236
DOC.
240 JULY
1916
The difficulties discussed arise from
the
fact
that
scalar
l
contains second
derivatives.[2] In
L
=
V-gl,
terms
in
the
form of
r,
d%9ik
iklm
a a
OX
i
OXm
occur.
(Giklm
only dependent
on
guv.)
However, ƒ
Ldr
can
be modified
through
partial integration
so
that[3]
f
LdT
=
f
L*
dr
+
surface
integral,
where
L*
just
contains
the
first
derivatives of
guv.
This
is
because
dgik
,
dldm
T
I
dGiklm
&9ik
j
Í
9
dxi
dxm
J
dxi
V
dr
That
is
why
convertible into surf.
integral.
contains
only
first
derivatives.
S{ ƒ
Ldr}
=
6{J
L*dr},
and
in
the
Hamiltonian
consideration,
L
is
replaceable by
L*.
Then
no
second
derivatives
appear
in
the
energy components
and
S{ ƒ (L*
+
M)dr}
=
0
can
safely
be set
up
as a
variation
principle,
where variations
are
to
be made
with
respect
to
guv (or
guv)
and
M
refers
to
matter. In
this
way you
will
undoubt-
edly
arrive
at
my
gravitational field
energy components.
I
did
not
perform
the
somewhat tedious
calculation of
tí
=
2k
\dg,aß
dL*
y
a
Qv
gf
-
KL’
It
seems
to
me
only
of
interest that
an
energy
law
thoroughly analogous
to
mine
also exists if the reference
system specialization according
to
the
condition V-g
=
1
is
omitted.[4]
In
any
case,
this
specialization
of
the coordinate
system
makes
the
formulas
clearer
without
impairing
the
generality
of
the
theory.
Best
regards, yours very truly,
A.
Einstein.