Lee considers in  the case where a non-linear scalar field is coupled to Vlasov matter. The form of the energy momentum tensor then reads
where , then future geodesic completeness is proven.
In  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  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  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.
Living Rev. Relativity 14, (2011), 4
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