On the Determinant Problem for the Relativistic Boltzmann Equation

This article considers a long-outstanding open question regarding the Jacobian determinant for the relativistic Boltzmann equation in the center-of-momentum coordinates. For the Newtonian Boltzmann equation, the center-of-momentum coordinates have played a large role in the study of the Newtonian non-cutoff Boltzmann equation, in particular we mention the widely used cancellation lemma [1]. In this article we calculate specifically the very complicated Jacobian determinant, in ten variables, for the relativistic collision map from the momentum p to the post collisional momentum $$p'$$ p ′ ; specifically we calculate the determinant for $$p\mapsto u = \theta p'+\left( 1-\theta \right) p$$ p ↦ u = θ p ′ + 1 - θ p for $$\theta \in [0,1]$$ θ ∈ [ 0 , 1 ] . Afterwards we give an upper-bound for this determinant that has no singularity in both p and q variables. Next we give an example where we prove that the Jacobian goes to zero in a specific pointwise limit. We further explain the results of our numerical study which shows that the Jacobian determinant has a very large number of distinct points at which it is machine zero. This generalizes the work of Glassey-Strauss (1991) [8] and Guo-Strain (2012) [12]. These conclusions make it difficult to envision a direct relativistic analog of the Newtonian cancellation lemma in the center-of-momentum coordinates.