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<p>We discuss in this paper thermodynamic properties of relativistic shock waves of arbitrary large magnitude. In the mathematical formulation, thermodynamic variables of the front and back sides of a shock front satisfy a single algebraic equation called the <i>Rankine-Hugoniot thermodynamic condition</i>. However a solution to the equation does not always represent an actual shock wave which has two particular properties: first, its wave front is supersonic relative to the flow ahead of it and subsonic behind it; second, the entropy of the medium behind the shock front exceeds the entropy ahead of it. Thus it is a mathematical problem to select physically admissible solutions. For general non-relativistic gases governed by general equations of state, this problem is solved by Bethe, H. A.: On the theory of shock waves for an arbitrary equation of state, Report No. 545 for the <i>Office of Scientific Research and Development</i>, Serial No. NDRC-B-237, May 4 (this report is available at )(1942) <a href="https://link.springer.com/chapter/10.1007/978-1-4612-2218-7_11">https://link.springer.com/chapter/10.1007/978-1-4612-2218-7_11</a> and Weyl, H.: Shock waves in arbitrary fluids, Comm. Pure Appl. Math., 5: 103-122 (National Defence Research Committee, Applied Mathematics Panel note, No. 12, Applied Mathematics Group – New York University, No. 46, 1944) (1949) in satisfying ways under reasonable conditions. The aim of this paper is to find thermodynamic conditions which provide an answer to the above problem for relativistic gases. All the conditions are described in terms of thermodynamic invariants. Since Weyl’s theorem is studied by Israel, W.: Relativistic theory of shock waves, Proc. Roy. Soc. London, Ser A <b>259</b>, 129–143 (1960) , we will focus on Bethe’s theorem. It is interesting to note that Bethe’s condition yields the <i>weak condition</i> of Menikoff, R., Plohr, B.J.: The Riemann problem for fluid flow of real materials. Rev. Modern Phys. <b>61</b>(1), 7–130 (1989) .</p>

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Bethe-Weyl theorems for relativistic Euler equations

  • 期刊论文

  • 发表日期：

    2026年3月28日

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  • Fumioki Asakura

摘要

We discuss in this paper thermodynamic properties of relativistic shock waves of arbitrary large magnitude. In the mathematical formulation, thermodynamic variables of the front and back sides of a shock front satisfy a single algebraic equation called the Rankine-Hugoniot thermodynamic condition. However a solution to the equation does not always represent an actual shock wave which has two particular properties: first, its wave front is supersonic relative to the flow ahead of it and subsonic behind it; second, the entropy of the medium behind the shock front exceeds the entropy ahead of it. Thus it is a mathematical problem to select physically admissible solutions. For general non-relativistic gases governed by general equations of state, this problem is solved by Bethe, H. A.: On the theory of shock waves for an arbitrary equation of state, Report No. 545 for the Office of Scientific Research and Development, Serial No. NDRC-B-237, May 4 (this report is available at )(1942) https://link.springer.com/chapter/10.1007/978-1-4612-2218-7_11 and Weyl, H.: Shock waves in arbitrary fluids, Comm. Pure Appl. Math., 5: 103-122 (National Defence Research Committee, Applied Mathematics Panel note, No. 12, Applied Mathematics Group – New York University, No. 46, 1944) (1949) in satisfying ways under reasonable conditions. The aim of this paper is to find thermodynamic conditions which provide an answer to the above problem for relativistic gases. All the conditions are described in terms of thermodynamic invariants. Since Weyl’s theorem is studied by Israel, W.: Relativistic theory of shock waves, Proc. Roy. Soc. London, Ser A 259, 129–143 (1960) , we will focus on Bethe’s theorem. It is interesting to note that Bethe’s condition yields the weak condition of Menikoff, R., Plohr, B.J.: The Riemann problem for fluid flow of real materials. Rev. Modern Phys. 61(1), 7–130 (1989) .