Lee considers in [107] the case where a non-linear scalar field is coupled to Vlasov matter. The form of the energy momentum tensor then reads
Here
where , then future geodesic completeness is proven.
In [187] the Einstein–Vlasov system with a linear scalar field is analyzed in the case of plane,
spherical, and hyperbolic symmetry. Here, the potential in Equations (53
) and (54
) is zero. A
local existence theorem and a continuation criterion, involving bounds on derivatives of the
scalar field in addition to a bound on the support of one of the moment variables, is proven. For
the Einstein scalar field system, i.e., when
, the continuation criterion is shown to be
satisfied in the future direction, and global existence follows in that case. The work [186] extends
the result in the plane and hyperbolic case to a global result in the future direction. In the
plane case when
the solutions are shown to be future geodesically complete. The past
time direction is considered in [188] and global existence is proven. It is also shown that the
singularity is crushing and that the Kretschmann scalar diverges uniformly as the singularity is
approached.
http://www.livingreviews.org/lrr-2011-4 |
Living Rev. Relativity 14, (2011), 4
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