Definition
- Minkowski metric
Using the Minkowski metric of metric signature (+−−−), its four components are given by:
where c is the speed of light, ρ is the charge density and j the conventional current density. The dummy index α labels the spacetime dimensions.
- Euclidean metric
Some authors prefer to work under an Euclidean spacetime rather than Minkowski's, in which case the Euclidean metric of signature (++++) is used, so that:
so the four-current is rewritten as
where i is the imaginary unit.
Read more about this topic: Four-current
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