Ask a Physicist: Slingshot Blueshift

Bruce, from the Netherlands, wants to know:
When manmade probes are sent out into our galaxy, they are sent in such manner that they take advantage of the ‘slingshot’ gravitational effect of large orbiting masses (planets) in order to accelerate. We know that that same gravitational force impacts upon light rays, bending them as they pass those large orbiting masses. Therefore, why is the velocity of light not also accelerated by the same ‘slingshot’ effect?

Thanks for writing in! This question actually had me scratching my head for a moment, but it’s a great way to dig into some of the more interesting consequences of general relativity. At its heart, this question is asking what happens when a photon feels the accelerating pull of a gravitational field, and what keeps it from going faster than c. To explore this idea, we don’t need to fuss with complicated slingshot trajectories, however—it can be demonstrated in a far simpler system.

Imagine you’re floating in orbit around the earth, and you throw a baseball downward, toward the surface. If you’ve got a good arm, you could maybe throw it at sixty miles per hour, or 27 meters per second. By the time it’s fallen for a few seconds, though, it’ll obviously be moving much faster than that—even high above the earth, it would still accelerate by about 9.8 m/s for each second that it falls. But what happens to a photon falling into a gravitational field like this, if it’s already moving literally as fast as possible?

When a particle—whether a baseball or a photon—falls into a “potential energy well” like Earth’s gravitational field, its gravitational potential energy decreases. Since we know that energy is always conserved, that particle must gain some other form of energy at an equal rate, making Bruce’s question a very logical one to ask—how is the photon gaining energy, if it’s not accelerating?

One of the fundamental tenets of general relativity, though, is that an electromagnetic wave’s energy doesn’t depend on its velocity, only on its frequency. All photons move at the same speed, but higher-energy ones oscillate more rapidly, meaning they have a higher frequency (and, correspondingly, a shorter wavelength.)

This image is intended to demonstrate gravitational
redshift, but could also be seen as a blueshift diagram.
Image Credit: Wiki users Vlad2i and mapos
Licensed under CC BY-SA 3.0

The upshot of all this is that if, instead of throwing a baseball, you shine a light of a certain color down from high gravitational potential to low, it will appear bluer—higher in frequency—to an observer on the ground. Conversely, if you shine a laser from Earth up to the international space station, it’ll be closer to the red end of the spectrum by the time it reaches its destination. This gravitational redshift is completely independent of the Doppler redshift that’s seen when a light source is moving away from its observer, and instead has to do with the fact that time appears to genuinely move slower in the vicinity of massive bodies.

But it’s the “slingshot” aspect of this question is what threw me for a loop at first. There’s a close equivalence between the gravitational blueshift that a photon experiences and the acceleration of more “classical” things like satellites; if a photon dips into a planet’s gravitational well but then comes back out, could it be blueshifted the same way a satellite is accelerated as it passes a massive planet on a carefully-tuned trajectory?

The answer is “yes”, it turns out, but it wouldn’t be the kind of gravitational blueshift we just discussed! When a satellite swings around a planet for a gravitational slingshot, it’s not just using the mass to make a sharper turn than it would ordinarily be able to; it’s also taking advantage of that planet’s motion around the sun. We don’t often think about it, but the planets of our solar system are moving at colossal speeds. To make it all the way around the sun in a year, Earth has to move at better than 18 miles per second, and this is the source of the speed that these satellites tap into—if the trajectory is plotted just right, a satellite can gain up to twice the speed of the planet it “slingshots” off of, if it’s going in the same direction as the planet.

The planet will actually be slowed down by the satellite, albeit imperceptibly.
Image Credit: Wiki user Leafnode.
Licensed under CC BY-SA 3.0

So given the proper trajectory around a compact enough object, a photon could indeed have energy imparted onto it by a similar process, but it wouldn’t speed up, it would be blueshifted. However, in order for it to be considered gravitational blueshift, the photon has to end up at a lower gravitational potential energy than it started at, which isn’t necessarily the case here. Instead, the mechanism that blueshifts a “slingshot photon” like this is the good old Doppler effect!

Doppler blueshift happens when a source is emitting light in the same direction it’s moving; since the waves can only travel at a finite speed, they end up “bunched up”, with a higher frequency and shorter wavelength. When the photon gets caught in the gravitational field of that planet, the planet can be thought of as that photon’s new “source”, sending it off with an energy boost—but still moving at c.

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