Transverse Diffeomorphisms (TDiff)

• TDiff is a subgroup of spacetime diffeomorphisms characterized by divergence-free vector fields that preserve volume and promote the metric determinant to a scalar.
• TDiff-invariant actions modify standard gravitational and gauge theories by allowing arbitrary functions of the metric determinant, leading to additional scalar modes and alternative cosmological dynamics.
• These models offer practical frameworks for addressing dark-sector interactions, super-Hubble magnetogenesis, and scalar-tensor equivalences, linking them to Weyl-transverse and unimodular gravity formulations.

Searching arXiv for recent and foundational papers on TDiff to ground the article. Transverse diffeomorphisms (TDiff) are the subgroup of spacetime diffeomorphisms generated by vector fields whose divergence vanishes, so that the coordinate transformation preserves volume and the metric determinant behaves as a scalar rather than as a density. In practice, the restriction

$\partial_\mu \xi^\mu=0$

or, in covariant form, $\nabla_\mu \xi^\mu=0$, weakens full diffeomorphism invariance while still allowing a broad class of generally covariant-looking models in which the matter or gravitational action may depend on arbitrary functions of $g\equiv |\det g_{\mu\nu}|$. This framework has been developed in pure gravity, scalar and gauge theories, cosmological model building, higher-spin systems, and Weyl-transverse formulations (Maroto et al., 2024, 0807.1293, Maroto, 2023).

1. Definition and subgroup structure

A general infinitesimal diffeomorphism acts as $x^\mu\to x^\mu+\xi^\mu(x)$. TDiff is the proper subgroup obtained by imposing the transversality or volume-preservation condition $\partial_\mu\xi^\mu=0$. Finite TDiff transformations therefore have unit Jacobian, $J=1$, and preserve the spacetime volume element. Algebraically, the generators close under the Lie bracket: $$[\xi,\eta]^\mu=\xi^\nu\partial_\nu\eta^\mu-\eta^\nu\partial_\nu\xi^\mu,$$ and the divergence-free condition is stable under this bracket. In four dimensions, the restriction leaves three local functional gauge degrees of freedom rather than the unrestricted Diff freedom (Maroto et al., 2024, Tessainer et al., 2024).

The central geometric consequence is the altered role of the determinant $g$. Under full Diff, $g$ is a density; under TDiff it behaves as a scalar. Equivalently, in formulations with a fixed background volume form $\bm\omega$, TDiff is defined by $\nabla_\mu \xi^\mu=0$0, which again implies $\nabla_\mu \xi^\mu=0$1. This observation underlies essentially all TDiff constructions: once $\nabla_\mu \xi^\mu=0$2 is promoted to a scalar under the reduced symmetry, arbitrary functions of $\nabla_\mu \xi^\mu=0$3 can appear in invariant actions (Odak et al., 22 Jan 2026, 0807.1293).

A closely related enlargement is Weyl-transverse symmetry, or WTDiff, in which TDiff is supplemented by local Weyl rescalings. In that case the tracelike determinant mode can be made gauge, and the combination $\nabla_\mu \xi^\mu=0$4 has unit determinant. WTDiff occupies a distinguished position because it removes the extra scalar mode that generic TDiff theories otherwise propagate (0807.1293, Bonifacio et al., 2015).

2. TDiff-invariant actions and symmetry restoration

The generic TDiff mechanism in the matter sector is simple. Instead of the Diff-enforced measure $\nabla_\mu \xi^\mu=0$5, one considers

$\nabla_\mu \xi^\mu=0$6

with arbitrary positive $\nabla_\mu \xi^\mu=0$7. Under a general diffeomorphism,

$\nabla_\mu \xi^\mu=0$8

so full Diff invariance requires $\nabla_\mu \xi^\mu=0$9, whereas TDiff invariance only requires $g\equiv |\det g_{\mu\nu}|$0. In scalar models this produces actions such as

$g\equiv |\det g_{\mu\nu}|$1

and in Abelian gauge theory the standard Maxwell action becomes

$g\equiv |\det g_{\mu\nu}|$2

For $g\equiv |\det g_{\mu\nu}|$3, the Diff-invariant limit is recovered (Jaramillo-Garrido et al., 2024, Maroto et al., 2024, Maroto, 2023).

The reduced spacetime symmetry does not by itself remove internal gauge symmetries. In the TDiff Maxwell theory, $g\equiv |\det g_{\mu\nu}|$4 remains a symmetry and commutes with TDiff; correspondingly, no new gauge modes appear. The field equation is

$g\equiv |\det g_{\mu\nu}|$5

For Dirac fields coupled to the gauge sector, the condition $g\equiv |\det g_{\mu\nu}|$6 is required in order to avoid violations of Einstein’s Equivalence Principle, while the leading WKB dynamics still yields the usual Lorentz-force law (Maroto et al., 2024).

A recurrent structural theme is “covariantization,” namely the restoration of full Diff by introducing compensator fields. For scalar TDiff matter one may define

$g\equiv |\det g_{\mu\nu}|$7

or $g\equiv |\det g_{\mu\nu}|$8, and rewrite the theory in a fully Diff-invariant form,

$g\equiv |\det g_{\mu\nu}|$9

In the TDiff gauge one sets $x^\mu\to x^\mu+\xi^\mu(x)$0 or, equivalently, fixes the associated scalar density $x^\mu\to x^\mu+\xi^\mu(x)$1 to unity, recovering the original TDiff action. This construction preserves locality and makes the energy-momentum tensor and perturbation theory accessible in standard covariant language (Jaramillo-Garrido et al., 2024, Pérez et al., 3 Apr 2025).

3. Gravitational realizations, extra modes, and equivalence results

In pure gravity, the most general two-derivative TDiff action in $x^\mu\to x^\mu+\xi^\mu(x)$2 dimensions can be written as

$x^\mu\to x^\mu+\xi^\mu(x)$3

Because $x^\mu\to x^\mu+\xi^\mu(x)$4 is a scalar under TDiff, the determinant mode is dynamical, and generic TDiff gravity propagates an additional scalar besides the usual spin-two graviton. After a conformal redefinition to Einstein frame, the theory becomes GR coupled to a scalar $x^\mu\to x^\mu+\xi^\mu(x)$5, making the extra degree of freedom explicit (0807.1293).

This extra scalar is central to both the classical and quantum structure. At one loop, generic TDiff gravity remains divergent, even when the cosmological constant is tuned to zero. The two exceptional on-shell finite cases are ordinary Einstein gravity, obtained for $x^\mu\to x^\mu+\xi^\mu(x)$6, and WTDiff, obtained for $x^\mu\to x^\mu+\xi^\mu(x)$7, where the scalar mode becomes pure gauge under the additional Weyl symmetry. These are precisely the cases in which the determinant mode does not survive as an independent propagating field (0807.1293).

A distinct but related line of work establishes that TDiff gravity is classically equivalent to a very general scalar-tensor theory, in close analogy with the relation between unimodular gravity and GR. Introducing an independent scalar $x^\mu\to x^\mu+\xi^\mu(x)$8 and then imposing the algebraic gauge

$x^\mu\to x^\mu+\xi^\mu(x)$9

maps the scalar-tensor parent action to the TDiff action. The same analysis yields a general result for gauge fixing without derivatives: under suitable boundary conditions, imposing the gauge before variation is equivalent to imposing it after deriving the equations of motion (Lopez-Villarejo, 2010).

Other gravitational realizations use additional constraints. In the curvature-density preserving construction, the condition

$\partial_\mu\xi^\mu=0$0

breaks Diff to TDiff. The Lagrange multiplier enforcing this condition becomes constant on shell when $\partial_\mu\xi^\mu=0$1, and the Einstein equations emerge with an effective Newton constant

$\partial_\mu\xi^\mu=0$2

In the limit $\partial_\mu\xi^\mu=0$3, the Einstein-Hilbert term disappears from the action but Einstein’s equations persist, realizing induced gravity with the Newton coupling appearing as an integration constant (Oda, 2016).

At the linearized level, TDiff is also the minimal symmetry used in several spin-2 constructions. Generic TDiff kinetic terms propagate massless spin-2 plus a scalar; the Diff and WTDiff special points remove the lower-spin sector. Kaluza-Klein reductions of TDiff- and WTDiff-invariant massless theories produce ghost-free massive scalar-tensor or massive spin-2 models with smooth massless limits, and the massive WTDiff formulation is physically equivalent to Fierz-Pauli after suitable Stückelberg transformations and gauge fixing (Dalmazi et al., 2020, Bonifacio et al., 2015).

4. Gauge fields, scalar fields, and super-Hubble dynamics

The cosmological significance of TDiff appears most sharply in theories whose matter sector would otherwise be conformally trivial. For an Abelian gauge field in a Robertson-Walker geometry

$\partial_\mu\xi^\mu=0$4

with $\partial_\mu\xi^\mu=0$5, the mode equation becomes

$\partial_\mu\xi^\mu=0$6

In the Diff limit $\partial_\mu\xi^\mu=0$7, one has $\partial_\mu\xi^\mu=0$8 and the usual conformal triviality of Maxwell theory is recovered. For $\partial_\mu\xi^\mu=0$9, conformal invariance is broken, and the super-Hubble evolution of magnetic fields is modified (Maroto et al., 2024).

The same gauge theory has a standard high-frequency limit. In geometric optics, photons satisfy $J=1$0, propagate along null geodesics, carry a conserved number current

$J=1$1

and have polarization vectors orthogonal to the propagation direction, with the transverse polarization parallel transported along the ray. Thus the TDiff deformation does not alter the leading eikonal propagation of the gauge bosons, even though it changes the infrared gravitational behavior of electric and magnetic sectors (Maroto et al., 2024).

For power-law couplings $J=1$2, the super-Hubble magnetic energy density scales as

$J=1$3

whereas in the sub-Hubble regime the total electromagnetic energy still scales as $J=1$4. The electric and magnetic contributions enter the energy-momentum tensor with different weights unless $J=1$5, so Diff breaking makes the electric and magnetic parts gravitate differently. This difference is the basis of the TDiff magnetogenesis scenario (Maroto et al., 2024, Maroto et al., 2024).

Matching an approximately scale-invariant inflationary spectrum at horizon exit yields the present magnetic power spectrum

$J=1$6

The parameter region giving intergalactic fields compatible with current observations is reported as $J=1$7 and $J=1$8, with $J=1$9 today. In particular, $$[\xi,\eta]^\mu=\xi^\nu\partial_\nu\eta^\mu-\eta^\nu\partial_\nu\xi^\mu,$$0 with $$[\xi,\eta]^\mu=\xi^\nu\partial_\nu\eta^\mu-\eta^\nu\partial_\nu\xi^\mu,$$1 can account for the lower blazar bounds without dynamo amplification, while CMB bounds exclude values too close to unity (Maroto et al., 2024).

Scalar TDiff cosmology shows a parallel pattern. In the geometric-optics limit, scalar particles follow geodesics as in Diff-invariant theories when $$[\xi,\eta]^\mu=\xi^\nu\partial_\nu\eta^\mu-\eta^\nu\partial_\nu\xi^\mu,$$2, but super-Hubble stress-energy contributions differ because the determinant enters explicitly. For purely kinetic models with $$[\xi,\eta]^\mu=\xi^\nu\partial_\nu\eta^\mu-\eta^\nu\partial_\nu\xi^\mu,$$3, the equation of state is

$$[\xi,\eta]^\mu=\xi^\nu\partial_\nu\eta^\mu-\eta^\nu\partial_\nu\xi^\mu,$$4

so $$[\xi,\eta]^\mu=\xi^\nu\partial_\nu\eta^\mu-\eta^\nu\partial_\nu\xi^\mu,$$5 gives dust, $$[\xi,\eta]^\mu=\xi^\nu\partial_\nu\eta^\mu-\eta^\nu\partial_\nu\xi^\mu,$$6 gives radiation, and $$[\xi,\eta]^\mu=\xi^\nu\partial_\nu\eta^\mu-\eta^\nu\partial_\nu\xi^\mu,$$7 gives accelerated expansion. Detailed analyses of the kinetic-dominated regime show that the fluid is adiabatic there, with an explicit effective sound speed determined by the coupling functions, while the potential-dominated regime yields cosmological-constant-type behavior (Maroto, 2023, Jaramillo-Garrido et al., 2023, Jaramillo-Garrido et al., 2024).

5. Multi-field TDiff cosmology, dark-sector exchange, and $$[\xi,\eta]^\mu=\xi^\nu\partial_\nu\eta^\mu-\eta^\nu\partial_\nu\xi^\mu,$$8CDM-like models

When several matter fields couple through $$[\xi,\eta]^\mu=\xi^\nu\partial_\nu\eta^\mu-\eta^\nu\partial_\nu\xi^\mu,$$9-dependent functions, TDiff breaking induces effective interactions even if the Lagrangian contains no explicit inter-field term. In a multi-scalar TDiff model,

$g$0

the Einstein-Hilbert term remains Diff-invariant, so the Bianchi identity enforces conservation of the total energy-momentum tensor,

$g$1

but the individual field contributions need not be separately conserved. For two homogeneous components one may write

$g$2

with the exchange term $g$3 fixed by the determinant-dependent couplings and the geometrical constraint relating $g$4 and $g$5 (Tessainer et al., 2024, Maroto et al., 22 Jul 2025).

For power-law couplings $g$6, each field has a constant native barotropic parameter

$g$7

but the subdominant component evolves with a shifted effective equation of state,

$g$8

This shift is a direct consequence of the TDiff constraint rather than an explicit interaction potential. In dark-sector applications, taking one field as dark matter with $g$9 and the other as dark energy with $g$0 can produce a transient phantom era with $g$1, followed by phantom crossing as the dark-energy component comes to dominate. Fits to Union2 supernovae and CMB distance priors yield a best-fit value around $g$2, with a statistical goodness-of-fit comparable to a $g$3CDM model (Tessainer et al., 2024).

A different unified dark-sector construction uses a single scalar with

$g$4

Then $g$5 gives an early dust-like phase, while $g$6 gives a late dark-energy phase. The model has the same number of free parameters as flat $g$7CDM and was confronted with Planck 2018 CMB distance priors, Pantheon+ and SH0ES supernovae, BAO, and cosmic chronometers. The reported best fit is $g$8, $g$9, and $\bm\omega$0, with $\bm\omega$1 and $\bm\omega$2 in favor of the TDiff model over $\bm\omega$3CDM in the joint fit (Alonso-López et al., 2023).

There also exist exact $\bm\omega$4CDM realizations. A single canonical scalar coupled through the TDiff-invariant volume element

$\bm\omega$5

can be rewritten in a Diff-restored scalar-vector form with $\bm\omega$6. In the special linear case, the on-shell energy-momentum tensor splits exactly into pressureless dust plus a cosmological constant,

$\bm\omega$7

and the sound speed is exactly zero in any background geometry. The resulting background equation of state coincides with that of $\bm\omega$8CDM (Pérez et al., 3 Apr 2025).

The mixed-regime multi-field models extend this picture by combining a potential-dominated field with a kinetic-dominated one. In the covariantized language, the exchange current takes the explicit form

$\bm\omega$9

For power-law couplings, under kinetic domination one obtains $\nabla_\mu \xi^\mu=0$00 and an effective dark-energy equation of state

$\nabla_\mu \xi^\mu=0$01

so $\nabla_\mu \xi^\mu=0$02 gives phantom behavior while $\nabla_\mu \xi^\mu=0$03 gives quintessence-like behavior. This provides a purely gravitational origin for dark-sector energy exchange (Maroto et al., 22 Jul 2025).

6. Perturbations, observational constraints, and recent structural developments

Cosmological perturbation theory in TDiff matter models differs from the Diff case because the determinant dependence produces new algebraic relations among perturbations. In single-field covariantized models, the linearized $\nabla_\mu \xi^\mu=0$04-equation eliminates $\nabla_\mu \xi^\mu=0$05 and yields an effective sound speed

$\nabla_\mu \xi^\mu=0$06

which generally differs from the adiabatic speed $\nabla_\mu \xi^\mu=0$07. In the kinetic-dominated limit, however,

$\nabla_\mu \xi^\mu=0$08

so perturbations are adiabatic there; in the potential-dominated limit one finds $\nabla_\mu \xi^\mu=0$09. Multi-field systems acquire additional non-adiabatic terms, and the total pressure perturbation takes the form

$\nabla_\mu \xi^\mu=0$10

Stability requires $\nabla_\mu \xi^\mu=0$11 and $\nabla_\mu \xi^\mu=0$12 (Maroto et al., 18 May 2026).

At the phenomenological level, the scalar-tensor equivalence of TDiff gravity implies the usual post-Newtonian and equivalence-principle bounds. The parameterized post-Newtonian result

$\nabla_\mu \xi^\mu=0$13

implies $\nabla_\mu \xi^\mu=0$14. The same analysis quotes

$\nabla_\mu \xi^\mu=0$15

and concludes that realistic models generally require either approximate metric coupling, a large scalar mass, or screening to satisfy observational constraints (Lopez-Villarejo, 2010).

In broken-diffeomorphism gravity with action

$\nabla_\mu \xi^\mu=0$16

the extra scalar graviton decouples from conserved matter sources at linear order, all propagating modes satisfy $\nabla_\mu \xi^\mu=0$17, and the local post-Newtonian parameters remain

$\nabla_\mu \xi^\mu=0$18

Cosmologically, the theory contains exact $\nabla_\mu \xi^\mu=0$19CDM backgrounds when the scalar mode is unexcited, but also admits tracker solutions, stiff-fluid behavior, late-time cosmological-constant-like freezing for soft Diff breaking, and expanding-to-contracting histories with recollapse (Bello-Morales et al., 2023).

Recent work on Weyl-transverse gravity with boundaries extends the formal structure of TDiff-type symmetry to the covariant phase space. In that setting the Noether surface charge coincides with the Wald-Iyer GR charge,

$\nabla_\mu \xi^\mu=0$20

but the Hamiltonian identity on a bifurcate Killing horizon yields an extended first law,

$\nabla_\mu \xi^\mu=0$21

so variations of the cosmological constant can contribute nontrivially unless further physical restrictions are imposed (Odak et al., 22 Jan 2026).

The same symmetry principle also appears beyond cosmology. For generalized TDiff-invariant massless spin-2 and spin-3 theories, the ghost-free region is characterized by positivity conditions such as $\nabla_\mu \xi^\mu=0$22, and within that region all models are related by local and mildly nonlocal field redefinitions to the simplest Maxwell-like representatives. The reported conclusion is that physically admissible TDiff models differ only by BRST-cohomologically trivial terms and are equivalent to the “doublet” actions appearing in the tensionless limit of open bosonic string field theory (Bittencourt et al., 8 Jan 2025).

Taken together, these results show that TDiff is not a single model but a broad organizing principle. Its defining move is always the same—restrict Diff to volume-preserving transformations—but its consequences range from determinant-driven scalar graviton dynamics and scalar-tensor equivalence to super-Hubble magnetogenesis, dark-sector energy exchange, exact $\nabla_\mu \xi^\mu=0$23CDM realizations, modified perturbation theory, and refined boundary charges.