Conservation Group: Invariance in Quantum Geometry

• Conservation Group is a family of group-theoretic constructions that extend coordinate invariance to preserve explicit conservation laws.
• The framework employs tetrad-based and gauge approaches to derive effective stress-energy tensors, naturally incorporating dark matter and dark energy.
• It unifies symmetry analysis in PDEs and continuum mechanics by linking Noether's theorem with geometric and physical conservation laws.

“Conservation group” denotes a family of group-theoretic constructions that tie invariance to conserved quantities. In one major line of work, it is an enlarged covariance group of coordinate transformations on a tetrad-based spacetime that strictly contains the diffeomorphism group as a proper subgroup and is proposed as a candidate foundational group for quantum geometry (Green, 2017). In adjacent literature, closely related structures appear as gauge groups for gravitational energy-momentum, as symmetry groups of systems of conservation laws, and as admitted Lie or quasigroup symmetries from which conservation laws are derived for nonlinear field equations (Wiesendanger, 2011, Sever, 25 Sep 2025, Robles et al., 24 Oct 2025). The unifying theme is that conservation is not treated as a purely kinematic byproduct, but as a consequence of an explicitly organized symmetry structure.

1. Enlarged covariance and the conservative-transformation definition

In the tetrad-based formulation associated with Pandres, spacetime is a real four-dimensional manifold $\mathcal{X}^4$ equipped with an orthonormal tetrad $h^i{}_\mu$, with metric

$$g_{\mu\nu}=\eta_{ij}h^i{}_\mu h^j{}_\nu,\qquad \eta_{ij}=\mathrm{diag}(-1,1,1,1).$$

The defining idea is to enlarge covariance beyond diffeomorphism invariance by requiring invariance of a conservation-law form,

$\tilde V^\alpha{}_{,\alpha}=0,$

or, equivalently in the later formulation, invariance of the scalar wave equation under the largest admissible coordinate group (Green, 2017, Green, 1 Apr 2026).

The corresponding transformations satisfy

$$x^\nu{}_{,\bar\alpha}\bigl(x^{\bar\alpha}{}_{,\nu,\mu}-x^{\bar\alpha}{}_{,\mu,\nu}\bigr)=0.$$

These are called conservative transformations, and the set of all such transformations is the conservation group. The construction is explicitly stated to contain the diffeomorphism group as a proper subgroup (Green, 2017). In this framework, a non-diffeomorphic but conservative transformation can be interpreted as an anholonomic change of coordinates or as a mapping between manifolds in the same “quantum family” (Green, 2017).

The conceptual claim attached to this enlargement is strong: the conservation group is presented as the “largest group of coordinate transformations” under which the relevant conservation-law structure remains invariant, and it is hypothesized to be a foundational group for quantum geometry (Green, 2017, Green, 1 Apr 2026).

2. Curvature vector, action principle, and effective stress-energy

The central geometric object is the curvature vector

$$C_\alpha \equiv h_i{}^\nu\bigl(h^i{}_{\alpha,\nu}-h^i{}_{\nu,\alpha}\bigr)=\gamma^\mu{}_{\alpha\mu},$$

where $\gamma^i{}_{\mu\nu}=h^i{}_{\mu;\nu}$ is the Ricci rotation coefficient (Green, 2017). In the extended formulation with internal fields $\Lambda^i{}_I$, one instead uses

$$\Upsilon^\alpha{}_{\mu\nu} \equiv h_I{}^\alpha h^I{}_{\mu;\nu}+h_i{}^\alpha h^I{}_\mu \Lambda^i{}_{I,\nu}, \qquad C_\alpha\equiv \Upsilon^\mu{}_{\alpha\mu},$$

which makes explicit that the free-field Lagrangian differs from the Einstein–Hilbert form by additional geometric terms (Green, 2017).

A defining property is that $C_\alpha$ is covariant under a transformation if and only if the transformation is conservative (Green, 2017). The free-field action is built from the scalar $h^i{}_\mu$0: $h^i{}_\mu$1 In the 2026 development, the identity

$h^i{}_\mu$2

is used to exhibit the deviation from general relativity through extra quadratic Ricci-rotation terms (Green, 1 Apr 2026).

Variation of the free-field action yields field equations of the form

$h^i{}_\mu$3

in the earlier formulation, and the same geometric structure is retained in the later source-augmented framework (Green, 2017, Green, 1 Apr 2026). In Einstein form, the purely geometric sector contributes a nonzero effective stress-energy tensor even where no explicit matter source is present. This is the mechanism by which the theory is said to bring in dark matter and dark energy “naturally” as geometric effects (Green, 2017).

With sources included, the total Lagrangian is

$h^i{}_\mu$4

and the total stress-energy is decomposed into $h^i{}_\mu$5, with only the sum satisfying

$h^i{}_\mu$6

in the usual covariant sense (Green, 1 Apr 2026).

In the weak-field limit, the tetrad is linearized as

$h^i{}_\mu$7

which leads, in harmonic coordinates and for the trace-reversed perturbation $h^i{}_\mu$8, to

$h^i{}_\mu$9

The stated consequence is that gravitational waves propagate at the speed of light (Green, 1 Apr 2026).

3. Spherical solutions, dark-sector interpretation, and galaxy models

A major application of conservation-group geometry is spherically symmetric gravitation. In the 2017 analysis, the free-field and source-supported equations are used to construct spherical metrics with nonzero effective density and anisotropic pressures even in regions with no explicit matter density (Green, 2017). For the free-field tetrad ansatz, one obtains $$g_{\mu\nu}=\eta_{ij}h^i{}_\mu h^j{}_\nu,\qquad \eta_{ij}=\mathrm{diag}(-1,1,1,1).$$0 while the Einstein tensor remains nonzero, so the geometry itself behaves as an effective source (Green, 2017).

The same paper introduces an idealized halo model with

$$g_{\mu\nu}=\eta_{ij}h^i{}_\mu h^j{}_\nu,\qquad \eta_{ij}=\mathrm{diag}(-1,1,1,1).$$1

for which the circular velocity obeys

$$g_{\mu\nu}=\eta_{ij}h^i{}_\mu h^j{}_\nu,\qquad \eta_{ij}=\mathrm{diag}(-1,1,1,1).$$2

independent of radius across the halo to first order in $$g_{\mu\nu}=\eta_{ij}h^i{}_\mu h^j{}_\nu,\qquad \eta_{ij}=\mathrm{diag}(-1,1,1,1).$$3. This yields flat rotation curves and is used to relate the model to the baryonic Tully–Fisher relation and to a modified radial-acceleration relation (Green, 2017). Matching the model to the McGaugh–Lelli–Schombert acceleration scale gives

$$g_{\mu\nu}=\eta_{ij}h^i{}_\mu h^j{}_\nu,\qquad \eta_{ij}=\mathrm{diag}(-1,1,1,1).$$4

interpreted there as a critical baryonic acceleration separating baryon-dominated from dark-matter-dominated behavior (Green, 2017).

The 2026 extension revises the spherical program by arguing that the free-field sector alone is not astrophysically acceptable. In that case, the weak-field limit gives

$$g_{\mu\nu}=\eta_{ij}h^i{}_\mu h^j{}_\nu,\qquad \eta_{ij}=\mathrm{diag}(-1,1,1,1).$$5

rather than the general-relativistic

$$g_{\mu\nu}=\eta_{ij}h^i{}_\mu h^j{}_\nu,\qquad \eta_{ij}=\mathrm{diag}(-1,1,1,1).$$6

so a source term in the Lagrangian is required (Green, 1 Apr 2026). The resulting galaxy model is partitioned into three spherically symmetric regions: a baryonic matter-dominated central bulge, a dark matter-dominated mesosphere, and an outside region where neither type dominates (Green, 1 Apr 2026).

In the bulge, the baryonic density is taken to have a central cusp,

$$g_{\mu\nu}=\eta_{ij}h^i{}_\mu h^j{}_\nu,\qquad \eta_{ij}=\mathrm{diag}(-1,1,1,1).$$7

while in the mesosphere an isothermal condition implies

$$g_{\mu\nu}=\eta_{ij}h^i{}_\mu h^j{}_\nu,\qquad \eta_{ij}=\mathrm{diag}(-1,1,1,1).$$8

This again yields flat rotation curves through

$$g_{\mu\nu}=\eta_{ij}h^i{}_\mu h^j{}_\nu,\qquad \eta_{ij}=\mathrm{diag}(-1,1,1,1).$$9

with a single free parameter in the mesosphere (Green, 1 Apr 2026). The outer region is matched to previously published asymptotically weak-field behavior, and continuity at $\tilde V^\alpha{}_{,\alpha}=0,$0 and $\tilde V^\alpha{}_{,\alpha}=0,$1 produces a continuous galaxy model. The radial acceleration relation is then used as the practical input for constructing rotationally supported galaxies within the framework (Green, 1 Apr 2026).

4. Related gauge and asymptotic symmetry constructions

A distinct but related use of conservation-group language appears in a gauge theory of gravity based on the independent conservation of inertial and gravitational energy-momentum. In that construction, inertial four-momentum $\tilde V^\alpha{}_{,\alpha}=0,$2 is conserved by spacetime translation invariance, while gravitational four-momentum $\tilde V^\alpha{}_{,\alpha}=0,$3 is conserved by inner translation invariance in an auxiliary inner space (Wiesendanger, 2011). Gauging the latter yields a theory of the non-compact group of volume-preserving diffeomorphisms of inner Minkowski space,

$\tilde V^\alpha{}_{,\alpha}=0,$4

with covariant derivative

$\tilde V^\alpha{}_{,\alpha}=0,$5

field strength

$\tilde V^\alpha{}_{,\alpha}=0,$6

and minimal gauge-field Lagrangian

$\tilde V^\alpha{}_{,\alpha}=0,$7

(Wiesendanger, 2011).

The declared payoff is that gravitational energy-momentum becomes a genuine Noether charge rather than a coordinate-dependent pseudotensor. The Hamiltonian formulation in axial gauge reduces to two times six unconstrained independent canonical variables and is nonnegative under a Lorentz-invariant support condition in inner momentum space (Wiesendanger, 2011). This is presented as a conservation-group-based gauge theory of gravity (Wiesendanger, 2011).

A further related development replaces asymptotic symmetry groups by quasigroups at future null infinity. The Newman–Unti group is reduced to a Poincaré quasigroup, and Noether charges linear on the generators of the Poincaré quasialgebra are defined so as to vanish identically in Minkowski spacetime and to be free of supertranslation ambiguity (Robles et al., 24 Oct 2025). In the center-of-mass reference frame, intrinsic angular momentum is given by a Komar-type expression,

$\tilde V^\alpha{}_{,\alpha}=0,$8

which is the null-infinity counterpart of selecting an intrinsic, rather than orbital, angular momentum (Robles et al., 24 Oct 2025).

5. Symmetry groups of conservation-law systems and Euler-type rigidity

In the mathematical theory of conservation laws, the symmetry group of a system is itself taken as a structural invariant. For quasi-linear weak-form systems

$\tilde V^\alpha{}_{,\alpha}=0,$9

the symmetry group $$x^\nu{}_{,\bar\alpha}\bigl(x^{\bar\alpha}{}_{,\nu,\mu}-x^{\bar\alpha}{}_{,\mu,\nu}\bigr)=0.$$0 acts by transformations of the independent variables, dependent variables, and the equation array, and every such system has at least translations in $$x^\nu{}_{,\bar\alpha}\bigl(x^{\bar\alpha}{}_{,\nu,\mu}-x^{\bar\alpha}{}_{,\mu,\nu}\bigr)=0.$$1 and uniform scaling of $$x^\nu{}_{,\bar\alpha}\bigl(x^{\bar\alpha}{}_{,\nu,\mu}-x^{\bar\alpha}{}_{,\mu,\nu}\bigr)=0.$$2 as trivial symmetries (Sever, 25 Sep 2025).

When an entropy extension exists, the flux matrix can be rewritten in gradient form

$$x^\nu{}_{,\bar\alpha}\bigl(x^{\bar\alpha}{}_{,\nu,\mu}-x^{\bar\alpha}{}_{,\mu,\nu}\bigr)=0.$$3

and the symmetry condition simplifies to

$$x^\nu{}_{,\bar\alpha}\bigl(x^{\bar\alpha}{}_{,\nu,\mu}-x^{\bar\alpha}{}_{,\mu,\nu}\bigr)=0.$$4

for linear infinitesimal actions (Sever, 25 Sep 2025). A particularly important subclass is formed by $$x^\nu{}_{,\bar\alpha}\bigl(x^{\bar\alpha}{}_{,\nu,\mu}-x^{\bar\alpha}{}_{,\mu,\nu}\bigr)=0.$$5-systems, characterized by

$$x^\nu{}_{,\bar\alpha}\bigl(x^{\bar\alpha}{}_{,\nu,\mu}-x^{\bar\alpha}{}_{,\mu,\nu}\bigr)=0.$$6

Within this class the nontrivial symmetry group acquires a linear decomposition, and its richness is quantified by

$$x^\nu{}_{,\bar\alpha}\bigl(x^{\bar\alpha}{}_{,\nu,\mu}-x^{\bar\alpha}{}_{,\mu,\nu}\bigr)=0.$$7

The main classification result is that, under the stated hyperbolicity, entropy, and reduction hypotheses, maximal symmetry leads precisely to Euler systems: the $$x^\nu{}_{,\bar\alpha}\bigl(x^{\bar\alpha}{}_{,\nu,\mu}-x^{\bar\alpha}{}_{,\mu,\nu}\bigr)=0.$$8 case yields isentropic Euler or extended Euler systems, while the $$x^\nu{}_{,\bar\alpha}\bigl(x^{\bar\alpha}{}_{,\nu,\mu}-x^{\bar\alpha}{}_{,\mu,\nu}\bigr)=0.$$9 case corresponds to a variant conserving entropy rather than energy (Sever, 25 Sep 2025). The paper uses this to support the conjecture that physically attractive higher-dimensional conservation-law models are strongly constrained to be Euler-like (Sever, 25 Sep 2025).

An earlier and complementary classification imposes invariance under the Galileo group on one-dimensional hyperbolic conservation laws,

$$C_\alpha \equiv h_i{}^\nu\bigl(h^i{}_{\alpha,\nu}-h^i{}_{\nu,\alpha}\bigr)=\gamma^\mu{}_{\alpha\mu},$$0

together with a strictly convex entropy $$C_\alpha \equiv h_i{}^\nu\bigl(h^i{}_{\alpha,\nu}-h^i{}_{\nu,\alpha}\bigr)=\gamma^\mu{}_{\alpha\mu},$$1 and entropy flux $$C_\alpha \equiv h_i{}^\nu\bigl(h^i{}_{\alpha,\nu}-h^i{}_{\nu,\alpha}\bigr)=\gamma^\mu{}_{\alpha\mu},$$2 (Dubois, 2011). Under those assumptions, there is no nontrivial scalar example, all two-component systems fall into hyperbolic or elliptic Galileo classes, and three-component systems fall into hyperbolic, elliptic, or nilpotent Galileo classes (Dubois, 2011). The classical Euler equations of gas dynamics are recovered as a particular nilpotent Galileo system (Dubois, 2011). This is a group-theoretic rigidity statement parallel to the Euler-type rigidity obtained in the $$C_\alpha \equiv h_i{}^\nu\bigl(h^i{}_{\alpha,\nu}-h^i{}_{\nu,\alpha}\bigr)=\gamma^\mu{}_{\alpha\mu},$$3-system framework.

6. PDE group analysis, self-adjointness, and Noether organization

A broader operational use of conservation-group ideas appears in the PDE literature, where admitted symmetry groups are the mechanism by which conservation laws are classified and constructed. For the generalized two-dimensional Kuramoto–Sivashinsky equation,

$$C_\alpha \equiv h_i{}^\nu\bigl(h^i{}_{\alpha,\nu}-h^i{}_{\nu,\alpha}\bigr)=\gamma^\mu{}_{\alpha\mu},$$4

the baseline symmetry algebra is

$$C_\alpha \equiv h_i{}^\nu\bigl(h^i{}_{\alpha,\nu}-h^i{}_{\nu,\alpha}\bigr)=\gamma^\mu{}_{\alpha\mu},$$5

no strictly self-adjoint subcase exists, and conservation laws are produced only for the quasi self-adjoint and nonlinear self-adjoint subclasses (Dimas et al., 2012). The paper emphasizes that conservation laws are tied to the intersection of self-adjointness and the admitted Lie symmetry algebra (Dimas et al., 2012).

For the Lin–Tsien equation,

$$C_\alpha \equiv h_i{}^\nu\bigl(h^i{}_{\alpha,\nu}-h^i{}_{\nu,\alpha}\bigr)=\gamma^\mu{}_{\alpha\mu},$$6

the full point symmetry group is generated by a modified Clarkson–Kruskal construction, and its infinitesimal form closes into an infinite-dimensional Kac-Moody-Virasoro algebra (Xi-Zhong, 2012). A second-order Lie–Bäcklund analysis then produces infinitely many conservation laws, with conserved densities and fluxes depending on arbitrary functions of second-order group invariants (Xi-Zhong, 2012).

The moving-frame formulation of Noether theory makes this symmetry–conservation relation algebraically explicit. For invariant variational problems, Noether laws can be written as

$$C_\alpha \equiv h_i{}^\nu\bigl(h^i{}_{\alpha,\nu}-h^i{}_{\nu,\alpha}\bigr)=\gamma^\mu{}_{\alpha\mu},$$7

or, in the one-dimensional case,

$$C_\alpha \equiv h_i{}^\nu\bigl(h^i{}_{\alpha,\nu}-h^i{}_{\nu,\alpha}\bigr)=\gamma^\mu{}_{\alpha\mu},$$8

with $$C_\alpha \equiv h_i{}^\nu\bigl(h^i{}_{\alpha,\nu}-h^i{}_{\nu,\alpha}\bigr)=\gamma^\mu{}_{\alpha\mu},$$9 a moving frame and $\gamma^i{}_{\mu\nu}=h^i{}_{\mu;\nu}$0 a vector of differential invariants (Goncalves et al., 2010). For Euclidean-invariant Lagrangians this becomes

$\gamma^i{}_{\mu\nu}=h^i{}_{\mu;\nu}$1

which is used to reduce extremal-curve problems for $\gamma^i{}_{\mu\nu}=h^i{}_{\mu;\nu}$2 and $\gamma^i{}_{\mu\nu}=h^i{}_{\mu;\nu}$3 to invariant curvature–torsion equations plus quadratures (Goncalves et al., 2011).

The same pattern extends to continuum models. The equations of a two-dimensional uniformly stratified rotating fluid are quasi-self-adjoint, which allows Lie symmetries to be converted into local conservation laws; in that model the dilation symmetry yields the energy density

$\gamma^i{}_{\mu\nu}=h^i{}_{\mu;\nu}$4

and the associated integral conservation law (Ibragimov et al., 2011). In plane one-dimensional MHD and one-dimensional polytropic gas flow, Lie classification in Lagrangian coordinates identifies the conductivity $\gamma^i{}_{\mu\nu}=h^i{}_{\mu;\nu}$5 or entropy profile $\gamma^i{}_{\mu\nu}=h^i{}_{\mu;\nu}$6 that permit symmetry extensions, and the variational cases yield mass, momentum, energy, magnetic-flux, angular-momentum, and center-of-mass conservation laws through Noether’s theorem (Dorodnitsyn et al., 2021, Dorodnitsyn et al., 2020).

Across these settings, the recurring structure is precise: conservation laws are organized by an admitted symmetry group, but the specific group varies with the model. In gravitational geometry the conservation group is an enlarged covariance principle; in gauge and asymptotic gravity it becomes an inner diffeomorphism group or a Poincaré quasigroup; in PDE analysis it is the symmetry algebra through which conserved vectors are selected, normalized, or classified.