# C-metric The C-metric is a solution to the vacuum Einstein's equation and describes a pair of black holes uniformly accelerating away from each other. The acceleration is provided by conical singularities extending out from each black hole (along the axis of symmetry). The conical deficits can be thought of as cosmic strings, or more precisely, cosmic strings can be used to source these defects. Then the solution physically represents two black holes being pulled away from each other by two infinitely long cosmic strings. By a certain choice of coordinates, the metric (in 3+1 dimensions) reads$d s^2=A^{-2}(x+y)^{-2}\left(F d t^2-F^{-1} d y^2-G^{-1} d x^2-G d z^2\right),$with$F=-1+y^2-2 m A y^3, \quad G=1-x^2-2 m A x^3,$where $A$ labels of the acceleration and $m$ labels the mass of the black hole. It's difficult to understand the geometry by reading this metric, so I recommend reading one of the references for details. It has been generalised to include the electric charge, rotation, and the NUT charge, as illustrated in the following diagram taken from [[2005#Griffiths, Podolsky]]: ![[GriffithsPodolsky2005_fig1.png]] Curiously, it was recently argued that conical singularities can be removed by uplifting in higher dimensions. <!-- (quoting Gauntlett)--> ## Refs - original: [[1970#Kinnersley, Walker]] - with rotation and charge: [[1976#Plebanski, Demianski]] - in AdS: [[2002#Dias, Lemos]] - with NUT charges: [[2005#Griffiths, Podolsky]] (a good review) - thermodynamics: [[2016#Appels, Gregory, Kubiznak]] ## Properties and applications - relation to [[0204 Quantum complexity|complexity]] - [[Nagasaki2022]][](https://arxiv.org/pdf/2205.00196.pdf) - [[0325 Quasi-normal modes|QNM]] from SCFT - [[2023#Lei, Shu, Zhang, Zhu]] - 3d C-metric and [[0181 AdS-BCFT|AdS/BCFT]] - [[2024#Tian, Lai]]