B. Deflection Of Light By Gravitational Field

B. Deflection Of Light By Gravitational Field

Section 22 mentioned that in General Relativity, a ray of light will experience a curvature of its path when passing through a gravitational field.

This curvature would be similar to that experienced by the path of a body which is projected through a gravitational field.

So we expect that a ray of light which is passing close to a heavenly body would be deviated towards the latter.

Aray of light which passes the sun at a distance of ∆ sun-radii from its centre, the angle of deflection ( α ) should amount to:

α = 1 . 7 seconds of arc _______ ∆

According to my theory:

• half of this deflection is produced by the Newtonian gravitational field of the sun
• the other half by the “curvature” of space caused by the sun.

This can be tested through the photograph of stars during a total eclipse of the sun which is needed because at every other time the atmosphere is so strongly illuminated by the light from the sun that the stars 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 D1, 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 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 with the standard photograph, the positions of the stars on the eclipse-photograph should appear displaced radially away from the centre of the sun by an angle α.

The Royal Society and the Royal Astronomical Society equipped 2 expeditions:

1. To Sobral (Brazil)
2. To the island of Principe (West Africa)

They sent several of Britain’s most celebrated astronomers (Eddington, Cottingham, Crommelin, Davidson), in order to obtain photographs of the solar eclipse of May 29, 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 General Relativity. 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:

C. Displacement Of Spectral Towards The Red Lines

Section 23 showed that in a system K’ which is in rotation with regard to a Galileian system K, identical clocks at rest with respect to the rotating reference-body, go at rates which are dependent on the positions of the clocks.

A clock at distance r from the centre of the disc, has a velocity relative to K which is given by :

v = ωr ,

where ω represents the * velocity of rotation of the disc K’ with respect to K. If ν 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 ( ν ) 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

…

or with sufficient accuracy by

..

This expression may also be stated in the following form:

…

If we represent the difference of potential of the centrifugal force between the position of the clock and the centre of the disc by φ , 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

φ = …

From this it follows that

…

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 also valid from the standpoint of an observer who is rotating with the disc.

Now, as judged from the disc, the latter is in a gravitational field of potential φ , 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 gravita- tional 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 φ = − K M , where K is Newton’s constant of r

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

….

For the sun, the displacement towards the red predicted by theory amounts to about two mil- lionths of the wave-length. A trustworthy cal- culation is not possible in the case of 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 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, 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 dis- cussed in detail from the standpoint of the question which has been engaging our attention here, in a paper by E. Freundlich entitled “Zur Prüfung der allgemeinen Relativitäts-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 gravita- tional potential, then the study of this displacement will furnish us with important information as to the mass of the heavenly bodies.