DOC.
71
PRINCETON LECTURES 337
THE GENERAL THEORY
This
skew-symmetrical tensor
of
the
second
rank,
faß,
characterizes the
surface
element bounded
by
the
curve
in
magnitude
and
position.
If the
expression
in
the
brackets
in
(85)
were
skew-symmetrical
with
respect to
the
indices
a
and
ß,
we
could conclude
its
tensor
character
from
(85).
We
can
accomplish
this
by interchanging
the
summation
indices
a
and
ß
in
(85)
and
adding
the
resulting
equation
to
(85).
We obtain
(86)
2AA»
=
-RLäA°faß
in
which
(87)
RU
=
“
+
ff
+
rÄ
“
W-
[94]
The
tensor
character of
Ruaaß
follows from
(86);
this
is
the
Riemann
curvature tensor
of
the fourth
rank,
whose
properties
of
symmetry
we
do
not
need
to
go
into. Its
vanishing is
a
sufficient
condition
(disregarding
the
reality
of
the chosen
co-ordinates)
that the continuum
is
Euclidean.
By
contraction
of
the
Riemann
tensor
with
respect to
the
indices
u, ß,
we
obtain
the
symmetrical
tensor
of
the
second rank,
(88)
npa
P
-
I
panßl
p0
I
01npa
___
pa p0aß-
-
-__
---r
1
,a
-T
1
The
last
two
terms
vanish
if
the
system
of co-ordinates
is
so
chosen that
g
=
constant.
From
Ruv
we can
form
[95]
the
scalar,
(89)
R
= guvRuv.
Straightest (Geodesic)
Lines.
A line
may
be
constructed
in
such
a
way
that
its
successive
elements
arise from
each
other
by parallel displacements.
This
is
the natural
[77]