132

DOC. 27 MAXWELL'S

EQUATIONS

Doc. 27

A New

Formal

Interpretation

of

Maxwell's Field

Equations

of

Electrodynamics

Plenary

session of

February

3,

1916

by

A. Einstein

[p. 184]

The current covariance-theoretical

interpretation

of

the

electrodynamic equations

originates

with

Minkowski.

It

can

be characterized

as

follows. The

components

of

the

electrodynamic

field form

a

six-vector

(antisymmetric

tensor

of

rank

two).

There

is

a

second six-vector associated to the first

one (and

is dual to

it)

whose

components

have in the

special case

of

the

original theory

of

relativity

the

same

values

as

the first

one,

but

are

distinct

in

the

way

the

components are

associated

with

the

four

coordinate

axes.

The two

systems

of

MAXWELLian

equations

are

obtained

by setting

the

divergence

of

the first

one equal

to

zero,

and the

divergence

of

the other

one

[1]

equal

to the four-vector of the electric current.

The introduction

of

the dual six-vector makes its covariance-theoretical

representation relatively

involved and

confusing. Especially

the derivations

of

the

conservation theorems

of

momentum and

energy are complicated, particularly

in the

case

of

the

general theory

of

relativity,

because it also considers the influence

of

the

gravitational

field

upon

the

electromagnetic

field. The

following

formulation avoids

the

concept

of

the dual six-vector and thus achieves

a

considerable

simplification

in

the

system.

Next,

we

will

immediately

treat the

case

in the

general theory

of

relativity.1

§1.

The

Field

Equations

Let

fv

be the

components

of

a

covariant

four-vector,

the four-vector

of

the

[p. 185]

electromagnetic potential.

We form from it the

components

Fpa

of

the covariant six-

vector

of

the

electromagnetic

field

according

to the

system

of

equations

[2]

1My

paper

"Die formale

Grundlage

der

allgemeinen

Relativitätstheorie"

(these

Sitzungsberichte

41

[1914],

p.

1030)

will in the

following

be assumed

as

known;

the codicil

"l.c."

in

the

following

text

always

refers to this

paper.