Kodama time: Geometrically preferred foliations of spherically symmetric spacetimes

In a general time-dependent ($3+1$)-dimensional spherically symmetric spacetime, the so-called Kodama vector is a naturally defined geometric quantity that is timelike outside the evolving horizon and so defines a preferred class of fiducial observers. However the Kodama vector does not by itself define any preferred notion of time. We first extract as much information as possible by invoking the “warped product” structure of spherically symmetric spacetime to study the Kodama vector, and the associated Kodama energy flux, in a coordinate-independent manner. Using this formalism we construct a general class of conservation laws, generalizing Kodama’s energy flux. We then demonstrate that a preferred time coordinate—which we shall call Kodama time—can be introduced by taking the additional step of applying the Clebsch decomposition theorem to the Kodama vector. We thus construct a geometrically preferred coordinate system for any time-dependent spherically symmetric spacetime, and explore its properties. We study the geometrically preferred fiducial observers, and demonstrate that it is possible to define and calculate a generalized notion of surface gravity that is valid throughout the entire evolving spacetime. Furthermore, by building and suitably normalizing a set of radial null geodesics, we can show that this generalized surface gravity passes several consistency tests and has a physically appropriate static limit.