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.

In the context of the teleparallel equivalent of general relativity, we establish the Hamiltonian formulation of the unimodular theory of gravity. Here we do not carry out the usual 3+1 decomposition of the field quantities in terms of the lapse and shift functions, as in the ADM formalism. The corresponding Lagrange multiplier is the timelike component of the tetrad field. The dynamics is determined by the Hamiltonian constraint H0$'${} and a set of primary constraints. The constraints are first class and satisfy an algebra that is similar to the algebra of the Poincar\'e group.

A brief discussion is made about the relevance of surface terms in the Lagrangian and Hamiltonian formulations of theories of gravity. These surface terms play an important role in the variation of the action integral and in the definition of field quantities such as the gravitational energy-momentum. Then we point out several inconsistencies of a recently proposed formulation of teleparallel theories of gravity with local Lorentz symmetry.

It is shown that in the passage of a short burst of nonlinear plane gravitational wave, the kinetic energy of free particles may either decrease or increase. The decrease or increase of the kinetic...