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Speaker 1 (00:00):
Correction for this chapter in mathematical formulae instead of I
here one. This is a LibriVox recording. All LibriVox recordings
are in the public domain. For more information or to volunteer,
please visit LibriVox dot org. Recording by mL Cullen, Cleveland, Ohio,
(00:22):
March two thousand and seven. Relativity the Special and General
Theory by Albert Einstein, Appendix three The experimental confirmation of
the general theory of relativity. From a systematic theoretical point
(00:44):
of view, we may imagine the process of evolution of
an empirical science to be a continuous process of induction.
Theories are evolved and are expressed in short compass as
statements of a large number of individual object in the
form of empirical laws from which the general laws can
(01:04):
be ascertained by comparison. Regarded in this way, the development
of a science bears some resemblance to the compilation of
a classified catalog. It is, as it were, a purely
empirical enterprise. But this point of view by no means
(01:25):
embraces the whole of the actual process, for it slurs
over the important part played by intuition and deductive thought
in the development of an exact science. As soon as
a science has emerged from its initial stages, theoretical advances
are no longer achieved merely by a process of arrangement
guided by empirical data. The investigator rather develops a system
(01:49):
of thought which in general is built up logically from
a small number of fundamental assumptions, the so called axioms.
We call such a system of thought a theory. The
theory finds the justification for its existence in the fact
that it correlates a large number of single observations, and
it is just here that the truth of the theory
(02:11):
lies corresponding to the same complex of empirical data. There
may be several theories which differ from one another to
a considerable extent, but as regards the deduction from the
theories which are capable of being tested, the agreement between
the theories may be so complete that it becomes difficult to
find such deductions in which the two theories differ from
(02:33):
each other. As an example, a case of general interest
is available in the province of biology, in the Darwinian
theory of the development of species by selection, in the
struggle for existence, and in the theory of development which
is based on the hypothesis of the hereditary transmission of
acquired characters. We have another instance of far reaching agreement
(02:59):
between the deductions from the two theories and Newtonian mechanics
on the one hand, and the general theory of relativity
on the other. This agreement goes so far that up
to the present we have been able to find only
a few deductions from the general theory of relativity which
are capable of investigation and to which the physics of
pre relativity days does not also lead, and this despite
(03:23):
the profound difference in the fundamental assumption of the two theories.
In what follows, we shall again consider these important deductions,
and we shall also discuss the empirical evidence appertaining to them,
which has hitherto been obtained. A motion of the perihelion
(03:46):
of Mercury. According to Newtonian mechanics and Newton's law of gravitation,
a planet which is revolving around the Sun would describe
an ellipse around the latter, or more correct, around the
common center of gravity of the Sun and the planet.
In such a system, the Sun or the common center
(04:08):
of gravity lies in one of the foci of the
orbital lips in such a manner as that, in the
course of a planet year, the distant Sun planet grows
from a minimum to a maximum and then decreases again
to a minimum. If instead of Newton's law, we insert
a somewhat different law of attraction into the calculation, we
(04:29):
find that according to this new law, the motion would
still take place in such a manner that the distant
Sun planet exhibits periodic variations. But in this case, the
angle described by the line joining Sun and planet during
such a period preentzes from perihelium closest proximity to the
Sun to perihelion, and prencees would differ from three hundred
(04:52):
and sixty degrees. The line of the orbit would not
then be a closed one, but in the course of
time it would fill up an an annular part of
the orbital plane, that is, between the circle of least
and the circle of greatest distance of the planet from
the Sun. According also to the general theory of relativity,
(05:13):
which differs, of course from the theory of Newton, a
small variation from the Newton Kepler motion of a planet
in its orbit should take place, and in such a
way that the angle described by the radius Sun planet
between one perihelion and the next should exceed that corresponding
to one complete revolution by an amount given by the
(05:33):
formula plus twenty four pi cubed a square divided by
t squared c squared times to quantity one minus e
squared nb. One complete revolution corresponds to the angle two
to the pie power in the absolute angle measure customary
(05:55):
in physics, and the above expression gives the amount by
which the radius un planet exceeds this angle during the
interval between one perihelium and the next close priens. In
this expression, A represents the major semi axis of the
ellipse e its eccentricity see the velocity of light, and
T the period of revolution of the planet. Our result
(06:18):
may also be stated as follows. According to the general
theory of relativity, the major axis of the ellipse rotates
around the Sun in the same sense as the orbital
motion of the planet. Theory requires that this rotation should
amount the forty three seconds of arc per century for
the planet Mercury, but for the other planets of our
(06:39):
solar system, its magnitude should be so small that it
would necessarily escape detection. Footnote, especially since the next planet
Venus has an orbit that is almost an exact circle,
which makes it more difficult to locate the perihelium with
precision end footnote. In point of fact, astronomers have found
(07:02):
that the theory of Newton does not suffice to calculate
the observed motion of Mercury with an exactness corresponding to
that of the delicacy of observation attainable at the present time.
After taking account of all the disturbing influences exerted on
Mercury by the remaining planets, it was found Prentzes Leverer
(07:23):
eighteen fifty nine. In Newcome eighteen ninety five close prenzes
that an unexplained perihelial movement of the orbit of Mercury
remained over the amount of which does not differ sensibly
from the above mentioned plus forty three seconds of arc
per century. The uncertainty of the empirical results amounts to
a few seconds only. B deflection of light by a
(07:46):
gravitational field. In section twenty two, it has been already
mentioned that according to the general theory of relativity, a
ray of light will experience the curvature of its path
when passing through a gravitational field, this curvature being similar
to that experience by the path of a body which
is projected through a gravitational field. As a result of
(08:09):
this theory, we should expect that a ray of light
which is passing close to a heavenly body would be
deviated towards the latter. For a ray of light which
passes the Sun at a distance of delta sun radii
from its center, the angle of deflection Prence's alpha closed
brend should amount to alpha equals one point seven seconds
(08:30):
of arc divided by delta. It may be added that,
according to the theory, half of this deflection is produced
by the Newtonian field of attraction of the Sun and
the other half by the geometrical modification prences quote curvature
end quote end preentheses of space caused by the Sun.
(08:51):
This result admits of an experimental test by means of
the photographic registration of stars during a total eclipse of
the Sun. The reason why we must wait for a
total eclipse is because at every other time the atmosphere
is so strongly illuminated by the light from the Sun
that the stars situated near the Sun's disc are invisible,
(09:12):
the predicted effect can be seen clearly from the accompanying
diagram Reader's annotation Fig. Five. The Earth is shown as
a dot at the bottom of the diagram. A straight
line proceeding from there, labeled D sub one, proceeds upward
(09:36):
and slightly to the right, passing the Sun at a
tangent Sun being represented by a circle. A second line,
label D sub two, starts at the Earth proceeds at
a relatively smaller angle, which results in its passing the
(09:56):
Sun at a greater distance than the the initial line
D one, which is signified by the symbol delta. After
passing the Sun, the line becomes parallel to D sub
one end of Reader's annotation. If the Sun prencees, s,
(10:21):
and preentzes were not present, a star which is practically
infinitely distant would be seen in the direction D sub
one as observed from the Earth, but as a consequence
of the deflection of light from the star by the Sun,
the star will be seen in the direction D sub two,
that is, at a somewhat greater distance from the center
of the Sun, then corresponds to its real position. In practice,
(10:46):
the question is tested in the following way. The stars
in the neighborhood of the Sun are photographed during a
solar eclipse. In addition, a second photograph of the same
stars is taken when the Sun is situated at another
position in the sky, that is a few months earlier
or later. As compared with the standard photograph, the positions
(11:09):
of the stars on the eclipse photograph ought to appear
displaced radially outwards prenzes away from the center of the
Sun close friends by an amount corresponding to the angle a.
We are indebted to the Royal Society and to Royal
Astronomical Society for the investigation of this important deduction. Undaunted
(11:33):
by the war and by difficulties of both the material
and a psychological nature aroused by the war, these societies
equipped two expeditions to Sobral, Brazil and to the island
of princeip West Africa, and sent several of Britain's most
celebrated astronomers Prenzies Eddington, Cottingham, Crommelin, Davidson and Prinz in
(11:56):
order to obtain photographs of the solar eclipse of twenty
ninth May nineteen nineteen. The relative discrepancies to be expected
between the stellar photographs of jeting during the eclipse, and
the comparison photographs amounted to a few hundreds of a
millimeter only. Thus, great accuracy was necessary in making the
adjustments required for taking of the photographs and in their
(12:18):
subsequent measurement. The results of the measurements confirmed the theory
in a thoroughly satisfactory manner. The rectangular components of the
observed and of the calculated deviation of the stars, prenzes
and seconds of an arc and prenzes are set forth
in the following table of results Reader's annotation. The table
(12:44):
consists of measurements on seven stars, which are then tabulated
in four additional columns, which are entitled first coordinate and
second coordinate, and then for each of those the observed
and calculated measurements are given end reader's annotation. Number of
(13:10):
the star eleven first coordinate observed minus zero point one
nine calculated minus zero point two two. Second coordinate observed
plus zero point one six, calculated plus zero point zero two.
Star number five first coordinate observed plus zero point twenty
(13:32):
nine calculated plus zero point three to one. Second coordinate
observed negative zero point four to six calculated minus zero
point four to three. Star number four observed zero point
one one calculated zero point one zero. Second coordinate observed
zero point eight three calculated plus zero point seven four.
(13:56):
Star number three observed plus zero point two zero calculated
plus zero point one two, second coordinate observed plus one
point zero zero calculated plus zero point eight seven. Star
number six observed at the first coordinate plus zero point
one zero calculated plus zero point zero four, second coordinate
(14:20):
observed plus zero point five seven calculated plus zero point
four zero. Number the star ten observed minus zero point
zero eight calculated plus zero point zero nine, second coordinate
observed plus zero point three five calculated plus zero point
(14:44):
three two number of the star two observed plus zero
point nine five calculated plus zero point eight five, and
at the second coordinate observed minus point two seven calculated
minus zero point zero nine c displacement of the spectral
(15:08):
line towards the red. In section twenty three, it has
been shown that in a system K prime which is
in rotation with regard to a Galalian system K clocks
of identical construction and which are considered at rest with
respect to the rotating reference body, go at rates which
are dependent on the position of the clocks. We shall
(15:31):
now examine this dependence quantitatively. A clock which is situated
at a distance R from the center of the disc
has a velocity relative k, which is given by V
equals omega R, where omega represents the angle of velocity
of rotation of the disc k prime with respect to k.
If the sub zero represents the number of ticks of
(15:54):
a clock per unit time prints these quote rate end
quote of the clock close brands relative to k. When
the clock is at rest, then the quote rate end
quote of the clock prince thes v close priends when
it is moving relative to kay with a velocity v
but at rest with respect to the disc, will, in
accordance with section twelve, be given by V equals v
(16:15):
sub zero times the square root of one minus V
squared over C squared, or with sufficient accuracy, buy v
equals v zero times the quantity one minus one half
V squared over C squared. This expression may be also
stated in the following form V equals v sub zero
(16:39):
times the quantity one minus one over C squared times
omega squared are squared over two. If we represent a
difference of potential of the centrifugal force between the position
of the clock and the center of the disc y five.
That is, the work considered negatively which must be performed
on the unit of mass against this trifical force in
(17:01):
order to transport it from the position of the clock
on the rotating disc to the center of the disc.
Then we have Pi equals minus omega squared r square
divided by two. From this, it follows that v equals
v sub zero times to quantity one plus five overse squared.
(17:24):
In the first place. We see from this expression that
the two clocks of identical construction will go at different
rates when situated at different distances from the center of
the disc. This result is also valid from the standpoint
of an observer who is rotating with the disc. Now,
as judge from the disc, the latter is in a
(17:46):
gravitational field of potential five. Hence, the result we have
obtained will hold quite generally for gravitational fields. Furthermore, we
can regard an atom which is emitting spectral lines as
a clock that the following statement will hold. An atom
absorbs or amidst light of a frequency which is dependent
(18:07):
on the potential of the gravitational field in which it
is situated. The frequency of an atom situated on the
surface of a heavenly body will be somewhat less than
the frequency of an atom of the same element which
is situated in free space prenzes or on the surface
of a smaller celestial body close Prentzes period now five
(18:28):
equals minus k times m over R, where K is
Newton's constant gravitation and M is the mass of the
heavenly body. Thus, a displacement towards the red ought to
take place for spectral lines produced at the surface of stars,
as compared with the spectral lines of the same element
produced at the surface of the Earth, the amount of
(18:48):
this displacement being v sub zero minus v divided by
v sub zero equals k over c squared times m
over R. For the Sun, the displacement towards the red
predicted by theory amounts to about two millions of the wavelength.
A trustworthy calculation is not possible in the case of
(19:10):
the stars, because in general, neither the mass M nor
the radius R is known. It is an open question
whether or not this effect exists, and at the present
time astronomers are working with great zeal towards the solution
owing to the smallness of the effect in the case
(19:30):
of the sun, it is difficult to form an opinion
as to its existence. Whereas Greb and Bacham, Prenzes, Bond
and Priends, as a result of their own measurements and
those of ever Sad and Schwartzchild on the cyanogen bands,
have placed the existence of the effect almost beyond doubt.
(19:51):
Other investigators, particularly Saint John, have been led to the
opposite opinion in consequence of their measurements. Mean displacements of
lines towards the less refrangible end of the spectrum are
certainly revealed by statistical investigation of the fixed stars, but
up to the present the examination of the available data
does not allow of any definite decision being arrived at
(20:13):
as to whether or not these displacements are to be
referred in reality to the effect of gravitation. The results
of observation have been collected together and discussed in detail
from the standpoint of the question which has been engaging
our attention here in a paper by E. Freundlich entitled
Sir Profunder alamingun rebtats Theori Prentzes dionito Zenshoften, nineteen nineteen,
(20:40):
Number thirty five, page five point twenty Julia Springer, Berlin,
Close Pranz period. At all events, a definite decision will
be reached during the next few years. If the displacement
of spectral lines towards the red by the gravitational potential
does not exist, then the general theory of relativity will
be untenable. On the other hand, if the cause of
(21:03):
the displacement of spectral lines be definitely traced to the
gravitational potential, then the study of this displacement will furnish
us with important information as to the mass of the
heavenly bodies. End of Appendix three
Correction for this chapter in mathematical formulae instead of I
here one. This is a LibriVox recording. All LibriVox recordings
are in the public domain. For more information or to volunteer,
please visit LibriVox dot org. Recording by mL Cullen, Cleveland, Ohio,
(00:22):
March two thousand and seven. Relativity the Special and General
Theory by Albert Einstein, Appendix three The experimental confirmation of
the general theory of relativity. From a systematic theoretical point
(00:44):
of view, we may imagine the process of evolution of
an empirical science to be a continuous process of induction.
Theories are evolved and are expressed in short compass as
statements of a large number of individual object in the
form of empirical laws from which the general laws can
(01:04):
be ascertained by comparison. Regarded in this way, the development
of a science bears some resemblance to the compilation of
a classified catalog. It is, as it were, a purely
empirical enterprise. But this point of view by no means
(01:25):
embraces the whole of the actual process, for it slurs
over the important part played by intuition and deductive thought
in the development of an exact science. As soon as
a science has emerged from its initial stages, theoretical advances
are no longer achieved merely by a process of arrangement
guided by empirical data. The investigator rather develops a system
(01:49):
of thought which in general is built up logically from
a small number of fundamental assumptions, the so called axioms.
We call such a system of thought a theory. The
theory finds the justification for its existence in the fact
that it correlates a large number of single observations, and
it is just here that the truth of the theory
(02:11):
lies corresponding to the same complex of empirical data. There
may be several theories which differ from one another to
a considerable extent, but as regards the deduction from the
theories which are capable of being tested, the agreement between
the theories may be so complete that it becomes difficult to
find such deductions in which the two theories differ from
(02:33):
each other. As an example, a case of general interest
is available in the province of biology, in the Darwinian
theory of the development of species by selection, in the
struggle for existence, and in the theory of development which
is based on the hypothesis of the hereditary transmission of
acquired characters. We have another instance of far reaching agreement
(02:59):
between the deductions from the two theories and Newtonian mechanics
on the one hand, and the general theory of relativity
on the other. This agreement goes so far that up
to the present we have been able to find only
a few deductions from the general theory of relativity which
are capable of investigation and to which the physics of
pre relativity days does not also lead, and this despite
(03:23):
the profound difference in the fundamental assumption of the two theories.
In what follows, we shall again consider these important deductions,
and we shall also discuss the empirical evidence appertaining to them,
which has hitherto been obtained. A motion of the perihelion
(03:46):
of Mercury. According to Newtonian mechanics and Newton's law of gravitation,
a planet which is revolving around the Sun would describe
an ellipse around the latter, or more correct, around the
common center of gravity of the Sun and the planet.
In such a system, the Sun or the common center
(04:08):
of gravity lies in one of the foci of the
orbital lips in such a manner as that, in the
course of a planet year, the distant Sun planet grows
from a minimum to a maximum and then decreases again
to a minimum. If instead of Newton's law, we insert
a somewhat different law of attraction into the calculation, we
(04:29):
find that according to this new law, the motion would
still take place in such a manner that the distant
Sun planet exhibits periodic variations. But in this case, the
angle described by the line joining Sun and planet during
such a period preentzes from perihelium closest proximity to the
Sun to perihelion, and prencees would differ from three hundred
(04:52):
and sixty degrees. The line of the orbit would not
then be a closed one, but in the course of
time it would fill up an an annular part of
the orbital plane, that is, between the circle of least
and the circle of greatest distance of the planet from
the Sun. According also to the general theory of relativity,
(05:13):
which differs, of course from the theory of Newton, a
small variation from the Newton Kepler motion of a planet
in its orbit should take place, and in such a
way that the angle described by the radius Sun planet
between one perihelion and the next should exceed that corresponding
to one complete revolution by an amount given by the
(05:33):
formula plus twenty four pi cubed a square divided by
t squared c squared times to quantity one minus e
squared nb. One complete revolution corresponds to the angle two
to the pie power in the absolute angle measure customary
(05:55):
in physics, and the above expression gives the amount by
which the radius un planet exceeds this angle during the
interval between one perihelium and the next close priens. In
this expression, A represents the major semi axis of the
ellipse e its eccentricity see the velocity of light, and
T the period of revolution of the planet. Our result
(06:18):
may also be stated as follows. According to the general
theory of relativity, the major axis of the ellipse rotates
around the Sun in the same sense as the orbital
motion of the planet. Theory requires that this rotation should
amount the forty three seconds of arc per century for
the planet Mercury, but for the other planets of our
(06:39):
solar system, its magnitude should be so small that it
would necessarily escape detection. Footnote, especially since the next planet
Venus has an orbit that is almost an exact circle,
which makes it more difficult to locate the perihelium with
precision end footnote. In point of fact, astronomers have found
(07:02):
that the theory of Newton does not suffice to calculate
the observed motion of Mercury with an exactness corresponding to
that of the delicacy of observation attainable at the present time.
After taking account of all the disturbing influences exerted on
Mercury by the remaining planets, it was found Prentzes Leverer
(07:23):
eighteen fifty nine. In Newcome eighteen ninety five close prenzes
that an unexplained perihelial movement of the orbit of Mercury
remained over the amount of which does not differ sensibly
from the above mentioned plus forty three seconds of arc
per century. The uncertainty of the empirical results amounts to
a few seconds only. B deflection of light by a
(07:46):
gravitational field. In section twenty two, it has been already
mentioned that according to the general theory of relativity, a
ray of light will experience the curvature of its path
when passing through a gravitational field, this curvature being similar
to that experience by the path of a body which
is projected through a gravitational field. As a result of
(08:09):
this theory, we should expect that a ray of light
which is passing close to a heavenly body would be
deviated towards the latter. For a ray of light which
passes the Sun at a distance of delta sun radii
from its center, the angle of deflection Prence's alpha closed
brend should amount to alpha equals one point seven seconds
(08:30):
of arc divided by delta. It may be added that,
according to the theory, half of this deflection is produced
by the Newtonian field of attraction of the Sun and
the other half by the geometrical modification prences quote curvature
end quote end preentheses of space caused by the Sun.
(08:51):
This result admits of an experimental test by means of
the photographic registration of stars during a total eclipse of
the Sun. The reason why we must wait for a
total eclipse is because at every other time the atmosphere
is so strongly illuminated by the light from the Sun
that the stars situated near the Sun's disc are invisible,
(09:12):
the predicted effect can be seen clearly from the accompanying
diagram Reader's annotation Fig. Five. The Earth is shown as
a dot at the bottom of the diagram. A straight
line proceeding from there, labeled D sub one, proceeds upward
(09:36):
and slightly to the right, passing the Sun at a
tangent Sun being represented by a circle. A second line,
label D sub two, starts at the Earth proceeds at
a relatively smaller angle, which results in its passing the
(09:56):
Sun at a greater distance than the the initial line
D one, which is signified by the symbol delta. After
passing the Sun, the line becomes parallel to D sub
one end of Reader's annotation. If the Sun prencees, s,
(10:21):
and preentzes were not present, a star which is practically
infinitely distant would be seen in the direction D sub
one as observed from the Earth, but as a consequence
of the deflection of light from the star by the Sun,
the star will be seen in the direction D sub two,
that is, at a somewhat greater distance from the center
of the Sun, then corresponds to its real position. In practice,
(10:46):
the question is tested in the following way. The stars
in the neighborhood of the Sun are photographed during a
solar eclipse. In addition, a second photograph of the same
stars is taken when the Sun is situated at another
position in the sky, that is a few months earlier
or later. As compared with the standard photograph, the positions
(11:09):
of the stars on the eclipse photograph ought to appear
displaced radially outwards prenzes away from the center of the
Sun close friends by an amount corresponding to the angle a.
We are indebted to the Royal Society and to Royal
Astronomical Society for the investigation of this important deduction. Undaunted
(11:33):
by the war and by difficulties of both the material
and a psychological nature aroused by the war, these societies
equipped two expeditions to Sobral, Brazil and to the island
of princeip West Africa, and sent several of Britain's most
celebrated astronomers Prenzies Eddington, Cottingham, Crommelin, Davidson and Prinz in
(11:56):
order to obtain photographs of the solar eclipse of twenty
ninth May nineteen nineteen. The relative discrepancies to be expected
between the stellar photographs of jeting during the eclipse, and
the comparison photographs amounted to a few hundreds of a
millimeter only. Thus, great accuracy was necessary in making the
adjustments required for taking of the photographs and in their
(12:18):
subsequent measurement. The results of the measurements confirmed the theory
in a thoroughly satisfactory manner. The rectangular components of the
observed and of the calculated deviation of the stars, prenzes
and seconds of an arc and prenzes are set forth
in the following table of results Reader's annotation. The table
(12:44):
consists of measurements on seven stars, which are then tabulated
in four additional columns, which are entitled first coordinate and
second coordinate, and then for each of those the observed
and calculated measurements are given end reader's annotation. Number of
(13:10):
the star eleven first coordinate observed minus zero point one
nine calculated minus zero point two two. Second coordinate observed
plus zero point one six, calculated plus zero point zero two.
Star number five first coordinate observed plus zero point twenty
(13:32):
nine calculated plus zero point three to one. Second coordinate
observed negative zero point four to six calculated minus zero
point four to three. Star number four observed zero point
one one calculated zero point one zero. Second coordinate observed
zero point eight three calculated plus zero point seven four.
(13:56):
Star number three observed plus zero point two zero calculated
plus zero point one two, second coordinate observed plus one
point zero zero calculated plus zero point eight seven. Star
number six observed at the first coordinate plus zero point
one zero calculated plus zero point zero four, second coordinate
(14:20):
observed plus zero point five seven calculated plus zero point
four zero. Number the star ten observed minus zero point
zero eight calculated plus zero point zero nine, second coordinate
observed plus zero point three five calculated plus zero point
(14:44):
three two number of the star two observed plus zero
point nine five calculated plus zero point eight five, and
at the second coordinate observed minus point two seven calculated
minus zero point zero nine c displacement of the spectral
(15:08):
line towards the red. In section twenty three, it has
been shown that in a system K prime which is
in rotation with regard to a Galalian system K clocks
of identical construction and which are considered at rest with
respect to the rotating reference body, go at rates which
are dependent on the position of the clocks. We shall
(15:31):
now examine this dependence quantitatively. A clock which is situated
at a distance R from the center of the disc
has a velocity relative k, which is given by V
equals omega R, where omega represents the angle of velocity
of rotation of the disc k prime with respect to k.
If the sub zero represents the number of ticks of
(15:54):
a clock per unit time prints these quote rate end
quote of the clock close brands relative to k. When
the clock is at rest, then the quote rate end
quote of the clock prince thes v close priends when
it is moving relative to kay with a velocity v
but at rest with respect to the disc, will, in
accordance with section twelve, be given by V equals v
(16:15):
sub zero times the square root of one minus V
squared over C squared, or with sufficient accuracy, buy v
equals v zero times the quantity one minus one half
V squared over C squared. This expression may be also
stated in the following form V equals v sub zero
(16:39):
times the quantity one minus one over C squared times
omega squared are squared over two. If we represent a
difference of potential of the centrifugal force between the position
of the clock and the center of the disc y five.
That is, the work considered negatively which must be performed
on the unit of mass against this trifical force in
(17:01):
order to transport it from the position of the clock
on the rotating disc to the center of the disc.
Then we have Pi equals minus omega squared r square
divided by two. From this, it follows that v equals
v sub zero times to quantity one plus five overse squared.
(17:24):
In the first place. We see from this expression that
the two clocks of identical construction will go at different
rates when situated at different distances from the center of
the disc. This result is also valid from the standpoint
of an observer who is rotating with the disc. Now,
as judge from the disc, the latter is in a
(17:46):
gravitational field of potential five. Hence, the result we have
obtained will hold quite generally for gravitational fields. Furthermore, we
can regard an atom which is emitting spectral lines as
a clock that the following statement will hold. An atom
absorbs or amidst light of a frequency which is dependent
(18:07):
on the potential of the gravitational field in which it
is situated. The frequency of an atom situated on the
surface of a heavenly body will be somewhat less than
the frequency of an atom of the same element which
is situated in free space prenzes or on the surface
of a smaller celestial body close Prentzes period now five
(18:28):
equals minus k times m over R, where K is
Newton's constant gravitation and M is the mass of the
heavenly body. Thus, a displacement towards the red ought to
take place for spectral lines produced at the surface of stars,
as compared with the spectral lines of the same element
produced at the surface of the Earth, the amount of
(18:48):
this displacement being v sub zero minus v divided by
v sub zero equals k over c squared times m
over R. For the Sun, the displacement towards the red
predicted by theory amounts to about two millions of the wavelength.
A trustworthy calculation is not possible in the case of
(19:10):
the stars, because in general, neither the mass M nor
the radius R is known. It is an open question
whether or not this effect exists, and at the present
time astronomers are working with great zeal towards the solution
owing to the smallness of the effect in the case
(19:30):
of the sun, it is difficult to form an opinion
as to its existence. Whereas Greb and Bacham, Prenzes, Bond
and Priends, as a result of their own measurements and
those of ever Sad and Schwartzchild on the cyanogen bands,
have placed the existence of the effect almost beyond doubt.
(19:51):
Other investigators, particularly Saint John, have been led to the
opposite opinion in consequence of their measurements. Mean displacements of
lines towards the less refrangible end of the spectrum are
certainly revealed by statistical investigation of the fixed stars, but
up to the present the examination of the available data
does not allow of any definite decision being arrived at
(20:13):
as to whether or not these displacements are to be
referred in reality to the effect of gravitation. The results
of observation have been collected together and discussed in detail
from the standpoint of the question which has been engaging
our attention here in a paper by E. Freundlich entitled
Sir Profunder alamingun rebtats Theori Prentzes dionito Zenshoften, nineteen nineteen,
(20:40):
Number thirty five, page five point twenty Julia Springer, Berlin,
Close Pranz period. At all events, a definite decision will
be reached during the next few years. If the displacement
of spectral lines towards the red by the gravitational potential
does not exist, then the general theory of relativity will
be untenable. On the other hand, if the cause of
(21:03):
the displacement of spectral lines be definitely traced to the
gravitational potential, then the study of this displacement will furnish
us with important information as to the mass of the
heavenly bodies. End of Appendix three
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