Catadores de la Palabra en Google+ To go to Einstein 5: http://fdocc.ucoz.com/index/090
As presented in the book "The
Meaning of Relativity”
"In 1921, five years
after the appearance of his comprehensive paper on general relativity and
twelve years before he left Europe permanently to join the Institute for
Advanced Study, Albert Einstein visited Princeton University,
where he delivered the Stafford Little Lectures for that year. These
four lectures constituted an overview of his then controversial theory of
relativity. Princeton University Press made the lectures available under
the title The Meaning of Relativity, the first book by Einstein to be
produced by an American publisher”.
"The General Theory of Relativity (Continued)” [Finiteness of the Universe]
… [I] shall give a brief
discussion of the socalled cosmological problem… our previous considerations,
based upon the field equations (96), had for a foundation the conception that
space on the whole is GalileanEuclidean, and that this character is disturbed
only by asses embedded in it. This conception was certainly justified as long
as we were dealing with spaces of the order of magnitude of those that
astronomy has to do with. But whether portions of the universe, however large
they may be, are quasiEuclidean, is a wholly different question. We can make
this clear by using an example from the theory of surfaces which we have
employed many times. If a portion of a surface is
observed by the eye to be practically plane, it does not at all follow that the
whole surface has the form of a plane; the
surface might just as well be a sphere, for example, of sufficiently large
radius. The question as to whether the universe as a whole is
nonEuclidean was much discussed from the geometrical point of view before the
development of the theory of relativity. But with the theory of relativity,
this problem has entered upon a new stage, for according to this theory the
geometrical properties of bodies are not independent, but depend upon the
distribution of masses [the field equation is represented as (96): R_{mn} – ½g_{mn
}R = kT_{mn
}]…
…The possibility seems to be
particularly satisfying that the universe is spatially bounded and… is of
constant curvature, being either spherical or elliptical; for
then the boundary conditions at infinity which are so inconvenient from the
standpoint of the general theory of relativity, may be replaced by the much
more natural conditions for a closed surface…
Thus we may present the following
arguments against the conception of a spaceinfinite,
and for the conception of a spacebounded, universe:
1. From the standpoint
of the theory of relativity, the condition for a closed surface is very much
simpler than the corresponding boundary condition at infinity of the
quasiEuclidean structure of the universe.
2. The idea that Mach expressed,
that inertia depends upon the mutual action of bodies, is contained, to a first
approximation, 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. 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,
Mach’s idea gains in probability. But this idea of
Match’s corresponds only to a finite universe, bounded in space, and not to a quasiEuclidean, infinite universe. From the standpoint of epistemology it is more satisfying
to have the mechanical properties of space completely determined by matter, and
this is the case only in a spacebounded 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 for the Second Edition.
On the "Cosmologic Problem”
…The mathematician Friedman found
a way out of this dilemma (the introduction of l (a
universal constant, the "cosmologic constant”) he showed that it is possible,
according to the field equations, to have a finite
density in the whole (threedimensional) space, without enlarging these
field equations ad hoc. Zeitschr. F. Phys. 10 (1922)). His result then
found a surprising confirmation by Hubble’s discovery of the expansion of the
stellar system (a red shift of the spectral lines which increases uniformly
with distance. The existence of the red shift of the spectral lines by the
(negative) gravitational potential of the place of origin. This demonstration
was made possible by the discovery of socalled "dwarf stars” whose average
density exceeds that of water by a factor of the order 10^{4}. For such
a star (e.g. the faint companion of Sirius), whose mass and radius can be
determined (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), this red shift was expected, by the theory,
to be about 20 times as large as for the sun, and indeed it was demonstrated to
be within the expected range). The following is essentially nothing but an
exposition of Friedman’s idea:
FOURDIMENSIONAL SPACE
WHICH IS ISOTROPIC WITH RESPECT TO
THREE DIMENSIONS
…The surfaces
of constant radius are then surfaces of constant (positive) curvature which are
everywhere perpendicular to the (radial) geodesics… There exists a
family of surfaces orthogonal to the geodesics. Each
of these surfaces is a surface of constant curvature…
CHOICE OF COORDINATES
(3c) A = (1/1 + cr^{2}); B
= 4c
c > 0 (spherical space)
c < 0
(pseudospherical space)
c = 0
(Euclidean space)
…we can further get in the first
case c = ¼, in the second case c = ¼
… In
the spherical case the "circumference” of the unit space (G = 1) is
∫ [dr/1+(r^{2}/4)]
= 2p
[ ∫ going from infinite (^{∞})
to infinite(_{∞}) ]
the "radius” of the unit space is
1. In all three cases the function G of time is a measure for the change
with time of the distance of two points of matter (measured on a spatial
section). In the spherical case, G is the
radius of space at the time x_{4}.
THE FIELD EQUATIONS
… Since G is in all cases
a relative measure for the metric distance of two material particles as
function of time, G’/G expresses Hubble’s expansion…
THE SPECIAL CASE OF VANISHING
SPATIAL CURVATURE
…The relation between Hubble’s
expansion… and the average density…, is comparable to some extent with
experience, at least as far as the order of magnitude is concerned. The expansion
is given as 432 km/sec for the distance of 10^{6} parsec…
…Can
the present difficulty, which arouse under the assumption of a practically
negligible spatial curvature, be eliminated by the introduction of a suitable
spatial curvature?…
SUMMARY AND OTHER REMARKS (1) The introduction of the "cosmologic member” (l) into the equation of gravity, though possible from
the point of view of relativity, is to be rejected from the point of view of
logical economy. As Friedman was the first to show
one can reconcile an everywhere finite density of matter with the original form
of the equations of gravity if one admits the time variability of the metric
distance of two mass points [If Hubble’s expansion had been discovered
at the time of the conception of the "general theory of relativity”, the
cosmologic member (l) would never have been
introduced. It seems now so much less justified to introduce such a member into
the field equations, since its introduction loses its sole original
justification,  that of leading to a natural
solution of the cosmologic problem]
(2) The demand for spatial isotropy
of the universe alone leads to Friedman’s form. It is therefore undoubtedly the
general form, which fits the cosmologic problem.
(3) Neglecting the influence of spatial
curvature, one obtains a relation between the mean density and Hubble’s
expansion which, as to order of magnitude, is confirmed empirically... (6) …It
seems to me… that the "theory of evolution” of the stars rests on weaker
foundations than the field equations.
…The
"beginning of the world” (the "beginning of the expansion”) really constitutes
a beginning, from the point of view of the development of the now existing
stars and systems of stars, at which those stars and systems of stars did not
yet exist as individual entities. (8) For
the reasons given it seems that we have to take the idea of an expanding
universe seriously, in spite of the short "lifetime” (10^{9}). If one
does so, the main question becomes whether space has
positive (spherical case) or negative (pseudospherical case) spatial curvature…
