Translational Invariance and the Space-time Lorentz Transformation with Arbitary Spatial Coordinates

Translational invariance requires that physical predictions are independent of the choice of spatial coordinate system used. The time dilatation effect of special relativity is shown to manifestly respect this invariance. Consideration of the space-time Lorentz transformation with arbitary spatial coordinates shows that the spurious 'length contraction' and 'relativity of simultaneity' effects —the latter violating translational invariance— result from the use of a different spatial coordinate system to describe each of two spatially separated clocks at rest in a common inertial frame PACS 03.30.+p Translational invariance is a mathematical expression of the homogeneity of physical space –the result of an experiment governed only by internal conditions does not depend on where, in space, it is performed. A corollary is that the prediction of the result of any such experiment does not depend on the choice of spatial coordinates used for its physical description. This is because moving the experiment to a different spatial position is mathematically equivalent to a change of the origins of coordinate axes x → x − x 0. In this letter, it is demonstrated that the space-time Lorentz transformation —when correctly interpreted— respects translational invariance, as just defined. As will be explained below, this is not the case in the conventional Einsteinian [1] interpretation of the transformation. It is instructive to first discuss the space transformation equation in the context of Galilean relativity. With a particular choice of coordinate axes, the Galilean space transformation for an object at rest in the inertial frame S', as observed in another such frame S, is: x ′ = x − vt = 0 (1) This equation describes an object at rest at the origin of S' that moves with uniform velocity, v, along the +ve x-axis in S. It is assumed that there is an array of synchronised clocks at rest in S and that t is the time recorded by any such clock. The spatial coordinate system in S is chosen so that x = 0 when t = 0. Introducing a more explicit notation and arbitary coordinate origins in S and S', Eqn(1) generalises to: x ′ (t) − x ′ (0) = x(t) − x(0) − vt = 0 (2)