Of Particular Significance

A Relatively Important Question from a Reader

POSTED BY Matt Strassler

POSTED BY Matt Strassler

ON 03/06/2024

Recently, a reader raised a couple of central questions about speed and relativity. Since the answers are crucial to an understanding of Einstein’s relativity in particular and of the cosmos in general, I thought I’d bring them to your attention, in case you’ve had similar questions.

The Questions

I understand that the vacuum speed of light [“c“] is constant throughout the Universe, and I’m familiar with the math that shows that the energy required to accelerate a particle becomes infinite as the speed approaches c. But what physical effect enforces this behavior? If a proton, for example, gets ejected in a supernova explosion, how does it “know” that it’s getting close to c and can’t go any faster?

And as a corollary to this question, what is the reference frame for measuring these relativistic velocities? For example, when a particle beam at CERN is said to be moving at 99.99% the speed of light, is that speed relative to the infrastructure at CERN? Or does it somehow account for the velocity components that arise from the rotation of the Earth, the orbital motion of the Earth around the Sun, the galactic motion of the Sun in the Milky Way, and so on?…

The Answers

The Oblivious Proton

How does a proton know that it is getting close to c and can’t go any faster? It doesn’t.

For the same reason, you, too, right here and now, have no sense of how fast you are moving.

  • If a proton is moving at a constant speed, and not accelerating, then as far as it can tell, it is stationary. It “feels” just as stationary, flying across our galaxy at nearly c, as you and I do, carried around the spinning galaxy at about 0.1% of c (150 miles per second).
  • If the proton is accelerating, it will indeed “feel” the acceleration, in that its internal structure will potentially respond to its changing speed. But even so, it has no idea how fast it is going, so it can’t possibly worry its little head about exceeding c. . .any more than you and I do.

Absolute Speed

Implicitly, in asking these questions, you are holding in your mind a notion of “absolute speed.” You are imagining that the proton has a real, honest, unambiguous true speed, and it is this absolute speed that can’t exceed c.

But even Galileo, nearly 400 years ago, would have suggested to you that your notion of “absolute speed” is suspect. After all, we are in extremely rapid motion relative to the Earth’s center, the Sun, the galaxy, and yet we feel nothing… and indeed, we only know about our various motions by a series of complex arguments and measurements in astronomy. There’s no way to go into a closed room and measure that we are moving around the galaxy at 150 miles per second. Absolute speed cannot be measured by any experiment. This is for a simple reason: absolute speed does not exist in our universe. The notion is meaningless.

The meaninglessness of absolute speed is nothing less than the principle of relativity, as stated by Galileo in 1632, and as preserved by Einstein in his work of the early 1900s.

The problem with the question, “what is the reference frame for measuring these relativistic velocities”, is that it presumes that there is a preferred reference frame relative to which absolute velocities have meaning. But both Galileo and Einstein tell you to banish the thought.

Relative speed

ALL velocities in our universe are relative. There are no absolute velocities. Einstein’s statement about “objects moving at speeds slower than or equal to c” is not about absolute velocities at all — because there are no such things. So what, in fact, does he mean?

Einstein’s statement is the following: when two objects pass one another, then the speed of the first object as measured by an experiment moving along with the second object — a relative speed!!! — must be less than c. There’s one sole exception: if the first object has zero rest mass and is moving through empty space, then the relative speed (of the first object as measured by the experiment moving with the second object) is exactly c.

You see how crucial it is to get the details exactly right. Otherwise paradoxes, such as those implicit in the reader’s original questions, loom everywhere. Indeed, if we try to make sense of Einstein’s statements using our familiar notions of space and time, we will immediately find contradictions that we cannot resolve. Einstein’s great leap of genius was to realize that once we understand that space and time can vary, in precisely the way that they do in his updated notion of Galileo’s relativity, all the apparent paradoxes are eliminated.

The Particle Beams at CERN

when a particle beam at CERN is said to be moving at 99.99% the speed of light, is that speed relative to the infrastructure at CERN?

Yes, typically that’s what a person making that statement implicitly means; the speed of the beam is typically measured relative to the CERN laboratory. But that’s a choice, made out of convenience. By contrast,

  • A person flying by CERN at a relative speed of 99.99% of c, moving in the same direction as the particle beam, would see the particles as stationary.
  • Meanwhile a person moving in the opposite direction of the beam at 99.99% of c relative to CERN would see the particles moving at 99.9999995% of c.

Who is right? Everyone. Speed is relative; and when something is relative, everyone disagrees, yet no one is wrong.

In summary, a proton does not need to know, and cannot know, its absolute speed. Instead, what Einstein’s view of relativity says is this: no matter how observers are moving around, they will always find that a passing proton’s speed, as measured relative to themselves, is less than c.

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26 Responses

  1. Matt Strassler wrote (03/06/2024):
    > […] Einstein’s statement is the following: when two objects pass one another, then the speed of the first object as measured by an experiment moving along with the second object — a relative speed!!! — must be less than c. There’s one sole exception: if the first object has zero rest mass and is moving through empty space, then the relative speed (of the first object as measured by the experiment moving with the second object) is exactly c.

    Could you please identify the original source of this statement in documents produced by Albert Einstein;
    preferably, if applicable, from the documents publicly available through https://einsteinpapers.press.princeton.edu/ ?

    Such a statement might be accompanied by a more detailed prescription of “an experiment moving along with” one object, by which to “measure the relative speed” of (or: wrt.) the other; which can be of wider interest.

    1. Are you serious? Absolutely not. Do you only believe statements from Newton if the original source can be found in Principia Mathematica?

      I do not teach physics by going back to the original sources, nor should I. In science, original sources are not the basis of knowledge; this is not history. Coherence, logic, and experiment are the basis of scientific knowledge. Many of those experiments and logical arguments were not available when Einstein wrote his first papers. Not all of Einstein’s statements in his original papers were correct, either.

      I teach science, as I should, by laying out the concepts and the equations, most of which were not, in fact, in Einstein’s original papers. They have been assembled over decades by Einstein and other scientists into a coherent web of understanding, which was not complete, even in Einstein’s head, when he wrote his original inspired work. At that time, he was making a guess. Today we have experiments that prove the statement.

      Simply put, the entire design of the Large Hadron Collider is built upon the assumption that my statement above is correct, because from that statement one derives all the equations for particle motion near the speed of light. The LHC would simply not work if this statement were false. That’s proof.

      Sure, you can go back to Einstein’s original papers and find that what he said is consistent with what I wrote, but the fundamental reason that it is true is Lorentz invariance, the mathematical basis of the whole theoretical structure. I don’t know when Einstein fully realized this, and I don’t care; I only care that it is true, which is proven by experiment. And I am certainly not going to spend a week or two reading through his papers, figuring out exactly what he thought and when, working out his mathematical notation and concepts compared to modern notation and concepts, checking also papers by people like Minkowski who may actually have understood it first, and finally writing an article proving this to you. This would be grounds for a publishable paper in history of physics, and is certainly not what physicists spend their time doing.

    1. It’s not a matter of “think”. 🙂

      In special relativity — i.e., in a world without gravity, acceleration is absolute and unambiguous.

      But in general relativity — i.e., in a world with gravity — the situation is far more subtle. Acceleration is relative **in the sense** that it cannot be distinguished, locally, from gravitation.

      In principle, one might always be able to exchange one for the other. But it is not always sensible, advisable, or entirely possible to do so. For instance, if ones feet are attached to a spinning turntable, there is an apparent outward force that seems to push one’s body outward, whose strength seeems to grow as one moves further out on the turntable. Newton would say this is a misinterpretation, caused by your acceleration, which in turn is due to the turntable pulling your feet inward. Locally, however, one can use a rotating frame of reference, and view this outward force as a form of gravity, as Einstein made clear.

      Globally, though — i.e. across all of spacetime — there are subtleties in defining such a frame. Moreover, one can check that there isn’t any meaningful gravity around the turntable in the sense that all measures of spacetime curvature are zero in this case.

      So the answer to your question is that acceleration isn’t absolute in the presence of gravity, but the story isn’t nearly as simple as your question would imply.

        1. Both you and I are being too glib here, by referring to “acceleration” as though it is unambiguous. One can define acceleration in multiple ways, and most of them are relative. (The same is true of mass.)

          However, just as with mass — there exists an “absolute” (or “intrinsic”) notion of “mass”, called “rest mass” or “invariant mass” — there exists an absolute notion of acceleration, whose magnitude is the same in all inertial reference frames. I believe this is usually called “proper acceleration”.

          In contrast to velocity, all forms of which can be set to zero by a suitable Lorentz transformation, the magnitude of this intrinsic acceleration cannot be changed, much less set to zero, by any constant spatial translation, time translation, rotation, or Lorentz transformation, or by any combination thereof. This shows that all inertial observers will measure that magnitude to be the same.

          Certainty comes, in the end, from studying the math and proving certain things about it.

    2. Hi Farhad and Matt Strassler,

      First, Matt Strassler, thank you for your excellent response to an interesting question!

      Both: Here are a few related observations that can help navigate such thought problems, ones with which I think (hope?) Matt Strassler concurs

      The first point is to agree fully with you and Einstein: No observer at any velocity, using tests within a closed environment (no cheating by looking out a window), can ever hope to detect that “they” are the ones in motion. The spacetime they see in their inertial frame is indistinguishable, all the way down the particle level, from any other inertial frame’s definition of spacetime.

      However, as you also noted, the energy transfer histories of such objects — their acceleration histories — are unique and cannot be altered by switching perspectives. This is just firecracker logic: Once you make a bang, that bang and its size both become history that looks the same to everyone in the future. Since and acceleration is nothing more than a slow bang, it, too becomes historical.

      Since these accelerations are historically unique energy transfers, one admittedly uncommon way to describe them is as energy “excitations,” that is, as “excited states” of the original pre-acceleration objects. While not common, this terminology is accurate and has certain advantages.

      Just as one can “excite” a gyroscope into a higher energy state by adding _angular_ momentum, one could, alternatively, excite a large object or proton into a higher energy state by adding _linear_ momentum. The fact that the physics inside the bullet is identical before and after this excitation process is a separate issue.

      I’m emphasizing the excitation issue because that is where the proton and accelerator differ dramatically. The total energy cost of accelerating one proton to near lightspeed relative to an accelerator is modest and easily doable. However, the inverse operation of pushing the _accelerator_ to near lightspeed relative to a stationary proton would require so much energy that you would consume or destroy the entire earth doing it.

      Following excitation energy distinction also provides a simple and surprisingly intuitive way to resolve the Twins’ Paradoxes: For any set of objects that begin in the same inertial motion state, it’s always the one that receives the most linear momentum energy whose clocks _immediately_ begin running slower than those of the system that did not undergo acceleration. The degree of slowing depends on the velocity imparted.

      It was Einstein who first predicted the slowing to be continuous in the argument by which he first predicted the Twins’ Paradox in 1911, that slowing effect is continual over the journey of the accelerated entity. How the acceleration is applied — quickly or slowly — only alters the profile of how long the object or proton spends at each level of slowdown. A rapid acceleration thus rapidly slows object or particle time to a well-defined level, while slow acceleration produces a more complicated mix of time requiring integration.

      One can see this continuous slowing of time in excited matter in particle accelerators, where, for example, protons experience slow time continuously slow in time as they circulate. It’s not just the centrifugal acceleration, either. When the same beams exit a collide on a straight path, different detectors see a total time dilation proportional to the distance the particles traveled.

  2. Dear professor Matt Strassler,
    James Webb new discoveries items about galaxy GN-z11. First discovered by Hubble with z = 11.1 red shift. Galaxy GN-z11 existed already when the universe was only about 430 million years old.
    Is it wrong to say, a photon of GN-z11 travelled about 13.4 billion years to reach the earth because of the absolute light velocity?
    If it is wrong, how must I define the discovery of galaxy GN-z11 in the early universe?

    with kind regards, Jo Pieters

    1. Time and space can be tricky in an expanding universe. Distances are tricky even on an expanding rubber sheet, so one should hardly be surprised that it might be tough to keep track of distances as space expands. The fact that time is tricky too is less obvious, but thank you, Einstein.

      The question you asked — but maybe it was a typo — was about *time*. “Has a photon traveled 13.4 billion years?” Yes (with a footnote.) Because our universe is approximately uniform (not exactly, of course; but the density of galaxies seen around the universe seems to be consistent with it being approximately so), it makes sense to synchronize clocks across the universe. For instance, we could each set our clocks at time=NOW to be when we measure the ever-decreasing temperature of the universe around us to be 2.72500 Kelvin. Or we could set our clocks to be time=1 second when the temperature around us is ten billion Kelvin. These different choices will give slightly different answers for time measurements, but in an inhomogeneous universe they would give wildly different answers, so we should not complain.

      In saying “the universe was 430 million years old when GN-z11 existed”, we are implicitly using synchronized clocks… and so, yes, a time of 430 million years at the location of GN-z11 is the same as a time of roughly 430 million years at the location of the Milky Way’s center. Now we just have to follow the Milky Way’s clock, and yes, about 13.4 billion years has elapsed since that time. So we may indeed say, because our universe is largely uniform, that the photon has traveled for roughly 13.4 billion years, meaningful because we have established a method for synchronizing clocks.

      In a highly inhomogeneous universe, we could not make sense of this. Nor could we use a clock that traveled with the photon itself; for objects that travel at the speed of light, there is no experience of time. A clock traveling with the photon will never tick.

      A question you didn’t ask, but you could have, is “did the photon travel 13.4 light-years to reach the Earth?”; i.e., how *far* did it travel. The answer is no, and the reason is already clear with an ordinary rubber sheet that is being stretched as an object tries to cross it. Spatial distances are changing with time. There is no sense in which GN-z11 was 13.4 billion light-years away when the photon started its journey, nor need it be 13.4 billion light years away now. Instead we must define our rulers with even more care than we did in synchronizing clocks, and decide how to define the “distance traveled”; is it the distance that was true 13.4 billion years ago, the distance now, or some notion of distance that averages in some way over the path that the photon took?

      1. The translation from red shift to time is not straightforward. It depends on the chosen cosmological model. So we can sat that the photon travelled 13.4 billions years supposing the lambda-CDM cosmological model is correct.

        That’s the reason why astronomers use red shift and not time/distance.

        1. Agreed, and thanks for the caveat. The commenter implicitly assumed that we knew the times, and so I ran with that. But you are correct that strictly what we know is redshift, and to go from that to time involves assumptions about the universe’s history, which is imperfectly known.

  3. When I went through Special and General Relativity (many years ago), my professor approached the idea as that each bit of matter inhabits its own frame in spacetime, self-defined by its own perspective of distances and relative velocities, all of which necessarily maintain a causal order with every other frame. Exchanges of energy simply connected frames by altering each by some equivalent quanta, say an acceleration in some direction that would result in a perspective of a spacetime distance being shortened. It wasn’t necessarily easy to calculate (especially relativistic rotation!), but it made sense to me. But it also seemed to imply something.

    Since anything “traveling” at c, such as photons, wouldn’t experience any space or time, they should from their own perspective merely connect and mutually alter two frames. So this has always left me with two questions. First, do they actually “exist”? Or is a “photon” merely a way of defining an amount of exchange between two frames? And secondly, can a photon ever *not* connect two frames, like a Feynman diagram going off into…? If not, it would seem to me that any interaction in spacetime, regardless of separation, is determined by a direct, causal connection between both frames. That is, a photon could only be emitted if some other spacetime frame will, with certainty, be available as a receiver (or, anti-emitter). My apologies if this is a somehow naive question, so I’ll leave it here. But I think you can probably see the implication.

    1. I’m afraid that your professor’s original perspective, as you’ve described it, is confusing to me, especially “Exchanges of energy simply connected frames by altering each by some equivalent quanta,” I don’t understand the words, nor have I encountered something that sounds like this. (There are many approaches to general relativity and I am not a world’s expert on it.) This makes it hard to answer your question. Maybe you can clarify this or point me to a discussion of this viewpoint?

      As for photons — they exist operationally, since we observe what we call “photons” in detectors… objects that are emitted through electromagnetism, travel at the cosmic speed limit, are absorbed electromagnetically, and satisfy the Planck Einstein relation E = h f relating their energy and frequency. Given these facts, you can try to define frames however you like. But whatever you do, you don’t want to define away photons, or view them as defining exchanges between frames… they exist in data! It’s important to start there, on an experimental foundation, before you try to put theory on top of it.

  4. I think I got that. But why then treat time as a onedimensional component of spacetime. Is it purely conventional? Or our common sense (?!) of an “arrow of time”? Or is there a true physical reason for that like entropy? Psychologically in any case I don’t experience time as an “arrow” .
    Thanks in any case for your enlightening comments that I read as an amateur, but with pleasure whenever I find time!

    1. These are important questions, though different ones than the one I’m addressing here. Sean Carroll has written much more about this than I have and he probably has more to offer you on issues of time and its physics and psychology.

      1) We treat time as a one-dimensional variable because of observation and experiment; we are able to describe the workings of the universe using a single time variable, whereas we need three of space. You can try doing it with two time variables; it doesn’t work. I can’t tell you why it doesn’t. But physics is about finding ways to understand observations and experiments, and multiple dimensions of time don’t agree with data. The universe has one time variable.

      2) The arrow of time is much more confusing, and may be more psychological than physical — or more likely, it involves an interplay of physics, physiology, evolution, psychology and consciousness. So we have left the realm of pure physics… and that means that I am probably out of my depth. In short, I don’t know the answers to the arrow-of-time questions… but maybe Sean’s book on the subject can at least delineate the problems more thoughtfully and carefully than I could do.

  5. Thanks very much for your helpful explanation of the points raised in these questions! But I’d like to come back to the concept of c as the “cosmic speed limit”, and try to understand more fully what physical interactions enforce this for various types of particles. To begin with, I do understand and accept that the speed of light in vacuum (c) is the same for all observers regardless of their motion (as shown by Michelson-Morley and others) and is governed by the self-induction behavior of the alternating electric and magnetic fields that make up electromagnetic radiation (following Maxwell’s equations). If I recall correctly, Einstein’s development of relativity was inspired by his desire to make the mathematics of physics compatible with the observed constancy of the speed of light.

    But what then prevents the speed of particles other than photons from exceeding, or even reaching, c? For the hypothetical proton mentioned above, blissfully cruising along unaware of its speed, why couldn’t it continue to be accelerated indefinitely, right up to c and beyond? Could it be that particles that feel the electromagnetic force, like protons, electrons or quarks, all experience an extension of the same self-induction mechanism described by Maxwell for photons? Or is some other interaction at work? For that matter, why should neutral particles like neutrinos or gravitons also be constrained by the same “speed limit” that applies to electromagnetic radiation? I realize that LIGO recently detected gravitational waves from (I believe) a neutron star merger more or less simultaneously with the observation of visible light from the same event, but I do not begin to understand why this should be so! Can you help me understand the physical phenomena that cause such a diverse range of particles to experience the same “cosmic speed limit”?

    Bob Schwerzel

    1. So this is indeed the point: what we call “the speed of light” is better called “the cosmic speed limit” precisely because it is built into space and time. All physical objects obey this limit (though things that aren’t physical objects don’t have to.) The reason is has nothing to do with objects and how they work. It has to do with how space and time work. That’s why this limit is unbreakable.

      This is crucially important to the universe and it is what makes it different from an ordinary material. Air has sound waves, and they have a speed, but even though it’s hard for airplanes to “break the sound barrier”, they can do it. The universe has light waves, and they have a speed; yet nothing can break the “light barrier”. What’s the difference? The atmosphere can’t play with space and time. The universe can… and does. That’s what makes a cosmic speed limit possible. Einstein’s conceptual breakthrough was to realize that the only way to save Galileo’s 300-year-old principle of relativity was to assume that this speed limit exists, and to show that the math that realizes this idea is completely consistent with all experiments.

      Again, these are central points in the book; I devote entire chapters to making sure this is conceptually clear. Of course, to really see how it works, one has to go a little deeper into the math, which I can help readers do at a later time.

      1. Thanks again for helping to sort all this out! I can tell it will take me a while to wrap my head around the concept of how the properties of space and time can limit the speed of physical particles, but I’ve recently ordered your book and will look forward to working through your explanations there. (And I enjoyed your pun in the title of this thread, with its nod to relativity!)

        1. You’re welcome (and it takes us all a while.) I should warn you that I discuss how this all works in the book, but only qualitatively. There are plenty of good books already out there on special relativity if you really want to explore this further.

          One perspective that you may find useful: from a physicist’s point of view, Einstein’s statements about special relativity boil down to a requirement. In order that all equations look identical from the point of view of any observer traveling at any steady velocity, all equations for waves of elementary fields must have very special forms. From the point of view of modern physics, all “particles” are waves in fields, and therefore they are described by equations that are subject to that requirement. Finally, from the forms of these equations, it can be derived mathematically that no information can travel faster than c, and that includes motion of particles. So the logic starts from a symmetry principle, and descends through equations for waves to the properties of particles. And that’s how it all works out.

  6. Is the Cosmic Microwave Background’s frame next best thing to the absolute frame? By extracting the dipole component any observer can measure its speed relative to that frame (which by Cosmological Principle is the same for all observers).

    1. The CMB is just another set of objects, and you can always measure your speed relative to objects. Do not confuse this with measuring your speed relative to empty space, which is impossible. To see this, just go inside a metal room; the CMB can’t enter, and no matter what experiment you do, you can’t measure your absolute speed anymore.

      There seems to be a very widespread confusion about this. On universe-wide scales, there is a preferred frame — but there is also a preferred frame on Earth, too. That’s just an accident of what is in the universe and what its overall shape is. Nevertheless, if you make a box, isolate it, and take everything out of it, and now do experiments inside it, you will discover that experiments inside the box are independent of the box’s motion and of the experiments’ motion as long as that speed is steady.

      1. Empty space in particle physics is absolutely mind blowing, like that every particle drags its own patch of the surrounding space while moving. Let’s make a thought experiment. Imagine an infinite universe having physical vacuum similar to ours and with just one stable elementary particle (say electron) within it. In that setting the notions of distance, time, energy are all undefined, so all space, time, energy, speed, movement are therefore relative, emergent phenomena.

        1. Now you are touching on the key point, I agree, but I don’t think you are focused on quite the right question.

          You can’t just have one particle, but you could have just one quantum field. Whatever the quantum state of that field, which could involve zero particles, one particle, a hundred particles, or a trillion trillion, you could find another quantum state that is overall shifted in speed in any direction, and that state would be indistinguishable from the one you started with. That’s the principle of relativity.

          The same would be true of any set of quantum fields.

          All evidence, from local experiments, is that the principle of relativity is true whenever you do small experiments inside (and far from the walls) of a medium-sized box that is itself isolated from the large universe. (The reason you can link the principle of relativity for the whole field to that appropriate for a small experiment is a principle known as cluster-decomposition that underlies quantum field theory.) That is already a truly remarkable statement about the cosmos and how it works; it would never be true, for example, inside a metal, for any experiment larger than nuclear scales.

          When we consider the very subtle effects of gravity, however, the in-principle questions potentially become more tricky. I’m still trying to understand them clearly.

  7. The corollary to “absolute speed does not exist” is absolute locations do not exist. Space is not made up of an infinite collection of individual points, although it can be parameterised in that way. But each parameterisation is observer dependent. I used to explain this while an undergraduate to friends, and argue that you therefore couldn’t properly define borders between, say, Klingon and Federation Space!

    1. 🙂 This is absolutely true! But you could still define interstellar borders operationally over a few thousand years, by drawing a shifting grid whose gridpoints were stars. You’d have to renegotiate the border every now and then as the stars move around. It’s not so different from borders defined by rivers or by distance from a shoreline, which can also change.

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