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 [142Jump To The Next Citation Point], cf. also [135Jump To The Next Citation Point, 156Jump To The Next Citation Point]. They chose to work in coordinates where the metric takes the form
ds2 = − e2μ(t,r)dt2 + e2λ(t,r)dr2 + r2(dπœƒ2 + sin2πœƒd φ2),

where t ∈ ℝ, r ≥ 0, πœƒ ∈ [0,π], φ ∈ [0, 2π]. These are called Schwarzschild coordinates. Asymptotic flatness is expressed by the boundary conditions

lr→im∞ λ(t,r) = lri→m∞ μ (t,r) = 0, ∀t ≥ 0.

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

λ(t,0) = 0 ∀t ≥ 0.

The coordinates (r,πœƒ,Ο•) give rise to difficulties at r = 0 and it is advantageous to use Cartesian coordinates. With

x = (r sin Ο•cos πœƒ,rsinΟ• sinπœƒ,r cosΟ• )

as spatial coordinates and

j j λ x-⋅ p-xj v = p + (e − 1) r r

as momentum coordinates, the Einstein–Vlasov system reads

( ∘ -------) ∂tf + eμ−λ∘---v-----⋅ ∇xf − λtx-⋅ v-+ eμ−λ μr 1 + |v|2 x-⋅ ∇vf = 0, (35 ) 1 + |v|2 r r −2λ 2 e (2rλr − 1) + 1 = 8πr ρ, (36 ) e−2λ(2rμr + 1) − 1 = 8πr2p, (37 ) λ = − 4πre λ+μj, (38 ) t e− 2λ( μ + (μ − λ )(μ + 1-)) − e− 2μ (λ + λ (λ − μ )) = 8 πp . (39 ) rr r r r r tt t t t T
The matter quantities are defined by
∫ ∘ ------2- ρ(t,x) = 3 1 + |v |f(t,x,v )dv, (40 ) ∫ℝ ( )2 p(t,x) = x-⋅ v f (t,x, v)∘--dv----, (41 ) ℝ3 r 1 + |v|2 ∫ x ⋅ v j(t,x) = ----f (t,x,v )dv, (42 ) ℝ3∫ r 1- ||x ×-v||2 pT(t,x) = 2 3 r f(t,x,v)dv. (43 ) ℝ
Here ρ is the energy density, j the current, p the radial pressure, and pT the tangential pressure. Let us point out that these equations are not independent, e.g., Equations (38View Equation) and (39View Equation) follow from (35View Equation)–(37View Equation).

As initial data we take a spherically-symmetric, non-negative, and continuously differentiable function f0 with compact support that satisfies

∫ ∫ ∘ ------2- r- 3 1 + |v| f0(y,v)dvdy < 2. (44 ) |y|<r ℝ
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. [149Jump To The Next Citation Point, 20Jump To The Next Citation Point, 24Jump To The Next Citation Point]. 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 [116Jump To The Next Citation Point] 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 [156Jump To The Next Citation Point]. These coordinates are also considered in [12Jump To The Next Citation Point]. The Einstein–Vlasov system is investigated in double null coordinates in [64Jump To The Next Citation Point, 63Jump To The Next Citation Point]. Maximal-areal coordinates and Eddington–Finkelstein coordinates are used in [21Jump To The Next Citation Point, 17Jump To The Next Citation Point], and in [24Jump To The Next Citation Point] respectively.


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