  We extend the BMS(4) group by adding logarithmic supertranslations. This is
done by relaxing the boundary conditions on the metric and its conjugate
momentum at spatial infinity in order to allow logarithmic terms of carefully
designed form in the asymptotic expansion, while still preserving finiteness of
the action. Standard theorems of the Hamiltonian formalism are used to derive
the (finite) generators of the logarithmic supertranslations. As the ordinary
supertranslations, these depend on a function of the angles. Ordinary and
logarithmic supertranslations are then shown to form an abelian subalgebra with
non-vanishing central extension. Because of this central term, one can make
nonlinear redefinitions of the generators of the algebra so that the pure
supertranslations ($\ell >1$ in a spherical harmonic expansion) and the
logarithmic supertranslations have vanishing brackets with all the Poincar\'e
generators, and, in particular, transform in the trivial representation of the
Lorentz group. The symmetry algebra is then the direct sum of the Poincar\'e
algebra and the infinite-dimensional abelian algebra formed by the pure
supertranslations and the logarithmic supertranslations (with central
extension). The pure supertranslations are thus completely decoupled from the
standard Poincar\'e algebra in the asymptotic symmetry algebra. This implies in
particular that one can provide a definition of the angular momentum which is
manifestly free from supertranslation ambiguities. An intermediate redefinition
providing a partial decoupling of the pure and logarithmic supertranslations is
also given.
