  In order to generalize the relativistic notion of boost to the case of non
inertial particles and to general relativity, we come back to the definition of
Lie group of Lorentz matrices and its Lie algebra and we study how this group
acts on the Minskowski space. We thus define the notion of tangent boost along
a worldline. This notion very general notion gives a useful tool both in
special relativity (for non inertial particles or/and for non rectilinear
coordinates) and in general relativity. We also introduce a matrix of the Lie
algebra which, together with the tangent boost, gives the whole dynamical
description of the considered system (acceleration and Thomas rotation). After
studying the properties of Lie algebra matrices and of their reduced forms, we
show that the Lie group of special Lorentz matrices has four one-parameter
subgroups. These tools lead us to introduce the Thomas rotation in a quite
general way. At the end of the paper, we present some examples using these
tools and we consider the case of an electron rotating on a circular orbit
around an atom nucleus. We then discuss the twin paradox and we show that when
the one who made a journey into space in a high-speed rocket returns home he is
not only younger than the twin who stayed on Earth but he is also disorientated
because his gyroscope has turned with respect to earth referential frame.
