184
DOC. 30
FOUNDATION OF GENERAL RELATIVITY
§
16. The General Form
of
the Field
Equations
of
Gravitation
The
field equations
for
matter-free
space
formulated
in
§
15
are
to
be
compared
with the
field equation
V2o
=
0
of
Newton's
theory.
We
require
the
equation corresponding
to
Poisson's
equation
V2£
=
47Tkp,
where
p
denotes the
density
of matter.
The
special
theory
of
relativity
has led to
the
conclusion
that inert
mass
is
nothing
more or
less
than
energy,
which
finds
its
complete
mathematical
expression
in
a
symmetrical
tensor of second rank,
the
energy-tensor.
Thus in
the
general theory
of
relativity
we
must
introduce
a
correspond-
ing energy-tensor
of
matter
Tao,
which,
like
the
energy-com-
ponents
ta
[equations
(49)
and
(50)]
of
the
gravitational field,
will
have
mixed
character,
but
will
pertain
to
a
symmetrical
covariant
tensor.*
The
system
of
equation
(51)
shows
how this
energy-tensor
(corresponding
to the
density
p
in Poisson's
equation)
is to
be
introduced into the
field
equations
of
gravitation.
For
if
we
consider
a complete system
(e.g.
the
solar
system),
the
total
mass
of
the
system,
and therefore its total
gravitating
action
as
well,
will
depend on
the total
energy
of
the
system,
and therefore
on
the
ponderable energy together
with the
gravitational
energy.
This
will allow
itself
to
be
expressed
by
introducing
into
(51),
in
place
of
the
energy-components
of
the
gravitational
field
alone,
the
sums
tau
+
Tau
of
the
energy-
components
of matter and
of
gravitational field.
Thus instead
of
(51)
we
obtain the
tensor
equation
ö
(sr=
-
4(C
+
T;)
-
hXit +
T],.
1)X
J
-
9
=
1
(52)
where
we
have set T
=
Tuu
(Laue's
scalar).
These
are
the
*garTaa
=
Tor
and
goBTao =
TaB
are
to
be
symmetrical
tensors.
[25]