DOC. 25 FIELD
EQUATIONS
OF GRAVITATION
117
Doc.
25
The Field Equations of Gravitation
Session of the physicalmathematical class
on
November 25,
1915
by
A.
Einstein
[p.
844]
In two
recently published
papers1
I have shown how to obtain field
equations
of
gravitation
that
comply
with the
postulate
of
general relativity,
i.e.,
which in their
general
formulation
are
covariant under
arbitrary
substitutions of
spacetime
variables.
Historically they
evolved
in
the
following sequence.
First,
I
found
equations
that
contain the
Newtonian
theory
as an
approximation
and
are
also covariant under
arbitrary
substitutions
of
determinant
1.
Then I found that these
equations are
equivalent
to
generallycovariant
ones
if
the scalar
of
the
energy
tensor
of
"matter"
vanishes. The coordinate
system
could then be
specialized by
the
simple
rule
that
v/g
must
equal
1,
which leads to
an
immense
simplification
of
the
equations
of
the
theory.
It has to be
mentioned, however,
that this
requires
the introduction of
the
hypothesis
that the scalar
of
the
energy
tensor
of
matter vanishes.
I
now quite recently
found that
one can get away
without this
hypothesis
about
the
energy
tensor of matter
merely by inserting
it
into the field
equations
in
a slightly
different
way.
The
field
equations
for
vacuum,
onto which
I
based the
explanation
of the
Mercury perihelion,
remain unaffected
by
this modification. In order not to
force the reader
constantly
to
consult the
previous publications,
I
repeat
here the
considerations in their
entirety.
One derives from the wellknown Riemanncovariant
of
rank four the
following
covariant
of
rank
two:
[2]
Gim
=
Rim
+
Sim
(1)
Rim =
El
d{iml}/dxl
+
Elp{ilp}{mpl}
(1a)
Sim
=
El
d{ill}/dxm

Elp{imp}{pll}
(1b)
1Sitzungsber.
44
p.
778
and 46,
p.
799 (1915). [1]