[ Pobierz całość w formacie PDF ]

of Maxwell s theory these cannot be conceived of as electro-
magnetic fields free from singularities. In order to be consistent
with the facts, it is necessary to introduce energy terms, not
contained in Maxwell s theory, so that the single electric par-
ticles may hold together in spite of the mutual repulsions
the general theory of relativity (continued) 109
between their elements, charged with electricity of one sign. For
the sake of consistency with this fact, Poincaré has assumed a
pressure to exist inside these particles which balances the elec-
trostatic repulsion. It cannot, however, be asserted that this pres-
sure vanishes outside the particles. We shall be consistent with
this circumstance if, in our phenomenological presentation,
we add a pressure term. This must not, however, be confused
with a hydro-dynamical pressure, as it serves only for the
energetic presentation of the dynamical relations inside matter.
Accordingly we put
dx± dx²
=
Tµ½ gµ± g½² Ã - gµ½p (122)
ds ds
In our special case we have, therefore, to put
=
Tµ½ ³µ½p (for µ and ½ from 1 to 3)
=
T44 Ã - p
= + =
T - ³µ½³µ½p à - p à - 4p
Observing that the field equation (96) may be written in the
form
=
Rµ½ - º(Tµ½ - 1 gµ½T)
2
we get from (96) the equations,
2
+ = - p ³µ½
³µ½ º
Ã
a2 2
= +
0 - º p
Ã
2
From this follows
110 the meaning of relativity
Ã
üø
=
p -
ôø
2
(123)
ýø
2
=
a
ºÃ ôø
þø
If the universe is quasi-Euclidean, and its radius of curvature
therefore infinite, then à would vanish. But it is improbable that
the mean density of matter in the universe is actually zero; this is
our third argument against the assumption that the universe is
quasi-Euclidean. Nor does it seem possible that our hypothetical
pressure can vanish; the physical nature of this pressure can be
appreciated only after we have a better theoretical knowledge of
the electromagnetic field. According to the second of equations
(123) the radius, a, of the universe is determined in terms of the
total mass, M, of matter, by the equation
Mº
=
a (124)
4À2
The complete dependence of the geometrical upon the physical
properties becomes clearly apparent by means of this equation.
Thus we may present the following arguments against the
conception of a space-infinite closed, and for the conception of a
space-bounded closed, universe:
1. From the standpoint of the theory of relativity, to postulate
a closed universe is very much simpler than to postulate the
corresponding boundary condition at infinity of the quasi-
Euclidean structure of the universe.
2. The idea that Mach expressed, that inertia depends upon
the mutual action of bodies, is contained, to a first approxima-
tion, in the equations of the theory of relativity; it follows from
these equations that inertia depends, at least in part, upon
mutual actions between masses. Thereby, Mach s idea gains in
the general theory of relativity (continued) 111
probability, as it is an unsatisfactory assumption to make that
inertia depends in part upon mutual actions, and in part upon an
independent property of space. But this idea of Mach s cor-
responds only to a finite universe, bounded in space, and not
to a quasi-Euclidean, infinite universe. From the standpoint of
epistemology it is more satisfying to have the mechanical prop-
erties of space completely determined by matter, and this is
the case only in a closed universe.
3. An infinite universe is possible only if the mean density of
matter in the universe vanishes. Although such an assumption is
logically possible, it is less probable than the assumption that
there is a finite mean density of matter in the universe.
APPENDIX I
On the  cosmologic problem
Since the first edition of this little book some advances have been
made in the theory of relativity. Some of these we shall mention
here only briefly:
The first step forward is the conclusive demonstration of the
existence of the red shift of the spectral lines by the (negative)
gravitational potential of the place of origin (see p. 95). This
demonstration was made possible by the discovery of so-called
 dwarf stars whose average density exceeds that of water by a
factor of the order 104. For such a star (e.g. the faint companion
of Sirius), whose mass and radius can be determined,* this red
shift was expected, by the theory, to be about twenty times as
* The mass is derived from the reaction on Sirius by spectroscopic means,
using the Newtonian laws; the radius is derived from the total lightness and
from the intensity of radiation per unit area, which may be derived from the
temperature of its radiation.
appendix i 113
large as for the sun, and indeed it was demonstrated to be within [ Pobierz całość w formacie PDF ]