By Dias, O J C
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Extra info for Black Hole Solutions and Pair Creation of Black Holes in Three, Four and Higher Dimensional Spacetimes
So, a straight string produces no gravitational potential in its vicinity, and suffers a tension along the z-axis that points inward. , it can be obtained from the Minkoswki space by suppressing a wedge (with a deficit angle proportional to the line mass density, δ = 8πGρ) and identifying its edges. So its line element is equal to the Minkowski one but the angle in the plane normal to the string varies in the range 0 ≤ φ < 2π − δ [see Fig. (a)]. A straight strut has similar properties, the only difference being the fact that its line mass density is negative and its tension along the z-axis points outward.
The issue of physical interpretation of the C-metric has been recovered by Bonnor , but now following a different approach. He transformed the C-metric into the Weyl form in which the solution represents a finite line source (that corresponds to the horizon of the black hole), a semi-infinite line mass (corresponding to a horizon associated with uniform accelerated motion) and a strut keeping the line sources apart. By applying a further transformation that enlarges this solution, Bonnor confirmed the physical interpretation given in .
One of the best examples of this statement is the C-metric solution, found by Levi-Civita  and by Weyl  in 1918-1919, that has been interpreted only fifty years after its discovery by Kinnersley and Walker , although some works during this gap period have dealt with it. From this solution one can construct another exact solution known as Ernst solution . The C-metric and the Ernst solution describe two uniformly accelerated black holes in opposite directions, and the acceleration source is perfectly identified.