Ask Ethan: Does mass or energy increase near the speed of light?

One of the most puzzling features of nature is this: as you approach the speed of light, everything you commonly understand about motion changes. If you’re on a train moving forward at 30 m/s (about 67 mph) and you throw a baseball forward at 30 m/s from it, to someone on the ground, they’ll see the baseball move forward at 60 m/s: much faster than any human could throw it from the ground. But if that train were moving at 60% the speed of light and the baseball were thrown at 60% the speed of light, that same observer on the ground wouldn’t see that baseball moving at 120% the speed of light, but rather only at 88% the speed of light. The familiar rules of how velocities add-or-subtract are different, and more complicated, at speeds close to the speed of light.

Other, familiar rules change as well: distances appear contracted, times appear dilated, and the energy of a fast-moving particle is greater than a slow-moving particle or one at rest, too. But, particularly when it comes to energy, how can we make sense of that? Does the particle’s mass change as well? That’s what Jerry Kaufmann wants to know, writing in to ask:

“Einstein’s equation, E = mc², says that a smaller value of c means that E will also be smaller. Yet we are told that as matter, that has a mass m (a fixed amount of protons, neutrons, and electrons) approaches the speed of light that mass increases and not E?”

If you learned physics a long time ago — or were taught it by someone who learned it a long time ago — you’ve likely encountered the concept of relativistic mass. Here’s what’s actually going on, and how to make sense of it all.

As shown in an episode of Mythbusters, a projectile fired backward from a forward-moving vehicle at the exact same speed will appear to fall directly down at rest; the velocity of the truck and the exit velocity from the ‘cannon’ exactly cancel each other out in this take. If you calculate, from the moment the ball is released, when it should hit the ground, you actually get two answers: a positive time and a negative time solution. Only physics can tell you which one (the positive solution) corresponds to reality.

The first thing you have to understand is the principle of relativity itself. What it states is very simple, but its implications are profound: the laws of physics are the same for all observers in constant (i.e., non-accelerated) motion. But which laws of physics are the ones that obey relativity: that are unchanged when you’re in motion?

There are a few, including:

• the law of gravity,
• the force laws of electromagnetism and the nuclear forces,
• the values of the fundamental constants, including Planck’s constant (h), the universal gravitational constant (G), and the speed of light in a vacuum (c),
• and the value of the rest mass energy (the E in E = mc²) of any particle.

However, there are a few omissions that came as a surprise, at least initially, to a great many people, including a great many physicists. A lot of people expected that time and space would be absolute, and that distances or the amount of time that elapsed would be the same for all observers; they are not. That’s where phenomena like length contraction and time dilation come into play: because if the laws of physics and the speed of light need to be the same for all observers, then the rate at which clocks tick or the distance between two points can’t simultaneously also be the same.

Different frames of reference, including different positions and motions, would see different laws of physics (and would disagree on reality) if a theory is not relativistically invariant. The fact that we have a symmetry under ‘boosts,’ or velocity transformations, tells us we have a conserved quantity: linear momentum. The fact that a theory is invariant under any sort of coordinate or velocity transformation is known as Lorentz invariance, and any Lorentz invariant symmetry conserves CPT symmetry. This notion of invariance under constant motion dates all the way back to the time of Galileo.

This means that we can’t use “distance” or “time” as a thing that all observers agree upon, and so it should come as no surprise that velocity — or the change in distance over the change in time — is something that differs between observers as well. However, the naive way you’d expect velocity to change between observers, in accordance with the laws of Galilean relativity (e.g., how the slow baseball appears when thrown from a slow-moving train), isn’t universal; it only applies at speeds that are negligibly small compared to the speed of light.

When you approach the speed of light, you have to abandon Galilean relativity and replace it with Einstein’s special relativity. Instead of time and space being agreed-upon and equivalent for all observers, as it is in Galilean relativity, distances and times are both observer-dependent in two particular ways.

• Distances contract, or shrink, by a factor of 1/γ, along the direction-of-motion for any observer (or observed thing) in motion.
• And times dilate, or lengthen, by a factor of γ, for any object-in-motion relative to their otherwise stationary surroundings.

Here, the factor of γ appears very frequently in relativity: as the Lorentz factor, which is the quantity (1-v/c)^(-½), where v is the relative speed of one object relative to another and c is the speed of light.

A “light clock” will appear to run differently for observers moving at different relative speeds, but this is due to the constancy of the speed of light. Einstein’s law of special relativity governs how these time and distance transformations take place between different observers. However, each individual observer will see time pass at the same rate as long as they remain in their own reference frame: one second-per-second, even though when they bring their clocks together after the experiment, they’ll find that they no longer agree.

You might then think to ask how other quantities change as an object approaches the speed of light? Sure, some quantities will remain constant: the speed of light itself, the value of rest-mass energy, other fundamental constants, the laws of physics themselves, etc.

But if our conception of distances, times, and velocities change due to their relativistic motion, then surely other physical quantities — particularly quantities that depend on things like distance, time, or velocity — will differ from the naive expectations of our familiar experience. Sure, Galilean relativity and motion under Newtonian mechanics might be good enough for everyday applications, but at speeds nearing the speed of light, they cannot be true any longer.

One such quantity is kinetic energy. You might be familiar with this as the “energy of motion,” which is true: an object in motion has more energy than an object at rest. This requires just a little bit of explanation in the context of one equation we brought up already: E = mc². That energy, E, doesn’t stand for the total energy of an object; only its rest-mass energy, or the energy inherent to it, that cannot be removed from it, so long as the object itself (with mass m) continues to exist. Kinetic energy, in Galilean or Newtonian terms, is just equal to one-half the object’s mass multiplied by its speed squared: KE = ½mv².

One revolutionary aspect of relativistic motion, put forth by Einstein but previously built up by Lorentz, FitzGerald, and others, is that rapidly moving objects appear to contract in space and dilate in time. The faster you move relative to someone at rest, the greater your lengths appear to be contracted, while the more time appears to dilate for the outside world. This picture, of relativistic mechanics, replaced the old Newtonian view of classical mechanics, but also carries tremendous implications for theories that aren’t relativistically invariant, like Newtonian gravity.

But this clearly cannot continue to be true as we approach the speed of light, for several reasons. For one, it implies that there’s a maximum energy-of-motion that a massive object can have, as speed (v) can never exceed the speed of light (c), and that maximum energy would be only half of the object’s rest mass energy. For another, it would make the creation of, say, proton-antiproton pairs impossible from the collision of two protons, and yet, that’s very clearly not how particle physics works, as high-energy proton-proton collisions are the primary means for the production of antimatter: yielding three protons and one antiproton, requiring much more energy (again, via E = mc²) to create those new particles.

We wouldn’t be able to understand cosmic rays if the energy of motion always obeyed KE = ½mv². We wouldn’t be able to explain particle accelerators or colliders. We wouldn’t be able to create top quarks and Higgs bosons at places like the Large Hadron Collider. And yet, clearly we can do all of these things:

• we can create particles with kinetic energies that far exceed their rest mass energies,
• we can create matter-antimatter pairs by colliding low-mass particles together,
• we can observe and even create ultra-energetic particles, and observe their decays and interactions with matter,
• and we can create every Standard Model particle and antiparticle predicted to exist in particle physics experiments.

None of that would be possible if the energy of motion were compelled to always be KE = ½mv².

The production of matter/antimatter pairs (left) from two photons is a completely reversible reaction (right), with matter/antimatter annihilating back to two photons. This creation-and-annihilation process, which obeys E = mc², is the only known way to create and destroy matter or antimatter. If high-energy gamma-rays collide with other particles, there is a chance to produce electron-positron pairs, diminishing the gamma-ray flux observed at great distances.

So what is kinetic energy, more generally?

The problem is that while E = mc² is always the rest mass energy of a particle, it’s only half of the full equation describing the total energy of a particle. Instead of that simple equation, which always applies to a massive particle at rest, we can write down a slightly more complex equation that applies to all massive particles, including ones at rest and ones in motion, even at relativistic (close to the speed of light) speeds. That equation is as follows:

E = √(m²c⁴ + p²c²),

where E is total energy, m is the particle’s rest mass, c is the speed of light, and p is the particle’s momentum.

We haven’t talked about momentum yet, so let’s use this opportunity to do so. If you were Galileo or Newton, you would define momentum as p = mv, or the object’s rest mass multiplied by the object’s velocity. But we just went through the thought process of recognizing that:

• distances,
• times,
• and velocities,

are not universally agreed-upon quantities for different observers, as they depend on the motion of the observer in question and their surroundings relative to them. Therefore, it shouldn’t come as a surprise that instead of momentum being p = mv, as it can be well-approximated at low speeds, we have to treat it as p = mγv, where γ is that same Lorentz factor that we saw earlier appearing in equations governing time dilation and length contraction.

A simulated relativistic journey toward the constellation of Orion at various speeds. As you move closer to the speed of light, not only does space appear distorted, but your distance to the stars appears contracted, and less time passes for you as you travel. StarStrider, a relativistic 3D planetarium program by FMJ-Software, was used to produce the Orion illustrations. You don’t have to break the speed of light to travel 1,000+ light-years in less than 1,000 years, but that’s only from your point of view.

So now, if we like, we can go back to the original question: what is it that increases when a particle moves close to the speed of light? Is it mass or is it energy?

Back in the early days of relativity — and I mean really early, like the first quarter of the 20th century — many physicists noticed how the classical version of momentum p = mv, needed to be replaced with a relativistic version of momentum, p = mγv, in order to have our equations work out properly. What they then did was define a new quantity that they called relativistic mass, M, and defined it so that M = mγ.

That doesn’t sound so hare-brained, does it? If p = mv worked great in low-speed situations, then wouldn’t it be nice to just replace that non-relativistic mass, m, with the relativistic mass, M, wherever it was necessary? And couldn’t that just fix things up really nicely?

Although Fermilab’s Don Lincoln has written about this before, and so have I, there’s a huge problem with that idea: whenever we go to measure an object’s mass, we always measure m, the non-relativistic rest mass. There isn’t any physical justification for putting the “m” and the “γ” together, as there’s no experiment that measures relativistic mass. The things that change are distances, times, velocities, and energies; mass, or rest mass, remains invariant.

The particle tracks emanating from a high energy collision at the LHC in 2012 show the creation of many new particles. By building a sophisticated detector around the collision point of relativistic particles, the properties of what occurred and was created at the collision point can be reconstructed, but what’s created is limited by the available energy from the kinetic energy of the colliding particles, with new particles capable of being created from that available energy, limited by Einstein’s E = mc². The maximum LHC energies are nearly a factor of a trillion (10^12) lower than the energies present at the start of the hot Big Bang.

So what, then, happens to energy?

We can see for ourselves, if we’re willing to apply the relativistic definition of momentum to the full equation for the energy of a massive object, and perform just a little bit of math. Remember, that the full equation for energy is:

E = √(m²c⁴ + p²c²).

Now, we’re going to use the relativistic formula for momentum, p = mγv, and put it into our formula for energy, getting:

E = √(m²c⁴ + m²γ²v²c²).

And now, we’re going to take one more step, and pull out a factor of mc² (from the original E = mc²) from both terms, out in front of the square root, giving us:

E = mc² * √(1 + (vγ/c)²).

Time dilation (left) and length contraction (right) show how time appears to run slower and distances appear to get smaller the closer you move to the speed of light. As you approach the speed of light, clocks dilate toward time not passing at all, while distances contract down to infinitesimal amounts.

Lo and behold, there’s no “relativistic mass” that shows up here anymore. We see that the total energy is only dependent on:

• the rest mass of the object, m, which never changes,
• the speed of light, c, which never changes,
• and the velocity of the object, v, which can change but is limited to be never greater than c,
• and γ, the Lorentz factor, which depends solely on v and can be very large if v approaches the speed of light, c.

That’s the key to the whole puzzle! When an object moves close to the speed of light, it experiences time and distance differently than objects do at lower speeds, but it also has a different energy-of-motion to it. All three of those quantities — time, distance, and kinetic energy — are dependent on a relativistic object’s properties, but it’s only one physical property, the object’s relative velocity, that determines all three of time, distance, and kinetic energy. The mass of the object is always the same, m (not M), and is always the m in the rest mass energy equation that Einstein gave us back in 1905: E = mc².

The results of the 1919 Eddington eclipse expedition showed, conclusively, that the general theory of relativity described the bending of starlight around massive objects, overthrowing the Newtonian picture. This was the first observational confirmation of Einstein’s theory of gravity. Eddington was a towering figure in early 20th century astronomy, and while he did great work, his ideas about “relativistic mass” have only served to confuse generations of physicists and laypersons alike.

So where did this notion of relativistic mass come from, and why did it stick around for so long?

Most science historians point the blame squarely at one man: Sir Arthur Eddington, one of the most famous astronomers of the early 20th century. Sure, Eddington did some great work:

• setting limits to how fast black holes can accrete matter,
• determining the maximum luminosity (intrinsic brightness) of stars and other objects,
• measuring the light deflection of starlight during a total solar eclipse, validating Einstein’s general relativity,
• as well as serving as a great explicator of Einstein’s work and the concepts underlying them.

Unfortunately, Eddington was also an incredibly dogmatic individual, and his vociferous adherence to the notion of relativistic mass ensured that it was a concept that was still taught all throughout the 20th century. (Including to me, when I was learning about relativity as an undergraduate in the late 1990s). It’s often said that “physics advances one funeral at a time,” and while many good scientific ideas that would later flourish were held back during Eddington’s lifetime, a number of bad ideas, like relativistic mass, have persisted despite his death. The point is that you don’t need it, and it isn’t mass that changes anyway; the rest mass of an object is independent of its motion, and it’s only kinetic energy, or the energy of motion, that’s dependent on an object’s speed. No further alterations are necessary, and I don’t recommend them unless you’re prepared to deal with an enormous amount of additional confusion!

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