Some consequences of the Kerr–Schild gauge

Abstract

Generalized Kerr–Schild nilpotent deformations of an arbitrary background spacetime g¯μν(x)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\overline{g}_{\\mu \ u }(x)$$\\end{document} are considered. Those are of the form g¯μν+lμlν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\overline{g}_{\\mu \ u }+l_\\mu l_\ u $$\\end{document} with l2=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$l^2=0$$\\end{document}. The relationship between the Ricci tensor of the background metric and the Ricci tensor of the deformed metric is found exactly. It consists of two terms, one is essentially the Fierz–Pauli operator, and the other is new. When the background is flat, the Kerr–Schild family is recovered. Novel examples for more general backgrounds (even including some simple sources) are discussed.