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In order to generalize the relativistic notion of boost to the case of noninertial particles and to general relativity, we come back to the definition ofLie group of Lorentz matrices and its Lie algebra and we study how this groupacts on the Minskowski space. We thus define the notion of tangent boost alonga worldline. This notion very general notion gives a useful tool both inspecial relativity (for non inertial particles or/and for non rectilinearcoordinates) and in general relativity. We also introduce a matrix of the Liealgebra which, together with the tangent boost, gives the whole dynamicaldescription of the considered system (acceleration and Thomas rotation). Afterstudying the properties of Lie algebra matrices and of their reduced forms, weshow that the Lie group of special Lorentz matrices has four one-parametersubgroups. These tools lead us to introduce the Thomas rotation in a quitegeneral way. At the end of the paper, we present some examples using thesetools and we consider the case of an electron rotating on a circular orbitaround an atom nucleus. We then discuss the twin paradox and we show that whenthe one who made a journey into space in a high-speed rocket returns home he isnot only younger than the twin who stayed on Earth but he is also disorientatedbecause his gyroscope has turned with respect to earth referential frame.

Introduction of the Lie group of Lorentz matrices in Special Relativity.  Tangent boost along a worldline and its associated matrix in the Lie algebra.  Applications