and separates points on a trajectory of a particle (think clock{?}). The same interval can be expressed in coordinates such that at each moment, the particle is at rest. Such a frame is called an instantaneous rest frame, denoted here by the coordinates ( c τ τ , x τ τ , y τ τ , z τ τ ) for each instant. Due to the invariance of the interval (instantaneous rest frames taken at different times are related by Lorentz transformations) one may write d s 2 = c 2 d τ τ 2 − − d x τ τ 2 − − d y τ τ 2 − − d z τ τ 2 = c 2 d τ τ 2 , since in the instantaneous rest frame, the particle or the frame itself is at rest, i.e., d x τ τ = d y τ τ = d z τ τ = 0 . Since the interval is assumed timelike (ie. d s 2 > 0 ), taking the square root of the above yields d s = c d τ τ , or d τ τ = d s c . Given this differential expression for τ, the proper time interval is defined as