80 DOC.
9
FORMAL FOUNDATION OF RELATIVITY
£
=
iK%av
a
dX:
(84)
rv
_
_
1 dA,
aß
~
2
dx'
(84a)
We
now
introduce
one
further
postulate
of
approximation by considering
in
Zvo
only
those
terms
that
correspond
to
ponderable matter,
while
terms
from
surface
forces
are
ignored.
Under these
postulates,
(48) represents
the
energy
tensor.
Since
the Zvo
remain finite according
to
(48),
one
already
obtains
a far-reaching approxima-
tion if
one
neglects
infinitesimals of first order in
(48).
In this
manner
one gets
[p. 1082]
T
=
«Po
dxa dxv
ds0 ds0
(84b)
Substituting
into
(84)
and
writing
the
left-hand side
Ohav,
one
obtains
dx
dx
l
=
kpo-^-
(IjCq
ds0
av
(85)
x1, x2, x3,
are spatial
coordinates in this
equation
and
x4
=
it
is
the
(imaginary)
{20}
time
coordinate,
while
dso =
dt
(.2
dx21
dx22
dx23
I
__
+
1~.
dt2 dt2
dt2
is the element of Min-
kowski's
"eigentime."
After
we
have
replaced equations
(81)
with
approximation equations,
their
similarity
with the poisson
equation
of
the
Newtonian
theory
of
gravitation
hits the
eye.
We shall
now
replace
the
equations
of
a
material
point, (50b)
and
(51), by
approximation equations.
One obtains
the
coarsest
approximation
if
one replaces
(51)
with
l°
=
~mH7~
dsQ
(86)
Introducing
the three-dimensional
velocity
vector
q
with
magnitude
q,
this
means
the
equations