SOME ASPECTS OF RELATIVITYTHE THIRD TESTBY HENRY L. BROSE, M.A.UP to the present, three methods of verifying Einstein's Theory of Relativity have been suggested.The first one, which was a direct outcome of the new gravitational field-equations proposed by Einstein, proved successful. The slow motion of Mercury's perihelion which long mystified astronomers was immediately accounted for. This result is the more remarkable as all other explanations of this phenomena were artificial in origin, consisting of a hypothesis formulatedad hocwhich could not be verified by observation.The second method involved the deflection of a ray of light in its passage through a varying gravitational field. The results of the total eclipse of the sun which occurred on 29th May, 1919, have become famous and were recorded as confirming Einstein's prediction. The results of a more recent expedition have proved finally conclusive.The third test, the results of which are still in abeyance, is perhaps the most important of the three, inasmuch as it depends upon a very simple calculation from Einstein's Principle of Equivalence, whichasserts that an observer cannot discriminate how much of his motion is due to a gravitational field and how much is due to an acceleration of his body of reference. Einstein illustrates his argument by supposing an observer situated in a closed box in free space. The observer has at first no sensation of weight, and need not support himself upon his feet. Now suppose an external agent to pull the box in a definite direction with constant force. The observer in the box performs experiments with masses of variable material, and as they all fall to the "floor" of the box at the same time, he concludes that he is in a gravitational field. He himself has acquired the sensation of weight. This result led Einstein to propound the equivalence of gravitational and accelerational fields. An immediate consequence of this principle is that the duration of an event depends upon the gravitational conditions at the place of the event.If we consider the light (of frequency) which is emitted by a distant star, and suppose it to traverse a practically invariable gravitational field in which bodies are assumed to fall with a constant acceleration, then an observer at a distancefrom the star will have attained a velocitywhereis the velocity of light and the distanceis small in comparison with the distance traversed by the observer in the time the light takes to reach him. By Doppler's Principle, the apparent frequencyis given byPotential of unit mass moved through a distanceis(say). This gives the work done in moving unit massfrom the source of light to the observer (the source of light is here the point to which the potential energy is referred in the field).Therefore, if we transform the accelerational field of the observer into the gravitational field, we get the result:This means that a spectral line of frequencywill appear to a distant observer to be displaced, if compared with the position of the same line, when produced by a source at a different point in the field. Each of these lines, produced by vibrating electrons, may be regarded as a clock, and this simple calculation shows how time-measurements are affected by the state of the gravitational field. This effect amounts to 0·008 Ångstroms, for a wavelength of 4000 Å. The same displacement would be produced as a Doppler effect by a velocity of 0·6 kms. per sec. When this test was put into practice, it was found difficult to discriminate it from the various superposed effects due to other causes such as the radial velocities of the stars, proper velocities of the gaseous envelopes, pressure, etc. The conditions of the emission of light by the sun have not been fully ascertained, nor is the light of the arc lamp free from disturbing elements. Dr. Erwin Freundlich, of the Neubabelsberg Observatory, has discussed, in conjunction with Professor Einstein, the possibility of recognizing this effect in spite of these obscuring influences. He points out three ways of establishing the result quantitatively. They may be briefly classified as being based on (1) statistical methods; (2) nebular spectra; (3) calcium lines in the spectra of the atmosphere surrounding double-stars.I. If we consider a great number of stars of about the same massevenly distributed over the heavens, and represent the spectral shift due to radial velocities (i.e. velocities in the line of sight) graphically, we should expect these velocities to be distributed according to the law of probability about the value zero, i.e. as depicted by Gauss's Error Curve, which resembles a vertical section of a bell. If, however, Einstein's gravitational effect really exists, we should expect these velocities to group themselves symmetrically about a positive velocity which would be that corresponding to this spectral shift. Gauss's Error Curve would thus appear displaced by precisely the amount of the radial velocity corresponding to this shift, as all the radial velocities would be falsified by just this amount.The values of the radial velocities have been plotted in the case of-stars, called Helium stars on account of the predominance of helium lines in their spectra. Other observations have led astronomers to infer that the-stars have unusually great masses but small densities. The result has been distinctly in favour of the Einstein shift on the basis of the foregoing discussion. The same was found to hold for the bright- and-stars, which are considered to be at a lower temperature and possessed of enormous surface extent, which accounts for their brilliance.If we indicate the mean shift of the lines towards the red by, then foris here expressed in terms of a Doppler shift as a velocity, i.e. as if the Einstein shift were due to an additional radial motion and hence expressible in kilometres.Alternative ways of accounting for this shift have been proposed.(a) It may be regarded as an ordinary Doppler effect. This would imply that the stars of the,, andtype suffer a general expansion to which stars of theandtype (yellow stars like the sun) and the-stars are not subject.This explanation does not seem very probable, as helium lines were used in determining the shift for the-stars, whereas quite different lines were used for measuring the effect for the- and-stars. It would be a strange coincidence if this shift, to which all the evidence points as arising from a common origin, should be manifested just in these cases which have been made the object of an investigation.(b) The general shift towards the red might be ascribed to pressure effects at the surfaces of the stars or to the presence of other lines which lie on the red side of the main lines, but which are very weak or even absent in the comparison spectrum of the sun. A detailed knowledge of conditions on the surfaces of stellar bodies could alone give a decision on this point.2. It is only possible to prove that the shiftis not due to a radial velocity if one can measure the ordinary Doppler effect arising from the radial velocity separately. Let us consider a single-star or group of-stars which happen to be embedded in a nebula of great extent which accompanies them in their motion. The Doppler effect due to the radial velocity would be the same for the star as the nebula, but the gravitational effect predicted by Einstein would not be the same, inasmuch as the gravitational field at the surface of the star will vary considerably from that at the outer edge of the nebula. Hence it wouldbe reasonable to attribute any difference in the magnitude of the spectral shifts in the case of the star and the nebula to the difference in gravitational fields at each place.The stars of the nebular group of Orion have hitherto offered the only possibility of applying this method. The results have fulfilled Einstein's expectations qualitatively, and it remains to be seen whether the agreement will hold quantitatively. A general shift of the star-spectrum as compared with the corresponding lines of the associated nebula was observed.Some very bright-stars in the constellation of Orion are considered to form an entity with their attendant nebula. This conclusion was reached as the result of independent research.The radial velocity of the Orion-nebula has been measured by various observers. The values obtained are: 17.7 (Wright), 17.4 (Vogel and Eberhardt), 18.5 (Frost and Adams). The mean value is 17.4 kms. per sec. This velocity is derived from the brightest part of the nebula, the so-called trapezium. The values obtained in the case of the stars almost all exceed 20 kms. per sec., and hence it seems likely that part of this radial velocity, viz. the excess over that of the nebula, is due to the Einstein effect. When the difference between the radial velocities of the stars and the associated nebula are tabulated for each star, we find that in the case of all members except two the difference is positive, i.e. indicative of a shift towards the red end, in agreement with the statistical investigation applied to the-,-, and M-stars. The difference amounts to, and is a little greater than that given by the statistical method.The two starsand 36 Orionis give a displacement towards the violet end. It has been suggested that they do not belong to the morelimited group of Orion stars, but are only projected into that portion of the celestial sphere. This is supported by the fact that both stars have only very small spherical proper motions, and that the radial velocities observed for them differ considerably from the mean of the radial velocities of the others.This method has not been successfully applied to other stellar systems inasmuch as the nebulæ of those which are available emit such feeble light that it has not been possible to establish the displacement to any degree of accuracy. Eddington recently pointed out that a very important factor had been neglected in the fundamental equations of the early theories concerning the equilibrium of stellar matter, viz. the pressure due to radiation. According to his theory, the equilibrium in the interior of the star (regarded as a gaseous sphere) is determined by three conditions. These are gaseous pressure, radiational pressure, and gravitational forces.Calculation shows that for very great masses the gravitational pressure is almost entirely balanced by radiational pressure. This implies that any additional force such as that due to a centrifugal field of rotation would lead to an unstable condition.It can, furthermore, be deduced from Eddington's theory that only stars whose masses exceed a certain minimum value can in the course of their evolution reach the very high surface-temperatures which have been observed in the case of the- and-stars.It therefore seems likely that the- and-stars have in the process of evolution passed through a stage of which the radiational pressure has brought about a condition of unstable equilibrium, and one might expect them to be surrounded by cosmic dust which has become dissociated from the nuclei of the system.In some cases this dissociated matter may be in a very fine stateof division, and may extend so far into space that the absorption lines they produce in the spectrum of the star they surround may originate from a gravitational field which differs perceptibly from that at the surface of the star. There are definite signs of the existence of such atmospheres. A high percentage of-stars are found to be spectroscopic double stars, i.e. their spectral lines fluctuate periodically about some mean position. Hartmann was the first to notice that in the spectrum of the-star. Orionis the absorption linesandof calcium, viz. 3933.82 and 3968.63 Ångstroms, occur, but that they do not share in the periodic movements of the other lines. A number of other stars belonging to early spectral types contain calcium absorption lines in their spectra, which exhibit a similar anomaly, inasmuch as they either remain immovable or execute periodic motions which are of feeble amplitude compared with the proper stellar lines. In view of the important rôle that calcium plays in the outermost layers of the gaseous atmosphere encircling the sun, and in view of the discussion above, the suggestion forces itself upon one that these calcium lines indicate the presence of an extensive atmosphere surrounding the star.It has often been put forward that these lines are due to the light from these stars being absorbed by vast interstellar clouds of calcium. Evershed considers that this is supported by the fact that when the motion of the solar system is subtracted from that calculated from the fixed calcium lines (owing to the ordinary Doppler effect), the remaining motion is very small. But this argument does not carry weight inasmuch as it is known that the-stars, in the spectrum of which these lines occur, themselves have very small radial velocities. As Young remarked, it seems very strange that these calciumclouds should so consistently choose to lie in front of stars of typeor earlier. An objection against this hypothesis is to be found in the fact that in the case of various systems these two calcium lines are not at rest but move, although with somewhat less amplitude than the other proper lines of the double star.An additional circumstance which lends support to the theory that calcium lines denote the presence of an atmosphere around the star is that a great number of helium-stars are enveloped in a nebulous atmosphere which is actually visible.Assuming then that the calcium absorption lines are due to such atmospheres, we may apply the same process as in the case of the Orion nebula, i.e. if the shifts of the spectral-lines of the stars be systematically falsified by a superposed gravitational effect, this should be expressed by the lines of the actual spectrum from a double star being displaced towards the red as compared with the fixed calcium lines.This phenomenon has been clearly observed. The result has not yet beenquantitativelyfixed, as the numbers taken are not regarded as final.All stars in the spectra of which theandlines of calcium occur have been used to test the conclusion, and all show a shift to the red end; the mean of the shifts corresponds to a velocity of + 6.3 kms. per sec.The results of this discussion have been formulated by Dr. Freundlich thus:—SUMMARY1. Statistical consideration gives us the means of separating the mean gravitational effect from the ordinary Doppler effect in the case of the helium-stars and the bright- and-stars, whichastronomical investigations compel us to regard as being of particularly great mass.A general shift of the spectra towards the red is exhibited with considerable certainty.2. It follows from a comparison of the displacement of the lines of the star-spectra that the above displacement which was found by a statistical examination is not an ordinary Doppler effect, but is due to the conditions of emission of light at the surfaces of the stars.3. The close connection of the- and-stars with nebulous matter in the heavens is a symptom that these stars are of great mass.4. If we regard the fixed calcium lines in the spectra of- and-stars as being caused by absorption in extended calcium atmospheres moving with each star in question, the shift towards the red which manifests itself may be regarded as the effect predicted by Einstein's theory, i.e. due to the different gravitational fields from which the absorption lines and the stellar lines have originated.
THE THIRD TEST
BY HENRY L. BROSE, M.A.
UP to the present, three methods of verifying Einstein's Theory of Relativity have been suggested.
The first one, which was a direct outcome of the new gravitational field-equations proposed by Einstein, proved successful. The slow motion of Mercury's perihelion which long mystified astronomers was immediately accounted for. This result is the more remarkable as all other explanations of this phenomena were artificial in origin, consisting of a hypothesis formulatedad hocwhich could not be verified by observation.
The second method involved the deflection of a ray of light in its passage through a varying gravitational field. The results of the total eclipse of the sun which occurred on 29th May, 1919, have become famous and were recorded as confirming Einstein's prediction. The results of a more recent expedition have proved finally conclusive.
The third test, the results of which are still in abeyance, is perhaps the most important of the three, inasmuch as it depends upon a very simple calculation from Einstein's Principle of Equivalence, whichasserts that an observer cannot discriminate how much of his motion is due to a gravitational field and how much is due to an acceleration of his body of reference. Einstein illustrates his argument by supposing an observer situated in a closed box in free space. The observer has at first no sensation of weight, and need not support himself upon his feet. Now suppose an external agent to pull the box in a definite direction with constant force. The observer in the box performs experiments with masses of variable material, and as they all fall to the "floor" of the box at the same time, he concludes that he is in a gravitational field. He himself has acquired the sensation of weight. This result led Einstein to propound the equivalence of gravitational and accelerational fields. An immediate consequence of this principle is that the duration of an event depends upon the gravitational conditions at the place of the event.
If we consider the light (of frequency) which is emitted by a distant star, and suppose it to traverse a practically invariable gravitational field in which bodies are assumed to fall with a constant acceleration, then an observer at a distancefrom the star will have attained a velocitywhereis the velocity of light and the distanceis small in comparison with the distance traversed by the observer in the time the light takes to reach him. By Doppler's Principle, the apparent frequencyis given byPotential of unit mass moved through a distanceis(say). This gives the work done in moving unit massfrom the source of light to the observer (the source of light is here the point to which the potential energy is referred in the field).
Therefore, if we transform the accelerational field of the observer into the gravitational field, we get the result:
This means that a spectral line of frequencywill appear to a distant observer to be displaced, if compared with the position of the same line, when produced by a source at a different point in the field. Each of these lines, produced by vibrating electrons, may be regarded as a clock, and this simple calculation shows how time-measurements are affected by the state of the gravitational field. This effect amounts to 0·008 Ångstroms, for a wavelength of 4000 Å. The same displacement would be produced as a Doppler effect by a velocity of 0·6 kms. per sec. When this test was put into practice, it was found difficult to discriminate it from the various superposed effects due to other causes such as the radial velocities of the stars, proper velocities of the gaseous envelopes, pressure, etc. The conditions of the emission of light by the sun have not been fully ascertained, nor is the light of the arc lamp free from disturbing elements. Dr. Erwin Freundlich, of the Neubabelsberg Observatory, has discussed, in conjunction with Professor Einstein, the possibility of recognizing this effect in spite of these obscuring influences. He points out three ways of establishing the result quantitatively. They may be briefly classified as being based on (1) statistical methods; (2) nebular spectra; (3) calcium lines in the spectra of the atmosphere surrounding double-stars.
I. If we consider a great number of stars of about the same massevenly distributed over the heavens, and represent the spectral shift due to radial velocities (i.e. velocities in the line of sight) graphically, we should expect these velocities to be distributed according to the law of probability about the value zero, i.e. as depicted by Gauss's Error Curve, which resembles a vertical section of a bell. If, however, Einstein's gravitational effect really exists, we should expect these velocities to group themselves symmetrically about a positive velocity which would be that corresponding to this spectral shift. Gauss's Error Curve would thus appear displaced by precisely the amount of the radial velocity corresponding to this shift, as all the radial velocities would be falsified by just this amount.
The values of the radial velocities have been plotted in the case of-stars, called Helium stars on account of the predominance of helium lines in their spectra. Other observations have led astronomers to infer that the-stars have unusually great masses but small densities. The result has been distinctly in favour of the Einstein shift on the basis of the foregoing discussion. The same was found to hold for the bright- and-stars, which are considered to be at a lower temperature and possessed of enormous surface extent, which accounts for their brilliance.
If we indicate the mean shift of the lines towards the red by, then foris here expressed in terms of a Doppler shift as a velocity, i.e. as if the Einstein shift were due to an additional radial motion and hence expressible in kilometres.
Alternative ways of accounting for this shift have been proposed.
(a) It may be regarded as an ordinary Doppler effect. This would imply that the stars of the,, andtype suffer a general expansion to which stars of theandtype (yellow stars like the sun) and the-stars are not subject.
This explanation does not seem very probable, as helium lines were used in determining the shift for the-stars, whereas quite different lines were used for measuring the effect for the- and-stars. It would be a strange coincidence if this shift, to which all the evidence points as arising from a common origin, should be manifested just in these cases which have been made the object of an investigation.
(b) The general shift towards the red might be ascribed to pressure effects at the surfaces of the stars or to the presence of other lines which lie on the red side of the main lines, but which are very weak or even absent in the comparison spectrum of the sun. A detailed knowledge of conditions on the surfaces of stellar bodies could alone give a decision on this point.
2. It is only possible to prove that the shiftis not due to a radial velocity if one can measure the ordinary Doppler effect arising from the radial velocity separately. Let us consider a single-star or group of-stars which happen to be embedded in a nebula of great extent which accompanies them in their motion. The Doppler effect due to the radial velocity would be the same for the star as the nebula, but the gravitational effect predicted by Einstein would not be the same, inasmuch as the gravitational field at the surface of the star will vary considerably from that at the outer edge of the nebula. Hence it wouldbe reasonable to attribute any difference in the magnitude of the spectral shifts in the case of the star and the nebula to the difference in gravitational fields at each place.
The stars of the nebular group of Orion have hitherto offered the only possibility of applying this method. The results have fulfilled Einstein's expectations qualitatively, and it remains to be seen whether the agreement will hold quantitatively. A general shift of the star-spectrum as compared with the corresponding lines of the associated nebula was observed.
Some very bright-stars in the constellation of Orion are considered to form an entity with their attendant nebula. This conclusion was reached as the result of independent research.
The radial velocity of the Orion-nebula has been measured by various observers. The values obtained are: 17.7 (Wright), 17.4 (Vogel and Eberhardt), 18.5 (Frost and Adams). The mean value is 17.4 kms. per sec. This velocity is derived from the brightest part of the nebula, the so-called trapezium. The values obtained in the case of the stars almost all exceed 20 kms. per sec., and hence it seems likely that part of this radial velocity, viz. the excess over that of the nebula, is due to the Einstein effect. When the difference between the radial velocities of the stars and the associated nebula are tabulated for each star, we find that in the case of all members except two the difference is positive, i.e. indicative of a shift towards the red end, in agreement with the statistical investigation applied to the-,-, and M-stars. The difference amounts to, and is a little greater than that given by the statistical method.
The two starsand 36 Orionis give a displacement towards the violet end. It has been suggested that they do not belong to the morelimited group of Orion stars, but are only projected into that portion of the celestial sphere. This is supported by the fact that both stars have only very small spherical proper motions, and that the radial velocities observed for them differ considerably from the mean of the radial velocities of the others.
This method has not been successfully applied to other stellar systems inasmuch as the nebulæ of those which are available emit such feeble light that it has not been possible to establish the displacement to any degree of accuracy. Eddington recently pointed out that a very important factor had been neglected in the fundamental equations of the early theories concerning the equilibrium of stellar matter, viz. the pressure due to radiation. According to his theory, the equilibrium in the interior of the star (regarded as a gaseous sphere) is determined by three conditions. These are gaseous pressure, radiational pressure, and gravitational forces.
Calculation shows that for very great masses the gravitational pressure is almost entirely balanced by radiational pressure. This implies that any additional force such as that due to a centrifugal field of rotation would lead to an unstable condition.
It can, furthermore, be deduced from Eddington's theory that only stars whose masses exceed a certain minimum value can in the course of their evolution reach the very high surface-temperatures which have been observed in the case of the- and-stars.
It therefore seems likely that the- and-stars have in the process of evolution passed through a stage of which the radiational pressure has brought about a condition of unstable equilibrium, and one might expect them to be surrounded by cosmic dust which has become dissociated from the nuclei of the system.
In some cases this dissociated matter may be in a very fine stateof division, and may extend so far into space that the absorption lines they produce in the spectrum of the star they surround may originate from a gravitational field which differs perceptibly from that at the surface of the star. There are definite signs of the existence of such atmospheres. A high percentage of-stars are found to be spectroscopic double stars, i.e. their spectral lines fluctuate periodically about some mean position. Hartmann was the first to notice that in the spectrum of the-star. Orionis the absorption linesandof calcium, viz. 3933.82 and 3968.63 Ångstroms, occur, but that they do not share in the periodic movements of the other lines. A number of other stars belonging to early spectral types contain calcium absorption lines in their spectra, which exhibit a similar anomaly, inasmuch as they either remain immovable or execute periodic motions which are of feeble amplitude compared with the proper stellar lines. In view of the important rôle that calcium plays in the outermost layers of the gaseous atmosphere encircling the sun, and in view of the discussion above, the suggestion forces itself upon one that these calcium lines indicate the presence of an extensive atmosphere surrounding the star.
It has often been put forward that these lines are due to the light from these stars being absorbed by vast interstellar clouds of calcium. Evershed considers that this is supported by the fact that when the motion of the solar system is subtracted from that calculated from the fixed calcium lines (owing to the ordinary Doppler effect), the remaining motion is very small. But this argument does not carry weight inasmuch as it is known that the-stars, in the spectrum of which these lines occur, themselves have very small radial velocities. As Young remarked, it seems very strange that these calciumclouds should so consistently choose to lie in front of stars of typeor earlier. An objection against this hypothesis is to be found in the fact that in the case of various systems these two calcium lines are not at rest but move, although with somewhat less amplitude than the other proper lines of the double star.
An additional circumstance which lends support to the theory that calcium lines denote the presence of an atmosphere around the star is that a great number of helium-stars are enveloped in a nebulous atmosphere which is actually visible.
Assuming then that the calcium absorption lines are due to such atmospheres, we may apply the same process as in the case of the Orion nebula, i.e. if the shifts of the spectral-lines of the stars be systematically falsified by a superposed gravitational effect, this should be expressed by the lines of the actual spectrum from a double star being displaced towards the red as compared with the fixed calcium lines.
This phenomenon has been clearly observed. The result has not yet beenquantitativelyfixed, as the numbers taken are not regarded as final.
All stars in the spectra of which theandlines of calcium occur have been used to test the conclusion, and all show a shift to the red end; the mean of the shifts corresponds to a velocity of + 6.3 kms. per sec.
The results of this discussion have been formulated by Dr. Freundlich thus:—
SUMMARY
1. Statistical consideration gives us the means of separating the mean gravitational effect from the ordinary Doppler effect in the case of the helium-stars and the bright- and-stars, whichastronomical investigations compel us to regard as being of particularly great mass.
A general shift of the spectra towards the red is exhibited with considerable certainty.
2. It follows from a comparison of the displacement of the lines of the star-spectra that the above displacement which was found by a statistical examination is not an ordinary Doppler effect, but is due to the conditions of emission of light at the surfaces of the stars.
3. The close connection of the- and-stars with nebulous matter in the heavens is a symptom that these stars are of great mass.
4. If we regard the fixed calcium lines in the spectra of- and-stars as being caused by absorption in extended calcium atmospheres moving with each star in question, the shift towards the red which manifests itself may be regarded as the effect predicted by Einstein's theory, i.e. due to the different gravitational fields from which the absorption lines and the stellar lines have originated.