In laboratory astrophysics, scaling laws serve as a crucial theoretical bridge that connects plasma dynamics at laboratory scales with astrophysical phenomena. Existing scaling relations are primarily formulated within the framework of non-relativistic ideal magnetohydrodynamics, which assumes that flow velocities are much smaller than the speed of light. However, many astrophysical processes involve relativistic or even ultra-relativistic motion, in which case non-relativistic scaling relations are no longer applicable. Therefore, developing scaling laws valid in the relativistic regime is of both fundamental and practical importance.
In this work, we systematically derive the scaling properties of the equations of ideal special relativistic magnetohydrodynamics (SRMHD) within the framework of special relativity. By applying conventional scaling transformations to the SRMHD system, we find that:(1) the invariance of the speed of light under scale transformations imposes a strict constraint on velocity scaling, requiring the velocity scaling factor to be unity. (2) Consequently, the Lorentz factor must remain strictly invariant between laboratory and astrophysical systems, forming the central constraint of relativistic scaling. (3) With the velocity scaling fixed, the remaining physical quantities, including length, time, density, and magnetic field can still be freely rescaled under dimensional consistency. (4) Since the SRMHD equations constitute a well-posed system of hyperbolic partial differential equations, possessing existence, uniqueness, and continuous dependence on initial data, this provides a theoretical foundation for applying the scaling laws to time-dependent problems.
In addition, based on existing experimental and theoretical analyses, we provide fitted scaling curves relating laser power density to the resulting plasma Lorentz factor. Several representative astrophysical phenomena are also marked on these curves, offering a direct mapping between laboratory-accessible laser conditions and relativistic astrophysical regimes. These results provide quantitative predictions for the laser power density required to reproduce astrophysical processes with different Lorentz factors in laboratory settings, thereby offering practical guidance for the design and interpretation of high energy density experiments.
By extending scaling laws from the non-relativistic to the relativistic regime, this work fills a key gap in the theoretical framework of laboratory astrophysics. The results not only clarify the fundamental constraints imposed by special relativity on scale transformations, but also provide quantitative estimates for the laser power required to simulate relativistic astrophysical phenomena in the laboratory. These findings are expected to offer important theoretical guidance for future high energy density experiments and the design of next generation high power laser facilities, thereby advancing the development of laboratory astrophysics.