152 DOC.
153
NOVEMBER
1915
153. To
Arnold Sommerfeld
Berlin, 28
November
[1915][1]
Dear
Sommerfeld,
You
must not be
cross
with
me
that
I
am
answering your
kind and
interesting
letter
only today.
But in
the last
month
I
had
one
of
the
most
stimulating,
exhausting
times
of
my life,
indeed
also
one
of
the
most
successful.
I
could
not
think
of
writing.
For
I
realized
that
my existing
gravitational
field
equations
were entirely
un-
tenable! The
following
indications led
to this:[2]
1)
I proved
that the
gravitational
field
on a
uniformly rotating system
does
not
satisfy
the
field
equations.[3]
2)
The motion of
Mercury’s perihelion
came
to 18"
rather than
45"
per
century.[4]
3)
The
covariance considerations
in
my
paper
of
last
year
do
not
yield
the
Hamil-
tonian function
H.
When
it
is
properly generalized,
it
permits
an
arbitrary H.[5]
From this it
was
demonstrated that
covariance with
respect
to
“adapted”
coor-
dinate
systems
was a
flop.[6]
Once
every
last bit
of
confidence
in
result and method
of
the
earlier theories
had
given way,
I
saw
clearly
that
it
was
only
through
a
link with
general
covariance
theory, i.e.,
with Riemann’s
covariant,
that
a
satisfactory
solution could be found.
Unfortunately,
I
have immortalized the final
errors
in this
struggle
in
the
Academy
contributions,
which
I
can
send
to
you
directly.[7]
The final result
is
as
follows.[8]
The
gravitational
field
equations
are
generally
covariant. If
(ik,
lm)
is
Christoffel’s
tensor of
the fourth
order,
then
Gim
=
]gk(ik,
lm)
is
a sym-
metrical
tensor of
the second
order.[9]
The
equations
read
Gim
-
-K
Tim
-
r9im
Ys^Tcxß)
¿
aß.
Scalar
derived from
the
energy
tensor
of
“matter,”
for which
I
write
“T”
in the
following.