Tensors and Relativity: Chapter 1

Minkowski spacetime

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Since ds is frame independent we can interpret this result geometrically by treating points tex2html_wrap_inline1393 in the four dimensional spacetime, with ds representing the four dimensional distance [ interval ] between neighboring points.

The expression for ds looks Euclidian if the time component is taken to be ict, tex2html_wrap_inline1401 :

equation823

The line element  allows one to classify spacetime intervals into three different classes: Consider a curve tex2html_wrap_inline1403 [ world- line of a particle, say] and an interval on that curve. If:

equation825

the curve is timelike  in that interval;

equation827

the curve is spacelike  in that interval; and

equation829

the curve is null  in that interval.

Note that the sign of tex2html_wrap_inline1405 is a matter of convention, invariance between frames is what is important.