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  • Print publication year: 2010
  • Online publication date: May 2011

3 - Space-time splitting


The concept of space-time brings into a unified scenario quantities which, in the pre-relativistic era, carried distinct notions like time and space, energy and momentum, mechanical power and force, electric and magnetic fields, and so on. In everyday experience, however, our intuition is still compatible with the perception of a three-dimensional space and a one-dimensional time; hence a physical measurement requires a local recovery of the pre-relativistic type of separation between space and time, yet consistent with the principle of relativity. To this end we need a specific algorithm which allows us to perform the required splitting, identifying a “space” and a “time” relative to any given observer. This is accomplished locally by means of a congruence of time-like world lines with a future-pointing unit tangent vector field u, which may be interpreted as the 4-velocity of a family of observers. These world lines are naturally parameterized by the proper time τu defined on each of them from some initial value. The splitting of the tangent space at each point of the congruence into a local time direction spanned by vectors parallel to u, and a local rest space spanned by vectors orthogonal to u (hereafter LRSu), allows one to decompose all space-time tensors and tensor equations into spatial and temporal components. (Choquet-Bruhat, Dillard-Bleick and DeWitt-Morette 1977).