The relativistic equation of motion in turbulent jets

The turbulent jets are usually described by classical velocities. The relativistic case can be treated starting from the conservation of the relativistic momentum. The two key assumptions which allow to obtain a simple expression for the relativistic trajectory and relativistic velocity are null pressure and constant density.

💡 Research Summary

The paper addresses the gap between classical descriptions of turbulent jets and the physics of jets that move at relativistic speeds, such as those found in high‑energy plasma experiments or astrophysical outflows. Starting from the conservation of relativistic momentum, the authors adopt two simplifying assumptions: the jet pressure is negligible (p ≈ 0) and the mass density remains constant throughout the flow. Under these conditions the energy‑momentum tensor reduces to (T^{\mu\nu}= \rho u^{\mu}u^{\nu}), where (u^{\mu}) is the four‑velocity and (\rho) the invariant rest‑mass density.

The analysis proceeds by applying the continuity equation (\partial_{\mu}(\rho u^{\mu})=0) and the momentum‑conservation equation (\partial_{\mu}T^{\mu x}=0) to a jet that expands axisymmetrically. The authors retain the empirical law that the jet’s cross‑sectional area grows linearly with distance, (A(x)=A_{0}(1+\alpha x)), a standard model for turbulent diffusion. From the continuity equation they obtain the invariant (\rho A(x)\gamma v = \text{const}), where (\gamma = (1-v^{2}/c^{2})^{-1/2}) is the Lorentz factor and (v) the axial velocity. This relation generalizes the familiar non‑relativistic result (v\propto 1/A(x)) by incorporating the Lorentz factor.

Differentiating the invariant yields a differential equation for the velocity profile:
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