Abstract
We show that the Kerr–Schild ansatz can be extended from the metric to the tetrad, and then to teleparallel gravity where curvature vanishes but torsion does not. We derive the equations of motion for the Kerr–Schild null vector, and describe the solution for a rotating black hole in this framework. It is shown that the solution depends on the chosen tetrad in a non-trivial way if the spin connection is fixed to be the one of the flat background spacetime. We show furthermore that any Kerr–Schild solution with a flat background is also a solution of f({mathcal {T}}) gravity.
Highlights
The Kerr–Schild ansatz [1,2,3,4], in which the full metric gμν = gμν +2Fkμkν is expressed as the sum of a background metric gμν and the tensor product of a null vector kμ, has been very successful in determining exact solutions of Einstein gravity
We show that the Kerr–Schild ansatz can be extended from the metric to the tetrad, and to teleparallel gravity where curvature vanishes but torsion does not
We have shown that the famous Kerr–Schild ansatz for the metric in Einstein gravity that effectively linearises the Einstein equations is useful in teleparallel gravity, and in the same way linearises the equations of motion for the torsion
Summary
The Kerr–Schild ansatz [1,2,3,4], in which the full metric gμν = gμν +2Fkμkν is expressed as the sum of a background metric gμν and the tensor product of a null vector kμ (rescaled by a scalar function F), has been very successful in determining exact solutions of Einstein gravity. This follows from the fact that linear perturbation theory with this ansatz is exact [5]: the inverse metric is given by gμν = gμν − 2Fkμkν, and the Ricci tensors of the full and background metric are related by.
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