Teleparallel Massive Gravity

• Teleparallel massive gravity is a torsion-based framework combining F(T) gravity with a dRGT massive graviton sector to enable novel wormhole solutions.
• The approach employs analytical reconstructions using the Morris–Thorne ansatz and varied redshift profiles to derive exact static, spherically symmetric geometries.
• The framework demonstrates that appropriate massive sector contributions can satisfy or mildly violate energy conditions, reducing reliance on exotic matter.

Searching arXiv for the cited paper and a few foundational related works to ground the article. Searching arXiv for ([2508.06290](/papers/2508.06290)). Teleparallel massive gravity, in the formulation studied in “Exact Analytical Traversable Wormhole Solutions in Teleparallel $F(T)$ Gravity with the de Rham–Gabadadze–Tolley Massive Graviton Sector” (Landry et al., 8 Aug 2025), is a torsion-based gravitational framework that combines covariant teleparallel $F(T)$ gravity with the non-linear de Rham–Gabadadze–Tolley (dRGT) massive graviton sector. Gravity is described through torsion rather than curvature, with the tetrad or coframe $e^a{}_{\mu}$ (also denoted $h^a{}_{\mu}$) as the fundamental variable, while the graviton mass enters through a dRGT potential built from $\sqrt{g^{-1}f}$ with a Minkowski fiducial metric. In the construction analyzed in (Landry et al., 8 Aug 2025), this combined theory admits exact, static, spherically symmetric traversable wormhole solutions, reconstructed analytically from the teleparallel field equations and the dRGT-modified conservation laws. The resulting geometries are asymptotically flat, horizon-free, and arranged so that the null and weak energy conditions are either satisfied or only mildly violated at the throat.

1. Conceptual and geometric framework

Teleparallel gravity employs the curvature-free Weitzenböck connection to describe gravity via torsion instead of curvature. The spacetime metric is assembled from the tetrad according to

$g_{\mu\nu}=g_{ab}\,e^a{}_{\mu} e^b{}_{\nu},$

with $g_{ab}=\eta_{ab}$ in the orthonormal gauge. The determinant is $e\equiv \det(e^a{}_{\mu})$, while the paper also uses $h\equiv \det(h^a{}_{\mu})$. In this setting, the torsion tensor and its contractions replace the Levi–Civita curvature invariants as the basic geometric objects.

The $F(T)$ extension generalizes the teleparallel equivalent of general relativity by replacing the torsion scalar $F(T)$0 with an arbitrary function $F(T)$1. The formulation used in (Landry et al., 8 Aug 2025) is explicitly covariant: rather than relying on a pure-tetrad prescription, it uses a coframe/spin-connection pair fixed by symmetry, isotropy, and the zero-curvature constraint. This avoids the local-Lorentz pathologies associated with non-covariant $F(T)$2 models and the use of “bad tetrads.”

For spherically symmetric configurations, the coframe is taken as

$F(T)$3

with $F(T)$4, and spin-connection components

$F(T)$5

Under the Morris–Thorne wormhole metric, one sets

$F(T)$6

The massive sector is the ghost-free dRGT construction, in which the graviton mass is generated by a non-linear potential built from

$F(T)$7

where $F(T)$8 is the fiducial metric, chosen here to be Minkowski. The theory then depends on the symmetric polynomials $F(T)$9 of $e^a{}_{\mu}$0, specialized in the spherically symmetric setting to diagonal ansätze for $e^a{}_{\mu}$1.

2. Action, torsion objects, and field equations

The teleparallel massive action with matter is written as

$e^a{}_{\mu}$2

and, in the normalization also given in the paper,

$e^a{}_{\mu}$3

The torsion tensor, superpotential, and torsion scalar are

$e^a{}_{\mu}$4

$e^a{}_{\mu}$5

$e^a{}_{\mu}$6

The equivalent index form quoted in the paper is

$e^a{}_{\mu}$7

$e^a{}_{\mu}$8

$e^a{}_{\mu}$9

Variation of the action yields symmetric and antisymmetric field equations: $h^a{}_{\mu}$0

$h^a{}_{\mu}$1

Here $h^a{}_{\mu}$2 is the Einstein tensor of the Levi–Civita connection built from $h^a{}_{\mu}$3, and $h^a{}_{\mu}$4 is the total energy-momentum including matter and the massive sector.

The dRGT contribution is taken as

$h^a{}_{\mu}$5

with

$h^a{}_{\mu}$6

and the explicit symmetric energy-momentum contribution used in the paper is

$h^a{}_{\mu}$7

Two diagonal massive-source realizations are distinguished.

For the general massive case,

$h^a{}_{\mu}$8

so that

$h^a{}_{\mu}$9

and

$\sqrt{g^{-1}f}$0

For the uniform-pressure massive case,

$\sqrt{g^{-1}f}$1

so that

$\sqrt{g^{-1}f}$2

and

$\sqrt{g^{-1}f}$3

The effective massive equation-of-state parameter is then

$\sqrt{g^{-1}f}$4

with the limits $\sqrt{g^{-1}f}$5 for $\sqrt{g^{-1}f}$6 and $\sqrt{g^{-1}f}$7 for $\sqrt{g^{-1}f}$8 (Landry et al., 8 Aug 2025).

3. Wormhole ansatz and torsion kinematics

The static, spherically symmetric geometries are built with the Morris–Thorne ansatz

$\sqrt{g^{-1}f}$9

where $g_{\mu\nu}=g_{ab}\,e^a{}_{\mu} e^b{}_{\nu},$0 is the redshift function and $g_{\mu\nu}=g_{ab}\,e^a{}_{\mu} e^b{}_{\nu},$1 is the shape function. The throat is located at $g_{\mu\nu}=g_{ab}\,e^a{}_{\mu} e^b{}_{\nu},$2, defined by

$g_{\mu\nu}=g_{ab}\,e^a{}_{\mu} e^b{}_{\nu},$3

and the flaring-out condition is

$g_{\mu\nu}=g_{ab}\,e^a{}_{\mu} e^b{}_{\nu},$4

For the diagonal tetrad and spin connection adopted in the paper, the torsion scalar is

$g_{\mu\nu}=g_{ab}\,e^a{}_{\mu} e^b{}_{\nu},$5

Three redshift profiles are treated explicitly:

• Constant:

$g_{\mu\nu}=g_{ab}\,e^a{}_{\mu} e^b{}_{\nu},$6

• Logarithmic:

$g_{\mu\nu}=g_{ab}\,e^a{}_{\mu} e^b{}_{\nu},$7

• Power-law:

$g_{\mu\nu}=g_{ab}\,e^a{}_{\mu} e^b{}_{\nu},$8

including $g_{\mu\nu}=g_{ab}\,e^a{}_{\mu} e^b{}_{\nu},$9 and $g_{ab}=\eta_{ab}$0, with $g_{ab}=\eta_{ab}$1 in the power-law cases studied to ensure well-behaved horizons.

For the general massive case, the teleparallel wormhole field equations are written in the paper as

$g_{ab}=\eta_{ab}$2

$g_{ab}=\eta_{ab}$3

$g_{ab}=\eta_{ab}$4

$g_{ab}=\eta_{ab}$5

The radial conservation law, for $g_{ab}=\eta_{ab}$6, is

$g_{ab}=\eta_{ab}$7

With a cosmological fluid equation of state $g_{ab}=\eta_{ab}$8, the total variables are split as

$g_{ab}=\eta_{ab}$9

In the general massive case, this gives

$e\equiv \det(e^a{}_{\mu})$0

In the uniform-pressure massive case, it becomes

$e\equiv \det(e^a{}_{\mu})$1

4. Exact analytical reconstructions

For constant redshift, $e\equiv \det(e^a{}_{\mu})$2, the torsion scalar reduces to

$e\equiv \det(e^a{}_{\mu})$3

The paper identifies a characteristic relation connecting $e\equiv \det(e^a{}_{\mu})$4 and $e\equiv \det(e^a{}_{\mu})$5: $e\equiv \det(e^a{}_{\mu})$6 hence

$e\equiv \det(e^a{}_{\mu})$7

Representative reconstructed $e\equiv \det(e^a{}_{\mu})$8 solutions are given for several shape-function families. For

$e\equiv \det(e^a{}_{\mu})$9

a cubic characteristic yields, in the limit $h\equiv \det(h^a{}_{\mu})$0,

$h\equiv \det(h^a{}_{\mu})$1

For

$h\equiv \det(h^a{}_{\mu})$2

various exact or approximate solutions arise. In the limit $h\equiv \det(h^a{}_{\mu})$3,

$h\equiv \det(h^a{}_{\mu})$4

$h\equiv \det(h^a{}_{\mu})$5

$h\equiv \det(h^a{}_{\mu})$6

$h\equiv \det(h^a{}_{\mu})$7

In the general massive case with constant redshift, the reconstructed teleparallel model is

$h\equiv \det(h^a{}_{\mu})$8

with

$h\equiv \det(h^a{}_{\mu})$9

The paper provides exact special-function evaluations for several families. Examples include

$F(T)$0

and, for $F(T)$1 with $F(T)$2,

$F(T)$3

$F(T)$4

These reconstructions yield closed-form $F(T)$5 containing polynomials, power laws, logarithms, and transcendental functions such as $F(T)$6 and $F(T)$7.

The paper also studies explicit teleparallel models under $F(T)$8. For the single-polynomial ansatz

$F(T)$9

one obtains

$F(T)$00

with $F(T)$01 again following from the characteristic relation. A two-term polynomial,

$F(T)$02

is handled analogously. For the Born–Infeld teleparallel model,

$F(T)$03

the field equations yield

$F(T)$04

For dust, $F(T)$05, the field-equation combination takes the integrating-factor form

$F(T)$06

where

$F(T)$07

$F(T)$08

$F(T)$09

For logarithmic redshift,

$F(T)$10

the torsion scalar becomes

$F(T)$11

The characteristic equation is

$F(T)$12

from which

$F(T)$13

Two special values simplify further: $F(T)$14

$F(T)$15

For the uniform-pressure massive case with $F(T)$16, the field equations yield

$F(T)$17

with

$F(T)$18

For the power-law redshift $F(T)$19 with $F(T)$20, the cases $F(T)$21 and $F(T)$22 simplify in the large-$F(T)$23 limit to

$F(T)$24

and

$F(T)$25

In the general massive case, the resulting $F(T)$26 contains terms such as

$F(T)$27

where the additional terms include $F(T)$28, incomplete gamma functions $F(T)$29, and exponential prefactors $F(T)$30 with fixed exponents $F(T)$31 (Landry et al., 8 Aug 2025).

5. Matter content, conservation, and energy conditions

The cosmological fluid contributes

$F(T)$32

The massive sector adds anisotropic or isotropic effective stresses depending on the diagonal choice of $F(T)$33. In the general case,

$F(T)$34

while in the uniform-pressure case,

$F(T)$35

For the general case, the null and weak energy conditions are assessed using the combinations

$F(T)$36

$F(T)$37

$F(T)$38

$F(T)$39

$F(T)$40

$F(T)$41

For the uniform-pressure case, the corresponding combinations are

$F(T)$42

$F(T)$43

$F(T)$44

$F(T)$45

The paper states that, for appropriate parameter ranges such as $F(T)$46, $F(T)$47, and $F(T)$48 determined by the conservation-law solutions, the null and weak energy conditions are either satisfied everywhere or undergo only localized, controlled violations at the throat (Landry et al., 8 Aug 2025). It also attributes an important role to the massive contribution in the conservation law: the $F(T)$49 profile can take Yukawa- or power-like forms, including $F(T)$50 for constant redshift and $F(T)$51, $F(T)$52, or $F(T)$53 depending on logarithmic or power-law choices. Within the paper’s interpretation, these effective stresses help stabilize the throat and reduce the need for exotic matter relative to general relativity or pure TEGR.

6. Regularity, limiting cases, and physical implications

The wormhole families are organized so that the throat conditions, asymptotic flatness, and absence of horizons are realized simultaneously. The redshift factor $F(T)$54 stays finite and non-zero throughout the domain, including the throat. This is automatic for constant $F(T)$55, and it is enforced for logarithmic and power-law profiles by parameter choices that avoid divergence, for example $F(T)$56 and $F(T)$57. In families with

$F(T)$58

one has $F(T)$59 as $F(T)$60, and the reconstructed $F(T)$61 behaves so that

$F(T)$62

which establishes asymptotic flatness.

The paper states that the flaring-out condition $F(T)$63 is verified in the explicit classes by choosing integration constants appropriately, for example by fixing $F(T)$64 and $F(T)$65 so that the reconstructed $F(T)$66 at the throat $F(T)$67 satisfies $F(T)$68 and $F(T)$69 (Landry et al., 8 Aug 2025). Regularity is also tied to the covariant teleparallel construction: the spin-connection components depend solely on $F(T)$70 and standard spherical geometry, and the zero-curvature condition is enforced from the outset.

Several limiting cases are emphasized. When $F(T)$71 or $F(T)$72, the massive sector disappears and one recovers massless teleparallel $F(T)$73 gravity. When $F(T)$74, the theory reduces to the TEGR limit, which is dynamically equivalent to general relativity. The comparison drawn in (Landry et al., 8 Aug 2025) is that traversable wormholes in GR or TEGR generically require exotic matter, whereas the addition of the dRGT massive sector allows teleparallel wormholes to satisfy the NEC and WEC in large regions of parameter space or to confine violations to controlled neighborhoods of the throat.

Within the framework analyzed in the paper, teleparallel massive gravity therefore functions as a self-consistent setting for asymptotically flat traversable wormholes without necessarily invoking an explicit cosmological constant. Torsion, through $F(T)$75, supplies geometric self-interactions that can mimic dark-energy-like behavior near the throat, while the dRGT sector modifies the effective stress-energy and the conservation equations through the profile $F(T)$76. A plausible implication is that the combined torsion–massive structure broadens the class of analytically tractable wormhole geometries beyond those available in curvature-based or massless teleparallel models, especially when one seeks exact closed forms for $F(T)$77, $F(T)$78, and $F(T)$79 involving logarithms, inverse hyperbolic functions, exponential integrals, and incomplete gamma functions.