Bending of Starlight by the Sun

In 1911, Einstein’s first prediction to verify general relativity was the bending of starlight as it passes near a massive object, like the sun. Bending was the same as the Newtonian model might have predicted for a light particle. Because it is not possible to see starlight while the sun is so bright, he suggested that the test be run during a solar eclipse. In 1912, efforts to test his prediction were prevented by cloudy weather at the observation site.

In the 1916 paper on general relativity, Einstein also discovered that his 1911 calculation for the bending of light by the sun was off by a factor of two, making the bending of light 1.745 seconds of arc for starlight just grazing the sun. After another failed attempt, in 1919, Sir Arthur Eddington on an island off the coast of Africa and another group in Brazil confirmed Einstein’s prediction. When announced, this prediction captured the imagination of the world and Einstein became an instant celebrity. Had the first attempt in 1912 to make this measurement not been called off by weather, Einstein’s prediction then would have been wrong. One could only wonder what effect that might have had on Einstein’s popularity.

GRAVITATIONAL EFFECT ON LIGHT

Photons moving in a gravitational field laterally to a massive body, are subject to two distinct, but related deflection mechanisms. (See the following figure) If viewed as a particle having a mass of value E/c 2 where E=h ν, it will experience a gravitational force downward orthogonal to it’s lateral motion, similar to all other matter, resulting in a deflection angle of 2GM/c 2R. This is the deflection mechanism that the equivalence principle addresses.

However, because gravity ‘pulls’ towards a point at the center of mass, there is also a horizontal gravitational component, which squeezes its wavefronts together. The force is greater for photons closer to the massive body, making the wavelength shorter and the front velocity larger. The bottom of the front compresses more than the top, moving faster, and producing bending towards the gravitational source. The bending causes an additional deflection angle numerically equal to 2GM/c 2R, making the total deflection 4GM/c 2R, as also predicted by general relativity. An analytical explanation is contained in the next section.

For those that don’t accept that photons in motion are affected by gravity, in Einstein's book, “Relativity”, when discussing the bending of light skimming the sun, he writes, “…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 ('curvature') of space caused by the sun.”

For those that still not convinced, the analysis could just as easily apply to other neutral sub-atomic particles traveling arbitrarily close to the speed of light, for example, neutrinos, in which some flavors are thought to have a non-zero rest mass. Also, because the gravitational mass and the inertial mass are numerically the same, the bending of the particle’s trajectory is the same independent of the mass value, and even would be the same as for photons. The fact that bending of a neutrino beam can’t be readily measured is irrelevant.

GRAVITATIONAL EFFECT ON LIGHT CALCULATION

As mentioned, there are two gravitational mechanisms that bend starlight. Imagine a stream of photons nominally moving horizontally at velocity “c” along the x- axis, skimming the sun and heading towards Earth. The Sun’s gravitational force can be separated into two components acting on the photon stream, one vertical and the other horizontal. Each causes the stream to bend. However, correction terms to the nominal paths are so small that they can be approximated without regard to the small changes of the nominal path.

Figure – Gravitational Deflection of Light

The vertical component deflects the stream downward toward the Sun as it heads towards earth. For simplicity, only the second half from the point of tangency to the Sun, onward to Earth need be considered. The section from the star to the sun is the mirror image, doubling the result. (See above diagram – Upper left))

The gravitational vertical acceleration component on each photon, treated as a particle, is: 2 (1) a y = [F g/m] [cos θ] = [GMm/mr ] [cos θ]

where: r 2 = [x 2+R 2] and: R is the distance from the sun’s center to the closest point of the photon stream. and: a y is vertical acceleration component and: m is the imputed mass (hf/c 2) of a photon in flight and: θ is the angle between r and the vertical y direction and: G is the gravitational constant and: M is the mass of the sun

The velocity in the vertical direction is the integral of acceleration over time: 2 2 (2) v y = ∫ a y dt = ∫GM/[x +R ] * [cos θ]dt

where: [cos θ] = R/[x 2+R 2]1/2 and: dt = [dx][dt/dx] = dx/c

Collecting terms: 2 2 -3/2 (3) v y = GMR/c ∫[x +R ] * dx

The angular deflection is: 2 2 2 -3/2 (4) α y = vy/c = GMR/c ∫[x +R ] * dx

Integrating: 2 2 2 1/2 (5) α y = GM/c * x/R[x +R ]

Evaluating between minus infinity (star) and plus infinity (earth): 2 2 2 (6) α earth = GM/Rc + GM/Rc = 2GM/Rc

If gravity pulled vertically and not to a point, this would be the total deflection, but that is not the case. As a “wave,” the photons are spread out comprising a wave front, and moving perpendicular to the front. Even though the signal speed of light doesn’t change, work is done, changing the frequency, and the energy of the photons accordingly. (See above diagram – Upper right)

Consequently, to derive the expression: 2 2 (7) E x = ∫ F x dx = ∫GMm/[x +R ] * [sin θ]dx

where: [sin θ] = x/[x 2+R 2]1/2 and: E x is the change in photon’s energy and: λx is the change in photon’s wavelegnth

Substituting and collecting terms: 2 2 -3/2 (8) E x = GMm/ ∫x[x +R ] * dx

The fractional change in energy: 2 2 2 2 -3/2 (9) u x = E x/E = E x/mc = GMm/mc ∫x[x +R ] * dx

Integrating and recognizing that the ratio of energy is approximately proportional to the ratio of wavelength: 2 2 2 -1/2 (10) u x = λx/λ = [GM/c ] [x +R ]

Evaluating between the star and tangent to the Sun: 2 (11) u s-e = -[GM/Rc ]

The wavelegnth gradient is : 2 2 (12) ∆ ux/∆R = GM/R c

But from the mid point it is only half as much: 2 2 (13) ∆ ux/∆R = GM/2R c

Notice that this differential is extremely small, but even such a small difference in the compression still causes considerable bending toward the increased front velocity, for example the compressed wave fronts. This phenomenon can be observed in other situations. For example, even small non-uniform shrinkage in a wood plank bends it greatly. For small deflections, the bending can be approximated by a right triangle, where the hypotenuse represents the mid-point photon path, the long side the compressed bottom path, and the small side the deflection.

Applying Pythagoras’ theorem, the deflection is: (14) β = − GM/c 2 [1 2 – (1 - 1/2R 2)2]1/2 = [1- 1- 2/R 2]1/2

Droping the R 4 term & doubling to account for the star to the sun: (15) β = 2GM/Rc 2

Finally, since each deflection is small, the total angular deflection can be approximated by the sum of the deflection due to its particle and wave behaviors: See preceding diagram – Lower illustration)

(16) δ = α + β = 4GM/Rc 2

If the bending contributions are compared, it is apparent that while both cause the same total bending, the profile for each is different. Particle bending is greatest while the ray is just grazing the sun. On the other hand, the wave contribution is less there, but larger off to each side. This is different from general relativity where both bending components are maximized near the center of mass.

Ironically, Einstein’s ubiquitous elevator would exhibit particle bending, with an insignificant wave contribution, resulting in about half as much as general relativity predicts, but the same amount as accelerating in outer space would cause. This is because general relativity’s predicted space distortion is also greatest at the point of tangency to the sun, where gravity is strongest, and would be expected to maximize in the elevator. Nevertheless, for the entire trip from the star to earth, both deflection contributions complement each other, together deflecting more uniformly than each one separately. Consequently, acceleration and gravity do not cause the same amount of deflection, because for acceleration, the wave component is missing. In the elevator, the bending would be the same for both, but for the wrong reason.

Interestingly, when Einstein, in 1911, originally proposed the equivalence principle, he predicted that light would bend, and proposed that it could be verified during the next eclipse of the sun. The calculation he presented used the wave property, not Newtonian reasoning, apparently assuming that both the wave and particle arguments were not separate and distinct, but just different views of the same phenomena. Consequently, he predicted half the deflection that was ultimately measured. Fortunately, for him, he corrected his prediction when he proposed general relativity in 1916, before the famous eclipse of the sun experiment was finally performed. In the GR theory, the deflection was attributed to space-time warping caused by the massive sun. However, space-time warping is not required to explain the bending of starlight as it grazes the sun on its way towards Earth.