Reading Help Relativity: The Special and General Theory
absolute angular measure customary in physics, and the above `
` expression giver the amount by which the radius sun-planet exceeds `
` this angle during the interval between one perihelion and the next.) `
` In this expression a represents the major semi-axis of the ellipse, e `
` its eccentricity, c the velocity of light, and T the period of `
` revolution of the planet. Our result may also be stated as follows : `
` According to the general theory of relativity, the major axis of the `
` ellipse rotates round the sun in the same sense as the orbital motion `
` of the planet. Theory requires that this rotation should amount to 43 `
` seconds of arc per century for the planet Mercury, but for the other `
` Planets of our solar system its magnitude should be so small that it `
` would necessarily escape detection. * `
` `
` In point of fact, astronomers have found 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 (Leverrier: 1859; and Newcomb: 1895) that an unexplained `
` perihelial movement of the orbit of Mercury remained over, the amount `
` of which does not differ sensibly from the above mentioned +43 seconds `
` of arc per century. The uncertainty of the empirical result amounts to `
` a few seconds only. `
` `
` (b) Deflection of Light by a Gravitational Field `
` `
` In Section 22 it has been already mentioned that according to the `
` general theory of relativity, a ray of light will experience a `
` curvature of its path when passing through a gravitational field, this `
` curvature being similar to that experienced by the path of a body `
` which is projected through a gravitational field. As a result of 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 D sun-radii from its `
` centre, the angle of deflection (a) should amount to `
` `
` eq. 42: file eq42.gif `
` `
` It may be added that, according to the theory, half of Figure 05 this `
` deflection is produced by the Newtonian field of attraction of the `
` sun, and the other half by the geometrical modification (" curvature `
` ") of space caused by the sun. `
` `
` This result admits of an experimental test by means of the `
` photographic registration of stars during a total eclipse of the sun. `
` The only 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. The predicted effect can be seen clearly from the `
` accompanying diagram. If the sun (S) were not present, a star which is `
` practically infinitely distant would be seen in the direction D[1], as `
` observed front 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[2], i.e. at a somewhat greater distance from the centre of the sun `
` than corresponds to its real position. `
` `
` In practice, the question is tested in the following way. The stars in `
` the neighbourhood 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, i.e. a few months `
` earlier or later. As compared whh the standard photograph, the `
` positions of the stars on the eclipse-photograph ought to appear `
` displaced radially outwards (away from the centre of the sun) by an `
` amount corresponding to the angle a. `
` `
` We are indebted to the [British] Royal Society and to the Royal `
` Astronomical Society for the investigation of this important `
` deduction. Undaunted by the [first world] war and by difficulties of `
` both a material and a psychological nature aroused by the war, these `
` societies equipped two expeditions -- to Sobral (Brazil), and to the `
` island of Principe (West Africa) -- and sent several of Britain's most `
` celebrated astronomers (Eddington, Cottingham, Crommelin, Davidson), `
` in order to obtain photographs of the solar eclipse of 29th May, 1919. `
` The relative discrepancies to be expected between the stellar `
` photographs obtained during the eclipse and the comparison photographs `
` amounted to a few hundredths of a millimetre only. Thus great accuracy `
` was necessary in making the adjustments required for the taking of the `
` photographs, and in their 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 deviations of the stars (in seconds of arc) are set `
` forth in the following table of results : `
` `
` Table 01: file table01.gif `
` `
` (c) Displacement of Spectral Lines Towards the Red `
` `
` In Section 23 it has been shown that in a system K1 which is in `
` rotation with regard to a Galileian 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 `
` positions of the clocks. We shall now examine this dependence `
` quantitatively. A clock, which is situated at a distance r from the `
` centre of the disc, has a velocity relative to K which is given by `
` `
` V = wr `
` `
` where w represents the angular velocity of rotation of the disc K1 `
` with respect to K. If v[0], represents the number of ticks of the `
` clock per unit time (" rate " of the clock) relative to K when the `
` clock is at rest, then the " rate " of the clock (v) when it is moving `
` relative to K with a velocity V, but at rest with respect to the disc, `
` will, in accordance with Section 12, be given by `
` `
` eq. 43: file eq43.gif `
` `
` or with sufficient accuracy by `
` `
` eq. 44: file eq44.gif `
` `
` This expression may also be stated in the following form: `
` `
` eq. 45: file eq45.gif `
` `
` If we represent the difference of potential of the centrifugal force `
` between the position of the clock and the centre of the disc by f, `
` i.e. the work, considered negatively, which must be performed on the `
` unit of mass against the centrifugal force in order to transport it `
` from the position of the clock on the rotating disc to the centre of `
` the disc, then we have `
` `
` eq. 46: file eq46.gif `
` `
` From this it follows that `
` `
` eq. 47: file eq47.gif `
` `
` In the first place, we see from this expression that two clocks of `
` identical construction will go at different rates when situated at `
` different distances from the centre of the disc. This result is aiso `
` valid from the standpoint of an observer who is rotating with the `
` disc. `
` `
` Now, as judged from the disc, the latter is in a gravititional field `
` of potential f, 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, so that the following `
` statement will hold: `
` `
` An atom absorbs or emits light of a frequency which is dependent 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 (or on the surface of a `
` smaller celestial body). `
` `
` Now f = - K (M/r), where K is Newton's constant of 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 this displacement being `
` `
` eq. 48: file eq48.gif `
` `
` For the sun, the displacement towards the red predicted by theory `
` amounts to about two millionths of the wave-length. A trustworthy `
` calculation is not possible in the case of the stars, because in `
` general neither the mass M nor the radius r are known. `
` `
` It is an open question whether or not this effect exists, and at the `
` present time (1920) astronomers are working with great zeal towards `
` the solution. Owing to the smallness of the effect in the case of the `
` sun, it is difficult to form an opinion as to its existence. Whereas `
` Grebe and Bachem (Bonn), as a result of their own measurements and `
` those of Evershed and Schwarzschild on the cyanogen bands, have placed `
` the existence of the effect almost beyond doubt, while other `
` investigators, particularly St. 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 investigations 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, 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 "Zur Pr�fung der allgemeinen `
` Relativit¨aut;ts-Theorie" (Die Naturwissenschaften, 1919, No. 35, `
` p. 520: Julius Springer, Berlin). `
` `
` 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 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. [5][A] `
` `
` `
` Notes `
` `
` *) Especially since the next planet Venus has an orbit that is `
` almost an exact circle, which makes it more difficult to locate the `
` perihelion with precision. `
` `
` The displacentent of spectral lines towards the red end of the `
` spectrum was definitely established by Adams in 1924, by observations `
`
` expression giver the amount by which the radius sun-planet exceeds `
` this angle during the interval between one perihelion and the next.) `
` In this expression a represents the major semi-axis of the ellipse, e `
` its eccentricity, c the velocity of light, and T the period of `
` revolution of the planet. Our result may also be stated as follows : `
` According to the general theory of relativity, the major axis of the `
` ellipse rotates round the sun in the same sense as the orbital motion `
` of the planet. Theory requires that this rotation should amount to 43 `
` seconds of arc per century for the planet Mercury, but for the other `
` Planets of our solar system its magnitude should be so small that it `
` would necessarily escape detection. * `
` `
` In point of fact, astronomers have found 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 (Leverrier: 1859; and Newcomb: 1895) that an unexplained `
` perihelial movement of the orbit of Mercury remained over, the amount `
` of which does not differ sensibly from the above mentioned +43 seconds `
` of arc per century. The uncertainty of the empirical result amounts to `
` a few seconds only. `
` `
` (b) Deflection of Light by a Gravitational Field `
` `
` In Section 22 it has been already mentioned that according to the `
` general theory of relativity, a ray of light will experience a `
` curvature of its path when passing through a gravitational field, this `
` curvature being similar to that experienced by the path of a body `
` which is projected through a gravitational field. As a result of 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 D sun-radii from its `
` centre, the angle of deflection (a) should amount to `
` `
` eq. 42: file eq42.gif `
` `
` It may be added that, according to the theory, half of Figure 05 this `
` deflection is produced by the Newtonian field of attraction of the `
` sun, and the other half by the geometrical modification (" curvature `
` ") of space caused by the sun. `
` `
` This result admits of an experimental test by means of the `
` photographic registration of stars during a total eclipse of the sun. `
` The only 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. The predicted effect can be seen clearly from the `
` accompanying diagram. If the sun (S) were not present, a star which is `
` practically infinitely distant would be seen in the direction D[1], as `
` observed front 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[2], i.e. at a somewhat greater distance from the centre of the sun `
` than corresponds to its real position. `
` `
` In practice, the question is tested in the following way. The stars in `
` the neighbourhood 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, i.e. a few months `
` earlier or later. As compared whh the standard photograph, the `
` positions of the stars on the eclipse-photograph ought to appear `
` displaced radially outwards (away from the centre of the sun) by an `
` amount corresponding to the angle a. `
` `
` We are indebted to the [British] Royal Society and to the Royal `
` Astronomical Society for the investigation of this important `
` deduction. Undaunted by the [first world] war and by difficulties of `
` both a material and a psychological nature aroused by the war, these `
` societies equipped two expeditions -- to Sobral (Brazil), and to the `
` island of Principe (West Africa) -- and sent several of Britain's most `
` celebrated astronomers (Eddington, Cottingham, Crommelin, Davidson), `
` in order to obtain photographs of the solar eclipse of 29th May, 1919. `
` The relative discrepancies to be expected between the stellar `
` photographs obtained during the eclipse and the comparison photographs `
` amounted to a few hundredths of a millimetre only. Thus great accuracy `
` was necessary in making the adjustments required for the taking of the `
` photographs, and in their 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 deviations of the stars (in seconds of arc) are set `
` forth in the following table of results : `
` `
` Table 01: file table01.gif `
` `
` (c) Displacement of Spectral Lines Towards the Red `
` `
` In Section 23 it has been shown that in a system K1 which is in `
` rotation with regard to a Galileian 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 `
` positions of the clocks. We shall now examine this dependence `
` quantitatively. A clock, which is situated at a distance r from the `
` centre of the disc, has a velocity relative to K which is given by `
` `
` V = wr `
` `
` where w represents the angular velocity of rotation of the disc K1 `
` with respect to K. If v[0], represents the number of ticks of the `
` clock per unit time (" rate " of the clock) relative to K when the `
` clock is at rest, then the " rate " of the clock (v) when it is moving `
` relative to K with a velocity V, but at rest with respect to the disc, `
` will, in accordance with Section 12, be given by `
` `
` eq. 43: file eq43.gif `
` `
` or with sufficient accuracy by `
` `
` eq. 44: file eq44.gif `
` `
` This expression may also be stated in the following form: `
` `
` eq. 45: file eq45.gif `
` `
` If we represent the difference of potential of the centrifugal force `
` between the position of the clock and the centre of the disc by f, `
` i.e. the work, considered negatively, which must be performed on the `
` unit of mass against the centrifugal force in order to transport it `
` from the position of the clock on the rotating disc to the centre of `
` the disc, then we have `
` `
` eq. 46: file eq46.gif `
` `
` From this it follows that `
` `
` eq. 47: file eq47.gif `
` `
` In the first place, we see from this expression that two clocks of `
` identical construction will go at different rates when situated at `
` different distances from the centre of the disc. This result is aiso `
` valid from the standpoint of an observer who is rotating with the `
` disc. `
` `
` Now, as judged from the disc, the latter is in a gravititional field `
` of potential f, 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, so that the following `
` statement will hold: `
` `
` An atom absorbs or emits light of a frequency which is dependent 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 (or on the surface of a `
` smaller celestial body). `
` `
` Now f = - K (M/r), where K is Newton's constant of 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 this displacement being `
` `
` eq. 48: file eq48.gif `
` `
` For the sun, the displacement towards the red predicted by theory `
` amounts to about two millionths of the wave-length. A trustworthy `
` calculation is not possible in the case of the stars, because in `
` general neither the mass M nor the radius r are known. `
` `
` It is an open question whether or not this effect exists, and at the `
` present time (1920) astronomers are working with great zeal towards `
` the solution. Owing to the smallness of the effect in the case of the `
` sun, it is difficult to form an opinion as to its existence. Whereas `
` Grebe and Bachem (Bonn), as a result of their own measurements and `
` those of Evershed and Schwarzschild on the cyanogen bands, have placed `
` the existence of the effect almost beyond doubt, while other `
` investigators, particularly St. 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 investigations 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, 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 "Zur Pr�fung der allgemeinen `
` Relativit¨aut;ts-Theorie" (Die Naturwissenschaften, 1919, No. 35, `
` p. 520: Julius Springer, Berlin). `
` `
` 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 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. [5][A] `
` `
` `
` Notes `
` `
` *) Especially since the next planet Venus has an orbit that is `
` almost an exact circle, which makes it more difficult to locate the `
` perihelion with precision. `
` `
` The displacentent of spectral lines towards the red end of the `
` spectrum was definitely established by Adams in 1924, by observations `
`