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 τ
u
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 LRS
u
), allows one to decompose all space-time
tensors and tensor equations into spatial and temporal components. (Choquet-
Bruhat, Dillard-Bleick and DeWitt-Morette 1977).
3.1 Orthogonal decompositions
Let g be the four-dimensional space-time metric with signature +2 and compo-
nents g
αβ
(α, β =0, 1, 2, 3), ∇ its associated covariant derivative operator, and η
the unit volume 4-form which assures space-time orientation (see (2.26)). Assume
that the space-time is time-oriented and let u be a future-pointing unit time-like
(u
α
u
α
= −1) vector field which identifies an observer. The local splitting of the
tangent space into orthogonal subspaces uniquely related to the given observer u
is accomplished by a temporal projection operator T(u) which generates vectors
parallel to u and a spatial projection operator P (u) which generates LRS
u
. These