DOC.
71
PRINCETON LECTURES 343
THE GENERAL THEORY
we
shall denote
this
in the
following
as
the
“energy tensor
of
matter.”
According to
our
previous results,
the
principles
of
momentum
and
energy
are
expressed by
the
statement
that the
divergence
of
this
tensor
vanishes
(47c).
In the
[100]
general theory
of
relativity,
we
shall
have
to
assume
as
valid the
corresponding general
co-variant
equation.
If
(Tuv)
denotes the co-variant
energy
tensor
of
matter,
Œvo
the
corresponding
mixed
tensor density,
then,
in
accord-
ance
with
(83),
we
must
require
that
(95)
0
=
dx
-
Trhd
be satisfied.
It
must
be
remembered that
besides
the
energy
density
of the
matter
there
must
also be
given
an
energy density
of the
gravitational field,
so
that there
can
be
no
talk of
principles
of conservation of
energy
and
momentum
for
matter
alone. This
is
expressed
mathe-
matically
by
the
presence
of
the
second
term
in
(95),
which
makes
it
impossible
to
conclude the
existence of
an
integral
equation
of
the
form of
(49).
The
gravitational
field
[101]
transfers
energy
and
momentum to
the
“matter,” in that
it
exerts
forces
upon
it and
gives
it
energy;
this
is
expressed
by
the
second
term
in
(95).
If there
is
an
analogue
of
Poisson’s
equation
in
the
general theory
of
relativity,
then
this
equation must
be
a
tensor
equation
for
the
tensor
guv
of
the
gravitational
potential;
the
energy
tensor
of
matter must
appear
on
the
right-hand
side of this
equation.
On the left-hand
side
of the
equation
there
must
be
a
differential
tensor
in the
guv.
We have
to
find this
differential
tensor.
It
is
completely
determined
by
the
following
three conditions:
[83]