00 8 Invariant space - time intervals , clock synchronisation and correct application of Lorentz transformations in special relativity

Space-like and time-like invariant space-time intervals are used to analyse measurements of spatial and temporal distances. The former are found to be Lorentz invariant –there is no 'relativistic length contraction', whereas the latter exhibit the experimentally confirmed 'time dilatation' effect, without, however, 'relativity of simultaneity'. It is shown how the spurious predictions of length contraction and relativity of simultaneity arise from the use of inconsistent synchronisation procedures for spatially separated clocks. PACS 03.30.+p With a particular choice of coordinate axes and clock synchronisation convention, to be discussed below, the space-time Lorentz Transformation a (LT) may be written as: x ′ = γ[x − vt] (1) t ′ = γ[t − vx c 2 ] (2) y ′ = y (3) z ′ = z (4) where γ ≡ 1 1 − (v c) 2 (5) In (1)-(4) the space-time coordinates x, y, z, t (E) and x ′ , y ′ , z ′ , t ′ (E') specify the same event as observed in two inertial frames S and S' respectively. The frame S' is in uniform motion relative to S with velocity v in the direction of the positive x-axis. The x,x ′-axes and the y,y ′-axes are parallel and c is the speed of light in vacuum. It was shown by Minkowski [6] that with every pair of events, E and E 0 in S, and E ′ and E ′ 0 in S', with space-time coordinates connected by the LT (1)-(5), may be associated space-like and time-like invariant intervals ∆s, ∆τ , respectively, according to the a The Lorentz Transformation (1)-(5) was first written down in the Literature by Larmor in 1900 [1]. See [2] for a discussion of the historical priority issues. It was independently derived by Einstein [3] and given by Poincaré [4] in 1905. The appellation 'Lorentz Transformation' is due to Poincaré. In Lorentz's last pre-relativity paper [5] another space-time transformation was given, from which the Lorentz Transformation (1)-(5) may be obtained by the substitution: x → x − vt.