 
Summary: arXiv:physics/00020093Feb2000
The singular points of Einstein's universe
by Marcel Brillouin
(translation by S. Antoci
)
1. Einstein's fourdimensional Universe is determined by the ten gµ of its ds2
. In order to
determine them it is not sufficient to know the six independent partial differential equations that
they must fulfil; one needs also to know the conditions at the boundaries of the Universe, which are
necessary for specialising the integrals in view of given problems. These boundary conditions are of
two sorts. One deals with the far away state of the Universe, completely outside the region where we
wish to study the events; it is the one whose choice, still in dispute, is translated into this question:
is the Universe infinite? Is it finite, although without limit? I do not bother with this here. The
other one deals with the singular lines that correspond to what, from the experimental viewpoint,
we call the attractive masses. In Newtonian gravitation the material point of mass m corresponds
to the point of the Euclidean space where the integral of Laplace's equation becomes infinite like
m/r, where r is (in the neighbourhood of this point) the distance from the material point to the
point where one studies the Newtonian potential. It is this kind of singularities, characteristic of
matter, that I come to consider.
We first remark that, in the present state of our experimental knowledge, nothing entitles us
