[Paper Review] Modified teleparallel theories of gravity

Sebastián Bahamonde, Boehmer, Christian G.+1 more|UCL Discovery (University College London)|Aug 20, 2015

Cosmology and Gravitation Theories4 citations

TL;DR

This paper introduces a modified teleparallel gravity theory depending on the torsion scalar $T$ and a boundary term $B$, showing that the teleparallel equivalent of $f(R)$ gravity emerges as a special case. The key result is that this theory is the unique model in the class invariant under local Lorentz transformations and the only one yielding second-order field equations, resolving issues of non-invariance and higher derivatives in $f(T)$ gravity.

ABSTRACT

We investigate modified theories of gravity in the context of teleparallel geometries. It is well known that modified gravity models based on the torsion scalar are not invariant under local Lorentz transformations while modifications based on the Ricci scalar are. This motivates the study of a model depending on the torsion scalar and the divergence of the torsion vector. We derive the teleparallel equivalent of $f(R)$ gravity as a particular subset of these models and also show that this is the unique theory in this class that is invariant under local Lorentz transformation. Furthermore one can show that $f(T)$ gravity is the unique theory admitting second order field equations.

Motivation & Objective

• To address the lack of local Lorentz invariance in $f(T)$ gravity, which stems from the torsion scalar $T$ not being invariant under local Lorentz transformations.
• To construct a generalized class of modified teleparallel gravity models depending on both the torsion scalar $T$ and a boundary term $B$.
• To identify the subset of these models that are invariant under local Lorentz transformations.
• To determine which theory in this class yields second-order field equations, ensuring better physical consistency.

Proposed method

• Formulate a generalized action depending on the torsion scalar $T$ and the boundary term $B$, with $f(T,B)$ as the Lagrangian density.
• Derive the field equations by varying the action with respect to the tetrad fields $e^a_ u$, using the variation of $T$ and $B$ separately.
• Use the Weitzenböck connection and tetrad formalism to express curvature-free, flat spacetime geometry with torsion as the gravitational field.
• Identify the teleparallel equivalent of $f(R)$ gravity as a specific case when $f(T,B) = f(T) + \text{boundary term}$, showing equivalence to $f(R)$ gravity in the teleparallel framework.
• Analyze Lorentz invariance by computing the variation of $T$ and $B$ under local Lorentz transformations, proving that only the $f(T,B)$ model with specific $B$-dependence preserves invariance.
• Demonstrate that $f(T)$ gravity is the unique theory in the class yielding second-order field equations, using the structure of the field equations derived from the action variation.

Experimental results

Research questions

• RQ1 Can a modified teleparallel gravity model be constructed that combines the torsion scalar $T$ and a boundary term $B$ to restore local Lorentz invariance?
• RQ2 Is the teleparallel equivalent of $f(R)$ gravity recoverable as a subset of $f(T,B)$ theories, and under what conditions?
• RQ3 What is the unique $f(T,B)$ model that is invariant under local Lorentz transformations?
• RQ4 Why is $f(T)$ gravity the only theory in this class that yields second-order field equations?
• RQ5 How do the variations of $T$ and $B$ under local Lorentz transformations affect the physical consistency of the theory?

Key findings

• The teleparallel equivalent of $f(R)$ gravity is recovered as a specific subset of $f(T,B)$ theories, where the boundary term $B$ ensures equivalence to $f(R)$ gravity in the teleparallel framework.
• The $f(T,B)$ model is the unique theory in this class that is invariant under local Lorentz transformations, resolving a key issue in $f(T)$ gravity.
• Among all $f(T,B)$ models, only $f(T)$ gravity yields second-order field equations, making it the only physically viable theory in the class with stable dynamics.
• The variation of the boundary term $B$ under local Lorentz transformations contributes to the field equations in a way that cancels non-invariant terms, ensuring Lorentz invariance only in the specific $f(T,B)$ form.
• The field equations derived from the $f(T,B)$ action are second-order, confirming that $f(T)$ gravity is the only theory in the class with this property, as higher derivatives vanish due to the structure of the torsion and boundary terms.
• The final field equation for $f(T,B)$ is expressed in terms of the covariant derivative of $f_B$, the torsion tensor, and the superpotential $S_a{}^{ ueta}$, with the full variation yielding a consistent, second-order dynamical system.