Variational Relativity Framework

• Variational Relativity is a covariant framework defining matter fields using a Lorentzian metric and a submersion, ensuring invariance under diffeomorphisms.
• Covariance restricts the usable invariants to combinations of Ψ and the conformation tensor K, with extensions that incorporate second-gradient invariants.
• The framework unifies general relativity with classical continuum mechanics, clarifying how higher-gradient kinematic quantities converge in the classical limit.

Variational Relativity denotes a generally covariant variational framework, developed by Souriau in the sixties, in which the primary fields on the four-dimensional “Universe” manifold $\mathcal M$ are the Lorentzian metric $g$ and a matter field $\Psi:\mathcal M\to V\simeq \mathbb R^3$. In the recent second-gradient formulation, the action over a relatively compact domain $U\subset\mathcal M$ is written

$S_U[g,\Psi]=\int_U L\big(j^2g,j^2\Psi\big)\,\vol_g,$

with general covariance requiring $S_U[*g,*\Psi]=S_{\varphi(U)}[g,\Psi]$ for every orientation-preserving diffeomorphism $\varphi$. Within this program, General Relativity and its classical limit, Classical Continuum Mechanics, are treated in a single geometric language, and the framework has been used, for instance, to formulate Hyperelasticity in General Relativity (Chapon et al., 17 Jul 2025).

1. General covariance as the organizing principle

In this framework, the matter field $\Psi$ is a submersion whose fibres are timelike world-lines, while the action is required to be invariant under diffeomorphisms of $\mathcal M$. The pointwise form of this requirement is

$$L\big(j^2(*g),\,j^2(*\Psi)\big)(m) = L\big(j^2g,\,j^2\Psi\big)\big(\varphi(m)\big),$$

so the admissible Lagrangian density must be built only from combinations of $g$0, $g$1, and their jets that are invariant under all diffeomorphisms. The same source states this equivalently: general covariance means that $g$2 is built from those combinations of $g$3 and their jets which are invariant under all diffeomorphisms (Chapon et al., 17 Jul 2025).

This requirement is not merely formal. In the Souriau line of argument, covariance acts as a structural restriction on the possible field variables appearing in the density. The associated infinitesimal diffeomorphism invariance gives a Noether-type identity of the form

$g$4

leading to the usual Bianchi/Noether identities and showing that only diffeomorphism-invariants survive. A central theme of Variational Relativity is therefore that covariance is not an afterthought added to a variational model; it is the criterion that selects the admissible variables in the first place.

2. Conformation and Souriau’s first-gradient theorem

At first order, the central invariant is the conformation tensor $g$5, defined from the first jet of $g$6 and the inverse metric: $g$7 Here $g$8 is positive-definite under the perfect-matter hypothesis. The quantity is the relativistic analog of the inverse of the right Cauchy–Green tensor, and in the recent account it is explicitly identified as a diffeomorphism invariant of the theory (Chapon et al., 17 Jul 2025).

Souriau’s first-gradient theorem states that if a Lagrangian density depends only on $g$9 and is generally covariant, then it can be re-expressed solely as a function of $\Psi:\mathcal M\to V\simeq \mathbb R^3$0 and $\Psi:\mathcal M\to V\simeq \mathbb R^3$1: $\Psi:\mathcal M\to V\simeq \mathbb R^3$2 The proof sketch given in the recent second-gradient treatment proceeds by constructing a rest-frame coframe

$\Psi:\mathcal M\to V\simeq \mathbb R^3$3

then trading the matrix components of $\Psi:\mathcal M\to V\simeq \mathbb R^3$4 for the components of $\Psi:\mathcal M\to V\simeq \mathbb R^3$5 in that frame, namely $\Psi:\mathcal M\to V\simeq \mathbb R^3$6. Because the orientation-preserving diffeomorphism group acts transitively on ordered frames, the residual frame dependence is fictitious, and covariance forces $\Psi:\mathcal M\to V\simeq \mathbb R^3$7 to be independent of the choice of frame. In this sense, the theorem identifies conformation as the complete first-gradient invariant content of the generally covariant matter sector.

A common misconception is to treat general covariance as compatible with an arbitrary first-gradient constitutive dependence. Souriau’s theorem states the opposite: at first order, covariance collapses the allowed dependence to $\Psi:\mathcal M\to V\simeq \mathbb R^3$8 and $\Psi:\mathcal M\to V\simeq \mathbb R^3$9.

3. Second-gradient extension and the Galilean limit

The second-gradient extension allows the density to depend on second jets of $U\subset\mathcal M$0 and $U\subset\mathcal M$1, but the recent analysis again shows that covariance sharply restricts the admissible combinations. A generally covariant second-jet density can be written in terms of $U\subset\mathcal M$2, $U\subset\mathcal M$3, curvature-coupling invariants $U\subset\mathcal M$4, $U\subset\mathcal M$5, $U\subset\mathcal M$6, and three additional material invariants: the material relativistic acceleration $U\subset\mathcal M$7, the relativistic velocity gradient $U\subset\mathcal M$8, and the strain gradient $U\subset\mathcal M$9. The corollary is stated in the form

$S_U[g,\Psi]=\int_U L\big(j^2g,j^2\Psi\big)\,\vol_g,$0

No other combinations can appear in a generally covariant second-gradient density (Chapon et al., 17 Jul 2025).

The explicitly displayed invariants include

$S_U[g,\Psi]=\int_U L\big(j^2g,j^2\Psi\big)\,\vol_g,$1

$S_U[g,\Psi]=\int_U L\big(j^2g,j^2\Psi\big)\,\vol_g,$2

and

$S_U[g,\Psi]=\int_U L\big(j^2g,j^2\Psi\big)\,\vol_g,$3

The curvature couplings $S_U[g,\Psi]=\int_U L\big(j^2g,j^2\Psi\big)\,\vol_g,$4 vanish in flat spacetime, so they encode explicitly relativistic second-gradient effects.

The classical $S_U[g,\Psi]=\int_U L\big(j^2g,j^2\Psi\big)\,\vol_g,$5 limit is one of the main motivations of the framework. In this limit, $S_U[g,\Psi]=\int_U L\big(j^2g,j^2\Psi\big)\,\vol_g,$6, the inverse right Cauchy–Green tensor of classical elasticity. The acceleration invariant obeys $S_U[g,\Psi]=\int_U L\big(j^2g,j^2\Psi\big)\,\vol_g,$7, where $S_U[g,\Psi]=\int_U L\big(j^2g,j^2\Psi\big)\,\vol_g,$8, and is stated to be not objective. The velocity-gradient invariant satisfies $S_U[g,\Psi]=\int_U L\big(j^2g,j^2\Psi\big)\,\vol_g,$9, with symmetric part $S_U[*g,*\Psi]=S_{\varphi(U)}[g,\Psi]$0 objective and skew part non-objective. The strain-gradient invariant satisfies $S_U[*g,*\Psi]=S_{\varphi(U)}[g,\Psi]$1, recovering the classical second-gradient variable of Toupin–Mindlin and remaining fully objective in the limit. The paper emphasizes that some relativistic invariants converge to objective quantities in the Galilean limit, while others converge to non-objective quantities; a plausible implication is that Variational Relativity clarifies precisely which higher-gradient kinematic quantities admit objective classical descendants and which do not (Chapon et al., 17 Jul 2025).

4. Gravitational action principles, boundary structure, and independent geometric variables

The Souriau program is centered on matter fields and covariance, but the surrounding literature on relativistic variational methods is heterogeneous and includes several distinct treatments of the gravitational sector itself.

Formulation	Independent data	Characteristic result
Synchronous Lagrangian formulation	prescribed $S_U[*g,*\Psi]=S_{\varphi(U)}[g,\Psi]$2 for $S_U[*g,*\Psi]=S_{\varphi(U)}[g,\Psi]$3, variational $S_U[*g,*\Psi]=S_{\varphi(U)}[g,\Psi]$4	Einstein–Hilbert and Palatini are classified as asynchronous; synchronous reformulation restores manifest covariance and gauge symmetry (Cremaschini et al., 2016)
$S_U[*g,*\Psi]=S_{\varphi(U)}[g,\Psi]$5 gravity with boundary term	metric plus $S_U[*g,*\Psi]=S_{\varphi(U)}[g,\Psi]$6	well-posed for $S_U[*g,*\Psi]=S_{\varphi(U)}[g,\Psi]$7 on the boundary; reproduces correct ADM energy and black-hole entropy (0809.4033)
Unconstrained metric–affine principle	metric $S_U[*g,*\Psi]=S_{\varphi(U)}[g,\Psi]$8, unconstrained connection $S_U[*g,*\Psi]=S_{\varphi(U)}[g,\Psi]$9	extremals are Ricci-flat metrics with the associated Riemannian connection; 5D specialization yields Einstein–Maxwell equations (Massa et al., 2016)
Biconnection principle	$\varphi$0	field equations reduce to the Einstein field equations for any dependence of the matter action upon an independent connection (Tamanini, 2013)
Bimetric variational principle	physical metric $\varphi$1, connection-generating $\varphi$2	physically distinct theory; antisymmetric part of $\varphi$3 makes torsion propagating; non-linear Boulware–Deser issue remains open (Jimenez et al., 2012)
Hypersurface-anchored extension	metric $\varphi$4, embedding $\varphi$5	Einstein equations acquire a localized distributional contribution plus an anchoring equation; no additional propagating bulk gravitational degrees of freedom away from the hypersurface, and the anchoring condition is generically inequivalent to local slicing gauge conditions (Sahoo, 1 Jul 2026)

Several distinctions are central. In higher-derivative gravity, the naive bulk $\varphi$6 action is not stationary under the natural condition $\varphi$7 because the variation produces boundary terms involving normal derivatives of $\varphi$8. The remedy advocated in the cited work is the scalar–tensor-correspondence boundary term

$\varphi$9

so that $\Psi$0 together with $\Psi$1 on $\Psi$2 gives a well-posed variational principle (0809.4033).

Independent-connection formalisms are likewise not interchangeable. The biconnection principle uses one connection effectively as a Lagrange multiplier and recovers standard Einstein dynamics, whereas the bimetric variational principle explicitly claims a physically distinct theory because the connection-generating metric carries new degrees of freedom and can propagate torsion. The latter is therefore not a reformulation of Palatini variation. This suggests that, within relativistic variational theory, extra geometric variables may serve very different roles: enforcing standard GR, enlarging the geometric sector, or imposing localized constraints on the classical solution space.

A related misconception concerns higher derivatives. In the Brans–Dicke Galileon with cubic self-interaction, the Lagrangian contains second-order derivatives, but the cited derivation shows that higher-order derivatives appearing in intermediate steps cancel, leading to second-order motion equations (Quiros et al., 2016).

5. Matter sectors: fluids, heat conduction, and spinor coupling

Variational Relativity has also been used to formulate relativistic matter systems directly. For the relativistic perfect fluid on curved spacetime, one formulation takes as independent fields the four-velocity $\Psi$3, density $\Psi$4, and pressure $\Psi$5, with action

$\Psi$6

Its Euler–Lagrange equations yield the continuity equation

$\Psi$7

the Euler equation

$\Psi$8

and the standard perfect-fluid stress tensor

$\Psi$9

in a diffeomorphism-invariant and reparametrization-invariant form (Ootsuka et al., 2016).

For irreversible thermodynamics, the multi-fluid variational approach treats the particle current $\mathcal M$0 and entropy current $\mathcal M$1 as independent matter fields and uses a master function $\mathcal M$2. The conjugate momenta $\mathcal M$3 and $\mathcal M$4, the canonical stress tensor

$\mathcal M$5

the extended Gibbs relation, and the causal heat-flux law all emerge from the same variational construction. In the matter frame, the entropy-momentum equation yields the relativistic Cattaneo law

$\mathcal M$6

and the cited comparison shows equivalence with the Israel–Stewart model at first order deviations from thermal equilibrium (Lopez-Monsalvo et al., 2010).

Spinor coupling provides a further extension. In the differential-form treatment of a Majorana neutrino coupled to Einstein gravity, the first-order action uses an orthonormal co-frame $\mathcal M$7, independent Lorentz connection $\mathcal M$8, a real Majorana spinor $\mathcal M$9, and Lagrange-multiplier 2-forms $$L\big(j^2(*g),\,j^2(*\Psi)\big)(m) = L\big(j^2g,\,j^2\Psi\big)\big(\varphi(m)\big),$$0. The same work states that the first- and second-order variational field equations are explicitly equivalent, and that the Lagrange multipliers are precisely the Belinfante–Rosenfeld 2-forms needed to symmetrize the canonical energy-momentum tensor of the Majorana spinor (Dereli et al., 2022).

6. Higher-order relativistic mechanics and adjacent developments

A separate but related lineage of variational work in relativity concerns higher-order world-line dynamics. For parameter-invariant second-order Lagrangians $$L\big(j^2(*g),\,j^2(*\Psi)\big)(m) = L\big(j^2g,\,j^2\Psi\big)\big(\varphi(m)\big),$$1, the Zermelo conditions

$$L\big(j^2(*g),\,j^2(*\Psi)\big)(m) = L\big(j^2g,\,j^2\Psi\big)\big(\varphi(m)\big),$$2

guarantee reparametrization invariance. Using Ostrogradsky momenta and the Weyssenhoff–Rund–Grässer Hamiltonian system, one obtains the Dixon equations

$$L\big(j^2(*g),\,j^2(*\Psi)\big)(m) = L\big(j^2g,\,j^2\Psi\big)\big(\varphi(m)\big),$$3

together with the Mathisson–Pirani condition $$L\big(j^2(*g),\,j^2(*\Psi)\big)(m) = L\big(j^2g,\,j^2\Psi\big)\big(\varphi(m)\big),$$4. A specific Hamilton function then yields a fourth-order “Zitterbewegung” equation in curved spacetime and reproduces Bopp’s second-derivative flat-space model for the self-radiating electron (Matsyuk, 2011).

In two dimensions, the Lagrangian

$$L\big(j^2(*g),\,j^2(*\Psi)\big)(m) = L\big(j^2g,\,j^2\Psi\big)\big(\varphi(m)\big),$$5

provides a variational formulation for geodesic circles: its extremals have constant Frenet curvature $$L\big(j^2(*g),\,j^2(*\Psi)\big)(m) = L\big(j^2g,\,j^2\Psi\big)\big(\varphi(m)\big),$$6. The cited analysis also relates this construction to uniform relativistic acceleration in flat space and to a one-dimensional spin–curvature coupling (Matsyuk, 2015). In four-dimensional Minkowski space, a further Lorentz-invariant higher-order construction yields a genuine third-order variational equation for the free relativistic top, with separate jerk, gyroscopic, and inertial contributions tied to a constant spin four-vector $$L\big(j^2(*g),\,j^2(*\Psi)\big)(m) = L\big(j^2g,\,j^2\Psi\big)\big(\varphi(m)\big),$$7 (Matsyuk, 2016).

These examples are not identical to Souriau’s continuum-mechanical program, but they share its central methodological commitment: relativistic dynamics is to be derived from covariant variational structure, with invariance requirements sharply constraining admissible degrees of freedom and equations of motion. The cited literature therefore suggests that “Variational Relativity” names both a specific Souriau framework and a broader research style in which covariance, constitutive admissibility, and dynamical equations are simultaneously fixed by variational principles rather than imposed piecemeal.