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 \(p'\); 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 (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.

本文考虑了动量中心坐标中相对论玻尔兹曼方程的雅可比行列式一个悬而未决的问题。对于牛顿玻尔兹曼方程，动量中心坐标在牛顿非截止玻尔兹曼方程的研究中起着重要作用，特别是我们提到了广泛使用的抵消引理[1]。在本文中，我们针对从动量p到碰撞后动量\（p'\）的相对论性碰撞图，专门计算了十个变量中非常复杂的Jacobian行列式；具体来说，我们为\（\ theta \ in [0,1] \）计算\（p \ mapsto u = \ theta p'+ \ left（1- \ theta \ right）p \）的行列式。然后，我们为该行列式给出一个上界，该上界在p和q变量中均不具有奇异性。接下来，我们给出一个示例，在该示例中，我们证明了雅可比行列式在特定的点极限内变为零。我们进一步解释了我们的数值研究结果，该结果表明，雅可比行列式具有大量不同的点，在该点处机器为零。这概括了Glassey-Strauss（1991）[8]和Guo-Strain（2012）[12]的工作。这些结论使得很难在动量中心坐标中设想牛顿对消引理的直接相对论模拟。