Abstract
The Kerr-Schild (KS) formalism is a powerful tool for constructing exact solutions in general relativity. In this paper, we present a generalization of the conventional KS formalism to double field theory (DFT) and supergravities. We introduce a generalized KS ansatz for the generalized metric in terms of a pair of null vectors. Applying this ansatz to the equations of motion of DFT, we construct the generalized KS field equation. While the generalized KS equations are quadratic in the fields, we show that it is possible to find solutions by considering linear equations only. Furthermore, we construct a Killing spinor equation under the generalized KS ansatz. Based on this formalism, we show that the classical double copy structure, which represents solutions of the Einstein equation in terms of solutions of the Maxwell equation, can be extended to the entire massless string NS-NS sector. We propose a supersymmetric classical double copy which shows that solutions of the Killing spinor equation can be realized in terms of solutions of the BPS equation of the supersymmetric Maxwell theory.
Highlights
Field equation to a set of linear differential equations
We present a generalization of the conventional KS formalism to double field theory (DFT) and supergravities
We construct a Killing spinor equation under the generalized KS ansatz. Based on this formalism, we show that the classical double copy structure, which represents solutions of the Einstein equation in terms of solutions of the Maxwell equation, can be extended to the entire massless string NS-NS sector
Summary
In general relativity (GR) the Kerr-Schild ansatz is a minimal extension of linear perturbation around a background metric g. It is achieved by introducing a null vector l, and the ansatz is given by gμν = gμν + κφlμlν , μ, ν, · · · = 0, 1, · · · d − 1 ,. The main advantage of the Kerr-Schild ansatz is that it preserves some features of the linearized perturbation. The form of the inverse metric and its determinant are gμν = gμν − κφlμlν , det(g) = det(g). We investigate Buscher’s rule and show that the form of the generalized Kerr-Schild ansatz is preserved under T-duality
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