On Analytic Solutions of (1+3)D Relativistic Ideal Hydrodynamic Equations

TL;DR

This paper derives various analytic solutions to (1+3)D relativistic ideal hydrodynamic equations by embedding known lower-dimensional solutions, exploring different velocity field scalings and their physical implications.

Contribution

It introduces a systematic method to find analytic (1+3)D solutions using embedding techniques and extends solutions beyond simple power law ansatz.

Findings

Derived solutions with 2D Hubble embedding and power law velocity fields

Found solutions with nontrivial longitudinal velocity fields

Classified all solutions with independently scaling velocity components

Abstract

In this paper, we find various analytic (1+3)D solutions to relativistic ideal hydrodynamic equations based on embedding of known low-dimensional scaling solutions. We first study a class of flows with 2D Hubble Embedding, for which a single ordinary differential equation for the remaining velocity field can be derived. Using this equation, all solutions with transverse 2D Hubble embedding and power law ansatz for the remaining longitudinal velocity field will be found. Going beyond the power law ansatz, we further find a few solutions with transverse 2D Hubble embedding and nontrivial longitudinal velocity field. Finally we investigate general scaling flows with each component of the velocity fields scaling independently, for which we also find all possible solutions. Possible physical relevance of these solutions will be discussed.