xvi INTRODUCTION TO VOLUME
4
found useful
was
Mach's
critique
of Newton's mechanics.
It
suggested to
Ein-
stein that
a
satisfactory generalization
of
the
theory
of
relativity
should allow
him to
consider rotational motion
as being equivalent
to
a
state
of
rest. A
crucial
insight,
which occurred
to
Einstein
probably
in
the
summer
of
1912,
was
the
recognition
of
an analogy
between
the
mathematical
problems
of
a
generalized relativity theory
and
Gauss's
geometry
of curved
surfaces. From
this
analogy
he
concluded
that the
gravitational
field
had
to be
represented
not
by a
scalar
potential
but
by a
ten-component
metrical
tensor,
a
crucial
step
toward
general relativity.
When Einstein returned
from
Prague
to
Zurich
in
August 1912,
his
friend Marcel
Grossmann,
professor
of mathematics
at the
ETH,
played
an
important
role in
extending
his
horizon
to
include
the
works
of
Riemann, Christoffel, Ricci, and Levi-Civita, and also in
assisting
him in
his
search for
gravitational
field
equations
based
on
tensor
calculus.
In
May 1913,
more
than
a
year
after
the
completion
of
the static
theory,
Einstein and Grossmann
published
a
preliminary
but
comprehensive
version
of
a
generalized theory
of
relativity
and
gravitation (Doc. 13). Although
they
had
made
important progress,
their results remained
unsatisfactory, not only
to
their critics but
also to
the
authors
themselves. The
reason was
that
they
had
searched for
a
generally
covariant
theory
but
had not
reached their
goal.
In
fact,
they were
not
even
sure
whether transformations
to
rotating
coordinates
were permitted by
their
equations so
that
it
was
not
clear whether
a
rotating
frame of reference could
be
considered
as being equivalent
to
a
rest
frame.
After
the
publication
of
the
paper,
Einstein
wrote to
H. A.
Lorentz:
"Thus,
if
not all
systems
of
equations
of
the
theory
...
admit transformations other than
linear
ones,
then
the
theory
violates its
own
point
of
departure;
it is then left
up
in the air."[5]
With
hindsight it
is
easy
to
see
how
close Einstein and Grossmann
came
to
the final
general
theory
of
relativity.
In
a
retrospective
account
Einstein recalled
how he
had
initially
considered
the
Riemann
curvature tensor
as a
possible
basis
upon
which
to construct generally
covariant
field
equations,
but had failed
to
understand that
it
was
indeed
applicable. Only
after
two years
of hard work
on
conceptual
as
well
as
mathematical
problems
had he
come
back
to
it and
based
his final
theory
on
it.
What made Einstein
reject
the
Riemann
tensor
is
a
question
that
has
been
widely
discussed in
the
literature.
Material
presented
in this
volume
provides important
clues toward
an answer.
Einstein's research
[5]"Wenn also nicht
alle
Gleichungssysteme
der Theorie
...
ausser
den linearen noch andere
Transformationen
zulassen,
so
widerlegt
die Theorie
ihren eigenen Ausgangspunkt;
sie steht
dann in
der Luft."
See
Einstein
to
H. A. Lorentz, 14 August
1913
(Vol. 5,
Doc.
467).