Abstract
We consider the C-metric as a gravitational field configuration that describes an accelerating black hole in the presence of a semi-infinite cosmic string, along the accelerating direction. We adopt the expression for the gravitational energy-momentum developed in the teleparallel equivalent of general relativity (TEGR) and obtain an explanation for the acceleration of the black hole. The gravitational energy enclosed by surfaces of constant radius around the black hole is evaluated, and in particular, the energy contained within the gravitational horizon is obtained. This energy turns out to be proportional to the square root of the area of the horizon. We find that the gravitational energy of the semi-infinite cosmic string is negative and dominant for large values of the radius of integration. This negative energy explains the acceleration of the black hole that moves towards regions of lower gravitational energy along the string.
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Notes
We remark that when \(C=1\), the line element (6), and consequently the set of tetrad fields (25), does not represent ordinary Minkowski space-time, but just a partition of the latter, delimited by the acceleration horizons. In Section 5 of Ref. [1], it is very clearly stated that test particles at the origin \(r=0\) acquire acceleration along the \(\pm z\) directions, which is certainly not a feature of ordinary, full Minkowski space-time.
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Carneiro, F.L., Ulhoa, S.C. & Maluf, J.W. On the Black Hole Acceleration in the C-Metric Space-Time. Gravit. Cosmol. 28, 352–361 (2022). https://doi.org/10.1134/S0202289322040077
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DOI: https://doi.org/10.1134/S0202289322040077