DOC. 32 INTEGRATION OF FIELD EQUATIONS 201
Session of the physical-mathematical class on June 22, 1916 [p. 688]
Approximative Integration of the Field Equations of Gravitation
by A. Einstein
For the treatment of the special (not basic) problems in gravitational theory one can
be satisfied with a first approximation of the guv. The same reasons as in the special
theory of relativity make it advantageous to use the imaginary time variable x4 = it.
By "first approximation" we mean that the quantities yuv, defined by the equation
guv = -8uv + Yuv, (1)
are small compared to 1, such that their squares and products are negligible compared
with first powers; furthermore, they have a tensorial character under linear,
orthogonal transformations. In addition, 8uv = 1 or 8uv = 0 resp. depending upon
u = v or u # v.
We shall show that these yuv can be calculated in a manner analogous to that of
retarded potentials in electrodynamics. From this follows next that gravitational fields
propagate at the speed of light. Subsequent to this general solution we shall
investigate gravitational waves and how they originate. It turned out that my
suggested choice of a system of reference with the condition g = |guv/ = - 1 is not
advantageous for the calculation of fields in first approximation. A note in a letter 
from the astronomer De Sitter alerted me to his finding that a choice of reference
system, different from the one I had previously given,1 leads to a simpler expression
of the gravitational field of a mass point at rest. I therefore take the generally
invariant field equations as a basis in what follows.
§1. Integration of the Approximated Equations of the Gravitational Field [p. 689]
The field equations in their covariant form are 
1Sitzungsber. 47 (1915), p. 833.