C-metric in a (nut)shell

Abstract

We present a comprehensive study of the C-metric in $2+1$ dimensions, placing it within a shell of stress energy and matching it to an exterior vacuum AdS metric. The $2+1$ C-metric is not circularly symmetric and hence neither are the constructed shells, which instead take on a cuspoidal or teardrop shape. We interpret the stress energy of the shells as a perfect fluid, calculating the energy density and pressure. For accelerating particles (Class I), we find the stress energy is concentrated on the part of shell farthest from the direction of acceleration and always respects the strong and weak energy conditions. For accelerating black holes (Class I$_{\mathrm{C}}$, II, and III), the shell stress energy may either respect or violate the energy conditions depending on the parameter of the exterior metric -- between the two regimes lies a critical value of the external parameter for which the shell stress energy vanishes, leading to new solutions of Einstein's field equations, which fall into three categories: an accelerated black hole pulled by a finite-length string with a point particle at the other end, an accelerated black hole pushed by a finite-length strut with a point particle at the other end, and an accelerated black hole pushed from one side by a finite-length strut and pulled from the other by a finite-length string, each with a point particle at the other end.