Episode Transcript
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Speaker 1 (00:00):
This is the LibriVox recording. All LibriVox recordings are in
the public domain. For more information or to volunteer, please
visit LibriVox dot org. Recording by Linda lew Relativity the
Special in General Theory by Albert Einstein, Part three, Considerations
(00:24):
on the Universe as a Whole, Sections thirty thirty one
and thirty two. Section thirty Cosmological difficulties of Newton's theory
afar from the difficulty discussed in section twenty one, there
is a second fundamental difficulty attending classical celestial mechanics, which,
(00:49):
to the best of my knowledge, was first discussed in
detail by the astronomer's Sealoker. If we ponder over the
question as to how the universe considered as a whole
to be regarded, the first answer that suggests itself to
us is surely this, as regards space and time, universe
is infinite. There are stars everywhere, so that the density
(01:13):
of matter, although very variable in detail, is nevertheless, on
the average everywhere the same. In other words, however far
we might travel through space, we should find everywhere an
attenuated swarm of fixed stars of approximately the same kind
and density. This view is not in harmony with the
(01:35):
theory of Newton. The latter theory rather requires that the
universe should have a kind of center in which the
density of the stars is a maximum, and that as
we proceed outwards from the center, the group density of
the stars should diminish until finally, at great distances it
is succeeded by an infinite region of emptiness. The stellar
(01:59):
universe to be a finite island in the infinite ocean
of space. Begin finite proof. According to the theory of Newton,
the number of lines of force which come from infinity
and terminate in a mass M is proportional to the
mass M. If on the average, the mass density row
(02:23):
sub zero is constant throughout the universe, then a sphere
of volume v will enclose the average man row sub
zero V. Thus, the number of lines of force passing
through the surface f of the sphere into its interior
is proportional to row sub zero V. For unit area
(02:45):
of the surface of the sphere. The number of lines
of force which enters the sphere is thus proportional to
row sub zero V over F or two row sub
zero R. Hence, the intensity of the field at the
surface would ultimately become infinite with increasing radius R of
(03:09):
the sphere, which is impossible. En footnote, this conception is
in itself not very satisfactory. It is still less satisfactory
because it leads to the result that the light emitted
by the stars and also individual stars of the stellar system,
are perpetually passing out into infinite space, never to return,
(03:34):
and without ever again coming into interaction with other objects
of nature. Such a finite material universe would be destined
to become gradually but systematically impoverished. In order to escape
this dilemma, se Lagur suggested a modification of Newton's law,
in which he assumes that for great distances, the force
(03:56):
of attraction between two masses diminished, which is more rapidly
than would result from the inverse square law. In this way,
it is possible for the mean density of matter to
be constant everywhere, even to infinity, without infinitely large gravitational
fields being produced. We thus free ourselves from the distasteful
(04:21):
conception that the material universe ought to possess something of
the nature of a center. Of course, we purchase our
emancipation from the fundamental difficulties mentioned at the cost of
a modification and complication of Neton's law, which is neither
empirical nor theoretical foundation. We can imagine innumerable laws which
(04:43):
would serve the same purpose without our being able to
state a reason why one of them is to be
preferred to the others. For any one of these laws
would be founded just as little on more general theoretical
principles as is the law of Newton. End of section
thirty Section thirty one the possibility of a finite and
(05:07):
yet unbounded universe. But speculations on the structure of the
universe also move in quite another direction. The development of
non Euclidean geometry led to the recognition of the fact
that we can cast out on the infiniteness of our
space without coming into conflict with the laws of thought
(05:29):
or with experience. Raymond Helmholtz. These questions have already been
treated in detail and with unsurpassablucidity by Helmholtz and point Clary,
whereas I can only touch on them briefly. Here. In
the first place, we imagine an existence in two dimensional space.
(05:50):
Flat beings with flat implements, and in particular flat rigid
measuring rots, are free to move in a plane. For them,
nothing exists outside of this plane. That which they observe
to happen to themselves and their flat things is the
all inclusive reality of their plane. In particular, the constructions
(06:13):
of plain Euclidean geometry can be carried out by means
of the rods, for example the lattice construction considered in
Section twenty four. In contrast to ours, the universe of
these beings is two dimensional, but like ours, it extends
to infinity. In their universe there is room for an
(06:36):
infinite number of identical squares made up of rods i e.
Its volume surface is infinite. If these beings say their
universe is quote Plaine end quote, there is sense in
the statement, because they mean that they can perform the
constructions of plain Euclidean geometry with their rods, and this
(06:59):
can the individual rods always represent the same distance independently
of their position. Let us consider now a second two
dimensional existence, but this time on a spherical surface instead
of on a play. The flat beings, with their measuring
rods and other objects, fit exactly on the surface. They
(07:22):
are unable to leave it. Their whole universe of observation
extends exclusively over the surface of the sphere. Are these
beings able to regard the geometry of their universe as
being plain geometry and their rods withal as a realization
of distance. They cannot do this, for if they attempt
(07:43):
to realize a straight line, they will obtain a cur
which we three dimensional beings designate as a great circle,
i e. A self contained line of definite finite length
which can be measured up by means of a measuring rod. Similarly,
(08:03):
this universe has a finite area that can be compared
with the area of a square constructed with rods. Great
charm resulting from this consideration lies in the recognition of
the fact that the universe of these beings is finite
and yet has no limits. But the spherical surface beings
(08:27):
do not need to go on a world tour in
order to perceive that they are not living in a
euclidian universe. They can convince themselves of this on every
part of their world, provided they do not use too
small a piece of it. Starting from a point, they
draw straight lines, arcs of circles as judged in three
(08:48):
dimensional space, of equal length in all directions. They will
call the line joining the free ends of these lines
a circle. For a plain surface. The ratio of the
circumference of a circle to its diameter, both lengths being
measured with the same rod is, according to Euclidean geometry
(09:10):
of the plane, equal to a constant value Pi, which
is independent of the diameter of the circle. On their
spherical surface, r flat beings would find for this ratio
the value equation twenty seven pie times sign parentheses little
(09:32):
r over big r unparentheses divided by parentheses little r
over big r unparentheses i e. Smaller value than pi,
the difference being the more considerable greater as the radius
of the circle in comparison with the radius R of
(09:55):
the world sphere. By means of this relation, spherical being
d can determine the radius of their universe quote world unquote,
even when only relatively small part of their world sphere
is available for their measurements. But if this part is
very small, indeed, they will no longer be able to
(10:17):
demonstrate that they are on a spherical world and not
on an Euclidean plane, For a small part of a
spherical surface differs only slightly from a piece of a
plane of the same size. Thus, if the spherical surface
beings are living on a planet of which the Solar
System occupies only a negligibly small part of the spherical universe.
(10:42):
They have no means of determining whether they are living
in a finite or an infinite universe, because a piece
of universe to which they have access is in both
cases practically plain or Euclidean. It follows directly from this
discussion that for our sphere beings, the circumference of a
(11:04):
circle first increases with the radius until the circumference of
the universe is reached, and that it thenceforward gradually decreases
to zero for still further increasing values of the radius.
During this process, the area of the circle continues to
increase more and more, until finally becomes equal to the
(11:27):
total area of the whole world sphere. Perhaps the reader
will wonder why we have placed our beings on a
sphere rather than on another closed surface, But this choice
has its justifications in the fact that of all closed surfaces,
the sphere is unique in possessing the property that all
(11:47):
points on it are equivalent. I admit that the ratio
of the circumference SEE of a circle to its radius
R depends on R, but for a given value of R,
it is the same for all points of the world sphere.
In other words, the world sphere is a surface of
constant curvature. To this two dimensional sphere universe, there is
(12:11):
a three dimensional analogy, namely the three dimensional spherical space,
which was discovered by Rimond. Its points are likewise all equivalent.
It possesses a finite volume, which is determined by its
radius two pi squared r cubed. Is it possible to
(12:31):
imagine a spherical space. To imagine a space means nothing
else than that we imagine an epitome of our space experience,
i e. Of experience that we can have in the
movement of rigid bodies. In this sense, we can imagine
a spherical space. Suppose we draw lines or stretch strings
(12:53):
in all directions from a point, and mark off from
each of these the distance are the measuring rock. All
the free ends of these lengths lie on a spherical surface.
We can specially measure up the area F of the
surface by means of a square made up of measuring rods.
(13:14):
If the universe is Euclidean, then F equals four pi
are squared. If it is spherical, then F is always
less than four pi are squared. With increasing values of r,
f increases from zero up to a maximum value, which
is determined by the world radius. But for still further
(13:37):
increasing values of r the area gradually diminishes to zero.
At first, the straight lines which radiate from the starting
point diverge farther and farther from one another, but later
they approach each other, and finally they run together again
at a quote counterpoint un quote to the starting point.
(14:00):
Under such conditions, they have traversed the whole spherical space.
It is easily seen that the three dimensional spherical space
is quite analogous to the two dimensional spherical surface. It
is finite i e of phinite volume, and has no bounds.
It may be mentioned that there is yet another kind
(14:21):
of curved space quote, elliptical space unquote. It can be
regarded as a curved space in which the two counterpoints
are identical indistinguishable from each other. An elliptical universe can
thus be considered to some extent as a curved universe
(14:42):
possessing central symmetry. It follows from what has been said
that closed spaces without limits are conceivable. From amongst these,
spherical space and the elliptical excels in its simplicity, since
all points on it are equivalent. As a result of
this discussion, a most interesting question arises for astronomers and physicists,
(15:07):
and that is whether the universe in which we live
is infinite, whether it is finite in the manner of
the spherical universe. Our experience is far from being sufficient
to enable us to answer this question, but the general
theory of relativity permits of our answering it with a
moderate degree of certainty, and in this connection, the difficulty
(15:30):
mentioned in section thirty finds its solution end of section
thirty one. Section thirty two. Structure a space according to
the general theory of relativity. According to the general theory
of relativity, the geometrical properties of space are not independent,
(15:52):
but they are determined by matter. Thus, we can draw
conclusions about the geometrical structure of the universe only if
we base our considerations on the state of the matter
as being something that is known. We know from experience
that for a suitably chosen coordinate system, the velocities of
the stars are small as compared with the velocity of
(16:15):
transmission of light. We can, thus, as a rough approximation,
arrive at a conclusion as to the nature of the
universe as a whole, if we treat the matter as
being at rest. We already know from our previous discussion
that the behavior of measuring rods and clocks is influenced
by gravitational fields, i e. By the distribution of matter.
(16:39):
This in itself is sufficient to exclude the possibility of
the exact validity of Euclidean geometry in our universe. But
it is conceivable that our universe differs only slightly from
a Euclidean one, and this notion seems all the more
probable since calculations show that the metrics of surrounding space
(16:59):
is influenced only to an exceedingly small extent by masses,
even of the magnitude of our sun. We might imagine that,
as regards geometry, our universe behaves analogously to a surface
which is irregularly curved in its individual parts, but which
nowhere departs appreciably from a plane, something like the rippled
(17:21):
surface of a lake. Such a universe might fittingly be
called a quasi euclidian universe. As regards its space, it
would be infinite. But calculation shows that in a quasi
euclidian universe, the average density of matter would necessarily be nil.
Thus such a universe could not be inhabited by matter everywhere.
(17:43):
It would present to us that unsatisfactory picture which we
betrayed in section thirty. If we are to have in
the universe an average density of matter which differs from zero,
however small may be that difference, then the universe cannot
be quasi euclide. On the contrary, the results of calculation
indicate that if matter be distributed uniformly, the universe would
(18:08):
necessarily be spherical or elliptical. Since in reality, the detailed
distribution of matter is not uniform, the real universe will
deviate in individual parts from the spherical i e. The
universe will be quasi spherical, but it will be necessarily
for night. In fact, the theory supplies us with a
(18:28):
simple connection. Begin footnote. For the radius R of the universe,
we obtain the equation R squared equals two over a
capa row. The use of the CGS system in this
equation gives two over k equals one to the eighth
power times ten to the twenty seventh power. P is
(18:52):
the average density of the matter, and K is a
constant connected with the Newtonian constant to gravitation en footnote
between the space expanse of the universe and the average
density of matter in it. End Section thirty two, end
(19:13):
of Part three D of Relativity Special in General Theory
by Albert Einstein.
This is the LibriVox recording. All LibriVox recordings are in
the public domain. For more information or to volunteer, please
visit LibriVox dot org. Recording by Linda lew Relativity the
Special in General Theory by Albert Einstein, Part three, Considerations
(00:24):
on the Universe as a Whole, Sections thirty thirty one
and thirty two. Section thirty Cosmological difficulties of Newton's theory
afar from the difficulty discussed in section twenty one, there
is a second fundamental difficulty attending classical celestial mechanics, which,
(00:49):
to the best of my knowledge, was first discussed in
detail by the astronomer's Sealoker. If we ponder over the
question as to how the universe considered as a whole
to be regarded, the first answer that suggests itself to
us is surely this, as regards space and time, universe
is infinite. There are stars everywhere, so that the density
(01:13):
of matter, although very variable in detail, is nevertheless, on
the average everywhere the same. In other words, however far
we might travel through space, we should find everywhere an
attenuated swarm of fixed stars of approximately the same kind
and density. This view is not in harmony with the
(01:35):
theory of Newton. The latter theory rather requires that the
universe should have a kind of center in which the
density of the stars is a maximum, and that as
we proceed outwards from the center, the group density of
the stars should diminish until finally, at great distances it
is succeeded by an infinite region of emptiness. The stellar
(01:59):
universe to be a finite island in the infinite ocean
of space. Begin finite proof. According to the theory of Newton,
the number of lines of force which come from infinity
and terminate in a mass M is proportional to the
mass M. If on the average, the mass density row
(02:23):
sub zero is constant throughout the universe, then a sphere
of volume v will enclose the average man row sub
zero V. Thus, the number of lines of force passing
through the surface f of the sphere into its interior
is proportional to row sub zero V. For unit area
(02:45):
of the surface of the sphere. The number of lines
of force which enters the sphere is thus proportional to
row sub zero V over F or two row sub
zero R. Hence, the intensity of the field at the
surface would ultimately become infinite with increasing radius R of
(03:09):
the sphere, which is impossible. En footnote, this conception is
in itself not very satisfactory. It is still less satisfactory
because it leads to the result that the light emitted
by the stars and also individual stars of the stellar system,
are perpetually passing out into infinite space, never to return,
(03:34):
and without ever again coming into interaction with other objects
of nature. Such a finite material universe would be destined
to become gradually but systematically impoverished. In order to escape
this dilemma, se Lagur suggested a modification of Newton's law,
in which he assumes that for great distances, the force
(03:56):
of attraction between two masses diminished, which is more rapidly
than would result from the inverse square law. In this way,
it is possible for the mean density of matter to
be constant everywhere, even to infinity, without infinitely large gravitational
fields being produced. We thus free ourselves from the distasteful
(04:21):
conception that the material universe ought to possess something of
the nature of a center. Of course, we purchase our
emancipation from the fundamental difficulties mentioned at the cost of
a modification and complication of Neton's law, which is neither
empirical nor theoretical foundation. We can imagine innumerable laws which
(04:43):
would serve the same purpose without our being able to
state a reason why one of them is to be
preferred to the others. For any one of these laws
would be founded just as little on more general theoretical
principles as is the law of Newton. End of section
thirty Section thirty one the possibility of a finite and
(05:07):
yet unbounded universe. But speculations on the structure of the
universe also move in quite another direction. The development of
non Euclidean geometry led to the recognition of the fact
that we can cast out on the infiniteness of our
space without coming into conflict with the laws of thought
(05:29):
or with experience. Raymond Helmholtz. These questions have already been
treated in detail and with unsurpassablucidity by Helmholtz and point Clary,
whereas I can only touch on them briefly. Here. In
the first place, we imagine an existence in two dimensional space.
(05:50):
Flat beings with flat implements, and in particular flat rigid
measuring rots, are free to move in a plane. For them,
nothing exists outside of this plane. That which they observe
to happen to themselves and their flat things is the
all inclusive reality of their plane. In particular, the constructions
(06:13):
of plain Euclidean geometry can be carried out by means
of the rods, for example the lattice construction considered in
Section twenty four. In contrast to ours, the universe of
these beings is two dimensional, but like ours, it extends
to infinity. In their universe there is room for an
(06:36):
infinite number of identical squares made up of rods i e.
Its volume surface is infinite. If these beings say their
universe is quote Plaine end quote, there is sense in
the statement, because they mean that they can perform the
constructions of plain Euclidean geometry with their rods, and this
(06:59):
can the individual rods always represent the same distance independently
of their position. Let us consider now a second two
dimensional existence, but this time on a spherical surface instead
of on a play. The flat beings, with their measuring
rods and other objects, fit exactly on the surface. They
(07:22):
are unable to leave it. Their whole universe of observation
extends exclusively over the surface of the sphere. Are these
beings able to regard the geometry of their universe as
being plain geometry and their rods withal as a realization
of distance. They cannot do this, for if they attempt
(07:43):
to realize a straight line, they will obtain a cur
which we three dimensional beings designate as a great circle,
i e. A self contained line of definite finite length
which can be measured up by means of a measuring rod. Similarly,
(08:03):
this universe has a finite area that can be compared
with the area of a square constructed with rods. Great
charm resulting from this consideration lies in the recognition of
the fact that the universe of these beings is finite
and yet has no limits. But the spherical surface beings
(08:27):
do not need to go on a world tour in
order to perceive that they are not living in a
euclidian universe. They can convince themselves of this on every
part of their world, provided they do not use too
small a piece of it. Starting from a point, they
draw straight lines, arcs of circles as judged in three
(08:48):
dimensional space, of equal length in all directions. They will
call the line joining the free ends of these lines
a circle. For a plain surface. The ratio of the
circumference of a circle to its diameter, both lengths being
measured with the same rod is, according to Euclidean geometry
(09:10):
of the plane, equal to a constant value Pi, which
is independent of the diameter of the circle. On their
spherical surface, r flat beings would find for this ratio
the value equation twenty seven pie times sign parentheses little
(09:32):
r over big r unparentheses divided by parentheses little r
over big r unparentheses i e. Smaller value than pi,
the difference being the more considerable greater as the radius
of the circle in comparison with the radius R of
(09:55):
the world sphere. By means of this relation, spherical being
d can determine the radius of their universe quote world unquote,
even when only relatively small part of their world sphere
is available for their measurements. But if this part is
very small, indeed, they will no longer be able to
(10:17):
demonstrate that they are on a spherical world and not
on an Euclidean plane, For a small part of a
spherical surface differs only slightly from a piece of a
plane of the same size. Thus, if the spherical surface
beings are living on a planet of which the Solar
System occupies only a negligibly small part of the spherical universe.
(10:42):
They have no means of determining whether they are living
in a finite or an infinite universe, because a piece
of universe to which they have access is in both
cases practically plain or Euclidean. It follows directly from this
discussion that for our sphere beings, the circumference of a
(11:04):
circle first increases with the radius until the circumference of
the universe is reached, and that it thenceforward gradually decreases
to zero for still further increasing values of the radius.
During this process, the area of the circle continues to
increase more and more, until finally becomes equal to the
(11:27):
total area of the whole world sphere. Perhaps the reader
will wonder why we have placed our beings on a
sphere rather than on another closed surface, But this choice
has its justifications in the fact that of all closed surfaces,
the sphere is unique in possessing the property that all
(11:47):
points on it are equivalent. I admit that the ratio
of the circumference SEE of a circle to its radius
R depends on R, but for a given value of R,
it is the same for all points of the world sphere.
In other words, the world sphere is a surface of
constant curvature. To this two dimensional sphere universe, there is
(12:11):
a three dimensional analogy, namely the three dimensional spherical space,
which was discovered by Rimond. Its points are likewise all equivalent.
It possesses a finite volume, which is determined by its
radius two pi squared r cubed. Is it possible to
(12:31):
imagine a spherical space. To imagine a space means nothing
else than that we imagine an epitome of our space experience,
i e. Of experience that we can have in the
movement of rigid bodies. In this sense, we can imagine
a spherical space. Suppose we draw lines or stretch strings
(12:53):
in all directions from a point, and mark off from
each of these the distance are the measuring rock. All
the free ends of these lengths lie on a spherical surface.
We can specially measure up the area F of the
surface by means of a square made up of measuring rods.
(13:14):
If the universe is Euclidean, then F equals four pi
are squared. If it is spherical, then F is always
less than four pi are squared. With increasing values of r,
f increases from zero up to a maximum value, which
is determined by the world radius. But for still further
(13:37):
increasing values of r the area gradually diminishes to zero.
At first, the straight lines which radiate from the starting
point diverge farther and farther from one another, but later
they approach each other, and finally they run together again
at a quote counterpoint un quote to the starting point.
(14:00):
Under such conditions, they have traversed the whole spherical space.
It is easily seen that the three dimensional spherical space
is quite analogous to the two dimensional spherical surface. It
is finite i e of phinite volume, and has no bounds.
It may be mentioned that there is yet another kind
(14:21):
of curved space quote, elliptical space unquote. It can be
regarded as a curved space in which the two counterpoints
are identical indistinguishable from each other. An elliptical universe can
thus be considered to some extent as a curved universe
(14:42):
possessing central symmetry. It follows from what has been said
that closed spaces without limits are conceivable. From amongst these,
spherical space and the elliptical excels in its simplicity, since
all points on it are equivalent. As a result of
this discussion, a most interesting question arises for astronomers and physicists,
(15:07):
and that is whether the universe in which we live
is infinite, whether it is finite in the manner of
the spherical universe. Our experience is far from being sufficient
to enable us to answer this question, but the general
theory of relativity permits of our answering it with a
moderate degree of certainty, and in this connection, the difficulty
(15:30):
mentioned in section thirty finds its solution end of section
thirty one. Section thirty two. Structure a space according to
the general theory of relativity. According to the general theory
of relativity, the geometrical properties of space are not independent,
(15:52):
but they are determined by matter. Thus, we can draw
conclusions about the geometrical structure of the universe only if
we base our considerations on the state of the matter
as being something that is known. We know from experience
that for a suitably chosen coordinate system, the velocities of
the stars are small as compared with the velocity of
(16:15):
transmission of light. We can, thus, as a rough approximation,
arrive at a conclusion as to the nature of the
universe as a whole, if we treat the matter as
being at rest. We already know from our previous discussion
that the behavior of measuring rods and clocks is influenced
by gravitational fields, i e. By the distribution of matter.
(16:39):
This in itself is sufficient to exclude the possibility of
the exact validity of Euclidean geometry in our universe. But
it is conceivable that our universe differs only slightly from
a Euclidean one, and this notion seems all the more
probable since calculations show that the metrics of surrounding space
(16:59):
is influenced only to an exceedingly small extent by masses,
even of the magnitude of our sun. We might imagine that,
as regards geometry, our universe behaves analogously to a surface
which is irregularly curved in its individual parts, but which
nowhere departs appreciably from a plane, something like the rippled
(17:21):
surface of a lake. Such a universe might fittingly be
called a quasi euclidian universe. As regards its space, it
would be infinite. But calculation shows that in a quasi
euclidian universe, the average density of matter would necessarily be nil.
Thus such a universe could not be inhabited by matter everywhere.
(17:43):
It would present to us that unsatisfactory picture which we
betrayed in section thirty. If we are to have in
the universe an average density of matter which differs from zero,
however small may be that difference, then the universe cannot
be quasi euclide. On the contrary, the results of calculation
indicate that if matter be distributed uniformly, the universe would
(18:08):
necessarily be spherical or elliptical. Since in reality, the detailed
distribution of matter is not uniform, the real universe will
deviate in individual parts from the spherical i e. The
universe will be quasi spherical, but it will be necessarily
for night. In fact, the theory supplies us with a
(18:28):
simple connection. Begin footnote. For the radius R of the universe,
we obtain the equation R squared equals two over a
capa row. The use of the CGS system in this
equation gives two over k equals one to the eighth
power times ten to the twenty seventh power. P is
(18:52):
the average density of the matter, and K is a
constant connected with the Newtonian constant to gravitation en footnote
between the space expanse of the universe and the average
density of matter in it. End Section thirty two, end
(19:13):
of Part three D of Relativity Special in General Theory
by Albert Einstein.
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