324 DOC. 71 PRINCETON LECTURES
THE GENERAL THEORY
with
by
any
choice of
co-ordinates
in
a
finite
region.
There
is,
therefore,
no
choice
of co-ordinates
for
which
the
metrical relations of the
special
theory
of
relativity
hold
in
a
finite
region.
But
the invariant
ds
always
exists
for
two
neighbouring points (events)
of
the
continuum.
This invariant
ds
may
be
expressed
in
arbitrary
co-ordinates.
If
one
observes
that the
local
dXv
may
be
expressed
linearly
in
terms
of the co-ordinate differentials
dxv,
ds2 may
be
expressed
in
the form
(55)
ds2
=
guvdxudxv.
The functions
guv
describe,
with
respect to
the
arbitrarily
chosen
system
of
co-ordinates,
the metrical relations
of
the
space-time
continuum and
also
the
gravitational
field.
As
in
the
special
theory
of
relativity,
we
have
to
discriminate
between time-like and
space-like
line
elements
in
the
four-
dimensional
continuum;
owing
to
the
change
of
sign
introduced,
time-like line
elements have
a
real,
space-like
line
elements
an
imaginary ds.
The
time-like
ds
can
be
measured
directly
by
a
suitably
chosen clock.
According to
what has been
said,
it
is
evident that the
formulation
of
the
general theory
of
relativity requires
a
generalization
of the
theory
of invariants and the
theory
of
tensors;
the
question
is
raised
as
to
the
form of the
equations
which
are
co-variant with
respect to arbitrary
point
transformations. The
generalized
calculus of
tensors
was
developed
by
mathematicians
long
before
the
theory
of
relativity.
Riemann
first
extended
Gauss’s
train of
thought
to
continua
of
any
number of
dimensions;
with
prophetic
vision he
saw
the
physical meaning
of
this
generalization
of Euclid’s
geometry.
Then
followed
the
development
of the
theory
in the form of
the calculus
of
[85]
tensors, particularly
by
Ricci and Levi-Civita. This
is
[64]