D O C . 1 3 C O M M E N T O N G E N E R A L R E L A T I V I T Y 1 7
13. “Comment on My Paper ‘On the General Theory of
Relativity’”[1]
[Einstein 1923h]
Presented 12 April 1923
Published 15 May 1923
In: Preußische Akademie der Wissenschaften (Berlin). Physikalisch-mathematische
Klasse. Sitzungsberichte (1923): 76–77.
In the following, the
formal[2]
essential aspects of the theory advanced in the cit-
ed paper are put forth once more in a clear overview, enriched by a new Hamilto-
nian form of expressing the theory, one that allows its relation to the general theory
of relativity in its original form to emerge particularly clearly.
One bases an affine connection in the space-time continuum according to the
formula
(1)
which is limited a priori only by the symmetry
condition[3]
.
From it follows the existence of the second-order Riemann tensor
(2)[4]
as well as the circumstance that the determinant has the character of the
square of a scalar density. From this follows the invariance of the fundamental law
of the theory
. (3)
In this case the ’s should be considered as being expressed by the Γ’s and should
be varied according to the Γ quantities. One thus obtains 40 equations that can be
solved for the quantities Γ. This yields
[p. 76]
Aμ
δ Γαβ
μ Aαdxβ
–=
Γαβ μ Γβα μ =
rkl
∂xα
∂
Γkl
α
– Γkβ
α Γlβ
α
∂xl
∂
Γkα
α
Γkl
α
Γαβ
β
– + + =
r rkl =
δ® –r rd
³
¯ ¿
¾
­ ½
0 =
rkl