INTRODUCTION
TO
VOLUME
8
li
nection
between
general
covariance and
energy-momentum
conservation
(Doc.
269).[35]
Although
Einstein
did not
use
this
argument
in his
response
to
Kretschmann in Einstein
1918f(Vol. 7,
Doc.
4),
this connection
suggests
that the
general
covariance
of
general relativity
is not
physically vacuous
after
all. The
exact
nature
of
the
relationship
between
general
covariance and
energy-momentum
conservation
was
subsequently
clarified
by
Hermann
Weyl,
and the
Göttingen
group
of
Hilbert and Felix Klein and their
assistants,
foremost
among
whom
was
Emmy
Noether. Einstein
was
involved
in
this clarification
mainly
through
his
cor-
respondence
with
Klein in
1918.[36]
Given
this
general background,
one
can
under-
stand
why
Einstein showed
no
interest in the
algebraic
identities known
as
the
Bianchi identities
or
in the related differential identities known
as
the
contracted
Bianchi
identities,
of
which at least two
of
his
correspondents
availed themselves
to derive
energy-momentum
conservation from the field
equations.[37]
Einstein
was
interested in how these identities
are
related
to
the invariance
of
the
action
integral.
Because
of
physical
considerations,
Einstein insisted
that
the
right-hand
side
of
the field
equations
can
be written
as
the
sum
of
quantities representing
the
energy-
momentum
of
matter and the
energy-momentum
of
the
gravitational
field,
and
that
the law
of
energy-momentum
conservation
can
be written
as
the
vanishing
of
the
ordinary
coordinate
divergence
of
that
sum.
This meant
that
he had to
represent
gravitational energy-momentum by a nongenerally
covariant
quantity. Virtually
all
his
correspondents
objected
to this
pseudo-tensor
since it
marred
the
general
covariance
of
the
theory.[38]
The
alternative-suggested
independently
by Lorentz,
Levi-Civita,
and
Klein[39]-was
to
define the
left-hand
side
of
the field
equations
as
the
gravitational energy-momentum
tensor and
to
accept
that the law
of
energy-
momentum conservation cannot be written in the
simple
form
on
which Einstein
insisted. Einstein
staunchly
defended his
usage
of
the
pseudo-tensor
tuv,
both in
correspondence
and in
print.[40]
From
a
modem
point
of
view,
Einstein
was right
in
this debate. Gravitational
energy-momentum
is
represented by a
pseudo-tensor
because
it
cannot
be
localized.[41]
It is thus
perfectly
acceptable
that,
at
any given
point,
tuv
can
be
zero or non-zero depending on
one’s
choice
of
coordinates.
This
interpretation
in terms
of
nonlocalizability
of
gravitational energy-momen-
tum
was
not articulated
at
the time and Einstein
himself
struggled
with
some
of
the
pseudo-tensor’s
counterintuitive
properties.
In Einstein
1916g (Vol. 6,
Doc.
32),
he
arrived at the
strange
result
that his
theory seems
to
allow
gravitational waves
that
do
not
transport energy.
He noticed that these
spurious waves can
be transformed
away by switching
to coordinates
satisfying
the
determinant
condition
V-g
=
1.
Initially,
he
thought
that this shows that such coordinates
are physically privileged
(Doc. 227).
The
following year,
however,
Nordstrom
showed that these coordinates
engender some
counterintuitive results
of their
own.
In coordinates
satisfying