Charged $C$-metric in conformal gravity

Using a $C$-metric-type ansatz, we obtain an exact solution to conformal gravity coupled to a Maxwell electromagnetic field. The solution resembles a $C$-metric spacetime carrying an electromagnetic charge. The metric is cast in a factorized form which allows us to study the domain structure of its static coordinate regions. This metric reduces to the well-known Mannheim-Kazanas metric under an appropriate limiting procedure, and also reduces to the (anti)de Sitter $C$-metric of Einstein gravity for a particular choice of parameters.

Figure 1

An example showing the construction of a domain structure for $\epsilon =-1$, $\mu =1$, $\alpha =0.3$, $q=0$, $a=-1$, $b=0.2$, and $c=1$. On the left is the sketch (not to scale) of the functions $P$ (solid) and $Q$ (dotted) showing the ordering of the roots. On the right is the two-dimensional plot where the horizontal (respectively vertical) direction represents the $x$-($y$-) coordinate. The shaded regions represent the static Lorentzian regions of interest.

Figure 2

Possible Class I domain structures for $\mu =1$ and various values of $\alpha$ and $q$. For (a)–(d) the roots are chosen to be $a=-1$, $b=-0.2$ and $c=1$, while for (e) the roots are $a=-1$ and $b=1$. The shaded regions correspond to static regions with Lorentzian signature $(-+++)$, one of which is the region of our primary interest that is shaded in dark gray. The diagonal line represents the conformal infinity $x=y$.

Figure 3

Possible Class II domain structures for $\mu =1$ and various values of $\alpha$ and $q$. For (a)–(d) the roots are chosen to be $a=-1$ and $b=0.2$, while for (e) the roots are $a=-1$ and $b=1$. The shaded regions correspond to static regions with Lorentzian signature $(-+++)$, one of which is the region of our primary interest that is shaded in dark gray. The diagonal line represents the conformal infinity $x=y$.