DOC.
71
PRINCETON LECTURES 345
THE GENERAL THEORY
In the
region surrounding
each
world-point
there
are
systems
of co-ordinates
for which,
choosing
the
x4-coordi-
nate imaginary,
at
the
given
point,
,
I
-
-1
if
u
=
v
g»
-
g"
-
s".
j
=
o
if
/ii
^ ^
and
for
which the
first
derivatives
of
the
guv
and the
guv
vanish.
We
shall
verify
the
vanishing
of
the
divergence
of
the left-hand
side
at
this
point.
At
this
point
the
components
TaoB
vanish,
so
that
we
have
to
prove
the
vanishing only
of
§¿\r=ir&~
-&,.*)]•
Introducing
(88)
and
(70)
into
this
expression,
we see
that the
only
terms
that remain
are
those in
which third
derivatives of
the
guv
enter.
Since
the
guv
are
to
be
replaced
[103]
by
-guv,
we
obtain,
finally, only
a
few
terms
which
may
easily
be
seen
to
cancel each other.
Since the
quantity
that
we
have formed
has
a
tensor character,
its
vanishing
is proved
for
every
other
system
of
co-ordinates
also,
and
naturally
for
every
other four-dimensional
point.
The
energy
principle
of
matter (97) is
thus
a
mathematical
consequence
of
the field
equations (96).
In order
to
learn
whether the
equations (96)
are con-
sistent
with
experience,
we
must,
above
all
else,
find
out
whether
they
lead
to
the Newtonian
theory
as
a
first
approximation.
For
this
purpose
we
must
introduce
various
approximations
into these
equations.
We
already
know
that Euclidean
geometry
and the
law of the
constancy
of the
velocity
of
light
are
valid,
to
a
certain
approxima-
tion,
in
regions
of
a
great
extent,
as
in
the
planetary
sys-
[85]