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24 Gravitational Light Bending 3Sep18

Introduction This essay continues the focus on the collection of experimental results involving light, gravity and General Relativity. It takes a closer look at the fact that the path of light is bent when it passes close to a massive object such as our Sun, and similar phenomena arising when light from distant sources passes close to a massive celestial body on its way to our telescopes.

History of the Bending of Light by Gravity In the latter part of the 17th century, just after the English Civil War, Sir Isaac Newton came up with an expression for the gravitational force between two massive objects. It was hard work - Newton almost had to invent integral calculus to arrive at his result, but it gives beautifully simple results that depend only on the masses of the two objects and the distance between their centers of mass. It then provides a good way to calculate the trajectory of planets around the Sun and these calculations accord almost perfectly with observations.

But what about the path of photons arriving at Earth from a distant star when a massive gravitating object like the Sun gets so close that the photons are forced to graze its edge?

Some people think that Newton’s Law of Gravitation implies that since the photon has no mass, there can be no gravitational force to deflect its path and hence it should not deflect at all.

But this makes several unjustified assumptions. Who says a photon has no mass? It may have no rest mass, but it does have something called relativistic mass. [The term relativistic mass is one which Einstein eventually came to regret, but the concept is still meaningful. Note that the famous Einstein equation is actually E2 = p2c2 + m2c4 and only becomes E = mc2 for an object which has rest mass m but no momentum p. A photon (or neutrino) does not have rest mass but does have momentum, so the equation becomes E2 = p2c2 and hence E = pc. It does not make much difference, but it does reveal that many commentators have only a poor grasp of what they are talking about.

Experiments on photons also show that E = hf where h is Planck’s constant and f is the frequency of the disturbance created by the photon when it is absorbed/observed, so it is easy to infer the momentum of a photon from its frequency effects when detected. It is also possible to feel the impact of photons when they are detected – (look up “solar sails” for example).

Newton did not know about relativistic mass and hence was silent about it. Nevertheless he thought of light as having a well defined path and robust nature and hence thought of light as being “corpuscular” in nature rather than some of sort of diffuse wave. So he fully expected it to be deflected by gravity (see below). In fact he even calculated the extent to which a massive object would deflect a passing “corpuscule”. Laplace went further and calculated the strength of gravity that would prevent light from escaping upwards – anticipating future considerations of back hole physics by about three hundred years.

Even if a photon has no mass of any sort then it is still not safe to assume that it cannot be deflected, for it would have absolutely no resistance to being deflected either. There is a popular conundrum that asks “what happens if an irresistible force meets an immoveable object?” Well the situation with light is a bit like the opposite conundrum – “what happens if a zero force applies to an entity with zero resistance to being deflected?”

Galileo (who died in the year Newton was born) had already realized that the rate of attractive acceleration towards a massive object is independent of the mass or composition of the attracted object. This is now called the Weak Equivalence Principle. Hence, in the absence of air resistance, a feather falls as quickly as a cannonball. Continuing the argument, an atom falls as quickly as a feather and a neutron falls as quickly as an atom. Whey then should not a neutrino or a photon fall just as quickly as anything else?

Here is a synopsis of how this issue has been handled over the centuries: • In 1704, Newton suggests the bending of light as an aside in his treatise, Opticks. • In 1784, Henry Cavendish calculates the bending of light due to Newtonian gravity but does not publish the result. The evidence of his calculation only surfaced in the 1900s. • In 1801, Johann von Soldner calculates the bending of light as it passes by a massive object in 1801 (taking 25 pages to do it!). The calculation uses Newton's theory of light as a stream of corpuscles with an unspecified mass. However, the mass of the corpuscle (photon) drops out of the calculation, and the angle only depends on the mass of the object and the closest approach to the massive object (e.g. our Sun). The angle of deflection turns out to be: angle ~ 2m/r, where m = GM/c2, M is the mass of the object and r is the closest approach distance of the photon to the object. This solution is an approximation, because it is the first term in a series. All of the other terms in the series are much smaller. Von Soldner's calculation is very close to Cavendish's, and to a first-order approximation, they are the same. • In 1911, Albert Einstein published a paper called "On the Influence of Gravitation on the Propagation of Light", which calculated the bending effect of gravity on light using his Equivalence Principle. This calculation was not based on the equations of General Relativity, since these had not yet been developed. It did rely on Einstein’s recent conclusion that gravity must have an effect on the speed of light. Einstein’s calculation in this paper was identical to von Soldner's approximation. • In 1915, Einstein finished his theory of General Relativity, and developed a full set of ten partial differential equations for the curvature of spacetime in a gravitational field. When Einstein used his full theory and recalculated of the deflection of starlight due to the Sun he obtained exactly twice the prediction he published in 1911. The additional bending was due to the curvature of space itself. [In mathematical terms it arises from non-zero off-diagonal terms in the 4x4 Riemannian metric tensor that describes the spacetime curvature]. Note that equal contributions are made by both the space and time perturbations of the metric. • In May 1919, Sir Frank Dyson and Sir Arthur Eddington and led an expedition to the equatorial African island of Principe and their colleague Andrew Crommelin led an expedition to Sobral in Brazil to observe what happened to the apparent position of stars in the constellation of Taurus when the Sun got in the way. They could see these stars because they situated their telescopes in the moving shadow path of the moon during a total eclipse of the Sun. They reported, and the Royal Astronomical Society announced, that the degree of starlight bending (‘aberration’) was exactly as predicted by General Relativity. This announcement became headlines in a world hungry for interesting good news and Einstein became a media superstar. Which may have helped him retain his job in Berlin in spite of the rise of anti-Semitism. However, it is interesting to note that there is some question as to whether or not the equipment and results of the 1919 eclipse expeditions really had the ability to conclusively determine the deflections as claimed. It is not a simple experiment. For example, the Sun’s corona has strong magnetic fields and emits a lot of plasma that can complicate the interpretation of the results, and the observed effect is tiny. It may be that the researchers injected some of their expectations into the reported results. However, subsequent and more robust observations have confirmed the deflection as predicted.

So there you have it. Modern physics interprets the bending of light by a massive object such as the Sun as being partly due to a combination of factors, neatly modeled by the spacetime solution to Einstein’s field equations for the spacetime region around the Sun.

The textbooks invariably show the stretched rubber sheet attempt to suggest what spacetime is like in Einstein’s model. There is a big depression caused by the ponderable mass of the Sun. Light comes in and is deflected slightly because of the curved topology. A bit like when you just miss a putt in golf. To be frank, it is not an analogy that I particularly admire.

A Heretical Alternative? Since the theme of these essays is to re-examine the foundations of modern physics and try to provoke some fresh thinking about them, I am going to suggest an alternative interpretation.

We already have a good explanation for half of the effect, due to classical luminaries like Sir Isaac Newton, Henry Cavendish and Simon Laplace. You can think of it as a kind of scattering effect due to the Sun’s gravity trying to pull the passing photons a bit closer. If you like you can think of the Sun’s gravity working on the mass equivalent to the photon’s energy, resisted by the inertia of the photon’s momentum.

It is the other half of the observed effect that presents the issue. I suggest that the earlier classical calculations do not get the full answer because they do not take into account the fact that gravity also slows down the speed of light. Classical physicists did not have the means to know this. But I suspect that if Isaac Newton knew that light travels more slowly when the presence of gravity is more intense, then he would have started to think about the phenomenon of refraction.

[Most transparent media have a refractive index higher than that of a vacuum, which we have assigned index value 1. This signifies that light travels more slowly in such media than it does in a vacuum. When light passes from one medium to another at an inclined angle the path it travels bends at the interface (see Snell’s Law, Fermat’s Principle etc).]

Gravity also slows down light so it is not unreasonable to conceive of “gravitational refraction”. We could say that the gravitational index of empty space in the absence of any discernable gravity fields is 1 and that near a back hole it is very high. Elsewhere it takes an intermediate value.

Note that ordinary refraction affects the speeds of photons, but not their energy level. We can deduce this by using a beam consisting of a large number of identical photons and sampling some of them at each stage of their collective journey. When the surviving photons emerge from the refractive medium they have the same energy level as those photons sampled as they attempt to enter the medium.

I suggest the same thing happens with gravitational refraction. Photons enter a region of high gravitational field and are slowed down and deflected (in accord with Fermat’s Principle, overlaid with a Newtonian gravity deflection) but when they emerge they speed up again. The photons emerge with the same energy level as when they went in.

Furthermore I suggest that if the gravitational refraction is added to the normal Newtonian scattering deflection the answer will be as observed experimentally. Curvature of spatial coordinate system not required.

Finally I would like to suggest that gravitational light bending and gravitational redshifts are closely related. In the case of gravitational light bending the photons traverse the gravitational field at high angles to the gravitational field but in typical gravitational redshift situations the photons travel more or less parallel to the direction of gravitational field. If the photons are travelling at an intermediate angle through a gravitational field, I suggest it is reasonable that the result would be a combination of the gravitational redshift effect and the gravitational light bending effect.

A spectrum? I have asked myself whether the gravitational bending of the light should create a spectrum effect. I think this is a reasonable question.

A spectrum effect might be expected if what is going on is similar to an ordinary refraction effect. Consider what happens if a beam of light passes through a medium with a refractive index greater than one, such as a glass prism. I will shun the usual explanation based on what I think is an old fashioned wave model of light and will talk about phots instead.

Consider a beam of light consisting of many phots, all travelling at the same speed in a vacuum. When they encounter a medium with a higher refractive index than 1, such as glass or water, they slow down, sometimes quite considerably. Some textbooks say this is because multiple scattering events makes their path longer, but I doubt this is true because well collimated beams of a single frequency, such as from a laser, remain well collimated and are not scattered. If the initial and final surfaces of the prism are perfectly parallel, the incident and emergent phot paths are also perfectly parallel.

If the phots emerge from a transparent medium they continue with the same energy they had before they entered it. This is a further reason to doubt the scattering explanation, for then it is reasonable to suppose that the medium itself would absorb some of the energy from individual phots. What I think happens is that the phots actually do slow down when they are in the denser medium. High energy photos slow down more than low energy phots. It would be interesting to explore the electromagnetic field reasons for this but let us not get distracted.

If the beam of light of light encounters the new medium at an angle, the path of the phots develops a kink at the interface. If the transition is more gradual the change in direction is a curved bend. Either way the light beam path is refracted. High energy phots are refracted more than low energy phots. The beam is spread out according to the energy of the phots. As far as visible light is concerned, a rainbow spectrum is formed, with red at the top and violet at the bottom.

If the light is passing through a prism say, then when it emerges at the final interface the opposite refraction occurs. Depending on the shape of the prism, the rainbow effect can be undone or not.

The whole process follows something called Fermat’s Principle. Fermat's Principle states that “light travels between two points along the path that requires the least time, as compared to other nearby paths.” From Fermat's principle, one can derive (a) the law of reflection [the angle of incidence is equal to the angle of reflection] and (b) the law of refraction [Snell's law].

In the case of refraction, think about cross country runners encountering a strip of boggy ground that slows down their running speed. If they want to minimize their overall effort and maximize their overall rate of progress, it makes sense to shorten their path across the boggy ground, even if this makes their overall path a little longer.

Returning to the case of a photon (to use the conventional term) traversing a gravity field around a spherical celestial object. The more energy the photon has, the more momentum it has and this might make it more difficult to bend its path. However, no spectrum effect is detected.

The Weak Equivalence Principle gives a naïve explanation. The more energy a photon has, the more momentum it has, and this offsets the bigger deflecting force. However, a modern physicist would probably prefer to say that all the photons, of whatever energy level, are following the same geodesic pathway as determined by the non-Euclidean four dimensional spacetime.

As a photon passes through a strong gravity field at a shallow angle. gravity slows down the speed of light (as viewed from a distance). More on the side closer to the Sun than on the other. Hence it is not unreasonable to think that the light will follow Fermat’s Principle and take the quickest overall path through the gravity field, even if this means bending away from its initial direction.

However, unlike ordinary refraction through glass or water, there is no rainbow effect. Gravitational slowdown applies equally to photons of all frequency/energy and the speed of all photons remains the same relative to each other. There is no dispersion and they all bend by the same amount.

Conclusion The bending of light by gravity is generally regarded as one of the key experimental results supporting Einstein’s theory of General Relativity, and its model of a spacetime with curvature in all of its time-time and space-time coordinate pairs. However, half of the effect was already predicted and explained in terms of classical physics.

In the next essay I will discuss the fact that gravity slows down both time and the speed of light.

We know that slowing the speed of light by passing it through a transparent medium that is not a vacuum causes the path of the light to bend in according with Fermat’s Principle. Why then is it not reasonable to consider that slowing down the speed of light by passing it through a gravity field might not also cause a degree of bending in accordance with Fermat’s Principle?

Adding the classical gravitational effect to a gravitational time-dilation refraction effect might give a satisfactory explanation for light bending in accord with experimental observation, without calling upon the full spacetime curvature model adopted by Einstein.

#Einstein GeneralRelativity LightBending hereticalphysics

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