ThatsMaths

Does light have weight? Newton thought that light was influenced by gravity and, using his laws of motion, we can calculate how gravity bends a light beam. The effect is observable during a total eclipse of the sun: photographs of the sky are compared with the same region when the sun is elsewhere and a radial displacement of the star images is found. But the amount predicted by Newton’s laws is only half the observed value.

Eclipse

A star near the limb of the sun appears to be slightly displaced radially outward compared to its position when the sun is absent. A remarkable series of observations during the solar eclipse in May 1919 confirmed that light passing close to the sun is deflected by its gravitation. Albert Einstein had calculated, based on his general theory of relativity, that the deflection of a sun-grazing beam would be 1.75 seconds of arc. This is a small angle but it was well within the then-current techniques of astronomy.

Two expeditions set out from Britain, one to the Isle of Principe off Africa, the other to Sobral in northern Brazil, both on the path of totality of the eclipse. Sir Arthur Eddington, who took part in the former expedition, described the observations and conclusions [1]. The observations at Principe gave an estimated deflection of 1.61+/−0.30 arc seconds, in excellent agreement with Einstein’s prediction. The Sobral results provided further confirmation.

Back to Newton

It is often stated that the bending of light is a purely relativistic phenomenon, but this is not correct. Classical Newtonian mechanics predicts such a deflection, but with half the magnitude of that observed in 1919. Newton himself considered that light has weight and should be influenced by the force of gravity. Eddington [1] quoted from Newton’s Opticks:

“Do not Bodies act upon Light at distance and by their action bend its Rays?”

Newton supported the corpuscular theory of light, regarding it as comprised of particles which would be influenced by a gravitational field.

The Classical Angle

It is a surprising fact that the motion of a test particle orbiting a massive body like the sun is independent of the mass of the particle. By Newton’s laws, the motion of the test mass is governed by the equation

m a = G M m / r²

where m is the test mass, a is its acceleration, M the solar mass, r the distance and G is Newton’s gravitational constant. On the left, m represents the inertial mass; on the right it represents the gravitational mass of the particle. But, due to the equivalence of inertial and gravitational mass, they are identical and cancel out, so the motion does not depend on the mass of the particle.

We now consider an asteroid approaching the sun from a great distance. For small values of the energy, the asteroid follows an elliptical orbit, returning repeatedly to its initial position. We assume that the asteroid has high energy, so that it traces out a hyperbolic orbit, with the equation

x² / a² – y² / b² = 1

where a and b are called the major and minor semi-axes of the hyperbola.

Let V be the limiting speed of the asteroid when it is remote from the sun and moving along a line whose perpendicular distance from the sun is R (see Figure above). We can express the energy and angular momentum in terms of these values. The energy (per unit mass) is E = ½V². The angular momentum, which is the moment of momentum about the sun, is L = R V .

These quantities can also be expressed in terms of geometric parameters, the major and minor semi-axes a and b of the hyperbolic orbit:

L² = G M b² / a      and      E = G M / 2 a

The proofs of these results are given in many text-books on orbital dynamics, for example Gregory [2]. The semi-angle α of the hyperbolic orbit (see Figure) is then given by

tan α = b / a = R V² / G M .

Then the deflection angle β = π – 2 α is tan (β / 2) = G M / R V² and since it is a small angle we can replace the tangent by its argument:

β = 2 G M / R V².

Recall that the mass of the asteroid does not influence its trajectory. Thus, the deflection angle β is the same for all bodies with given R and V values. If light is influenced by gravity, as Newton thought, we set V = c and obtain the bending angle for a ray of light:

β = 2 G M / R c² .

This is the formula for the deflection of sun-grazing starlight.

Assuming that the light ray passes close to the sun, we take R to be the solar radius. Using values (all in SI units) for the universal constant of gravitation G = 6.67 x 10^-11, the solar mass M = 2 x 10^30, solar radius R = 7 x 10^8 and the speed of light c = 3 x 10^8, this gives

β ≈ 0.423 x 10^-5 radians = 0.87 arc seconds

So Newton’s theory predicts that starlight passing close to the Sun will be bent by an angle of 0.87 arc seconds.

Fast Forward to Einstein

Clearly, the astronomical observations in Principe and Sobral could not be reconciled with Newton’s theory, and a scientific revolution ensued. General relativity showed that there is an additional effect, the distortion of spacetime due to the presence of the massive sun. The details of this are found in many works on general relativity (e.g. [3]) and also in Einstein’s original paper on general relativity (1916), and will not be repeated here. The final result is that the bending angle is

β = 4 G M / R c² ≈ 0.847 x 10^-5 radians = 1.75 arc seconds

which is twice the value of the deflection angle predicted by classical mechanics. The astronomical observations in 1919 were a splendid confirmation of this prediction. They catapulted Einstein into world fame and he has remained an icon of science ever since.

Acknowledgement: Thanks to Edward Thornley, Sandymount, Dublin for reminding me about the Newtonian bending of light.

(*) This is a preliminary version of an article published in Plus Magazine.
Sources

[1] Eddington, Arthur, 1920: Space, Time and Gravitation. Camb. Univ. Press, 218pp. ISBN: 0-521-33709-7.

[2] Gregory, R Douglas, 2006: Classical Mechanics. Camb. Univ. Press, 596pp. ISBN: 978-0-521-53409-3.

[3] McVittie, G. C., 1965: General Relativity and Cosmology. 2nd Edn. Chapman and Hall, 241pp.

 

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