with the last form of this using the Einstein summation convention.
One motivation for introducing both upper and lower indices is that special
relativity is a limiting case of general relativity, which is a fully geometrical
theory based on taking space-time to be a manifoldM with a metricgthat
varies from point to point. In such a theory it is important to distinguish between
elements of the tangent spaceTx(M)at a pointx∈Mand elements of its dual,
the co-tangent spaceTx∗(M), while using the fact that the metricgprovides an
inner product onTx(M)and thus an isomorphismTx(M)'Tx∗(M). In the
special relativity case, this distinction betweenTx(M)andTx∗(M)just comes
down to an issue of signs, but the upper and lower index notation is useful for
keeping track of those.
A second motivation is that position and momenta naturally live in dual
vector spaces, so one would like to distinguish between the vector spaceM^4
of positions and the dual vector space of momenta. In the case though of a
vector space likeM^4 which comes with a fixed inner productημν, this inner
product gives a fixed identification ofM^4 and its dual, an identification that is
also an identification as representations of the Lorentz group. Given this fixed
identification, we will not here try and distinguish by notation whether a vector
is inM^4 or its dual, so will just use lower indices, not both upper and lower
indices.
The coordinatesx 1 ,x 2 ,x 3 are interpreted as spatial coordinates, and the
coordinatex 0 is a time coordinate, related to the conventional time coordinate
twith respect to chosen units of time and distance byx 0 =ctwherecis the
speed of light. Mostly we will assume units of time and distance have been
chosen so thatc= 1.
Vectorsv∈M^4 such that|v|^2 =v·v >0 are called “space-like”, those with
|v|^2 <0 “time-like” and those with|v|^2 = 0 are said to lie on the “light cone”.
Suppressing one space dimension, the picture to keep in mind of Minkowski
space looks like this: