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.
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