350 DOC.
71
PRINCETON LECTURES
THE
GENERAL THEORY
From
the known
numerical value
of
K,
it
therefore
follows
that
(105a)
k
8irK
87t
.
6‘67
.
10
8
=
9
.
102u
1-86
.
10-27.
From
(101)
we see
that
even
in
the
first
approximation
the
structure
of the
gravitational
field differs
fundamentally
from that which
is
consistent
with
the
Newtonian
theory;
this
difference lies in the fact that the
gravitational potential
has
the character
of
a
tensor
and
not
a
scalar. This
was
[106]
not
recognized
in
the
past
because
only
the
component
g44,
to
a
first
approximation, enters
the
equations
of
motion
of
material
particles.
In order
now
to
be
able
to
judge
the behaviour
of
measuring
rods
and
clocks from
our
results,
we
must
observe
the
following.
According
to
the
principle
of
equivalence,
the metrical relations
of the
Euclidean
geom-
etry
are
valid
relatively to
a
Cartesian
system
of
reference
of
infinitely
small
dimensions,
and
in
a
suitable
state
of
motion
(freely falling,
and without
rotation).
We
can
make
the
same
statement
for local
systems
of
co-ordinates
which,
relatively
to these,
have
small accelerations,
and
therefore
for such
systems
of
co-ordinates
as are
at rest
relatively to
the
one we
have
selected.
For
such
a
local
system,
we
have, for
two neighbouring point
events,
ds2
=
-
dX12
-
dX22
-
dX32
+
dT2
=
-
dS2
+
dT2
where
dS
is
measured
directly
by
a
measuring
rod
and
dT
by
a
clock
at rest
relatively
to
the
system:
these
are
the
naturally
measured
lengths
and
times. Since
ds2, on
the other
hand,
is
known in
terms
of
the co-ordinates
xv
employed
in finite
regions,
in
the
form
ds2
= guvdxudxv
[90]
Previous Page Next Page