### 3.1 Set up and choice of coordinates

The study of the global properties of solutions to the spherically-symmetric Einstein–Vlasov system was initiated two decades ago by Rein and Rendall [142], cf. also [135, 156]. They chose to work in coordinates where the metric takes the form

where , , , . These are called Schwarzschild coordinates. Asymptotic flatness is expressed by the boundary conditions

A regular center is also required and is guaranteed by the boundary condition

The coordinates give rise to difficulties at and it is advantageous to use Cartesian coordinates. With

as spatial coordinates and

as momentum coordinates, the Einstein–Vlasov system reads

The matter quantities are defined by
Here is the energy density, the current, the radial pressure, and the tangential pressure. Let us point out that these equations are not independent, e.g., Equations (38) and (39) follow from (35)–(37).

As initial data we take a spherically-symmetric, non-negative, and continuously differentiable function with compact support that satisfies

This condition guarantees that no trapped surfaces are present initially.

The set up described above is one of several possibilities. The Schwarzschild coordinates have the advantage that the resulting system of equations can be written in a quite condensed form. Moreover, for most initial data, solutions are expected to exist globally in Schwarzschild time, which sometimes is called the polar time gauge. Let us point out here that there are initial data leading to spacetime singularities, cf. [149, 20, 24]. Hence, the question of global existence for general initial data is only relevant if the time slicing of the spacetime is expected to be singularity avoiding, which is the case for Schwarzschild time. We refer to [116] for a general discussion on this issue. This makes Schwarzschild coordinates tractable in the study of the Cauchy problem. However, one disadvantage is that these coordinates only cover a relatively small part of the spacetime, in particular trapped surfaces are not admitted. Hence, to analyze the black-hole region of a solution these coordinates are not appropriate. Here we only mention the other coordinates and time gauges that have been considered in the study of the spherically symmetric Einstein–Vlasov system. These works will be discussed in more detail in various sections below. Rendall uses maximal-isotropic coordinates in [156]. These coordinates are also considered in [12]. The Einstein–Vlasov system is investigated in double null coordinates in [64, 63]. Maximal-areal coordinates and Eddington–Finkelstein coordinates are used in [21, 17], and in [24] respectively.