172
DOC.
13
GENERALIZED THEORY OF RELATIVITY
II
Mathematical Part
by
Marcel Grossman
The
mathematical
tools for
developing
the
vector
analysis
of
a gravitational
field,
which
is
characterized
by
the invariance of the line element
v
-
/XV
derive from Christoffel's fundamental
paper on
the transformation
of
quadratic
differential
forms.1
Taking
Christoffel's results
as
their
starting point,
Ricci and Levi-
Civita2
developed
their methods of the absolute differential calculus-i.e.,
a
differential calculus
that
is
independent
of the
coordinate
system-which
permit
our
[47] giving an
invariant form
to
the differential
equations
of mathematical
physics.
But
since the
vector
analysis
of
a
Euclidean
space
referred
to
arbitrary
curvilinear
coordinates
is
formally
identical with the
vector
analysis
of
an
arbitrary
manifold
specified by
its line
element,
the extension of the
vector-analytical conceptions
that
[48]
Minkowski, Sommerfeld, Laue, et al.
worked
out
for the
theory
of
relativity
in
recent
years
to
the
general theory
of Einstein's
expounded
above does
not
present any
difficulty.
With
some
practice,
the
general vector
analysis
obtained
in
this
way
is
as
simple
to
handle
as
the
special
vector
analysis
of three-
or
four-dimensional
Euclidean
space;
in
fact,
the
greater generality
of
its
conceptions
lends
it
a
clarity
that
is
lacking
often
enough
in
the
special
case.
The
theory
of
special
tensors
(§3)
has been
treated to
the
full
in
a
paper by
Kottler,3 published
while this work
was
in
progress;
the
treatment is
based
on
the
theory
of
integral
forms,
something
that is
not
possible
in
the
general
case.
Since
more
detailed mathematical
investigations
will have
to
be done in
connection with Einstein's
theory
of
gravitation,
and
especially
in connection
with
the
[50]
problem
of the differential
equations
of the
gravitational
field,
a
systematic
presentation
of the
general
vector
analysis
might
be in order.
I
have
purposely
not
employed geometrical
aids
because,
in
my opinion, they
contribute
very
little
to
an
intuitive
understanding
of the
conceptions
of
vector
analysis.
1Christoffel,
"Über die Transformation der
homogenen
Differentialausdrücke zweiten
[45]
Grades,"
J.
f.
Math. 70
(1869):
46.
2Ricci
et
Levi-Civita,
"Methodes de calcul differentiel absolu
et leurs
application."
[46]
Math. Ann. 54
(1901):
125.
[49]
3Kottler,
"Über die Raumzeitlinien der Minkowskischen Welt."
Wien. Ber. 121
(1912).