in its own rest frame (S). Time dilation and length contraction are not merely appearances. Time dilation is explicitly related to our way of measuring time intervals between events that occur at the same place in a given coordinate system (called "co-local" events). These time intervals (which can be, and are, actually measured experimentally by relevant observers) are different in another coordinate system moving with respect to the first, unless the events, in addition to being co-local, are also simultaneous. Similarly, length contraction relates to our measured distances between separated but simultaneous events in a given coordinate system of choice. If these events are not co-local, but are separated by distance (space), they will not occur at the same spatial distance from each other when seen from another moving coordinate system. Lorentz transformation of velocities Consider two frames S and S in standard configuration. A particle in S moves in the x direction with velocity vector What is its velocity in frame S ? We can write Substituting expressions for and from into , followed by straightforward mathematical manipulations and back-substitution from yields the Lorentz transformation of the speed to : The inverse relation is obtained by interchanging the primed and unprimed symbols and replacing with For not aligned along the x-axis, we write: The forward and inverse transformations for this case are: and can be interpreted as giving the resultant of the two velocities and and they replace the formula which is valid in Galilean relativity. Interpreted in such a fashion, they are commonly referred to as the relativistic velocity addition (or composition) formulas, valid for the three axes of S and S being aligned with each other (although not necessarily in standard configuration). We note the following points: If an object (e.g., a photon) were moving at the speed of light in one frame then it would also be moving at the speed of light in any other frame, moving at . The resultant speed of two velocities with magnitude less than c is always a velocity with magnitude less than c. If both |u| and |v| (and then also |u| and |v|) are small with respect to the speed of light (that is, then the intuitive Galilean transformations are recovered from the transformation equations for special relativity Attaching a frame to a photon (riding a light beam like Einstein considers) requires special treatment of the transformations. There