DOC. 13
GENERALIZED
THEORY
OF RELATIVITY
341
[19]See Einstein 1912c
(Doc.
3), §3,
for
an
earlier discussion of
the
effect of
the
gravitational
field
on
time
measurement;
see
also Einstein's consideration of the effect of rotational motion
on
the
geometry
of
a
rigidly rotating
disk
in the
same
paper
on p.
356.
[20]An outline of the
following
calculations
is
found in Einstein's
research
notes
on a
gen-
eralized
theory
of
relativity (Doc. 10),
[p.
40].
[21]This
formula
appears on
[p.
10],
[p.
40],
and
[p.
51]
of Einstein's research
notes
on a
generalized theory
of
relativity (Doc. 10).
On
the
first two
pages
listed
one
also
finds
expressions
equivalent
to
the
momentum
and the force
density given
here.
[22]This
equation appears
on [p. 10]
and
on
[p.
51]
of Einstein's research
notes
on a
gener-
alized
theory
of
relativity
(Doc.
10).
See
also
[p. 26],
where Einstein
uses a
gravitational
field
equation
to replace
the
stress-energy tensor
by a
(not generally covariant) gravitation
tensor.
He discusses
a
general
formulation of
energy-momentum
conservation
in terms
of
a sum
of
tensors
in
his
manuscript
on
special relativity
(Doc.
1), [p.
63].
[23]In
a no
longer
available letter
to Einstein, H. A.
Lorentz
pointed
out
that
the
general
form
of
equation
(10)
is
only
covariant
in
the
case
of
a symmetric stress-energy
tensor.
See Einstein
to H. A. Lorentz,
16 August
1913
(Vol. 5,
Doc.
470),
for Einstein's
reply.
[24]See
the editorial
note,
"Einstein
on
Gravitation
and
Relativity:
The Collaboration with
Marcel
Grossmann,"
pp.
297-298,
for
a
discussion of the evolution of Einstein's views
on
the
lack of
general
covariance of the "Entwurf"
theory.
[25]Early
in
1914,
Einstein found "most
general
transformations"
("höchst
allgemeine
Transformationen")
which transform
the
gravitational equations
into
themselves
(see
Einstein
to
Paul
Ehrenfest,
before
10
March
1914
[Vol. 5,
Doc. 512]). See
also Einstein and Grossmann
1914b,
for
a more
detailed discussion of the covariance
properties
of the "Entwurf"
field
equations.
[26]Einstein's
unpublished manuscript
on
special relativity
(Doc. 1)
includes
an
outline of
tensor
calculus
on
[pp.
46-54];
differential
operations
are
discussed
on [pp.
53-54].
[27]This
expression played
an
important
role
in
Einstein's search for
a
gravitational
tensor.
For
a
historical
discussion,
see
Norton
1984,
sec.
3.
[28]The
above
expression, together
with
equation
(11),
generally
does
not
lead
to
the weak
field
equations as they
follow from Einstein's
final
theory
of
gravitation
(see
Einstein
1915e;
see
also Norton
1984,
sec. 4,
for
a
historical
discussion).
For
an
analysis
of
the
role of Einstein's
preconceptions
on
the static
case
for the
development
of the "Entwurf"
theory developed in
this
paper,
see
Norton
1984,
sec.
3,
and
also the editorial
note,
"Einstein's
Research Notes
on
a
Generalized
Theory
of
Relativity,"
sec.
IV.
[29]Various
examples
of the method
employed in
the
following paragraphs
are
found
in
Einstein's research
notes
on a
generalized
theory
of
relativity (Doc. 10)
(see,
e.g.,
[p.
41]).
See
also Einstein 1913c
(Doc. 17),
p.
1254,
for
an application
of
this
method
to
Nordström's
theory
of
gravitation.
[30]On
[pp.
38-41]
of Einstein's research
notes
on a
generalized theory
of
relativity
(Doc.
10),
this
method
is
applied
to
the weak
field
case.
An outline of the derivation of the
equation
below
is
found
on [p. 51]
of these
notes;
see
also
[pp.
49-50]
for related calculations. Einstein's
approach,
and
in
particular
the conditions under
which
it
leads
to
a
unique
solution,
are
dis-
cussed
in
Norton
1984,
sec.
4.
[31]The
divergence
of
this tensor is
evaluated for
the
special
case
of
a
rotating
frame of
reference
in
Einstein's research
notes
on a
generalized theory
of
relativity
(Doc. 10), [p. 48].
[32]Einstein
discussed the transformation
properties
of
this
equation
in
a
letter
to
Lorentz
(see
Einstein
to
H.
A. Lorentz, 14
August
1913
[Vol. 5,
Doc.
467]).
[33]See
also the discussion of
this
similarity
between
the two tensors in
Einstein 1913c
(Doc.
16),
p.
1259,
where it
is
interpreted
as
expressing
the
equality
of
inertial
and
gravitational mass.
See
Einstein 1914c
(Doc. 24)
for Einstein's further
analysis
of the role of
gravitational energy
in
his
theory.
[34]For
a
discussion of the role
this
equation
later
played
in
the
analysis
of the covariance
properties
of the "Entwurf"
field
equations, see
the editorial
note,
"Einstein
on
Gravitation
and
Relativity:
The Collaboration with Marcel
Grossmann,"
p.
297.