1. Introduction

The Riemann-Silberstein (RS) vector is defined as the complex sum of the electric and
magnetic field vectors: **E**+ *i*
**B**. It appeared in 1907 in an article by Silberstein [^{1}], and was applied many years later by various
authors to different problems [^{2}-^{4}] (see, in particular, Ref. [^{3}] for a historical account and full
bibliography).

The RS vector appears conspicuously in electromagnetism because it describes the
electromagnetic field in a particular representation of the Lorentz group, namely an
irreducible representation of the *SL*(2*,C*) group.
This is evident if the method of spin coefficients is applied to the Maxwell
equations.

As for the applications of spinor algebra, several formulation of the Maxwell and
Einstein equations have been proposed following the pioneering article of Newman and
Penrose [^{5}]. A particulary compact formulation
was worked out by Plebanski [^{6}] in the
seventies, based on the use of a null tetrad as a system of reference (see Ernst
[^{7}] for its relation with other authors
formulations).

The aim of the present article is to further elucidate the role of the RS vector in
the context of spinorial calculus. For this purpose, the null tetrad formalism of
general relativity is used in combination with Dirac spinors, *i.e.*
four-components spinors, and the related matrices of the Dirac algebra. The RS can
thus be identified as the spinorial image of the electromagnetic field in this
particular representation. Being fully covariant, our approach is valid in any
Riemannian spacetime. Furthermore, it generalizes to the Maxwell equations a
previous work on the Dirac equation in curved space-time [^{8}]. The result is a particularly compact and covariant form of
the Maxwell equations that can be used in combination with the Dirac equation in
problems of general relativity.

2. Maxwell equations and Dirac matrices

The Dirac matrices *γ*
^{
α
} are such that

where *g*
^{
αβ
} is the metric tensor (signature {− + ++} and *c* = 1, in the
following)

In the chiral gauge, for instance, they take the form

where *σ*
^{
i
} are the usual Pauli matrices.

Let *A*
_{
µ
} be the electromagnetic potential and *f*
_{
αβ
} = *∂*
_{
β
}
*A*
_{
α
} − *∂*
_{
α
}
*A*
_{
β
} the electromagnetic tensor. In flat space and Cartesian coordinates
(*f*
_{01} = *E*
_{
x
}
*, f*
_{12} = −*B*
_{
z
} , etc.), we have

and it follows from the definition of **E** =
−(*∂/∂t*)**A** − ∇*A*
^{0} and **B** = ∇ × **A** in terms of *A*
^{
µ
} -which are equivalent to the homogeneous Maxwell equations-:

if the Lorentz gauge,

is used.

Accordingly, the inhomogeneous Maxwell equations take the compact form

where *J*
^{
µ
} = (*ρ,*
**J**) is the electromagnetic current

We thus see that the RS vector, defined as

appears naturally in the above representation of the electromagnetic field.

Defining

it simply follows that

The components of *f*
_{
µν
} will be identified in the context of the null-tetrad formalism (see
below).

2.1. The RS vector

Defining the invariants of the field:

we have **F**
^{2} = (*Ɛ + iB*)^{2}, and it follows that

Thus 𝔽^{2} is totally diagonal.

In the particular case of a null-electromagnetic field, *Ɛ* = 0 =
*B*, the matrix 𝔽 turns out to be nilpotent of degree
2: 𝔽^{2} = 0.

In the general case, the matrix 𝔽 satisfies the equation

implying that the eigenvalues *λ* of 𝔽 are

Accordingly we have in general 4 eigenvalues, *λ*
_{
(i)
} = ±(*Ɛ* ± *iB*), with 4 eigenfunctions
*ψ*
_{
(i)
} such that

Furthermore, since 𝔽^{2} is completely diagonal, its eigenvectors
can be taken as any set of four linearly independent spinors *u*
_{
(i)
} , namely

It then follows that, in general,

which can be interpreted as a generalization to Dirac spinors of the
two-components Bloch spinors (if the *u*
_{
(i)
} are chosen as constant units spinors).

3. Null tetrad formalism

The null-tetrad is a set of null-vectors *e*
^{1} and *e*
^{2} are complex conjugates to each other, *e*
^{3} and *e*
^{4} are real, and

where

As shown in Ref. [^{8}], a convenient choice of
the Dirac matrices in the null-tetrad formalism is

satisfying the condition

With the above choice of Dirac matrices, it follows that in (standard) Cartesian coordinates

where the directional derivatives *∂*
_{
n
} are

The associated matrices *σ*
^{
ab
} = (1*/*2)(*γ*
^{
a
}
*γ*
^{
b
} −*γ*
^{
b
}
*γ*
^{
a
} ) were given in [8]; here we repeat them in the appendix for the sake of
completeness. From their explicit form, it follows that for the electromagnetic
tensor *f*
_{
ab
} , in particular, and for *any* antisymmetric tensor,
*f*
_{
ab
} = −*f*
_{
ba
} , in general,

Comparing with (5), we see that the cartesian components of the RS vector
**F** are

3.1. General coordinates system

In a general system of coordinates, the directional derivatives
*∂*
_{
n
} must be replaced by the covariant directional derivative ∇_{
n
} . The Dirac equation in a general coordinates system can thus be written
as

in terms of the covariant derivative

where Γ_{
n
} are the Fock coefficients [^{9},^{10}]. In a tetradial representation, they
are defined as

where

with _{
abc
} = −Γ_{
bac
} (see, *e.g*. Refs. [^{6},^{7}]).

In the null-tetrad formalism, their forms follow from (15) (see also [^{8}]).

For a rank two tensor, in particular, we have

Applying this formula to our matrix 𝔽, one finds after some lengthy but straightforward algebra that

a formula that can also be checked by direct substitution.

Accordingly, the inhomogeneous Maxwell equations take the form

valid in general.

4. Concluding remark

The above analysis clarifies the role of the RiemannSilberstein vector in the context of a spinorial approach to classical electromagnetism. Given the full covariance of all the formulas obtained in this paper, the present formulation can be applied in future publications to problems in general relativy involving electromagnetism and Dirac fields.