DOC.
13
GENERALIZED THEORY OF RELATIVITY
169
___
dy dz
dt
x
acy
,
djt,
dx
dy
dz
Up
to
the choice of the
units,
these
equations
coincide with Maxwell's first
system.
In
constructing
the second
system,
one
has first
to
bear
in
mind that
to
the
components
#
§^
%
_
6_
of
v
there
correspond
the
components
- ®a;?
®y
®*'
of the
complement
fuv
(Part
II, §3,
formulas
41a).
For the
case
where
no gravita-
tional field
is
present,
this
yields
the second
system, i.e., equation
(24)
in the form
dx
dz
c*
dt
1
d§x
1
c§y
1
a$.
0.'
c
2
dx
c*
dt
c*
dz
This
proves
that the
equations
we
have
set
up really
constitute
a
generalization
of the
equations
of the
ordinary theory
of
relativity.
§7.
Can
the
Gravitational
Field
Be
Reduced to
a
Scalar?
[42]
In view of the undeniable
complexity
of the
theory
of
gravitation propounded here,
we
must
ask
ourselves in
earnest
whether
the
conception
that
has,
until
now,
been the
only
one
advanced,
according
to
which the
gravitational
field
is
reduced
to
a
scalar
0,
is the
only
one
that
is
reasonable and
justified.
I
will
briefly explain why we
think that this
question
must
be answered
in
the
negative.
When
one
characterizes the
gravitational
feld
by
a
scalar,
a path presents
itself
that
is
completely analogous
with that which
we
followed
in
the
foregoing.
One
sets
up
the
equation
of motion of the material
point
in
Hamiltonian form
SlJOcfe} =
0,
where ds
is
the four-dimensional line element from the
ordinary theory
of
relativity
and
O is
a
scalar,
and then
proceeds
wholly
by analogy
with the
foregoing,
without
having
to
give up
the
ordinary theory
of
relativity.
[41]