Numerical Relativistic Gravitational Collapse with Spatial Time Slices

Abstract

In the usual flatspace hydrodynamics, there is a unique split between Eulerian and Lagrangian observers. An Eulerian observer is one at rest in space, while the Lagrangian observer is one at rest in the fluid. We can define an observer at an event in spacetime by giving his 4-velocity at that event. The path through spacetime taken by the observer is called the timeline of that observer. In flat Minkowski spacetime (the spacetime of special relativity), the Eulerian observer’s 4-velocity is exceedingly simple, because the time slices (3-spaces representing a given instant of time) are parallel flat 3-planes. Since the Eulerian observer is at rest, his 4-velocity is normal to the time slice. He is not shearing, converging, accelerating, or rotating relative to nearby Eulerian observers. In contrast, the Lagrangian observer at the same event is doing all these things, relative to nearby Lagrangian observers. While the timelines of the Lagrangian observers, following the fluid particles, are complicated curved trajectories through spacetime, the Eulerian observer’s timelines are straight and parallel to each other.