Perfect fluids revisited: an action principle approach

Abstract: We revisit the variational principle for relativistic perfect fluids in a manifestly covariant formulation based on differential forms, with particular attention to the boundary data required for a well-posed action principle. For timelike flows, the formalism is largely a geometric reformulation of the Schutz action principle for perfect fluids. We then analyse the extension of the same variational principle to null flows. In that case, the system is not a generic perfect fluid: the equations of motion force the enthalpy density to vanish, $ρ+P=0$. The resulting stress-energy tensor decomposes into a vacuum energy-like term with variable pressure and a null dust contribution. This shows that the obstruction to the naive fluid extension is dynamical rather than kinematical. Since the matter action is formulated independently of any gravitational field equations, the construction can be generalised to first-order or non-metric theories of gravity.

• The paper introduces a covariant Eulerian action based on differential forms that unifies the treatment of timelike and null flows for perfect fluids.
• The formulation employs Lagrange multipliers and explicit boundary data to enforce conservation and dynamical constraints, avoiding degeneracies in null flows.
• The analysis delineates how vacuum energy and null dust emerge from the variational dynamics, offering insights for extended gravitational theories.

Action Principle Formulation of Relativistic Perfect Fluids

Introduction and Context

The paper "Perfect fluids revisited: an action principle approach" (2606.17084) revisits the variational characterization of relativistic perfect fluids using a fully covariant, differential-form-based formalism that makes explicit the role of boundary data in the well-posedness of the action principle. The analysis unifies the treatment of timelike and null flows within the same variational structure and establishes the precise dynamical and kinematical constraints that arise, particularly when extending the formalism to null congruences. This yields a clear delineation between standard perfect-fluids, vacuum energy, and null dust, as emergent from the variational dynamics.

Geometric Reformulation and Boundary Data

A principal contribution is the manifestly covariant Eulerian action, based on the differential forms approach. The formulation elucidates the separation of variables: fluid velocity $u$, particle number density $n$, and entropy density $s$ are treated as independent, with the Lagrange multipliers and auxiliary potentials specifying the constraints. This extends the geometric approach of Schutz and underscores the importance of Lagrangian labels as boundary data, leading naturally to a Dirichlet problem for the velocity potentials and flow labels.

Standard treatments typically adopt the particle current as primary and reconstruct $n$ and $u$ through its decomposition. This is structurally efficient for unit timelike flows but becomes degenerate for null currents, where the current's norm vanishes and the decomposition breaks down. The present formalism avoids this pitfall and, consequently, remains well-defined for null flows.

Covariant Action and Euler-Lagrange Structure

The fluid action is formulated as

$$\mathcal{L} = -\left[\star (\rho + \mathcal{A}) + \mathcal{C}\wedge\star u^\flat \right],$$

where $\rho$ is the energy density, $\mathcal{A}$ encodes the normalization constraint on $u$, and $\mathcal{C}$ implements the conservation of particle number, entropy, and invariance of flow labels. The Lagrangian multipliers $n$0 enforce these constraints. The boundary conditions are rendered explicit: the action is extremized with Dirichlet data for the velocity potentials and Lagrangian coordinates on $n$1.

The system admits internal symmetries representing relabelling invariances of the Lagrangian coordinates and potentials, yielding a large algebra of advected Noether charges transported by the flow.

Timelike and Null Flows: Dynamical Implications

The variational analysis yields, for unit timelike flows, the canonical Euler system:

• Conservation of the particle number and entropy currents: $n$2, $n$3,
• Clebsch-type representation for the velocity,
• Stress-energy tensor of the standard perfect fluid: $n$4.

For null flows, imposing $n$5 renders the definition of the "perfect fluid" dynamically restricted. The equations of motion force the enthalpy density to vanish: $n$6. The stress-energy tensor thus generically decomposes as

$n$7

which comprises a term reminiscent of a variable vacuum energy (with equation of state $n$8) and a null dust piece. This result demonstrates that the "obstruction" to a naive null extension is not just a degeneracy of the kinematical variables but encoded in the equations of motion imposed by the variational principle.

The system lacks a rest frame for null flows, so $n$9 and $s$0 must be regarded as scalar degrees of freedom in the action, without the thermodynamic interpretation that applies in the timelike case.

General Covariance and Stress-Energy Structure

The stress-energy tensor is derived via tetrad variation, facilitating generalization to settings with non-metricity or torsion. In metric theories, energy-momentum conservation $s$1 emerges, but the formalism also accommodates first-order (Einstein-Palatini, etc.) or non-metric gravitational theories via generalized contracted Bianchi identities, expressed in terms of torsion and non-metricity contributions.

The explicit separation between kinematic and thermodynamic variables ensures that the matter action remains independent of the dynamical content of gravity; it can thus be coupled consistently in a wide class of gravitational frameworks.

Implications and Future Directions

This geometric variational approach clarifies the precise conditions under which perfect fluid actions can be extended to null congruences and shows that "null fluids" obtained in this way are not generic perfect fluids but must satisfy $s$2. The resulting stress tensor interpolates between the structures of cosmological constant and null dust.

An immediate implication is that attempts to model matter as a "null fluid" using classical hydrodynamics must respect this constraint; otherwise, the matter model lacks a consistent action principle origin. This impacts work on effective matter models in the context of gravitational radiation, early universe, and modified gravity theories. The covariant action is immediately applicable to first-order and non-metric geometries, opening the possibility for systematic treatments of fluids in extended gravity and in the presence of microphysical fields that select null directions.

Whether this variational "null fluid" arises as a limit of coarse-grained microphysical models remains an open problem. Future analyses may explore such connections, for example, by considering the effective description of quantum field excitations or in the study of null surfaces (e.g., event horizons, causal boundaries) and their hydrodynamic analogues.

Conclusion

The paper provides a rigorous, boundary-conscious action principle for relativistic perfect fluids that systematically includes both timelike and null flows, clarifying the dynamical distinction between perfect fluids, vacuum energy, and null dust under the same geometric apparatus. The formulation is broadly extensible to alternative theories of gravity and offers a foundation for developing consistent variational matter models in generalized relativistic contexts.