Of Particular Significance

# “Moving” Faster than the Speed of Light?

#### POSTED BY Matt Strassler

ON 02/20/2024

Nothing goes faster than the speed of light in empty space, also known as the cosmic speed limit c. Right? Well, umm… the devil is in the details.

Here are some of those details:

1. If you hold two flashlights and point them in opposite directions, the speed at which the two beams rush apart, from your perspective, is indeed twice the cosmic speed limit.
2. In an expanding universe, the distance between you and a retreating flash of light can increase faster than the cosmic speed limit.
3. The location where two measuring sticks cross one another can potentially move faster than the cosmic speed limit.

I addressed issue #1 in a blog post last year.

Today I’ve just put up an article on issue #2. (This is a part of my effort to expand on loose ends raised in the footnotes from my upcoming book).

As for issue #3, can you see why it might be true?

If you aren’t sure and want the answer, click here:

The “location where two measuring sticks meet” is not itself an object. Only objects — localized material things with energy and momentum — are constrained to have relative speeds below the cosmic speed limit — and even that statement needs to be made more precisely (see detail #1 above!) A meeting point between objects is not itself an object, and may move faster than c even if the objects move slower than c. The figure below illustrates this.

One stick is shown in black, the other in red. The red one then moves downward; the dotted line shows where it was initially. Although the red stick moves only a short distance during the animation, the meeting point of the two sticks crosses the entire animation from left to right. If the red stick is moving at half the cosmic speed limit, which is perfectly consistent with Einstein’s relativity, the meeting point, which covers much more ground in the same amount of time, is clearly moving faster than the cosmic speed limit. No problem: no thing moved faster than the speed of light.

Note added: in materials, the speed of light is generally slower than the cosmic speed limit. For this reason, objects inside materials can move faster than the speed of light but slower than the cosmic speed limit. The result is a very interesting and useful effect: Cerenkov radiation, widely used in particle physics experiments such as Ice Cube!

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### 44 Responses

1. Pavel says:

Another example of superluminal velocity (and I think one of the most confusing) is phase velocity of el-mag waves. It is typically superluminal and accelerators constructors have to build complex-shaped accelerating cavities to make it slower then speed of light.

1. Interesting, I don’t remember hearing about that [meaning the need to design special cavities for that purpose.] Do you have a link where I and others might read up on this?

1. The phase velocity of electromagnetic waves in neutral plasma is superluminal (neutral plasmas are dispersive, assuming the frequency is above the Langmuir plasma frequency and propagation is possible at all). Nothing physical travels at that speed, and it cannot be be used for superluminal communication. Group delays in a plasma are not superluminal, and in radio data travels at the group delay speed*. Waveguides can also have superluminal phase velocities, and that has to be taken into account.

This review article goes into this and a lot more. https://arxiv.org/abs/1111.2402

* There is an old spacecraft navigation hack called DRVID (Differenced Range versus Integrated Doppler) where the change in the group delay and phase delay over some period are summed, removing the plasma delay to a very good approximation (the delay is proportional to 1/f^2, and the next order is 1/f^4, much smaller at GHz frequencies).

1. I should have been clearer in my comment; I did not know about the special acceleration cavities or why it is essential to use them.

1. Aaron Sher says:

Centuries of experimental evidence to the contrary…

2. mls says:

A little off -topic…

Was reading your Higgs FAQ today and ran into this…

“it is possible for there to be fields that are at rest with respect to all observers!”

It caught my attention because I recently picked up a copy of Newton’s “Principia.” I did not interpret his explanation of “absolute space” as in the common literature. I interpreted it as requiring that “rest” and “motion” have to be in binary opposition to be applicable to any physical theory whatsoever (rotating buckets of water notwithstanding!).

Your quoted statement seems very close to this.

Thank you for all of the work you do to deconstruct oft-repeated misconceptions.

1. The issue of what it means to be at rest with respect to all observers is a key issue… after all, space itself is at rest with respect to all observers. When you consider that space can warp, stretch and ripple, that’s quite counter-intuitive. (This is a central topic in my upcoming book, by the way.)

1. mls says:

I am looking forward to reading it.!

3. ADM says:

Is the “spin” of an electron up or down because we don’t have the resolution of observing or conceptualizing deeper into the composition of an “electron?

Same comment for all the fundamental particles, i.e. they are all composites of bosons.

4. Vincent Kinequon says:

Have you ever imagined Einstein’s special theory of relativity, and placed that upon the history of our universe. I was somewhat aww struck by this. I wonder if that’s why the inflationary model was proposed. But I believe I have difficulty with understanding relativity. At the beginning of the universe, surely time went by at one second per second.

5. Vincent Kinequon says:

I was just thinking about the speed of light today. Could it be that the speed of light has nothing to do with light, but everything to do with spacetime.

1. It does, in fact have everything to do with spacetime, and spacetime’s cosmic speed limit. The speed of light in empty space equals the cosmic speed limit for a very simple reason: light is made of particles with zero rest mass, and all objects with zero rest mass must travel at the limit.

6. Paul says:

Thought experiment:
You’re on a merry-go-round traveling at the speed of light and you stick your hand out. Your hand is now traveling faster than the speed of light because it’s traveling in a larger circumference than you are traveling at the speed of light.
One more (similar concept):
You’re holding a long, a very long stick, and the far end of the stick is touching Alpha Centauri. You swing the stick ever so slightly and now the end of it is touching Sirius. Although you have moved your hand a few inches, the other end of the stick moved 9.5 light years in a couple of seconds. Just for fun… Instead of a stick, you’re shining a laser light from one star to the other. Now your laser light is traveling faster than light!

1. Good homework problems for undergraduates! (i.e. — “what’s wrong with this question?”)

Einstein’s relativity (like quantum physics) is quite subtle, full of little apparent paradoxes like these. It’s easy to formulate what sounds like a reasonable question but in fact is not.

I won’t deal with the first question, because it would take too long; the problem assumes something is possible that isn’t, and draws incorrect conclusions from it.

The second question, part 1: no, you can’t swing the stick like that. The stick is not a rigid object that will instantly know that you are swinging it. Even the information that you are trying to move it will take many years to reach the end of the stick, and your efforts to move its far end at extreme speeds will cause acceleration that becomes greater and greater as the information moves down the stick, inevitably causing it to break somewhere.

The second, part 2: no, the laser light is not traveling faster than light. Each individual light wave is traveling at the speed of light, just as always. Only the location where the light passes one star or the other is moving, and that, since it is not an *object*, is perfectly free to travel faster than light. In fact this is exactly the same idea as point #3 in this post.

1. Paul says:

Thank you for an explanation!

2. Given that, depending on the type of wood, the longitudinal wave speed in wood is about 3 – 5 km / second (i.e., 10^-5 c or so), it would take several hundred thousand years for ” the information that you are trying to move it ” to get to Alpha Centauri, and unless your induced swinging of it was very, very slow then yes, it would break long before the transverse velocity got to be anywhere near relativitistic.

7. Dave B says:

The difference between two light beams moving in opposite directions isn’t the same as something moving faster than c.

8. Saharsh says:

Neutrinos are faster than Speed of light…. And btw search for Light Body Vallalar . He is also known as Swami Ramalingam and he was a devotee of Lord Shiva . And he turned his body to light through prayer and meditation many times during his life but it lasted for a short time period. Finally on January 30, 1874,he entered the(his meditation) room and locked himself and told his followers not to open it. He said that even if they did open it they would find nothing. His seclusion spurred many rumors, and the Government finally forced the doors open in May. The room was empty, with no clues. The Madras District Gazetteer published by the South Arcot District in 1906 records his disappearance.

9. Jason Stanidge says:

How dependent is life emerging in our universe dependent upon the value of c being what it is?

What happens to the idea of there being a cosmic speed limit c in quantum gravity?

10. Wouter says:

hi Matt,
at what level of accuracy would any deviation from Newtonian mechanics be observable for an observer in low earth orbit, say in the ISS?
I ask this to get a conversion of “noticable, even important at Cosmic Distances” to “extremely small, unmeasurable with current techniques at Laboratory Scale”. Isn’t there a log-log plot hiding in there?

1. Gotta run right now, but here’s a suggestion: try dimensional analysis! You need something with zero dimensions that involves comparing the cosmic speed limit to the speed of an object in a known orbit of the Earth (whose mass is known). There’s only one possible combination.

Then you need one additional small argument to convince yourself that the answer should involve the square of the speed, not the speed alone.

11. rain says:

Search “the physical reason time slows at the speed of light” on YouTube. 2 minutes will make it all make sense!

12. Max Wood says:

The speed of light is incredibly high. Because the speed of light is squared in Einstein’s equation, tiny amounts of mass contain huge amounts of energy. Another result of the theory of special relativity is that as an object moves faster, its observed mass increases. This increase is negligible at everyday speeds. But as an object approaches the speed of light, its observed mass becomes infinitely large. As a result, an infinite amount of energy is required to make an object move at the speed of light. For this reason, it is impossible for any matter to travel faster than light speed.

https://www.energy.gov/science/doe-explainsrelativity

Office of
Science
U.S. Department of Energy
1000 Independence Ave., SW
Washington, DC 20585
(202) 586-5430

1. Unfortunately, this quote from the DOE itself is extremely unwise. This paragraph refers to relativistic mass, and at no point is it mentioned that this is different from rest mass. Rest mass (a) is what we mean by “mass” when we talk about “the mass of the electron” or “the mass of the proton”, (b) does not increase with speed, and (c) is what the Higgs field provides to the elementary particles that we’ve discovered so far.

Specifically, the DOE’s statement “Another result of the theory of special relativity is that as an object moves faster, its observed mass increases.” is true only of relativistic mass; for rest mass, the type of mass obtained from the Higgs field, it is false.

The difference between relativistic mass and rest mass must be made clear to non-experts if massive confusion (pardon the pun) is to be avoided.

(End rant. These issues form a sub-theme in my book, which is all about what rest mass is and where it comes from.)

1. Scott Stillwell says:

Mass is a scalar that always has the same underlying physics, even if v=0, at T~0K and m(0)=rest mass. m(v)=m(0)*Sqrt(1-v^2/c^2) or m(0)*f(0,c). “m” or m(v) is always due to Higgs. m(0) is the energy of the lowest state of the particle/system at ~0K where the kinetic E term of the total E is 0; but it’s always Higgs I believe.

2. Max Wood says:

So, I find this “photons have zero rest mass but contribute to the inertia (and weight in a gravitational field) of any system containing them.”

Later in your discussion here you refer to light as Waves, how do we measure the “mass” of Waves?

There is a disjunction between classical Newtonian physics and Einsteinian Relativity, often referred to but in effect, never explained.

Mixing the ideas of light as “lumps” (photons) and Waves undermines at least one of these explanations.

Non-experts struggle to gauge the veracity of science when explained using two different and contradicting models.

Needless to say either position can, in varying circumstances fail to adequately describe “the real thing”

One other observation, a good friend of mine said of String theory… “when science becomes philosophy, I prefer not to be involved…”

I looked for clarification of “The difference between relativistic mass and rest mass” and found myself in the maze that is “Physics.Stackexchange.com” read it if you dare!

Not convinced by your argument Dr Strassler, it seems to refer only to sub-atomic particles and not, for example the Heart of Gold.

Best regards,

1. Excellent questions. These issues are very confusing to the non-expert — even to the beginning physics student — and I have made them central in my book for precisely this reason. (It’s not an accident that the first word in the book is “Waves”.)

Specifically, photons are not “lumps”. They are particulate waves — waves that come in discrete little ripples, sometimes called “wavicles” — and not lumps. Lumps, after all, would not have frequency, but photons do. (How? Well, that’s how quantum physics works. Why? I don’t know; nobody knows why quantum physics is true, but it’s central to our universe.) The same is true of electrons; they are not lumps either. They too are particulate waves — wavicles — and they have frequency also.

As for rest mass, relativistic mass, gravitational mass — this is also central to the book, or rather, rest mass is central, and distinguishing it from other forms of mass is crucial to making sense of it.

I totally agree that, on the whole, these subjects are mangled in the popular science literature, and that’s something that needs to change.

1. Max Wood says:

Sounds like I better order my copy of the book!

1. It was written, in part, for people like you… frustrated with all the inconsistencies and not able to get to the heart of the matter. You will no doubt tell me where I succeeded and failed — this website will host many opportunities for feedback and follow-up.

2. What do you think of Willis Lamb’s 1995 paper titled “Anti-photon,” where he thinks “photon” is a terrible term. “Photons cannot be localized in any meaningful manner, and they do not behave at all like particles, whether described by a wave function or not.”

1. Joseph says:

Dr.Strassler:
Absolutely love the book. Im about half way done, two questions if I may:

I really enjoyed the review on relativity , specifically how Galileo put forth the concept, and this is what Newton & Einstein built on. With that said, the earth is used as a convenient reference frame, as its motion (rotation) is small enough that it is “inertial enough” (I’m not including big scale motions as associated with the coriolis effect)
With that said, the “distant stars” are often used as a reference frame….coordinate system as to the motion of the earth, indeed you often see gyroscopes aligned with distant stars to show the rotation of the earth. However, as you state in your book, there is no grid system attached to the universe. So, the convenience of using the distant stars is really based on the fact that they are so very, very, very far away, that any motion they have relative to us is negligible, and the distant stars can, for practical purposes, be used as an anchor point for a coordinate system?

Sean Carrol has a video on you tube where he says the term “relativistic mass” should be banned (Sean Carrol / relativistic mass). He says that what physicists call mass is the energy content of an object when it is at rest….period.
Any corrections due to the objects relative velocity are corrections to the energy
I’m assuming this is just a preference in terminology?

1. About the distant stars as a “reference frame” — both reference for speed and reference for position — yes, it’s sometimes a convenient choice. But saying “the distant stars” isn’t entirely precise — which of the stars are you considering? and over what time frame? Depending on your answer, you’ll choose a slightly different grid.

The key point of the book, though, is that the distant stars, like the cosmic microwave background [CMB] (which defines another convenient reference frame, quite different from that of the distant stars) only tells your motion relative to the distant stars, or to the CMB. Neither one tells you your motion with respect to space itself, which, if it made any sense, you could measure directly, without reference to something far away. You can measure your motion through water just by looking at your wake, without looking at the distant houses on the shore, or through air by feeling the wind, without looking at the distant trees. There is no analogous measurement for empty space itself.

On relativistic mass; Sean and I and all professional physicists agree on the equations. We do not always agree on the best concepts, interpretations, and terminology — and this causes potential problems for non-expert readers. This is one of the reasons I highlight these issues in the book — in a perhaps hopeless effort to reduce some of the confusion caused by these disagreements. [I hope my book will inspire physicists and journalists to define their terminology and concepts more carefully.]

2. Joseph says:

Dr.Stassler:
Yes, I understand the point of the choice of a coordinate system, as in which “distant stars” am I picking to anchor my coordinate system to. However, isn’t one of the key concepts of physics that the laws of physics are reference frame invariant? In that two observers, viewing the same event, but choosing different coordinates, will disagree about the components (total momentum, total energy….etc) but they will agree on the results?
For instance, I read your blog about how we know the earth is rotating. Using gyroscopes you state that we can point the axis of the gyroscope towards a distant star, OR just a black point in space ( no star reference) and the results are the same. Wouldn’t choosing a star to aim at, or a black part of space, be a simple example of choice of reference frame?

BTW: your book is fantastic, I read every night, I have had it three days, and I’m half way done.

1. Choosing a star to aim at is not a reference frame choice. Reference frame is about choosing coordinates over a region of space and time, and a single star does not a reference frame make.

The laws of physics are the same in every reference frame, yes [or more precisely, they are if you do general relativity carefully, and if gravity is irrelevant and you restrict yourself to inertial observers, then they are in special relativity too.] But that is not the same issue as whether different observers disagree or agree about events.

For the latter, the issue is that some quantities in nature are relative, while others are not (and I call them “intrinsic” in the book.) Observers will generally disagree about the “results” of measurements of relative quantities, and agree on the results of measurements of intrinsic quantities. Some of those relative quantities may indeed be different even for observers in the same reference frame; some may only differ for observers in different reference frames. But intrinsic quantities have a special status, since those are the ones on which we all agree.

2. Joseph says:

Dr.Stassler:
I guess I should have been more specific and said I anchor one axis of my coordinate system to a “distant star” and anchor another axis to another point…etc
But, I understand your point.

3. A couple of comments, because this is actually something I helped to set up.

With modern technology, the “distant stars” are not so distant, and in fact move around a lot. The current International Celestial Reference Frame (ICRF) is based on Very Long Baseline Interferometry (VLBI) observations of quasars (and other objects, but they are all supermasssive black holes, so I’ll call them quasars), mostly at redshifts from 0.5 to 3 (light travel times of 5.1 to 11.5 billion years). The ICRF underpins all modern navigation, including the maintenance of the GPS system everyone uses. VLBI observations actually determine (with a General Relativistic propagation model) the angular distance between different sources and also the angles between those sources and the VLBI radio telescope network here on the ground (the propagation model thus defines the reference _standard_), together with an arbitrary orientation (rotation) of the set of positions of all of the sources (that defines the reference _system_). In a similar fashion, the angles between the quasars and the ground station network (i.e., the latitude and longitude of those stations) are given in the ITRF (International Terrestrial Reference Fame).

In one sense then, the ICRF reference system is arbitrary – we could rotate it. In another sense, it is not, it was chosen to be as close as possible with previous optical observations of stars and galaxies, in other words, with previous (optical) reference systems. Each of those (GAIA, Hipparcos, FK5) was itself chosen to be consistent with a previous catalog, so the ICRF reference system ultimately traces all the way back to the stellar positions provided in the Almagest of Ptolemy. (Since we chose to align celestial coordinates with the Earth’s spin axis, and since that slowly changes due to precession, the celestial reference system has rotated significantly since Ptolemy, but in a known fashion.)

2. Well, he doesn’t like “photon” because it seems to make it into a dot-like particle. I agree with his objections in part, but I focus instead (both on this website and in the new book) on the word “particle” instead.

Lamb says “electron” is okay because in the non-relativistic limit it really does act like a “particle”. I disagree; it’s not like a particle, in that a stationary electron spreads out, whereas a stationary “particle”, as usually defined, would be located at a point. [I can’t be sure if Lamb’s intuition is overly shaped by the old quantum mechanics and not sufficiently by quantum field theory, but I suspect so.] In short, his objections about “photon” apply, in my view, to “electron” as well. And so I’m fine with both “photon” and “electron”, as long as we understand both of them as a “wavicle” rather than “particle”. [Discussed in detail in the book’s chapters 16 and 17.]

2. mls says:

Angela Collier made a YouTube video which I found helpful,

https://m.youtube.com/watch?v=6HlCfwEduqA

She speaks of “intrinsic mass” (I presume to be the correlate of Dr. Strassler’s rest mass) and presents an equation that prevents one from concluding that a massless photon has zero energy. Whatever any professional might criticize, she makes a herculean effort here to sort out good physics from the many inaccurate statements found in simplified explanations.

1. Looks like it could be okay; I don’t know her. The term “rest mass” is also called “invariant mass” and it is indeed intrinsic to an object, and so, while “intrinsic mass” is not commonly in use, I have no problem with the use of that term. No matter what you call it, it is this type of mass that all electrons have the same amount of, no matter what they are doing. And this is the type of mass that comes from the Higgs field.

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