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 As initial data we take a spherically-symmetric, non-negative, and continuously differentiable function
with compact support that satisfies
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.
http://www.livingreviews.org/lrr-2011-4 |
Living Rev. Relativity 14, (2011), 4
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