DOC. 28 NORDSTRÖM'S THEORY OF GRAVITATION 295
the
momentum
density,
fx
etc.
the
components
of
the
energy
flux
density,
and n
denotes the
energy density.
In the
general
theory,
the mentioned conservation
laws have the
generally
covariant
form
(3)
E
^
-
j£
%Vs-
v Uy*v
^
\i\r
Uj*o
The
right-hand
side of this
equation expresses
the fact that the
process
under
consideration does
not,
by
itself,
satisfy
the conservation
laws,
since momentum and
energy
are
transferred from the
gravitational
field to
the material
system.
In
general,
the
components
2ov
refer
to
all
physical processes
in
space except
for those that
concern
the
gravitational
field itself.
We
know from the
original theory
of
relativity
that the
energy
tensor
alone
determines the inertial
properties
of
a
system.
The
right-hand
side of
(3) implies
that
[9]
the
effect
of
a
gravitational
field is also determined
solely by
the
components
of the
energy
tensor.
This
is
in
complete agreement
with the
empirical
laws
of
equality
of
inertial and
gravitational
mass.
In what follows
we
will
assume
that the creation
of
a gravitational
field
by
a
material
system
is
also determined
by
the
energy
tensor
alone.
§2.
The
Differential
Equation
for
the Gravitational Field
in
the Case
of Nordström's
Theory
[10]
What
we
have said
heretofore holds
just
as
well for Nordström's
theory
as
for the
Einstein-Grossmann
theory;
the difference between the
two
theories consists in the
following:
The
gravitational
field
is
determined
by
ten
quantities guv.
According
to
the
Einstein-Grossmann
theory,
ten
formally equivalent equations
are
given
for these
ten
quantities.
But Nordström's
theory
is
based
on
the
assumption
that it
is
possible
to
satisfy
the
principle
of
the
constancy
of the
velocity
of
light by
an
appropriate
choice
of
the reference
system.
We will
now
show that this
amounts to
the
assumption
that
with
an
appropriate
choice of the reference
system
one
can
reduce the
ten
quantities
guv
to
a single quantity
Q2.
For in order that the
principle
of
the
constancy
of the
velocity
of
light
be
satisfied,
the
equation determining
the
propagation
of
light
£
«,v^v
=
0
/XV