is a contravariant 4-vector if 
The repeated index b is
summed over. A contravariant 4-vector is a set of 4 quantities
which transform under a Lorentz transformation like
is a contravariant 4-vector if The repeated index b is
summed over.
A covariant 4-vector is a set of 4 quantities which
transform under a Lorentz transformation like 
is a contravariant 4-vector if 
If the primed coordinate system moves with velocity 
with respect to the unprimed one, then
.
.
A contravariant tensor of second rank is a set of
16 quantities which transform under a Lorentz transformation according to 
A covariant tensor of second rank transforms under
a Lorentz transformation according to 
and a mixed tensor transforms according to 
is the Kroneker delta
extended to 4 indices.
is the metric
tensor.
The dot product between two contravariant 4-vectors is defined as
It is invariant under a Lorentz
transformation, it is a Lorentz scalar.
4-vector
velocity 4-vector momentum
4-dimensional gradient
(SI units),
4-vector current 4-vector potential
(Gaussian
units).
The divergence of a 4-vector 
is a Lorentz scalar.
Examples:
is the D’Alaembertian.
is the statement of charge
conservation.
(SI),
(Gaussian),
is the Lorentz condition.
(SI units),
(Gaussian units),
is the inhomogeneous wave equation for the potentials.
All contravariant 4-vectors transform as
.
The antisymmetric field strength tensor is defined
through 
. It is a second rank contravariant
tensor.
(SI units),
(Gaussian),
This yields
(SI units),
(Gaussian units),
(SI units),
(Gaussian units).
For a charge distribution moving with velocity v with respect to an observer, we have in the frame K’ of the observer
, (SI units),
, (Gaussian units),
(SI units),
(Gaussian units),
or 
(SI units)
, (Gaussian units),
or 
(SI units),
(Gaussian units).
E2-c2B2 (SI units) or E2-B2 (Gaussian units) and (E·B)2 are invariant under a Lorentz transformation.