We show that the 1D planar ultrarelativistic shock tube problem with an ultrarelativistic polytropic equation of state can be solved analytically for the case of a working surface, i.e. for the case when an initial discontinuity on the hydrodynamical quantities of the problem form two shock waves separating from a contact discontinuity. The procedure is based on the extensive use of the Taub jump conditions for relativistic shock waves, the Taub adiabatic, and performing Lorentz transformations to present the solution in a system of reference adequate for an external observer at rest. The solutions are found using a set of very useful theorems related to the Lorentz factors when transforming between systems of reference. The energy dissipated inside the working surface is relevant for studies of light curves observed in relativistic astrophysical jets and so, we provide a full analytical solution for this phenomenon assuming an ultrarelativistic periodic velocity injected at the base of the jet.

中文翻译：

---

平面工作表面的全解析超相对论一维解

我们证明了具有超相对论多方状态方程的一维平面超相对论激波管问题可以在工作表面的情况下解析求解，即问题的流体动力学量的初始不连续形成两个激波分离的情况接触不连续。该程序基于广泛使用相对论冲击波的 Taub 跳跃条件、Taub 绝热条件和执行洛伦兹变换，以在一个足以让外部观察者静止的参考系统中呈现解决方案。在参考系统之间转换时，使用一组与洛伦兹因子相关的非常有用的定理找到了解决方案。