Lorentz covariant formulation of the Navier-Stokes equations by Laplace-Beltrami differential operators

Though the Navier-Stokes equations of fluid mechanics have proven useful in multiple dis- ciplines of physics and engineering, they are incapable of describing flows where relativistic effects are present due to their Galilean invariance. For a Navier-Stokes equation set to be mathematically capable of modeling relativistic effects on the temporal evolution of its velocity field in curved space, all of its differential operators must instead exhibit Lorentz covariance. Many attempts have been made at formulating a theory of viscous relativistic fluid mechanics, but none have emphasized Lorentz covariance. This work therefore seeks to construct a Lorentz covariant version of the Navier-Stokes model of fluid motion so as to support relativistic vis- cous flows. To accomplish this task, the Galilean invariant gradient, divergence, Laplacian, and material derivative operators present in the Navier-Stokes equations are reconstructed in terms of the covariant derivative operator for curved space. The velocity and external force fields are also modeled as four-vectors to capture the effects of time dilation and length contraction on turbulent flow patterns, which helps to properly account for the relativistic scaling and skewing of important observed phenomena such as temporal mixing scales and Komolgorov microscales.