The Kerr--Schild structure behind the Painlevé-Gullstrand representation
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We first review the Painlevé-Gullstrand representation of the Schwarzschild solution and its generalization to the Kerr geometry. We then extend this construction to the broader class of Carter metrics admitting separable Hamilton-Jacobi equations. Within this framework, we show how a Painlevé-Gullstrand-type representation can be constructed for separable Carter metrics and identify the conditions under which the resulting metric admits a Kerr-Schild representation. In particular, we prove that certain subclasses of Carter metrics generically possess a Kerr-Schild structure. This establishes a direct connection between Hamilton-Jacobi separability, Painlevé-Gullstrand coordinates, and the Kerr-Schild form, and provides a systematic way of identifying Kerr-Schild geometries within the class of separable Carter metrics.