f(T,T) Theories in Teleparallel Gravity
- f(T,T) theories are teleparallel gravity extensions that add an extra argument to couple torsion with matter or scalar fields.
- They maintain second-order field equations while introducing novel degrees of freedom and frame-dependence challenges.
- Empirical constraints from f(T) models, including weak-field tests and cosmological data, critically shape the feasibility of f(T,T) theories.
theories are best understood, in the literature considered here, as teleparallel modified gravities that extend beyond a function of the torsion scalar alone. The supplied sources do not present a single canonical action for the notation ; instead, they place “-type theories” among related extensions that modify the torsional sector, often by adding couplings between torsion and matter or other scalars, and alongside mixed torsion-matter constructions such as (Chattopadhyay et al., 2011). In that sense, the subject belongs to the broader deformation of the teleparallel equivalent of general relativity (TEGR), where the tetrad and the Weitzenböck connection replace the metric-Levi-Civita pair, the torsion scalar replaces curvature as the basic gravitational invariant, and the decisive technical questions are second-order dynamics, frame dependence, remnant local Lorentz symmetry, and cosmological and strong-field viability (Ferraro, 2012).
1. Conceptual scope and notation
In the supplied literature, -type models do not appear as a uniquely standardized theory. Rather, the notation functions as a family label for teleparallel extensions beyond pure . One source states that related extensions such as -type theories modify the torsional sector, often by adding couplings between torsion and matter or other scalars, while the paper itself studies only (Chattopadhyay et al., 2011). Another source places 0 beside 1 and other mixed torsion-matter theories as natural generalizations of teleparallel phase-space constructions (Feng et al., 2014).
This usage is significant because it distinguishes two levels of discussion. At the first level lies explicit 2 gravity, with concrete actions, field equations, cosmological reductions, and observational constraints. At the second lies a broader class of torsion-based deformations for which the 3 notation is suggestive rather than canonical. A plausible implication is that the second argument is not fixed by the notation alone; what matters is that the theory retains nonlinear dependence on the teleparallel torsion scalar while coupling it to an additional quantity.
Review literature reinforces this reading. It does not develop an 4 theory explicitly, but it identifies the ingredients needed to imagine such extensions: a torsion scalar built from tetrads, second-order field equations in pure 5, extra degrees of freedom tied to loss of local Lorentz invariance, and applications to cosmology, singularity resolution, and defect structure (Ferraro, 2012). Accordingly, 6 theories are most naturally situated as generalized teleparallel Lagrangians rather than as a single universally fixed model.
2. Teleparallel geometric basis
The geometric substrate is teleparallel gravity. The fundamental variable is the tetrad 7, related to the metric by
8
Teleparallelism employs the Weitzenböck connection rather than the Levi-Civita connection; curvature vanishes identically, while torsion is nonzero. The torsion tensor is
9
and the torsion scalar is built from the torsion tensor and superpotential as
0
In TEGR the action is linear in 1, and the teleparallel and Einstein-Hilbert descriptions are related by the identity
2
so the two theories differ by a boundary term [(Yang, 2010); (Ferraro et al., 2011)].
The standard 3 deformation replaces the linear torsion scalar by an arbitrary function. Depending on convention, the action is written either as
4
or as
5
For a spatially flat FRW universe,
6
so the torsion scalar is algebraically tied to the Hubble rate [(Yang, 2010); (Wei et al., 2011)].
This teleparallel foundation is the part of the framework that an 7-type theory would necessarily inherit. A plausible implication is that any such extension remains tetrad-based, with the additional argument entering a Lagrangian already organized around 8.
3. Dynamical structure inherited from 9
A central attraction of 0 gravity is that its field equations remain second order. In FRW form they can be written as
1
2
or, in the alternative normalization used in some papers,
3
These equations show how nonlinear torsion terms generate an effective dark sector without introducing a separate dark-energy fluid [(Yang, 2010); (Wei et al., 2011)].
The same structure also produces an effective gravitational coupling,
4
If 5 is linear, 6, then 7 is constant and 8 is a constant rescaling of 9. For genuinely nonlinear 0, 1 depends on time through 2, and 3 varies. The present-day drift obeys
4
which makes cosmological evolution directly sensitive to the nonlinearity of the torsion Lagrangian (Wei et al., 2011).
The conformal properties of the theory are equally important. Unlike 5, generic 6 gravity is not dynamically equivalent to teleparallel action plus a scalar field via conformal transformation. An unavoidable scalar-torsion derivative coupling remains,
7
so there is no clean Einstein-frame analogue of the familiar 8 scalar-tensor map (Yang, 2010). This strongly suggests that 9-type theories should not be expected to simplify into an ordinary Einstein-frame scalar theory merely by adding an additional argument to the Lagrangian.
4. Tetrads, local Lorentz symmetry, and covariance
The decisive structural complication of nonlinear teleparallel gravity is tetrad sensitivity. In TEGR, local Lorentz transformations change the Lagrangian only by a boundary term, but in 0 gravity generic local Lorentz invariance is broken. Two tetrads that generate the same metric are therefore not generally dynamically equivalent [(Ferraro et al., 2011); (Ulhoa et al., 2013)].
This is why the literature on 1 repeatedly emphasizes “good” tetrads. In FRW cosmology, the naive diagonal tetrad works for the spatially flat case, but for closed and open FRW one must construct nontrivial rotated tetrads adapted to the global parallelization of the spatial slices; otherwise the torsion scalar acquires spurious coordinate dependence and the cosmological reduction becomes inconsistent (Ferraro et al., 2011). In spherically symmetric settings the same issue reappears: reconstructing the Lagrangian is frame-sensitive, and in one tetrad a solution may force
2
even though another tetrad reproduces a broader solution space (Wang, 2011).
The symmetry breaking is not total. The remnant-group analysis shows that 3 theories retain a nontrivial on-shell subgroup of local Lorentz transformations characterized by
4
Minkowski space, being a 6-CAF, admits all infinitesimal local Lorentz transformations in the remnant group; flat FRW and Bianchi type I are 3-CAFs with substantial residual symmetry; curved FRW has a much smaller remnant set (Ferraro et al., 2014). The practical lesson is that teleparallel nonlinearities do not merely reduce gauge freedom; they reorganize it into a solution-dependent remnant symmetry.
A further development is the covariant formulation of 5, which introduces an inertial spin connection and thereby restores local Lorentz covariance. In that framework the diagonal spherical tetrad becomes admissible, but Schwarzschild is not a vacuum solution for 6, so the spherically symmetric vacuum must be recomputed rather than imported from GR (DeBenedictis et al., 2016). A plausible implication is that any explicit 7 theory must decide from the outset whether it is formulated in pure-tetrad language or in covariant teleparallel language; otherwise its field content and admissible frames remain ambiguous.
5. Cosmological constructions and phase structure
The model-building repertoire developed for 8 is extensive and forms the immediate background for any 9-type extension. One reconstruction program rewrites
0
and matches the effective torsion fluid to holographic dark-energy prescriptions. In that framework HDE reconstruction is restricted to the dark-energy-dominated era because the event horizon 1 is a global quantity; NADE reconstruction is carried out in the radiation era for the same reason; RDE, depending only on 2 and 3, can be reconstructed in radiation, matter, and dark-energy eras. The same analysis concludes that an improved boundary condition is needed for a more precise reconstruction of 4 theory from holographic dark energy models (Huang et al., 2013).
Phase-space methods show a similarly rich structure. If the effective torsion density variable
5
can be inverted to 6, then the system becomes autonomous. For the power-law model
7
the radiation, matter, and accelerating points are present, and the accelerating point is a de Sitter attractor with 8. For the logarithmic model
9
the only fixed point is a stable de Sitter state, with no radiation- or matter-dominated critical point (Zhang et al., 2011). Global nullcline analysis strengthens this result by showing that singular lines such as
0
can reverse the flow and produce bifurcation phenomena, so local stability alone does not determine the cosmic fate (Feng et al., 2014).
Other teleparallel constructions broaden the phenomenology. Under an emergent-universe ansatz with
1
the effective torsion fluid satisfies 2, so the model behaves as a phantom cosmology (Chattopadhyay et al., 2011). In mimetic 3 teleparallel gravity, the extra conformal degree of freedom behaves exactly like pressureless dust, and the dynamical-system analysis exhibits fixed points representing inflation, radiation, matter, mimetic dark matter, and dark-energy-dominated eras (Mirza et al., 2017). These results do not define 4 directly, but they indicate the kinds of cosmological sectors, attractors, and effective fluids that generalized teleparallel theories are expected to explore.
6. Empirical and strong-field constraints
The best-developed constraints in the supplied literature are for explicit 5 models, but they are already severe enough to delimit the plausible parameter space of wider teleparallel generalizations. The varying-gravitational-constant analysis uses
6
together with the lunar-laser-ranging bound
7
For the power-law model 8, the allowed range shrinks to
9
and for the exponential model 0, to
1
In both cases the theory is forced into a narrow band around the 2CDM limit (Wei et al., 2011).
Constraints from the varying fine-structure constant are even tighter. For
3
the allowed deviation is
4
while for
5
the bound is
6
The analysis concludes that these simple one-parameter 7 theories become almost indistinguishable from 8CDM (Wei et al., 2011).
Background-data fits give a more model-dependent picture. In the “dark torsion” power-law ansatz
9
combined Union2 + BAO/CMB + GRBs + OHD gives
0
and the 1CDM point 2 lies outside the 3 confidence region in that data combination, although the AIC comparison leaves 4 and 5CDM with comparable support (Bengochea, 2010).
Weak-field and compact-object tests also constrain nonlinear torsion. In covariant
6
the perturbative spherical vacuum differs from Schwarzschild at order 7, and Mercury’s perihelion shift gives
8
while PSR J0045-7319 yields an independent estimate
9
These bounds imply that nonlinear torsion effects are tightly suppressed in weak fields (DeBenedictis et al., 2016).
For 00-type theories, the immediate inference is straightforward: any explicit model that induces time-dependent effective couplings, modified post-Newtonian dynamics, or extra weak-field degrees of freedom is likely to face similarly stringent local and low-redshift constraints.
7. Open structural issues
Several unresolved issues in 01 become even more central for 02-type theories. First is the degree-of-freedom problem. Review literature states that 03 theories are not exempted from additional degrees of freedom, and discusses this as an open issue; a Hamiltonian analysis cited there finds five degrees of freedom in four dimensions (Ferraro, 2012). Since the nonlinearity in 04 already enlarges the dynamical content relative to TEGR, adding a second argument plausibly increases the need for a careful canonical analysis.
Second is screening and Einstein-frame control. The conformal-transformation study shows that 05 does not reduce to teleparallel gravity plus a minimally coupled scalar because of the surviving scalar-torsion coupling (Yang, 2010). The varying-06 analysis adds that, unlike 07, 08 does not seem to admit the usual conformal transformation to an Einstein-frame scalar-tensor theory, so screening mechanisms such as the chameleon mechanism may not be available (Wei et al., 2011). This places teleparallel extensions in a less forgiving phenomenological position than many curvature-based modified gravities.
Third is the status of energy, vacuum structure, and noncosmological solutions. A tetrad-based gravitational energy-momentum vector exists in 09,
10
but it is not invariant under local Lorentz transformations, and in FRW the gravitational energy of the Universe vanishes for the good tetrad used in that analysis (Ulhoa et al., 2013). Static spherical solutions remain highly frame-sensitive, with some solution branches collapsing back to TEGR plus a cosmological constant and others supporting wormhole- or black-hole-like geometries only under restrictive assumptions [(Wang, 2011); (Daouda et al., 2011)].
The encyclopedic status of 11 theories is therefore intermediate. The notation is established as part of the teleparallel extension landscape, but the supplied literature treats it primarily as a natural generalization of 12, not as a settled standalone formalism. What is already clear is the structural inheritance: tetrad-based dynamics, torsion as the basic invariant, sensitivity to frame choice, nontrivial remnant Lorentz symmetry, second-order equations in the pure 13 sector, and stringent viability requirements from cosmology and local tests. What remains open is the precise definition of the second argument, the covariant completion, the degree count, and the mechanism—if any—by which such theories can evade the observational severity already seen in nonlinear 14 gravity.