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with approximately the same density in all directions. Thereby
we are moved to the assumption that the spatial isotropy of the
system would hold for all observers, for every place and every
time of an observer who is at rest as compared with surrounding
matter. On the other hand we no longer make the assumption
that the average density of matter, for an observer who is at rest
relative to neighbouring matter, is constant with respect to time.
With this we drop the assumption that the expression of the
metric field is independent of time.
We now have to find a mathematical form for the condition
that the universe, spatially speaking, is isotropic everywhere.
Through every point P of (four-dimensional) space there is the
path of a particle (which in the following will be called  geo-
desic for short). Let P and Q be two infinitesimally near points of
* He showed that it is possible, according to the field equations, to have a finite
density in the whole (three-dimensional) space, without enlarging these field
equations ad hoc. Zeitschr. f. Phys., 10 (1922).
116 the meaning of relativity
such a geodesic. We shall then have to demand that the expres-
sion of the field shall be invariant relative to any rotation of the
co-ordinate system keeping P and Q fixed. This will be valid for
any element of any geodesic.*
The condition of the above invariance implies that the entire
geodesic lies on the axis of rotation and that its points remain
invariant under rotation of the co-ordinate system. This means
that the solution shall be invariant with respect to all rotations of
the co-ordinate system around the triple infinity of geodesics.
For the sake of brevity I will not go into the deductive deriva-
tion of the solution of this problem. It seems intuitively evident,
however, for the three-dimensional space that a metric which
is invariant under rotations around a double infinity of lines
will be essentially of the type of central symmetry (by suitable
choice of co-ordinates), where the axes of rotations are the radial
straight lines, which by reasons of symmetry are geodesics. The
surfaces of constant radius are then surfaces of constant (posi-
tive) curvature which are everywhere perpendicular to the
(radial) geodesics. Hence we obtain in invariant language:
There exists a family of surfaces orthogonal to the geodesics.
Each of these surfaces is a surface of constant curvature. The
segments of these geodesics contained between any two surfaces
of the family are equal.
Remark. The case which has thus been obtained intuitively is
not the general one in so far as the surfaces of the family could
be of constant negative curvature or Euclidean (zero curvature).
The four-dimensional case which interests us is entirely
analogous. Furthermore there is no essential difference when
the metric space is of index of inertia 1; only that one has to
choose the radial directions as timelike and correspondingly
* This condition not only limits the metric, but it necessitates that for every
geodesic there exist a system of co-ordinates such that relative to this system
the invariance under rotation around this geodesic is valid.
appendix i 117
the directions in the surfaces of the family as spacelike. The axes
of the local light cones of all points lie on the radial lines.
CHOICE OF CO-ORDINATES
Instead of the four co-ordinates for which the spatial isotropy of
the universe is most clearly apparent, we now choose different
co-ordinates which are more convenient from the point of view
of physical interpretation.
As timelike lines on which x1, x2, x3 are constant and x4 alone
variable we choose the particle geodesics which in the central
symmetric form are the straight lines through the centre. Let
x4 further equal the metric distance from the centre. In such
co-ordinates the metric is of the form:
=
ds2 dx42 - dÃ2
(2)
= =
dÃ2 ³ik dxi dxk (i, k 1, 2, 3)
dÃ2 is the metric on one of the spherical hypersurfaces. The ³ik
which belong to different hypersurfaces will then (because of
the central symmetry) be the same form on all hypersurfaces
except for a positive factor which depends on x4 alone:
=
³ik ³ikG2 (2a)
where the ³ depend on x1, x2, x3 only, and G is a function of x4
alone. We have then:
= =
dÃ2 ³ik dxi dxk (i, k 1, 2, 3) (2b)
0 0
is a definite metric of constant curvature in three dimensions,
the same for every G.
Such a metric is characterized by the equations:
118 the meaning of relativity
=
Riklm - B³il³km - ³im³kl 0 (2c)
0 0 0 0 0
We can choose the co-ordinate system (x1, x2, x3) so that the line
element becomes conformally Euclidean:
= + + =
dÃ2 A2(dx12 dx22 dx32) i.e. ³ik A2´ik (2d)
0 0
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