Reaching the speed of light: the first 99% is the easy part
Modern particle accelerators can boost electrons to 99% the speed of light in a few centimeters: so why are some accelerators a mile long?
If you’ve ever heard of “particle accelerators”, I would bet that one of the things you’ve heard about them is that they are big. If you know of any specific accelerators, you probably know the Large Hadron Collider (LHC). The LHC is a circular accelerator whose circumference is a whopping 27 kilometers/17 miles! You can see its outline drawn with tan lines in the picture below.
You are probably less likely to have heard about the Linac Coherent Light Source (LCLS). The LCLS is a linear accelerator in California, where I work (it sits just next to Stanford University at SLAC National Accelerator Laboratory). On it’s own, it’s about 1 kilometer long, and it sits in a line with two other kilometer long accelerators, for a total length of 3 kilometers/2 miles. Here’s a picture of it from above: the accelerator is housed under that long tan line.
These accelerators are very cool to look at, but the question I want to talk about today is why are they so big? Particle accelerators of this sort — the kind with price tags that measure in the hundreds of millions of dollars and sizes measured in kilometers — accelerate particles to nearly the fundamental speed limit of the universe: the speed of light. That’s 299,792,458 meters per second. This sounds like quite the task, so it doesn’t seem too surprising that it would take a kilometer to do it.
But the funny thing is: modern particle accelerators are able to accelerate light particles like electrons up to 99% the speed of light in just a few centimeters. The rest of the accelerator — the other kilometer minus a few centimeters — is just to get the particles from 99% to 99.9999%, or whatever other number between 99 and 100 might be the goal. If you were to plot the speed of the particles against the accelerator length, you would find something like this:
When I first realized this during my early research in this field, my mind was absolutely blown. I had always thought of the speed of light as some mind boggling concept that surely we could never get that close to. And yet in just a few centimeters we achieve 99% of it, and we still bother to add another kilometer onto the end of that just to get to 99.9999%!
The question I want to answer is why. Really two why’s:
- Why is it so “easy” to accelerate to 99% but so hard to get to 99.9999%?
- Why is that extra 0.9999% or 0.99999% worth hundreds of millions of dollars?
To answer these questions we’ll have to talk about what happens when something starts trying to move close to that fundamental speed limit. We’ll learn why it gets harder and harder to accelerate more the closer you get to the speed of light. On the other hand, we’ll learn that the effort is worth it: the closer we get to the speed of light, the more interesting effects we’re able to make happen. Let’s get started.
Life near the speed of light: special relativity
As soon as anything starts moving close to the speed of light, it starts to experience the world in a very different way than we’re used to. The reason for that is one of Einstein’s original key assumptions of relativity:
“The speed of light is the same in any inertial reference frame”.
There’s a few things that could be unpacked in that sentence, but I don’t want to turn this post into a “special relativity from the ground up” kind of thing. I can’t do that justice as an afterthought to our main topic. Instead I’ll just focus on the results that matter for our discussion.
One of the major side effects of this assumption, that the speed of light is the same regardless of who you ask, is that no object with mass can ever move at or above the speed of light. No matter how much you push something, you’ll never get to that number.
That’s a little surprising, because nothing stops me from inventing a scenario in which I take something and keep pushing it forward for as long as I want. And when I push something, I am giving energy to it: where does that energy go if we never reach the speed of light?
In an introductory physics class, you learn that a moving object has something called “kinetic energy”: when we push something, the energy we put into it with our push is converted into its movement as kinetic energy. In particular, for an object with a mass m and speed v, its kinetic energy is mv²/2. That makes it seem like if I just give something an energy of mc²/2 (where c is the speed of light, 299,792,458 meters per second), it’ll end up moving at the speed of light. Something must be wrong here.
It turns out that in order to avoid this simple argument, the universe plays a dirty trick. It changes the formula for the energy of a moving object, so that mv²/2 is no longer the right kinetic energy if you’re close to the speed of light! Instead, it makes it so that the closer you get to the speed of light, the more energy you have to put into it to get it to move even a little bit faster! It turns out that the total energy of a moving object has to be written as γmc², where
γ is called the Lorentz factor of the object. If you speak math you’ll notice that when v=0, meaning the object is not moving, γ=1 and its total energy is given by Einstein’s famous formula, E=mc². If hypothetically you had v=c, meaning the object moved at the speed of light, γ would be infinity! Meaning that it would take an infinite amount of energy to push something up to the speed of light. You can see a cartoonish picture of how γ changes with v here:
In between v=0 and v=c something funny happens. γ grows somewhat slowly early on, but as you get closer to c it starts to grow very rapidly. What does that mean? It means that as you get closer to the speed of light it takes more and more energy to increase the speed any further, exactly like we said earlier.
This is a little bit like if you wanted to get to the other side of a long treadmill, but the closer you got to the other side the faster the treadmill pushed you back. You can keep getting closer and closer with more and more effort, but it’ll never let you reach the other side.
To give some perspective, imagine that you have accelerated something from rest to 99% of the speed of light. Good job! How much more energy would it take to get to 99.9%, 99.99%, 99.999%, and 99.9999%? It turns out it takes:
- 2.5 times more energy to get to 99.9%
- 10 times more energy to get to 99.99%
- 35 times more energy to get to 99.999%
- 115 times more energy to get to 99.9999%!
I know it can be easy to get lost when we start throwing around numbers, but take a second to realize how insane this is. If it takes you some amount of energy to get to 99% the speed of light, you have to more than double all of that energy just to get an extra 0.9%!
That’d be like if accelerating your car from 40 miles per hour to 40.4 miles per hour took twice as much gas as it took to get from 0 to 40! And it only gets worse the higher you try to go! Nobody in their right mind would buy a car with that kind of twisted fuel efficiency.
To give you an example from the accelerator where I work, the LCLS that I mentioned before, we accelerate electrons to 99% the speed of light in a few centimeters. At the end of the kilometer-long machine we get to 99.9999995% (depending on the day). Relative to the energy needed to get to 99%, the energy to get to that final speed is 1600 times larger. So hopefully it isn’t too surprising that it takes a whole lot more than a few centimeters to get there.
Hopefully I’ve convinced you that it is way harder to accelerate from 99% the speed of light to anything larger, and yet all around the world governments spend good money to build that extra kilometer for that extra 0.9999%. What’s it worth?
What 0.9999% buys you
At long last we get to talk about the exciting stuff: some of the things you can do with such large accelerators. Don’t get me wrong, there are tons of interesting things you can do with accelerators that “only” get to 99% the speed of light and even lower. But there are certain things you can only do if you go the extra mile (literally). There are two primary applications of accelerators at these giant scales: colliders and light sources, so let’s talk about what that extra bit of speed gets us in each case.
Colliders
Colliders are exactly what they sound like: devices that collide particles into things. Often times those “things” are other particles. I mentioned the LHC above, easily the most famous example of a particle accelerator. Most of the time it collides bunches of protons into each other.
The accelerator where I work, the LCLS, used to be a collider as well. I mentioned before that the LCLS sits in a line with two other kilometer long accelerators, well back in the 70s those three accelerators were just one big accelerator called the Stanford Linear Collider (SLC). The SLC collided bunches of electrons with bunches of positrons (the antimatter equivalent of electrons).
The purpose of a collider is generally to study very fundamental physics: most notably, they are used to discover new particles. The way this works is shown in the cartoon below: when two particles collide with each other at high energy, they can cause an explosion of other particles to be sent in all directions. Sometimes, one of those particles is something the world has never observed before, and in seeing it we learn more about how the universe works.
What determines what sorts of particles you can produce in this way? Well there are lots of rules in particle physics that I don’t want to get into, but there’s one very simple one: you can only make a particle if you had enough energy in the first place to do so. According to Einstein every particle has a “rest energy” corresponding to its mass: E=mc². If you want to make a certain particle, you had better have at least its rest energy in your colliding bunches.
So for example, though 99% the speed of light sounds quite high, the energy it corresponds to is pretty low on the scale of particle rest energies. An electron moving at 99% the speed of light has an energy of 3.5×10⁶ eV (eV or “electron volts” is a funny unit for energy we use in accelerator physics and high energy physics). We can compare that to one of the hottest particles in physics, the Higgs boson, whose rest energy is 125×10⁹ eV — that’s 100,000 times larger!
For that reason you’ll generally hear the energies of colliders quoted in “Teraelectronvolts” or “TeV”, which is 10¹² eV. For reference, that’s an electron moving at 99.9999999999875% the speed of light.
Unfortunately this means that if you want to discover particles that have never been discovered before, you probably need to build colliders at even higher energies than have been built before. This constant seeking of higher and higher energies leads us to talk about the “energy frontier” — the energies we haven’t yet accessed in our machines. You can see this type of consideration in action in what we accelerator physicists call “Livingston plots” — plots of energies achieved in different machines over time, like the one I reproduce here:
But what happens when a collider has discovered everything it can discover without going to a higher energy? You could upgrade it, but that can be very costly and inefficient — alternatively you can find another use for it. Often, you turn it into a light source (as the “Free-Electron Laser” section of the plot above hints).
Light sources
Onto my preferred flavor of giant accelerator: the light source. Light sources do exactly what their name implies: they use particles moving near the speed of light to produce waves of light. These accelerators take advantage of a concept called the Doppler effect to generate short wavelength light like x-rays. But let’s not get ahead of ourselves.
I’ve discussed light sources, how they work and their uses, in some other posts already, but I’ll summarize the basics here. First of all, why are we interested in light sources? We use light with our eyes to observe our everyday world. To see the world on even smaller scales, we need to use light with a shorter wavelength than visible (if you want a refresher on the electromagnetic spectrum, I discuss it in that post about the Doppler effect).
Essentially, accelerator-based light sources are sort of like microscopes that give us the ability to study the world on tiny length scales and extremely short time spans. Light sources that make x-rays enable us to view the world with a resolution of Angstroms (that’s 0.0000000001 meters, the size of the Hydrogen atom), and a specific kind of light source called the x-ray free-electron laser let’s us see those small things happen over attoseconds (that’s 0.000000000000000001 seconds).
These light sources generally work by a process called “synchrotron radiation” that I’ve described in more detail in this post. In a nutshell, by wiggling a beam of electrons up and down you can make them emit light. And if those electrons emit the light while moving near the speed of light they emit extremely short wavelength light. Specifically, the wavelength they would emit if they weren’t moving near the speed of light gets shrunk by 2γ² (that’s the same Lorentz factor we talked about before). For accelerators like the LCLS, which have γ≃10,000, that’s a 200,000,000 times shrinking in the wavelength! That’s enough to get you from a centimeter scale wavelength down to an Angstrom.
Unlike colliders, where new machines are only relevant if they hit huge energy thresholds, x-ray light sources are more than happy to rest around γ=10,000. By contrast, recent collider proposals seek γ closer to 1,000,000 and higher.
That’s why we’re able to repurpose used up colliders for light sources — in transitioning from the SLC I mentioned before to the “LCLS” x-ray free-electron laser, we only used 1/3 of the original accelerator. That left another two kilometers of accelerator to do other things with! Recently we modified one of the other kilometers into yet another x-ray free-electron laser. That’s two light sources for the price of one old collider, pretty good!
X-ray light sources most often come in two flavors these days: circular machines called “synchrotrons” and linear machines called “free-electron lasers”, but they both work by the same basic principle of wiggling electrons up and down and letting the Doppler effect work its magic. You can see the difference between their layouts in the graphics below.
This post has already gotten too long so I’ll save discussions of their particular features and uses for another time. The key takeaway for us is that they are an excellent use of electrons that are quite a bit faster than 99% the speed of light but not as fast as those for colliders.
Conclusions
I will never forget how shocked I was to learn that we accelerate electrons to 99% the speed of light in just a few centimeters. And yet this feat, which truly is as impressive as it sounds, pales in comparison to the daunting task of getting any fraction of a percent faster. We’ve learned that this is due to a fundamental change in the way particles move when they approach the speed of light, causing them to use much more energy to make small changes in speed.
But we’ve also seen that the effort to get from 99% to anything higher is justified. The difference between 99 and 99.99999 is enough to enable colliders to discover particles we could only theorize about prior, and gives electrons in light sources the Doppler boost they need to emit powerful bursts of x-rays to study the world at its smallest and fastest scales.
I hope that some of the things in this post have surprised and/or amazed you as much as they amaze me, and that they make you curious about what else is going on in these wonderful devices.
