DOC.
13
GENERALIZED THEORY OF RELATIVITY
173
§1.
General Tensors
Let
(1)
ds2
=
/XV
be the
square
of the line
element,
which is viewed
as
the invariant
measure
of the
distance between
two
infinitely
close
space-time points. Except
where otherwise
noted,
the
following developments
are
independent
of the number of
variables;
let
us
denote this number
by
n.
In
a
transformation
(2)
Xi
=
xi(x1',
X2',
. .
.
Xn') (i
=
1, 2,
...
n)
of the
variables,
or a
transformation
(3)
dx
=
E
=
üPik
^k
k
axk
k
=
E
^-^k
=
E
17ki
dXk
of their
differentials,
the coefficients of the line element transform
according
to
the
formulas
(4)
Srs =
E
P"rPvs
'
/XV
Let
g
be the discriminant of the differential form
(1), i.e.,
the determinant
8
~
IS/xv I*
If
Yuv
denotes
the subdeterminant of
g
divided
by
the discriminant
("normal-
ized")
and
adjoined
to the element
guv,
then these
quantities
yuv
transform
according
to
the formulas
(5) y
rs'
=
E
•
/XV
We
now
introduce the
following
definitions:
I.
The
totality
of
a
system
of
functions
Ti1,
i2,...i^
of
the
variables
x
shall
be
called
a
covariant
tensor
of rank
X
if these
quantities transform according to
the
formulas
(6)
Trxr2...rx
=
E
Pi^Pi2r2
'
•"
Pixrx
'
Tili2...ix'
h h
- -
-
II. The
totality
of
a system
of
functions
. •
of
the
variables
x
shall
be
called
1
2'''
A
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