Conformally invariant teleparallel theories of gravity

We analyze the construction of conformal theories of gravity in the realm of teleparallel theories. We first present a family of conformal theories which are quadratic in the torsion tensor and are constructed out of the tetrad field and of a scalar field. For a particular value of a coupling constant, and in the gauge where the scalar field is restricted to assume a constant value, the theory reduces to the teleparallel equivalent of general relativity, and the tetrad field satisfies Einstein’s equations. A second family of theories is formulated out of the tetrad field only, and the theories are not equivalent to the usual Weyl Lagrangian. Therefore, the latter is not the unique genuinely geometrical construction that yields a conformally invariant action. The teleparallel framework allows more possibilities for conformal theories of gravity.