4-vector norm names

[speeding electron]

Are there names for the Lorentz invariant norm of the four-potential and four-current? I assume that they are invariant under the transformations. Also, is it true that any physical quantities which form a four-vector have an invariant quantity associated with them (i.e. the norm of the corresponding four-vector)? Thanks.

[Atheist]

Vectors are invariant under changes of the coordinate system (what you probably mean by tranformation). Only their representation as a linear combination of the base vectors changes (as an inverse to the changes of the base => contravariant vectors). As the vectors (do not confuse a vector with it´s representation (t,x,y,z)) remain unchanged so is their norm - unless you switch to another norm/metric for some reason.
Or in short: Yes, the norm of every vector is an invariant.

About the name: I´m not sure if there´s a special name for it but if you talk about "Lorentz metric" (note that the norm derives from the metric) I think people will know what you talk about - but beware that the sign-convention for this metric might vary.

[pmb_phy]

Are there names for the Lorentz invariant norm of the four-potential and four-current? I assume that they are invariant under the transformations. Also, is it true that any physical quantities which form a four-vector have an invariant quantity associated with them (i.e. the norm of the corresponding four-vector)? Thanks.No. There are no names for them. The main reason being is that their physical meaning will depend on the particular distributions of charges/currents. I.e. if there exists a frame in which there are no charges present but there is a non-zero current density then the norm is proportional to that current density. If, on the otherhand, there is a frame in which there is no current density but there is a charge density then the norm is proportional that charge density.

Pete