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Abstract

The Kerr-Schild double copy is celebrated for producing exact gravitational spacetimes from gauge fields, yet the preservation of symmetry content has been less explored. We investigate the fate of residual symmetries in the Kerr-Schild double copy, focusing on the Schwarzschild solution. On the gauge theory side, we derive the residual transformations that preserve the Kerr-Schild potential, finding they form an infinite-dimensional parameterized by arbitrary null functions. On the gravity side, we analyze the resulting residual diffeomorphisms of the Kerr-Schild Schwarzschild metric. Restricting our focus to the Killing vector class of solutions, we find that the only surviving diffeomorphisms are the finite-dimensional global isometries of Schwarzschild, reducing the residual algebra to a subalgebra of Poincar\'e. This finding confirms a fundamental structural mismatch: the infinite-dimensional algebra of the gauge side admits no analogous structure in this gravitational sector. We formalize this by showing that the BRST operator for the residual symmetry is trivialized under the Killing condition, establishing the consistency of this algebraic reduction within a quantum field theoretic framework. This paper is the first of a two-part series. In Part II, we complete this analysis by examining the more complex proper conformal Killing vector class of solutions and formulating a unified BRST framework.