176
DOC.
13
GENERALIZED THEORY OF RELATIVITY
or, respectively,
which in the
case
of the
current
theory
of
relativity
becomes5
§2.
Differential
Operations
on
Tensors
[57]
We introduce the
following general
definitions:
1. We designate
as
the
extension
of
a
covariant
(contravariant)
tensor of
rank
X
the
covariant
(contravariant)
of
rank
X
+
1
that
is
obtained
from
the
former
by
"covariant
(contravariant) differentiation."
According
to
Christoffel
(l.c.),
(16)
is
a
covariant
tensor
of rank
X
+
1
that
comes
from the covariant
tensor
of rank
X.
Ricci
and Levi-Civita
call
the differential
operation performed on
the
right-hand
side
of this
equation
the "covariant differentiation" of the
tensor
Tr1
r2
^.
The
following
notations
has
been used
here:
[58] (17)
(18)
Y
ik
^ik
ik
£
8ik
®ik
ik
Txx
+ J1
+ J1
+
Tn.U
yy
zz
dT
r\r2•••
r\r2---
rX
s
=
dx
/
E
r\s
+
r2s
T
k
+ +
rxs
kr2...
rk r\K
rx •
• r,r2...
k
k \
/
rs rs
ut
u
=
Ey
rs
1
/
^Srt
+
rs
t
2
dx
dx
t /
5In
what
follows,
we
do not indicate
every
time the
particular
form that
our
formulas
take
in
the
case
of
the
customary theory
of
relativity;
instead,
we
content
ourselves with
referring
to
the
following
presentations:
1. Minkowski,
"Die
Grundgleichungen
für die
elektromagnetische Vorgänge
in
bewegten "Körpern," Göttinger
Nachrichten
(1908).
2.
Sommerfeld,
"Zur Relativitätstheorie
I,"
Ann.
d. Physik
32
(1910)
and "Zur
Relativitätstheorie
II,"
Ann.
d.
Physik
33
(1910).
3.
Laue,
Das
Relativitätsprinzip (2d ed.),
Die
Wissenschaft,
no.
38 (1913).
[54]
[55]
[56]