194
DOC.
16 FOUNDATIONS OF
GRAVITATION
ds2
=
Y,
ik
as
the
absolute invariant
(scalar),
then
one
sees
at
once
how
one
attains
a generaliza-
tion of the
theory
of
relativity
that
encompasses gravitation
on
the basis of the
equivalence hypothesis.
While in the
original theory
of
relativity
the
independence
of
the
physical equations
from the
special
choice
of
the
reference
system
is based
on
the
postulation
of
the fundamental invariant ds2
=
e dxi2,
we are
concerned with
i
constructing a theory
in which the
most
general
line element
of
the form
ds2 =
£
gikdxidxk
ik
plays
the role of the fundamental
invariant. The
concepts
of
vector analysis
needed
for that
purpose
are
provided by
the method of the absolute differential
calculus,
which will be
explained
in the lecture
by
Grossmann which
is to
come
next.
It follows from the idea
outlined above that the
ten
quantities
gik
characterize
the
gravitational
field;
they replace
the scalar
gravitational potential
(p
of Newtonian
gravitation
theory,
and form the
second-rank fundamental covariant
tensor
of the
gravitational
field. The fundamental
physical significance
of
these
quantities
gik
consists, i.a.,
in the fact that
they
determine the
behavior of
measuring
rods and
clocks.
The method of
the absolute differential calculus allows
us
to
generalize
the
systems
of
equations
of
any physical process,
as
they occur
in the
original theory
of
relativity,
in such
a
way
that
they
fit into the scheme of the
new
theory.
The
components
gik
of
the
gravitational
field
always appear
in these
equations.
The
physical meaning
of
this
is that
the
equations
provide
information about the influence
of the
gravitational
field
on
processes
in the
region
under
study.
The
previously
indicated law of
motion of the material
point may serve as
the
simplest example
of
this kind.
Otherwise,
we
shall confine
ourselves
to
the formulation
of
the
most
general
law known
to
physics,
namely,
the law that
corresponds
to
the
momentum
and
energy
conservation law in the
original theory
of
relativity.
As
is
well
known,
one
has there
a
symmetric
tensor
Tuv,
the
components
of
which,
the
stress
components, yield
the
components
of
the
momentum,
and the
components
of
energy
flux
density
and
energy density.
These
quantities
can
be
specified
for
phenomena
in
any
domain. The laws
of
momentum and
energy
conservation
are
contained in the
equations
r)
T
(1)
£
=
0,
(v,o=1,2,3,4)
since
by integrating
with
respect
to
the
spatial
coordinates
over
the whole
system,
one
can
obtain from these
equations
the conservation
equations
[7]