Rigid Bodies, Flat Space, Relativity and Einstein’s Summation Convention

Einstein introduced his summation convention saying that when the same script appeared just twice in a term, once as a superscript and once as a subscript, then a summation sign for that script running over the four dimensions was to be imagined before the term. In the context of a curved manifold with a metric tensor, adherence to this index pattern avoids many bugs.

In a flat space things are simpler. For a standard Cartesian coördinate system the metric tensor is gij = δij which is Kronecker’s delta. An expression like xjkgjk in the context of a curved manifold becomes xjkδjk in a flat space with conventional Cartesian coördinates. In turn xjkδjk = xkk but at the cost of abandoning the rule of upper and lower scripts. The upper-lower rule is not needed when gij = δij.

The upper and lower rule ensured that only vectors from a space and its dual were “multiplied”. The distinction is unnecessary in Cartesian coördinates.