DOC.
42 SPECIAL AND GENERAL RELATIVITY 413
174
Relativity
This
train
of
ideas
is
based
essentially
on
the field
as an
independent
concept.
For
the
conditions
prevailing
with
re-
spect to
S2
are
interpreted
as a
gravitational field,
without
the
question
of the
existence
of
masses
which
produce
this field
being
raised.
By
virtue
of
this train
of
ideas
it
can
also
be
grasped why
the
laws
of
the
pure gravitational
field
are more
directly
linked with the idea
of
general relativity
than the
laws
for
fields
of
a
general
kind
(when, for instance,
an
electromag-
netic field
is
present).
We
have,
namely, good ground
for
the
assumption
that
the "field-free"
Minkowski-space represents
a
special
case
possible
in natural
law,
in
fact,
the
simplest
conceivable
special case.
With
respect
to
its
metrical charac-
ter,
such
a space
is
characterised
by
the
fact
that
dx12
+
dx22
+
dx32 is
the
square
of
the
spatial separation,
measured with
a
unit
gauge,
of
two infinitesimally neighbouring points
of
a
three-dimensional
"space-like"
cross
section
(Pythagorean
theorem),
whereas
dx4
is
the
temporal separation,
measured
with
a
suitable time
gauge,
of
two events
with
common
(x1, x2,
x3).
All
this
simply
means
that
an
objective
metrical
signifi-
cance
is
attached
to
the
quantity
ds2
=
dx12
+
dx22
+
dx32
-
dx42
(1)
as
is
readily
shown with the aid
of
the Lorentz transforma-
tions.
Mathematically,
this fact
corresponds to
the condition
that
ds2
is
invariant with
respect to
Lorentz transformations.
If
now,
in the
sense
of
the
general principle
of
relativity,
this
space (cf. eq.
(1))
is
subjected
to
an arbitrary
continuous
transformation
of
the
co-ordinates,
then the
objectively
sig-