4 8 D O C . 5 2 O N A F F I N E F I E L D T H E O R Y
52. “On the Affine Field Theory”
[Einstein 1923k]
Presented 31 May 1923
Published 28 June 1923
In: Preußische Akademie der Wissenschaften (Berlin). Physikalisch-mathematische Klasse.
Sitzungsberichte (1923): 137–140.
Further reflection led me to a perfecting of the field theory of gravitation and
electricity treated in two earlier communications. In what follows I would like to
present the theory briefly in its new
form.[1]
Let the affine connection be described by the 40 functions . Riemann’s cur-
vature tensor of second order is split up into a symmetrical part and an
antisymmetrical part , so that one has
, (1)
. (2)
The Hamiltonian function H (scalar density) shall be a preliminarily un-
known function
1)
of the ’s and ’s. The Hamilton integral is varied ac-
cording to the (= ). One initially obtains
,
(3)[3]
where to abbreviate we set
1)
The assumption that H depended only on is abandoned. I would like to
remark furthermore that Mr. Droste in Leyden already thought of similar considerations
two years before but did not publish
them.[2]
[p. 137]
Γμν
α
Rμν γμν
φμν
γμν
∂xα
∂
Γμν α – Γμβ α Γνα β
1
2©
--§
-
∂xν
∂
Γμα α
∂xμ
∂
Γνα¹ α +
·
Γαν α Γαβ β – + + =
φμν
1
2©
--§
-
∂xν
∂
Γμα α
∂xμ
∂
Γνα¹ α –
·
=
γμν φμν
γμν φμν +
Γμν α Γνμ α
gμνδγμν fμνδφμν) + dτ
³(
0 =