  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 by Alexandre et al. (Arch. Ration. Mech. Anal.
152(4):327-355, 2000). 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 $p'$;
specifically we calculate the determinant for $p\mapsto u = \theta
p'+\left(1-\theta\right)p$ for $\theta \in [0,1]$. 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 (Transport Theory Statist. Phys. 20(1):55-68, 1991) and
Guo-Strain (Comm. Math. Phys. 310(3):649-673, 2012). These conclusions make it
difficult to envision a direct relativistic analog of the Newtonian
cancellation lemma in the center-of-momentum coordinates.
