DOC.
21
GENERAL RELATIVITY
103
serve as a
counterargument
because
the first
term
on
its
right-hand
side
can
be
brought
into the form
EvT{avT}TvT.
Therefore,
from
now on we
shall call the
quantities
Tmv
=
-{mv}
=
-Egav
)U,V
a
=
-^s
**
n
dgfja
dgv d8,TiV
dx" dx"
dx
(15)
the
components
of
the
gravitational
field.
Kv
vanishes when
Tva
denotes the
energy
tensor
of
all "material"
processes,
and the conservation theorem
(14)
takes the form
E
aTav/SXa
=
-E
TaaB
TBa.
(14a)
We note
that
the
equations
of motion
(23b)
l.c. of
a
material
point
in
a
gravitational
field take
the
form
d2x
,__._
dxfl dx
T
_
V
t-iT
lb2 =
(15)
{2}
2.
The considerations in paragraphs
10
and
11
of the quoted paper remain
unchanged, except that the structures which were there called
V-scalars
and
V-tensors
are now ordinary scalars and tensors, respectively.
§3.
The Field
Equations
of Gravitation
From what has been
said,
it
seems
appropriate
to
write the field
equations
of
gravitation
in
the
form
Ruv=
-kTmv
(16)
since
we already
know that these
equations
are
covariant under
any
transformation
of
a
determinant
equal
to
1. Indeed,
these
equations satisfy
all conditions
we can
demand. Written out in
more
detail,
and
according
to
(13a)
and
(15),
they are
£
If
*
E
(16a)
a
oxa
"ß.
We wish to show
now
that these field
equations can
be
brought
into the
Hamilto-
nian
form
[p.
784]