### 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?

Bruce,
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.

Would the same rules apply to a macroscopic object attempting to exceed c. by gravitational slingshot?
Go with your baseball, used earlier (assuming it survived reentry). Imagine that, (flung by an amazingly energetic arm and through a series of superbly calculated earlier slingshots), your baseball is now at around 99.99% of c., and is now approaching, say, Jupiter for a final catapult. It swings in and picks up energy, increasing speed until...

What? It gains the equivalent of an infinite mass, preventing it from passing c.? It ignores the extra energy it can't use, and continues in a straight line? It turns blue...? :)

1. Rob,
It might start realizing postiron/electron pairs in its wake, shedding its energy to the vacuum a bit like turbulent drag.
Matter is, by and large, analogous to eddies, the vortical dimples that show up briefly on the surface of a pond when a current catches its tail. One of those eddies isn't going to be able to move faster than the speed of sound in water, and by the same token matter can't move faster than the speed of light in vacuum.
(Curiously enough, an electron CAN be made to move faster than the speed of light in water, which will cause it to produce Cherenkov radiation!)

2. Thanks for the reply :)
Of course, drag *facepalm*
Would it then shed mass in this process, instead of gaining speed?

2. Hi, I would like to know where is the centre of the universe. Did the universe expand in all directions at the big bang or did it just expand in one direction, like the flash coming out of a gun barrel. The latter suggests to me that the it is possible that it may not be a big BANG! but a big GUSH! if we are to replace the gun with a water hose, therefore planets and galaxies are continually being formed into our own universe.

3. This all leads me to a prior questioning as it were ...
Why is it that a satalite can indeed slingshot off a planet and actually accelerate thereafter as my mind gets stuck with the reasoning that surely the same gravitational attraction upon the satalite causing it to accelerate towards said planetary mass is the same gravitational pull restraining the satalite and thus holding the satalite back , slowing it down again as it moves away from said planetary gravitational mass .
Shouldn't the two effectual forces cancel out any resultant acceleration ? I know it doesn't but why not ??
Email thetruthispriceless@gmail.com if you can answer my small minds dilemma please as I would dearly love to appreciate this why .

Have a wonderful Christmas everyone .
God bless x

Mat

1. Mat,
You're right—the planet's gravitational field isn't the source of the acceleration. Instead, it's the planet's kinetic energy that's being transferred to the satellite. The planet has to be moving for it to work, and the strength of the slingshot depends on the planet's speed. Take a close look at the second image in the post—when a satellite slingshots around a planet like that, it actually slows the planet down some. When the satellite is at the right-most point in its trajectory, it's pulling the planet in the opposite direction of its motion! Of course, the satellite is always much less massive than the planet, so the change in the planet's speed is imperceptible, but the satellite gets accelerated to keep pace with the planet (and then some), which is where the "slingshot" effect comes from.

4. I know this may not be the right place to post something for consideration but I have a question.
If a star of sufficient mass (red hypergiant?) and a small black hole, let's say less than one stellar mass, were travelling toward each other at high enough velocity, would it be possible for the black hole end up completely inside the star (however briefly) before the star was completely consumed?

5. Not sure if I got the gist of this one?

What you seem to say is that you expect a photon to 'accumulate' energy that it then will keep making it 'blueshift' at all times, even when outside the gravitational 'slingshot field' that then gave it its 'new energy'? How would you go about proving it experimentally?

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