Relativistic Fluid Dynamics in Curved Spacetime: a Novel Effective Hamiltonian Approach

TL;DR

This paper develops a Hamiltonian framework for relativistic fluid dynamics in curved spacetime, especially Schwarzschild geometry, enabling analysis of stability and flow properties from a first-principles perspective.

Contribution

It introduces a novel relativistic Eulerian Poisson algebra and Hamiltonian formulation for fluids in curved spacetime, extending non-relativistic methods to general relativity.

Findings

Derived exact stationary solutions for radial flows in Schwarzschild spacetime

Performed stability analysis of relativistic flows in curved backgrounds

Explored the role of generalized vorticity in flow stability

Abstract

We present a comprehensive Eulerian (Hamiltonian) framework for relativistic fluid dynamics in curved spacetimes, with emphasis on Schwarzschild geometry. The key innovation lies in the consistent use of density and three-velocity fields, all defined in coordinate (Newtonian) time, while fully incorporating relativistic and curvature effects. {\it We stress that the entire dynamics is developed in coordinate (Eulerian) time, from the viewpoint of a static Eulerian observer.} In the non-relativistic regime, it is well known that Poisson brackets between the density $\rho(\vec{x}, t)$ and velocity fields $v^i(\vec{x}, t)$, together with an appropriate Hamiltonian, yield the continuity and Euler equations as Hamiltonian equations of motion. These field-theoretic Poisson brackets can be derived from canonical phase-space brackets defined on Lagrangian variables $(x_i, p_j)$, mapped to Eulerian fields $(\rho, v^i)$. We extend this framework to curved spacetime by constructing a relativistic version of the Eulerian Poisson algebra and proposing a corresponding Hamiltonian. This yields relativistic continuity and Euler equations valid in generic spacetimes, with evolution in coordinate time. Applying this to the Schwarzschild metric, we obtain exact stationary background solutions for radial flows and perform a perturbative stability analysis. We also introduce a generalised vorticity and examine its effect on perturbations over irrotational backgrounds, revealing its role in flow stability. This work offers a unified, first-principles Hamiltonian formulation of relativistic fluid dynamics in curved geometries, forming a foundation for further study of astrophysical and cosmological fluid phenomena.