This paper achieves, among other things, the following:
• It frees the main result of  from the hypothesis of determinant class and extends this result from unitary to arbitrary representations.
• It extends (and at the same times provides a new proof of) the main result of Bismut and Zhang  from finite dimensional representations of Γ to representations on an A-Hilbert module of finite type (A a finite von Neumann algebra). The result of  corresponds to A = C .
• It provides interesting real valued functions on the space of representations of the fundamental group Γ of a closed manifold M. These functions might be a useful source of topological and geometric invariants of M.
These objectives are achieved with the help of the relative torsion ℛ, first introduced by Carey, Mathai and Mishchenko  in special cases. The main result of this paper calculates explicitly this relative torsion (cf. Theorem 1.1).