dy=dr/(Ar^2)$ , so the radial term becomes $dr^2/(\Omega^2Q)$ . Both rescalings are required.

References

Kenneth Hong and Edward Teo, “A New Form of the C-Metric,” Classical and Quantum Gravity 20, 3269–3277 (2003). DOI.