Light & Gravity (Script) (Patreon)

We know that gravity exerts its pull on light, and we have an explanation for why.  Actually, we have multiple explanations that all predict the same thing. And at first glance, these explanations seem to describe completely different causes. So what is the true connection between light and gravity, or is truth, in fact, entirely relative?

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Gravity bends the path of light. That fact is guaranteed by Einstein’s general theory of relativity. But for some reason we - scientists - have been convinced of the fact with no good evidence for hundreds of years. In 1783, the English clergyman John Mitchel proposed that a particle of light gripped by the gravitational field of a sufficiently massive star would slow down, stop, and fall back - and so was the first to predict the existence of black holes. Mitchel wrote to his good friend Henry Cavendish on the subject, and Cavendish followed similar reasoning to predict that a particle of light would be deflected in its path as it passed near a massive object.

Mitchell and Cavendish used the same utterly wrong assumptions to do their calculations. They assumed that light could be slowed down, and that light experiences a force of gravity in the same way that a massive object does. They assumed that Isaac Newton’s theory of gravity was the full picture, and that light behaves like any other particle in response to Newtonian gravity. But the weird thing is that despite these incorrect assumptions, the effects these guys predicted have proved very real.

We now know that both gravity and light are much weirder than Newton thought. We explored some of this in recent episodes, when we saw that what we experience as the force of gravity is mostly due to the way mass warps the flow of time. But the photon doesn’t experience the flow of time - it doesn’t even have any mass. And yet general relativity demands that gravity does affect light, in ways eerily close to the predictions of Mitchell and Cavendish. The really hard part is understanding the why of it - what is really happening when light interacts with gravity. Let’s see if we can’t figure it out.

In general relativity, the best place to start is always the equivalence principle. This is Einstein’s great insight that there’s no experiment that can distinguish between the backwards pull due to being in an accelerating reference frame and the downward pull of gravity of the same strength. Or of the sense of weightlessness in freefall in a gravitational field versus the weightlessness felt in the absence of gravity.

Imagine that you’re in a rocket ship, accelerating rapidly. I dunno, to escape giant alien spiders or something. A spider breaks through the front hull

and you fire your laser at it. By the time the beam reaches its target, the ship is moving a little faster than when it fired. The spider still observes the laser traveling at the speed of light - because the speed of light is invariant to all observers. But something else does change.

See, light is a wave. The distance between the peaks of that wave is its wavelength.

After the front peak of your laser pulse reaches the spider the ship continues to accelerate - which means the second peak has to travel a little further than the first, and the third peak further still. The distance between the peaks gets drawn out.

Wavelength increases, which means frequency and energy drop.

Your laser’s power rating goes from kill to tickle - good news for the giant alien spider, not so much for you.

The equivalence principle tells us that we must experience all the same physics if at rest in a gravitational field - say, in a fake rocket ship in a Hollywood set. That for some reason is also being attacked by giant alien spiders. Light emerging from a gravitational field is stretched out - it experiences gravitational redshift.

And we get exactly the same prediction if we use the fact that time runs slow in gravitational fields. Any process that generates a photon can be thought of as a clock - be it an electric charge pulsing up and down a radio antenna, or an atom vibrating back and forth in a glowing light filament due to its heat. These are the ticks of a clock,

and the frequency dictates the frequency and the wavelength of the photon produced by that motion. But from a great distance, those clocks run slow, and so the frequency of light emerging from within a gravitational field is lower.

And if the density of the gravitating body is large enough, light emerging from can be sapped of ALL energy - redshifted so the wavelength is effectively infinite. At the event horizon of the black hole, gravitational time dilation is so strong that clocks stop and the frequency of photons trying to escape is brought to zero. And the bizarre thing is that the density of mass required to produce this infinite redshift is exactly the same as is required to turn a light-speed particle around and have it fall back, as calculated by Michell from totally wrong assumptions.

Let’s tackle an even trickier problem - the solving of which shot Einstein to his international fame - the deflection of the path of light by gravity. Let’s start with the good-old equivalence principle again, and a spaceship attacked by giant alien spiders. This time you fire your laser across the deck perpendicular to the direction of acceleration.

If you aim straight do you hit? Let’s look at it from the perspective of a non-accelerating observer outside the ship. They have to see the light travel in straight line in the absence of any gravity. But for that to be the case, light must travel a curved path from the point of view of the accelerating frame. You aim straight but miss your mark.

These giant alien spiders are no joke. And the equivalence principle tells us we must see the same bending of the light ray in our stationary rocketship set in our gravitational field.

OK, so that’s the prediction. In the case of gravitational redshift, we could come up with a physical explanation for the prediction - the difference in the flow of time changes the frequency of outgoing light. Can this gravitational time dilation also explain the bending of a ray light traveling horizontally? Well, tricky - because there’s no change in clock speed in the horizontal direction - only the vertical. But we saw in our previous episode that a gradient in clock speed across the vertical extent of the object is enough to drag it downwards. To get technical: any massive object has a component of its 4-dimensional spacetime velocity - its 4-velocity in the time direction. That temporal component is rotated into the downward spatial direction, which translates into falling. But light is frozen in time from its point of view. It has no time-component to its velocity. It has, in fact, no time to lose.

So if we imagine light as a perfectly narrow ray, or even as a massless, timeless particle, none of our intuitive explanations say that it should be deflected by gravity. So let’s be smart and do what the smartest guy in the world did - let’s think about light in an entirely different way. Einstein was in the business of overturning the ideas of Newton, so it’s not surprising that he turns to one of Newton’s great rivals. The Dutch physicist Christiaan Huygens disagreed with Newton on many things - including the idea that light is a particle. Huygens’ wave theory of light advanced the field of optics enormously. A big part of that was his Huygens or Huygens-Fresnel principle. The idea is that any wave can be described as an infinite number of point-like oscillations, each of which produces new waves. The sum of all those waves perfectly describes the future evolution of the original wave.

For example, imagine a single circular ripple on a pool of water. At any point in time, the expanding ripple can be thought of as an infinite number of sources of new circular ripples, or wavelets. Those wavelets also expand outwards, but they cancel each other out everywhere except in the outward direction of the original ripple.

The result looks exactly like the continuation of the original ripple.

We can model light in the same way. A plane wave of light is just an infinite number of new sources of light that generate the next step in the plane wave. This explains why light can turn corners around opaque objects in the phenomenon of diffraction, and why the path of light is bent when passing between substances - refraction. Our plane wave reaches the boundary to a new medium with a slower speed of light. If it arrives at some angle, then the next set of  wavelets forming at that boundary will be more closely packed - the fronts of those ripples don’t travel as far in the time it takes to make each new wavelet. So now if we connect the wavelets to reconstruct the overall wavefront, we see that the path of the light has bent.

So that’s refraction. In a sense, light gets refracted by gravitational fields - or at least you can model it that way. You just need two completely crazy assumptions - first that light acts like a very classical, 17th-century style plane wave so you can use Huygen’s principle. And also that the speed of light changes in gravitational fields, which sounds counter to everything I’ve told you. But this is exactly what Einstein did. Let me explain why this isn’t so crazy after all.

As you might expect from a theory called “relativity”, stuff is relative. The only thing that’s not relative is the speed of light - everyone observes the same local speed of light - 300,000 km/s in a vacuum. But notice that I said LOCAL speed of light - that means everyone measures the same speed of light passing through their own local patch of space. But viewed from a distance, the speed of light can at least appear to change.

Imagine you’re looking at the earth from a distance. A photon passes by, and the amount of time it takes to cross that space is larger than if Earth wasn’t there. That’s because of two effects: your clock is ticking faster that clocks in the gravitational field, and space within the gravitational field is stretched.

The photon has to travel further through a region of slowed time - and both conspire in the same direction to slow the apparent speed. Of course for someone actually inside the gravitational field, the photon is still traveling at the speed of light as it whizzes past them.

And even if this change in velocity isn’t “real”, it’s enough to let us use Huygen’s principle. At each location perpendicular to a gravitational field, the wavefront of light can be thought of as a vertical column new wavelets. But for you, tracking this from a distance, the effective speed of light decreases downwards, because time slows and space stretches. So those lower wavelets become bunched up from your perspective. If you track the path of the wavefront by connecting the ripples, you see it bends.

Einstein used this approach to calculate the deflection expected when light passed by a massive object. The number he got was exactly twice the deflection calculated by Cavendish based on Newton’s gravity.

And Einstein’s deflection angle was famously verified by Sir Arthur Eddington, who voyaged to the west coast of Africa to watch a solar eclipse. That enabled him to measure the slight offset in the apparent positions of stars around the sun, due to their light rays being “refracted” in the Sun’s gravitational field.

So I guess the question I’d be asking at this point is why did Einstein’s calculation even work. Light isn’t really a simple plane wave - it’s a much weirder quantum wave-particle thing. The fact is, none of these pictures really represent the “ultimate” reality. If you solve Einstein’s equations in a different way, you can show that gravity is due to a free-falling waterfall of space, rather than a gradient in time. It’s a matter of perspective, and how you define your reference frames.

Relativity is relative - because so is our universe. We come up with our “physical explanations” and then find that seemingly contradictory explanations work just as well. Light is a wave and a particle; time slows or space flows in gravitational fields. But our guiding star is that the universe is deeply self-consistent, our explanatory stories are all “true” in their own regard, but are just narrow slices of some more fundamental reality underlying this generally relative spacetime.