The Proca equation is a linear partial differential equation for a massive four-vector
field 
.
In flat spacetime, with units chosen so that 
and metric signature 
, it is
 |
(1)
|
where 
,
is the mass parameter, and 
is a source (Gondran 2009, Schambach
and Sanders 2018).
Applying 
to both sides gives
 |
(2)
|
If the source is conserved, so that 
, the field also satisfies 
. Substitution into the defining equation yields
 |
(3)
|
Thus each component satisfies an inhomogeneous Klein-Gordon equation. Setting 
in the defining equation gives the source form of Maxwell's
equations, but the divergence condition above follows only for 
(Schambach and Sanders 2018).
