DOC.
13
GENERALIZED THEORY OF RELATIVITY
155
x'=
x'(x,
y,
z,
t)
y'=
y'(x, y,
z,
t)
z'=
z'(x,
y,
z,
t)
t'
=
t'(x,
y,
z,
t),
and if the
gravitational
field
in the
original system
K
was
static, then,
upon
this
substitution, equation
(1) goes over
into
an
equation
of the form
«{.m
.
"•
where
ds'2
=
gudx'A + gzidy'1 + ...+
2gndx'dy'
+
...,
'2 '2
and
where
the
quantities guv are
functions
of
x',
y',
z',
t'.
If
we
put
x1, x2, x3, x4
in
place
of
x'
y',
z'
t',
and write
again
ds instead of
ds',
then the
equations
of
motion
of the material
point
with
respect
to
K' take the form
(1")
Sj
dsl
=
0,
where
ds2=
E
S^dxdxv.
XV
We thus arrive
at
the view that
in
the
general
case
the
gravitational
field is
characterized
by
ten space-time functions
(guv
=
gvu)
g11
g12
gl3
g14
g21
g22
g23
g24
g31
g32
g33
g34
g41
g42
g43
g44
which in the
case
of the
customary theory
of
relativity
reduce
to
-1
0
0 0
0 -1
0 0
0
0 -1 0
0
0 0
+c2,
where
c
denotes
a
constant.
The
same
kind of
degeneration occurs
in
the
static
gravitational
field of
the
kind considered
above,
except
that
in
the latter
case
g44 =
c2
is
a
function of
x1, x2, x3.
The
Hamiltonian
function
H thus has the
following
value
in
the
general
case:
ds
(5)
H
=
-m-
=
-m\lguxf +
•
+ •+
2g12x1x2
+ •+ •+
2
g14x\
+ •+ •+
g44.
dt
The
corresponding Lagrangian
equations
(6)
_d
dt
/
\
dH
dx
dH
dx
=
0
yield directly
the
expression
for the
momentum
J
of the
point
and for the force
S
[11]
[12]
[13]