Kodama-Schwarzschild versus Gaussian Normal Coordinates Picture of Thin Shells

Abstract EN

Geometry of the spacetime with a spherical shell embedded in it is studied in two coordinate systems: Kodama-Schwarzschild coordinates and Gaussian normal coordinates.

We find explicit coordinate transformation between the Kodama-Schwarzschild and Gaussian normal coordinate systems.

We show that projections of the metrics on the surface swept by the shell in the 4D spacetime in both cases are identical.

In the general case of time-dependent metrics we calculate extrinsic curvatures of the shell in both coordinate systems and show that the results are identical.

Applications to the Israel junction conditions are discussed.