Narrow Results

Whether Jordan’s and Einstein’s frame descriptions of $F(R)$ theory of gravity are physically equivalent, is a long standing debate. However, practically none questioned on true mathematical equivalence, since classical field equations may be translated from one frame to the other following a transformation relation. Here, we show that, neither Noether symmetries, Noether equations, nor may quantum equations be translated from one to the other. The reason being, — conformal transformation results in a completely different system, with a different Lagrangian. Field equations match only due to the presence of diffeomorphic invariance. Unless a symmetry generator is found which involves Hamiltonian constraint equation, mathematical equivalence between the two frames appears to be vulnerable. In any case, in quantum domain, mathematical and therefore physical equivalence cannot be established.

Classical equivalence between Jordan’s and Einstein’s frame counterparts of $F(R)$ theory of gravity has recently been questioned, since the two produce different Noether symmetries, which could not be translated back and forth using transformation relations. Here we add the Hamiltonian constraint equation, which is essentially the time–time component of Einstein’s equation, through a Lagrange multiplier to the existence condition for Noether symmetry and show that all the three different canonical structures of $F(R)$ theory of gravity, including the one which follows from Lagrange multiplier technique, admit each and every available symmetry independently. This establishes classical equivalence.