论文标题
相对论鲍尔茨曼方程的逆问题
An Inverse Problem for the Relativistic Boltzmann Equation
论文作者
论文摘要
我们认为，在全球双曲线洛伦兹时空$（m，g）$的玻尔兹曼方程中，带有未知度量$ g $的逆问题。我们考虑在附近的$ V \子集M $中进行的测量，该$ $ $ $将$ x^ - $连接到点$ x^+$。测量值是通过源到解决方案的映射建模的，该图将$ V $支持的源映射到将解决方案限制到Boltzmann方程的限制到集合$ v $。我们表明，源到解决方案映射唯一地确定了lorentzian时空，直到等轴测图，在集合$ i^+（x^ - ）\ cap i^ - （x^+）\ subset m $中。集合$ i^+（x^ - ）\ cap i^ - （x^+）$是点$ x^ - $的未来的相交，而点$ x^+$的过去，因此是从$ x^ - $可以传播并返回到point $ x^+$ x^+$的最大因果信号。结果的证明是基于使用Boltzmann方程的非线性作为解决反问题的有益特征。
We consider an inverse problem for the Boltzmann equation on a globally hyperbolic Lorentzian spacetime $(M,g)$ with an unknown metric $g$. We consider measurements done in a neighbourhood $V\subset M$ of a timelike path $μ$ that connects a point $x^-$ to a point $x^+$. The measurements are modelled by a source-to-solution map, which maps a source supported in $V$ to the restriction of the solution to the Boltzmann equation to the set $V$. We show that the source-to-solution map uniquely determines the Lorentzian spacetime, up to an isometry, in the set $I^+(x^-)\cap I^-(x^+)\subset M$. The set $I^+(x^-)\cap I^-(x^+)$ is the intersection of the future of the point $x^-$ and the past of the point $x^+$, and hence is the maximal set to where causal signals sent from $x^-$ can propagate and return to the point $x^+$. The proof of the result is based on using the nonlinearity of the Boltzmann equation as a beneficial feature for solving the inverse problem.
