Lagrange coordinates for the Einstein-Euler equations

We derive a new symmetric hyperbolic formulation of the Einstein-Euler equations in Lagrange coordinates that are adapted to the Frauendiener-Walton formulation of the Euler equations. As an application, we use this system to show that the densitized lapse and zero-shift coordinate systems for the vacuum Einstein equations are equivalent to Lagrange coordinates for a fictitious fluid with a specific equation of state.