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{\it Kerr-Schild} form. The four-metric is written  %  \begin{array}  g_{\mu \nu} &= \eta_{\mu \nu} + 2H\ell_{\mu}\ell_{\nu}, 2H\ell_{\mu}\ell_{\nu},\\  \Rightarrow g^{\mu \nu} &= \eta^{\mu \nu} - 2 H \ell^{\mu} \ell^{\nu}  \end{array}  %