6
DOC. 2 COVARIANCE PROPERTIES
Doc. 2
Covariance
Properties
of the
Field
Equations
of the
Theory
of Gravitation Based
on
the
General
Theory
of
Relativity
by
Albert Einstein in Berlin and Marcel Grossmann in Zurich
[p. 215]
In
a paper1
published
in
1913
we
based
a
generalized theory
of
relativity upon
absolute differential
calculus in
a manner
such that it also embraces the
theory
of
[3]
gravitation.
Two
basically
different kinds of
systems
of
equations occur
in this
theory.
For
a
gravitational
field considered
as
given,
we
first established
systems
of
equations
for material
(e.g.,
mechanical,
electrical) processes.
These
equations are
covariant
under
arbitrary
substitutions
of
the
space-time
variables
("coordinates")
and
can
be
considered
as
generalizations
of the
corresponding equations
of
the
original theory
of
relativity.
Second,
we
established
a system
of
equations
that determines the
gravitational
field insofar
as
the
quantities
that determine the material
processes are
given;
and this
system can
be considered
a
generalization
of
the Poisson
equation
of
Newton's
theory
of
gravitation.
In the
original theory
of
relativity,
there is
no
corresponding system
of
equations
for this. In contrast
to
the
equations
mentioned
above,
we
could not demonstrate
general
covariance for those
"gravitational
equations."
The
reason
is that their derivation
was
based
(besides
the conservation
theorems) only upon
the covariance with
respect
to
linear
transformations,
and thus
left it
an
open question
as
to
whether
or
not
there exist other substitutions that would
transform
the
equations
into themselves.
There
are
two
reasons why
the resolution
of
this
question
is of
particular
importance
to
the
theory.
The
answer
to
this
question
gives,
first,
information
on
how
far the basic idea
of
relativity theory
can
be
developed;
and this is
of
great import
to
[p. 216]
the
philosophy
of
space
and time. And
second,
the
judgment
about the value of the
theory
from the
point
of view of
physics depends
to
a
high degree upon
the
answer
to
this
question,
as
is shown
by
the
following
consideration.
The entire
theory
evolved from the conviction that all
physical processes
in
a
gravitational
field
occur
just
in
the
same way as
they
would without
it,
if
an
appropriately
accelerated
(three-dimensional)
coordinate
system
would be introduced
("hypothesis
of
equivalence").
This
hypothesis,
which is based
upon
the
experimental
fact
of
the
equality
between
gravitational
and inertial
mass,
gains
additional
convincing
force if the
"apparent" gravitational
field-which
exists relative
to
the
[1]
[2]
1"Entwurf
einer
verallgemeinerten
Relativitätstheorie
und einer Theorie der Gravitation"
(Leipzig:
B. G. Teubner,
1913).
[In
the
following
it
is
abbreviated
as
"Outline." The
paper
is
printed
in this
journal, Zeitschrift
für
Mathematik
und
Physik,
vol.
62, pp.
225-261.]