本研究首先回顧Joel Smoller 與 Blake Temple 合著的論文《Global Solutions of the Relativistic Euler Equations》，以探討在狀態方程 p(ρ) = σ2ρ 下的相對論效應下的歐拉方程。我們著重於不同初始條件下，震波曲線與稀疏波曲線的圖形結構。透過MATLAB，我們在 v–ρ 平面上繪製這些曲線，並觀察其隨初始條件變化的圖形。此外，我們亦參考Joel Smoller 的著作，回顧震波與稀疏波的基本理論。

We review the paper entitled by ”Global Solutions of the Relativistic Euler Equations”by Joel Smoller and Blake Temple in order to study the Relativistic Euler Equations. The equation of state is given by p(ρ) = σ2ρ. The main focus of interest is the geometric structure of shock curves and rarefaction curves corresponding to different initial data. We discuss the Riemann problem and plot these curves on the v–ρ plane by using MATLAB.

[1] A.M. Anile. Relativistic Fluids and Magneto-Fluids. Cambridge University Press, 1989.
[2] David Arnett. Supernovae and Nucleosynthesis. Princeton University Press, 1996.
[3] Donald D. Clayton. Principles of Stellar Evolution and Nucleosynthesis. University of Chicago Press, 1983.
[4] J. B. Hartle, Gravity: An Introduction to Einstein’s General Relativity, Addison Wesley, 2003.
[5] J. Smoller and B. Temple, “Global solutions of the relativistic Euler equations”, Comm. Math. Phys., vol. 156, pp. 67–99, 1993.
[6] J. Smoller, Shock Waves and Reaction–Diffusion Equations, 2nd ed., vol. 258, Grundlehren der mathematischen Wissenschaften, Springer, 1994.
[7] Steven Weinberg. Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. Wiley, 1972.