Thomas–Whitehead Gravity

• Thomas–Whitehead gravity is a projectively invariant extension of Einstein–Hilbert gravity, promoting unparameterized geodesics to dynamic variables.
• It employs the Thomas cone and projective connection to integrate new angular-momentum scales and a cosmological constant into gravitational dynamics.
• The action’s inclusion of a projective Gauss–Bonnet term enables analysis of radiative sectors, matter couplings, and potential dark matter interactions.

Searching arXiv for papers on Thomas-Whitehead Gravity to ground the article in the relevant literature. Thomas–Whitehead gravity is a projectively invariant extension of Einstein–Hilbert gravity in which the geometry of unparameterized geodesics, rather than affine parametrization alone, is promoted to dynamical significance. Its central gauge-geometric object is the projective connection, realized on spacetime through the rank-two diffeomorphism field $\mathcal D_{ab}$, and on the $(d+1)$-dimensional Thomas cone through an affine connection $\tilde\Gamma^\alpha{}_{\beta\gamma}$. In this framework, projective Gauss–Bonnet terms endow the projective sector with dynamics, the theory reduces to general relativity when the projective fields vanish and the connection becomes Levi-Civita, and in four dimensions it admits a formulation in which the cosmological constant is tied to a new angular-momentum coupling scale $J_0$ (Brensinger et al., 2019, Brensinger et al., 2020, Grover et al., 2024).

1. Projective-geometric basis

Thomas–Whitehead gravity begins from the classical fact that torsion-free affine connections related by a projective shift define the same unparameterized geodesics. In the form emphasized in the literature,

$$\hat{\Gamma}^i{}_{jk} = \Gamma^i{}_{jk} + \delta^i{}_k A_j + \delta^i{}_j A_k,$$

or equivalently

$$\Gamma^a{}_{bc}\to \Gamma^a{}_{bc}+\delta^a{}_b v_c+\delta^a{}_c v_b,$$

the connection changes, but the geodesic paths are preserved up to reparameterization. Projective geometry therefore isolates the “paths” of free-fall rather than their affine parameterization (Brensinger et al., 2019, Grover et al., 2024).

This geometric starting point is tied, in the TW literature, to reparameterization invariance in string theory and to the coadjoint-orbit structure of the Virasoro algebra. In one dimension, the diffeomorphism field reduces to an object in one-to-one correspondence with a Virasoro coadjoint element, with transformation law

$$\delta \mathcal D = 2\xi'\mathcal D+\mathcal D'\xi-\frac12\xi'''.$$

The higher-dimensional theory is presented as the geometric realization of this structure on a $d$-dimensional manifold, with the projective connection data no longer fixed but dynamical (Brensinger et al., 2020).

A recurrent misconception is that TW gravity is simply a higher-dimensional reformulation of ordinary affine geometry. The formalism instead treats projective equivalence classes of paths as primary. This is why the independent fields in the general gauge-invariant formulation are $g_{ab}$, $\Pi^a{}_{bc}$, and $(d+1)$0, rather than the metric alone (Brensinger et al., 2020).

2. Thomas cone, projective connection, and the diffeomorphism field

To convert projective data into ordinary affine data, TW gravity uses the Thomas cone, also described as the volume bundle $(d+1)$1, with coordinates $(d+1)$2. The extra coordinate $(d+1)$3 transforms as a volume coordinate. The literature stresses that this extra direction is not a physical extra spacetime dimension in the usual Kaluza–Klein sense; it functions more like a gauge direction, and after integrating it out one obtains a $(d+1)$4-dimensional theory on spacetime (Grover et al., 2024, Brensinger et al., 2020).

The fundamental projective invariant on spacetime is the Thomas projective invariant

$(d+1)$5

which is traceless and projectively invariant, but not itself an affine connection under general coordinate transformations. The Thomas–Whitehead connection on the cone restores affine covariance by combining $(d+1)$6 with a new rank-two object: $(d+1)$7 The field $(d+1)$8 is the diffeomorphism field. In the original formulation it is projectively invariant but not tensorial; later work constructs tensorial, projectively invariant replacements such as $(d+1)$9 and a projectively invariant affine connection $\tilde\Gamma^\alpha{}_{\beta\gamma}$0 with associated covariant derivative $\tilde\Gamma^\alpha{}_{\beta\gamma}$1 (Brensinger et al., 2020, Grover et al., 2024).

The spacetime tensor that appears prominently in the effective theory is often the projective Schouten-type object

$\tilde\Gamma^\alpha{}_{\beta\gamma}$2

In the projectively invariant covariant reformulation,

$\tilde\Gamma^\alpha{}_{\beta\gamma}$3

This converts many expressions that were previously built from non-tensorial objects into ordinary tensor equations (Grover et al., 2024).

A structurally important decomposition used in the cosmological analysis is

$\tilde\Gamma^\alpha{}_{\beta\gamma}$4

where $\tilde\Gamma^\alpha{}_{\beta\gamma}$5 is the dynamical part and $\tilde\Gamma^\alpha{}_{\beta\gamma}$6 is the bare cosmological constant. The field $\tilde\Gamma^\alpha{}_{\beta\gamma}$7 is further decomposed as

$\tilde\Gamma^\alpha{}_{\beta\gamma}$8

so that the trace sector becomes explicit (Brensinger et al., 2019).

3. Action principle and field equations

TW gravity is built from a projective Einstein–Hilbert term together with a projective Gauss–Bonnet term on the Thomas cone. In the general coordinate-invariant formulation, the action is written as

$\tilde\Gamma^\alpha{}_{\beta\gamma}$9

with a projective Einstein–Hilbert part constructed from the projective scalar curvature and a curvature-squared projective Gauss–Bonnet part. In the covariant reformulation, the Lagrangian retains the form

$J_0$0

with

$J_0$1

The projective Gauss–Bonnet sector is the mechanism that gives dynamics to the diffeomorphism field while avoiding higher-derivative equations in the Lovelock sense (Grover et al., 2024, Brensinger et al., 2020).

A central reduction result is that, after choosing a cone metric and integrating over the cone coordinate, the higher-dimensional projective action yields a $J_0$2-dimensional action containing an Einstein–Hilbert term, a cosmological constant term, a Gauss–Bonnet term, and a dynamical sector for $J_0$3: $J_0$4 In this sense, ordinary gravity with cosmological constant emerges from a pure projective Gauss–Bonnet theory (Brensinger et al., 2019).

The field equations can be reorganized into an Einstein-equation form. In the older formulation the metric equation reads

$J_0$5

with $J_0$6 computed from the $J_0$7 sector. In the covariant reformulation the metric equation can be written as

$J_0$8

The effective stress tensor contains the projective corrections from the Palatini sector, the projective Gauss–Bonnet sector, and the diffeomorphism field (Brensinger et al., 2019, Grover et al., 2024).

An important technical development is that the covariant, manifestly projective invariant formulation is classically equivalent on-shell to the original formulation. Off-shell, the earlier variables were manifestly projective invariant but not covariant in the usual tensorial sense; the later reformulation supplies a “Rosetta Stone” between the two descriptions without changing the classical dynamics (Grover et al., 2024).

4. Four-dimensional reduction, cosmological constant, and the GR limit

In four dimensions, the TW construction acquires a particularly sharp form. The literature states that the theory collapses to Einstein–Hilbert gravity when the diffeomorphism field vanishes and the connection becomes Levi-Civita, and also that in four dimensions the action collapses to Einstein–Hilbert gravity with cosmological constant when $J_0$9 is proportional to the Einstein metric (Grover et al., 2024, Brensinger et al., 2020).

For the decomposition

$$\hat{\Gamma}^i{}_{jk} = \Gamma^i{}_{jk} + \delta^i{}_k A_j + \delta^i{}_j A_k,$$0

the $$\hat{\Gamma}^i{}_{jk} = \Gamma^i{}_{jk} + \delta^i{}_k A_j + \delta^i{}_j A_k,$$1 analysis yields

$$\hat{\Gamma}^i{}_{jk} = \Gamma^i{}_{jk} + \delta^i{}_k A_j + \delta^i{}_j A_k,$$2

Using

$$\hat{\Gamma}^i{}_{jk} = \Gamma^i{}_{jk} + \delta^i{}_k A_j + \delta^i{}_j A_k,$$3

the paper finds

$$\hat{\Gamma}^i{}_{jk} = \Gamma^i{}_{jk} + \delta^i{}_k A_j + \delta^i{}_j A_k,$$4

Here $$\hat{\Gamma}^i{}_{jk} = \Gamma^i{}_{jk} + \delta^i{}_k A_j + \delta^i{}_j A_k,$$5 is a new coupling constant with dimensions of angular momentum, and the authors interpret it as a cosmic angular momentum constant. They further argue that this is compatible with plausible upper bounds on the angular momentum of the observable Universe, $$\hat{\Gamma}^i{}_{jk} = \Gamma^i{}_{jk} + \delta^i{}_k A_j + \delta^i{}_j A_k,$$6 (Brensinger et al., 2019).

The four-dimensional case is also special because the Gauss–Bonnet term is topological at the classical level and does not contribute to local metric equations in the standard way. When $$\hat{\Gamma}^i{}_{jk} = \Gamma^i{}_{jk} + \delta^i{}_k A_j + \delta^i{}_j A_k,$$7, the vacuum equations reduce to

$$\hat{\Gamma}^i{}_{jk} = \Gamma^i{}_{jk} + \delta^i{}_k A_j + \delta^i{}_j A_k,$$8

with

$$\hat{\Gamma}^i{}_{jk} = \Gamma^i{}_{jk} + \delta^i{}_k A_j + \delta^i{}_j A_k,$$9

This is the cleanest sense in which TW gravity reproduces standard Einstein gravity with a small positive cosmological constant (Brensinger et al., 2019).

The cosmological interpretation proposed in the 2019 analysis is that $$\Gamma^a{}_{bc}\to \Gamma^a{}_{bc}+\delta^a{}_b v_c+\delta^a{}_c v_b,$$0 is not an independent mysterious input but is tied to a cosmic rotation or patch angular-momentum scale, with the net angular momentum of many patches vanishing in accordance with the cosmological principle. This interpretation is explicitly presented as the authors’ physical picture rather than a universally established consequence (Brensinger et al., 2019).

5. Matter couplings, torsion, and graded extensions

TW gravity couples fermions to projective geometry through the spin connection on the Thomas cone. In the four-dimensional analysis of the dynamical projective-connection model, requiring projective invariance of spinors gives

$$\Gamma^a{}_{bc}\to \Gamma^a{}_{bc}+\delta^a{}_b v_c+\delta^a{}_c v_b,$$1

and the Dirac Lagrangian becomes

$$\Gamma^a{}_{bc}\to \Gamma^a{}_{bc}+\delta^a{}_b v_c+\delta^a{}_c v_b,$$2

Because $$\Gamma^a{}_{bc}\to \Gamma^a{}_{bc}+\delta^a{}_b v_c+\delta^a{}_c v_b,$$3 appears, the coupling is axial and CP-violating, and the trace $$\Gamma^a{}_{bc}\to \Gamma^a{}_{bc}+\delta^a{}_b v_c+\delta^a{}_c v_b,$$4 couples universally to fermions. The 2019 paper therefore suggests that the trace sector may serve as a dark matter portal for non-standard model fermions (Brensinger et al., 2019).

The broader gauge-invariant formulation also derives a chiral structure from the cone geometry itself. In four dimensions,

$$\Gamma^a{}_{bc}\to \Gamma^a{}_{bc}+\delta^a{}_b v_c+\delta^a{}_c v_b,$$5

so the extra cone direction mixes naturally with chirality. The literature emphasizes that no gauge choice removes the $$\Gamma^a{}_{bc}\to \Gamma^a{}_{bc}+\delta^a{}_b v_c+\delta^a{}_c v_b,$$6-fermion interaction, even though certain induced density or chiral terms can be simplified by a specific density-weight choice (Brensinger et al., 2020).

A substantial extension introduces torsion into the TW connection. The torsionful theory contains additional fields $$\Gamma^a{}_{bc}\to \Gamma^a{}_{bc}+\delta^a{}_b v_c+\delta^a{}_c v_b,$$7, $$\Gamma^a{}_{bc}\to \Gamma^a{}_{bc}+\delta^a{}_b v_c+\delta^a{}_c v_b,$$8, $$\Gamma^a{}_{bc}\to \Gamma^a{}_{bc}+\delta^a{}_b v_c+\delta^a{}_c v_b,$$9, and $$\delta \mathcal D = 2\xi'\mathcal D+\mathcal D'\xi-\frac12\xi'''.$$0, and it modifies matter couplings. In the torsionfree case, Yang–Mills fields decouple from the TW projective sector $$\delta \mathcal D = 2\xi'\mathcal D+\mathcal D'\xi-\frac12\xi'''.$$1 and $$\delta \mathcal D = 2\xi'\mathcal D+\mathcal D'\xi-\frac12\xi'''.$$2, a result described as the “Yang-Mills Miracle” and, in the abelian case, the “Maxwell Miracle.” With torsion, new couplings appear through $$\delta \mathcal D = 2\xi'\mathcal D+\mathcal D'\xi-\frac12\xi'''.$$3 and $$\delta \mathcal D = 2\xi'\mathcal D+\mathcal D'\xi-\frac12\xi'''.$$4, while the key decoupling from $$\delta \mathcal D = 2\xi'\mathcal D+\mathcal D'\xi-\frac12\xi'''.$$5 and $$\delta \mathcal D = 2\xi'\mathcal D+\mathcal D'\xi-\frac12\xi'''.$$6 in the relevant interaction term remains (Brensinger et al., 2024).

The torsionful Dirac analysis yields a geometrically sourced chiral mass term. The paper identifies the term

$$\delta \mathcal D = 2\xi'\mathcal D+\mathcal D'\xi-\frac12\xi'''.$$7

as the chiral mass contribution and emphasizes that it is sourced by the TW connection and the lifting structure rather than by a Higgs-like mechanism (Brensinger et al., 2024).

The graded extension, called Super TW Gravity, places the theory on a DeWitt supermanifold and upgrades the projective invariant, the Thomas–Whitehead connection, and the action to their graded analogues. In the bosonic truncation with $$\delta \mathcal D = 2\xi'\mathcal D+\mathcal D'\xi-\frac12\xi'''.$$8, in four dimensions, and with $$\delta \mathcal D = 2\xi'\mathcal D+\mathcal D'\xi-\frac12\xi'''.$$9, the theory reduces to ordinary TW gravity and then to Einstein gravity. The graded construction is presented as a prelude to a future understanding of TW-Supergravity and its relation to the Super-Virasoro algebra (Mera et al., 2022).

6. Radiative sectors, two-dimensional anomalous gravity, and current issues

The radiative content of the diffeomorphism field has been analyzed in Minkowski space. Choosing

$d$0

with $d$1, the fluctuation $d$2 admits a York-style decomposition into tensor, vector, and scalar sectors. In $d$3,

$d$4

and the field-equation constraint relating the scalar and trace reduces the propagating content to

$d$5

The separated mode equations become

$d$6

$d$7

$d$8

The paper states that the trace is massive, while the tensor and solenoidal sectors are tachyonic in the free-field sense (Fiedorowicz et al., 2024).

The same analysis studies geodesic deviation on the Thomas cone. At zeroth order in metric fluctuations, the background diffeomorphism field gives only trivial or reparameterized straight-line separation. Once linearized metric backreaction is included, however, scalar, solenoidal, and TT projective modes produce nontrivial geodesic deviations, including longitudinal effects and memory-like residuals. The antenna response of a LIGO-like interferometer is described by

$d$9

and the scalar mode has a pattern proportional to

$g_{ab}$0

The literature therefore presents the projective sector as potentially relevant to detector responses beyond standard GR tensor polarizations (Fiedorowicz et al., 2024).

A distinct but related development occurs in two dimensions, where TW gravity is used as a geometric extension of the anomalous Polyakov action. In the background-field treatment, both the dynamical light-cone gauge and the ADM formalism retain the familiar vanishing-Hamiltonian structure of the effective Polyakov action. When the diffeomorphism field itself becomes dynamical through the projective Gauss–Bonnet sector, the Minkowski-space action acquires genuine kinetic terms and a nontrivial constrained Hamiltonian density. The paper’s central claim is that adding dynamics to the diffeomorphism field subsequently removes the vanishing Hamiltonians (Biedke et al., 7 Mar 2026).

Several current issues follow directly from the published formulations. One is formal: the original manifestly projective formulation was not covariant off-shell, and the later tensorial reformulation addresses precisely that limitation (Grover et al., 2024). Another is physical: the Minkowski radiative analysis identifies tachyonic free-field sectors in the tensor and vector modes, while the scalar trace mode is the one previously emphasized in matter couplings (Fiedorowicz et al., 2024). A further issue is interpretive: the Thomas cone should not be read as an ordinary extra spacetime dimension, because the formalism treats it as projective or volume-bundle structure rather than as Kaluza–Klein geometry (Grover et al., 2024).

Taken together, these developments define Thomas–Whitehead gravity as a projective gauge theory in which the diffeomorphism field $g_{ab}$1 is elevated from compensating projective data to a genuine dynamical component of gravitation, with consequences for the GR limit, cosmological constant structure, fermion couplings, torsion, supergeometric extensions, radiative mode content, and the Hamiltonian structure of anomalous two-dimensional gravity (Brensinger et al., 2019, Brensinger et al., 2024, Biedke et al., 7 Mar 2026).