  The ability to deal with complex geometries and to go to higher orders is the
main advantage of space-time finite element methods. Therefore, we want to
develop a solid background from which we can construct appropriate space-time
methods. In this paper, we will treat time as another space direction, which is
the main idea of space-time methods. First, we will briefly discuss how exactly
the vectorial wave equation is derived from Maxwell's equations in a space-time
structure, taking into account Ohm's law. Then we will derive a space-time
variational formulation for the vectorial wave equation using different trial
and test spaces. This paper has two main goals. First, we prove unique
solvability for the resulting Galerkin--Petrov variational formulation. Second,
we analyze the discrete equivalent of the equation in a tensor product and show
conditional stability, i.e. a CFL condition. Understanding the vectorial wave
equation and the corresponding space-time finite element methods is crucial for
improving the existing theory of Maxwell's equations and paves the way to
computations of more complicated electromagnetic problems.
