Yeah, but WHY is c the speed limit of the universe?

[Pasta]

To follow on from my first post above (which I know is long, but you should read that before reading this) –

While I’ve tried to show concretely that c doesn’t enter fundamentally, it’s also worth expanding on the “it’s just units” way of viewing the question. This approach may remain unsatisfactory (in which case, ignore this and work from my first post), but hopefully it helps.

Consider points in 3D space relative to some origin. We could say that some point is at an (x, y, z) position of (3 m, 5 m, 10 m). The distance D from the origin to this point is calculable via Pythagoras. In particular:

D = \sqrt{x^2+y^2+z^2}
D = \sqrt{(3~\mathrm{m})^2+(5~\mathrm{m})^2+(10~\mathrm{m})^2} = 11.6~\mathrm{m} .

We could have decided that the “up” (say, z) direction is privileged in some way, maybe because we are alien beings that live on a smooth planet with strong gravity, so we consider “up” very different in our lives than the other directions. So, maybe we have always measured “up” distances in units of shmeters (shm) instead of meters. 1 shmeter was established long ago to be the height of a particular tree. The conversion between shmeters and meters is k=70~\mathrm{m/shm}, say.

Our point in space is thus stated routinely and without hesitation by these aliens to be at position (3 m, 5 m, 0.143 shm).

We recognize, though, that the 3D concept of distance is a real thing anyway, so we plow ahead and teach all our students a distance formula:

D = \sqrt{x^2+y^2+(kz)^2}

Note the presence of the constant k. In fact, in this system, k will show up in all kinds of formulas. No big deal. It’s what we (the aliens on this smooth planet) are used to.

Regardless of the clumsy k showing up, the fundamental “sameness” of the three dimensions remains. For instance, we can rotate an object and its length remains unchanged, or we can equivalently rotate our coordinate system (our “perspective” on the object), and we will always calculate the same length for it (i.e., the distance from end to end).

We jump, now, to spacetime. In 4D spacetime, distances are a very real thing, too. As are rotations. However, the geometry of spacetime is not strictly Euclidean, so calculations look slightly different. Consider a point in space time at (x, y, z, w), where I’m using w for the “time” dimension at the moment, but I don’t want to call it time yet. Instead, w is on exactly the same footing as the other quantities (units-wise). In this 4D spacetime, distance is calculated via:

D = \sqrt{x^2+y^2+z^2-w^2} .

Let’s actually talk solely about squared distances:

D^2 = x^2+y^2+z^2-w^2 .

Notice the minus sign. That’s just how spacetime works. The cosmic speed limit is simply this: An object can’t move between two points unless the spacetime distance (squared) between the two points is negative. You can flip the sign convention, swapping all + and - signs, and it’s a matter of taste (and to some degree the physics subfield) which convention one adopts.

Rotations in spacetime lead to a jumbling of the space and time dimensions, so it’s extra important that we are treating them coherently and consistently. If you are comfortable with complex numbers, you can cast the w dimension as imaginary, and then rotations in 4D spacetime look just like rotations in 3D space. The details of the rotation (axis of rotation and amount of rotation) are related to the differences in two observers’ reference frames (in the special relativity sense). And, just like distances in 3D, the distances in 4D are unchanged through such rotations (i.e., through changes in reference frame). It’s all quite elegant.

And all of physics sits on top of this basic structure of 4D spacetime.

If we continue without breaking this clean picture, one could say that a person might be 2 meters tall and might live for 8x10¹⁷ meters. Of course, if we view them from a different reference frame (e.g., with some relative velocity), we are effectively “rotating” our 4D viewpoint, and we would measure different values for the person’s size and lifespan. This is the familiar length contraction and time dilation of special relativity, but really we are just “rotating” space and time into one another. The inter-rotatability of space and time is paramount in special relativity. And the maximum rotation you can have is how and why we can sensibly talk about a “meter” in time, just as in 3D I can take a horizontal meter stick and rotate it to become a vertical meter stick in order to establish the correspondence between horizontal and vertical distances (and use “meters” for both).

If we were used to this scheme for talking about time, this thread would not exist. However, a long time ago, someone measured a random w (time-like) distance that corresponded to 2.6x10¹³ m, called that “1 day”, chopped that day into 86,400 pieces labeled “seconds”, and then insisted that we talk not about w but about t=w/c, where c has a not-at-all fundamental, unit converting constant value of 3x10⁸ m/s. This is exactly as arbitrary as k is in the 3D alien example above. The w axis and the t axis are the same axis, but with an arbitrarily constructed conversion introduced.

In terms of t, distances can be written as:

D^2 = x^2+y^2+z^2-(ct)^2 .

Now, the cosmic speed limit rule that used to be (change in spatial position)/(change in w position) < 1 becomes (change in spatial position)/(change in time) < c. And, rotations pick up lots of c's all over the place, and things like energy and mass get messier relationships. But it’s all artificial.

[Measure_for_Measure]

My underlying question is, “How fine tuned is the fine-tuned universe?”

@Stranger_On_A_Train

Fundamentally, the speed of light, c is related to two other natural constants, the electric permittivity of vacuum, ε0, and the magnetic permeability of vacuum, μ 0 in a specific ratio . And if either of these values were different by even a very tiny proportion, we would have a very different universe from the cosmological scale right down to the subatomic level even if the relative behavior of spacetime was the same. “Why” is the physics of the universe this way is a kind of open ended philosophical question for which we have no answer, other than that if it were different we wouldn’t be here to ask that question, and indeed, there might not be any kind of stable chemistry or the ability to even form elements.

What is “a very tiny proportion”? x10, only 1 order of magnitude? 10%? 1%? 0.1%? 10^-4%? Etc.?

At what point does stable chemistry break down? At what point does the ability to form elements break down?

Also relevant:

@Stranger_On_A_Train

However, if it were much slower, we would (likely) have a very empty visible universe as it would be outpaced by the expansion of spacetime, and if it were much faster we would have a vastly more energetic universe in which chemistry (at least, as we know it) would not be possible. There are also a vast number of other physical mechanisms that would be impacted that would make the universe not inhabitable for us, although this should not be taken as any evidence of an “anthropocentric universe”, since if the fundamental constants were not the way they are we just wouldn’t be here to notice.

How much does the speed of light outpace the expansion of the universe? (Bonus question, given a high but plausible estimate of spacetime’s acceleration, when does the universe go dark?)

[Pasta]

@Saint_Cad

I suppose another way to put it is: what other constants could I change that would a result in a change in c?

You’ve probably had a chance to see my simul-post with yours that serves as a useful intro to this question.

To talk about “a change in c”, we need to be sure what is meant by c. If we mean “a change in a fundamental constant of the universe”, then we must immediately halt because it has (hopefully?) been established that there is no such constant to change.

If instead we mean “a change in the arbitrary unit conversion factor c”, then we can talk about that. Any numerical change we effect on c implies that we have changed something about the physics governing our distance and/or time standards of measurement. Such a numerical change does not mean we have changed anything about how relativity works. We would only be changing what a “meter” means, say. Of course, changing real parameters will have knock-on effects that could be interesting.

I’ll return to the question at hand in a moment. But first:

It is worth noting here that c is in no way unique in its arbitrariness. Any “fundamental” constant that you know of that has units is not actually fundamental. The universe does not know and does not care what a meter is, or a joule, or a kelvin, or any combination of these things. Unitless quantities (often ratios) can be discussed in absolute terms. Unitful quantities are intrinsically tied to artificial conventions somewhere along the way.

A perhaps more tangible case of all this is the Boltzmann constant, k_B, which equals 1.38\times 10^{-23}~\mathrm{J/K} and relates energy to temperature. It feels like a constant of the universe, but it’s just as made up as c, in that we never needed to introduce separate “energy” and “temperature” units in the first place. In statistical mechanics and thermodynamics, literally everywhere that temperature T appears, there is a secret k_B hiding somewhere just to get back to energy units. In formal treatments, one usually works with the “fundamental temperature” \tau=k_B T, where \tau is just temperature in energy units directly, eliminating the need for k_B entirely. And I mean entirely. It doesn’t exist in the equations anymore. The only time you need it is to convert some answer back to whatever the hell a “kelvin” is so that it can be compared to a thermometer that reports “kelvin” instead of “ergs” or something, because humans are used to temperature having different units than energy.

“What if k_B had a different value?” isn’t a question about anything intrinsic. It’s about how a joule and a kelvin were independently defined even though they didn’t need to be.

And so with c. Let’s get back to that question.

In the hydrogen measurement standard I introduced above, changing the fundamental (unitless) constant of the universe \alpha by 1%, reestablishing our unit standards, and then measuring c would yield a different numerical value for c by 1%.

However, we could use instead a nuclear-based measurement standard, say the radius of a carbon nucleus for distance and the oscillation period of radiation from a carbon nuclear transition for time. If those were our standards, then changing \alpha by 1% would not change the numerical value of c by 1% but instead by something much less. That’s because the physics we are basing our standard on now is driven primarily by the strong interaction rather than the electromagnetic interaction. But if we change the strong interaction’s coupling constant (often labeled \alpha_s), we would then alter the numerical value of c directly under this unit standard.

However, in both these cases, if we change something about gravity, we would introduce no perceivable difference in the numerical value for c.

What if we used “rotation period of Earth” and “circumference of Earth” as our standards, as was done historically. Well, now changing something about gravity absolutely would influence c since our standards would be based on physical systems that are heavily influenced by gravity.

It is hopefully clear there remains no sensible or consistent answer to “How can we change c?” because the numerical value is 100% a reflection of the unit standards.

You might ask: What if we just adopt our modern unit standards, then? How could we change the numerical value of c in today’s conventional unit system?

You cannot, because c is defined as a particular string of numbers in meters per second (299,792,458 m/s, exactly). It does not depend on any measurements or any physics. If you change \epsilon_0 or G_N or \alpha or whatever, c will remain unchanged in the current unit system. And this makes sense! It is not artificial to do this, and it is in fact manifestly encoding the artificiality of c. This string of numbers says nothing about physics and everything about the system of units we maintain so that we can talk about two equivalent things on nonequivalent footing. We like talking about meters and seconds separately, and also joules and kelvin separately, etc. These separations are artificial, and so we must suffer the presence of physically meaningless constants to convert between our choices.

If you defined c to be a different number (say doubled) as input to the unit system, you would be redefining what a meter is, and so your car would travel along the highway at 120 mph instead 60 mph. But no physics would be changing, just the size of the units. (Drawing back to the concept of unitless quantities: the (unitless!) ratio of your car’s speed to the speed of light – something we can talk physics about – would remain unchanged here.)

[Chronos]

@Exapno_Mapcase

I remember when “what is the speed of dark?” was commonly asked, not always humorously.

Oh, that one’s easy. If I have a hallway that’s 3*10^8 meters long, lit by a lightbulb at one end, then when I turn off the light, it takes one second for the dark to reach the other end. In other words, the speed of dark is the same as the speed of light.

[Francis_Vaughan]

@Measure_for_Measure

How much does the speed of light outpace the expansion of the universe?

It doesn’t. The universe outpaces light over a sufficiently large extent of universe. Hence how we get an “observable universe”. The speed of light is how fast causality goes within space time. Space time itself isn’t limited by c.

[The_Librarian]

Light thinks it travels faster than anything but it is wrong. No matter how fast light travels, it finds the darkness has always got there first, and is waiting for it.

Terry Pratchett

[HMS_Irruncible]

@DPRK

The concept of a 4-dimensional spacetime was introduced to describe Lorentz-invariant physics, and of course this symmetry is experimentally tested, but if the OP’s question is why the universe has such a law, I am not sure there is a succinct answer. You cannot merely say “that is what the equations say” because that is begging the question.

The way I think of it is this: the definition of the universe is the set of time, space, and objects that we can observe, measure, or infer causal relationships about. This is determined by a set of equations that were chosen because they are mutually consistent.

Since those are the rules of the universe,anything that falls out of those equations isn’t in the universe, nothing can ever interact with it, so functionally they don’t exist. You could imagine them maybe existing in a different universe, but that’s purely a guess since we can never actually know or calculate those things.

This is a prelude basically to what Buck_Godot wrote about why certain things can’t be in our universe. Maybe they exist in some other universe, but since they’re outside direct observation and not calculable by any theory, they’re simply unknowable and may as well not exist.

@Buck_Godot

Yes basically, there is a constant c, that can be expressed as a velocity. And according to the relativity, for particles with mass, their energy increases asymptotically as their speed approaches this value c. Thus in order to reach c it would require infinite energy, so no massive particle can move at c or faster

Meanwhile a zero mass particle moving less than c, has zero energy and zero momentum which means it has no way to affect other matter and so for all intents and purposes it doesn’t exist. Photons are massless particles that have energy, therefor they have to move at this constant c. So we call c the speed of light.

Meanwhile if you want to imagine tachyons that move faster than c, you find that according to their reference frame you can have events which reverse their order, so that effect happens before cause, which if they were able to transmit information would lead to all sorts of paradoxes. So if such particles “exist” they would not affect our universe and so for all intents and purposes might as well not exist.

[Chronos]

@The_Librarian

Light thinks it travels faster than anything but it is wrong. No matter how fast light travels, it finds the darkness has always got there first, and is waiting for it.

Terry Pratchett

That just means that the dark had a head start, not that it was faster.

[Hakushon]

@CC I’m just curious: have any of these various explanations answered your question in a way that feels satisfactory to you? If so, which explanation, might you say, gave you an “ah-ha” moment?

[CC]

No “aha!” moment. At some point, when the discussion turned either philosophical or mathematical, my eyes glazed over. (This is the problem with pinheads like me asking questions of people who really know this stuff.) But then I realize my question really is “Why should there be a speed limit in the universe at all”? And that kind of question can’t really have an answer, any more than most “why” questions, beyond “that’s just the way it is.”

[Chronos]

If it helps, in a sense, there isn’t a maximum speed to the Universe; we just measure speed in a funny way (except it doesn’t seem funny to us, at low speeds). If, instead of defining speed as distance over time, you define speed as proper distance over proper time, then you get something called “proper velocity”, or u. Which is, in some ways, a weird sort of quantity, since proper distance between two points is measured by someone at rest relative to those points, while proper time is measured by someone traveling between those points. But nonetheless, the combined quantity turns out to be useful, relevant, and easy in many contexts.

You can find the proper velocity of an object by multiplying its normal velocity by gamma, the relativistic dilation factor (\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}). For low speeds, gamma is very close to 1, and so proper velocity is almost the same thing as velocity, while for high speeds, gamma is large but v is very close to 1, so proper velocity is almost the same thing as gamma.

Proper velocities do not, in fact, have any upper bound. And a lot of physics becomes a lot easier when you use them: For instance, relativistic momentum is given by P = m*u, just like it is in Newtonian physics, and force is still mass times proper acceleration.

[Lumpy]

The speed of light causes/produces something I recently read about: the space-time interval: Time: Yes, it's a dimension, but no, it's not like space - Big Think
In brief, mathematically describing 4D spacetime requires the use of hyperbolic coordinates, which in turn requires that the time interval include the imaginary i in it’s value.

• the speed of light is a constant for all observers,
• and the only truly invariant quantity is neither distance nor duration, but the spacetime interval,

[Hakushon]

Beyond saying “that’s just the way it is,” would it help if we add, “because it is not only a theoretical speed limit, but has been observed to be true, and all our efforts to disprove it have failed”?

[CC]

I’d prefer “As far as we know.”

[Exapno_Mapcase]

What does that get you in addition to “all our efforts to disprove it have failed”? If an experiment or discovery disproved it, then we would know differently. Until then, everything in science and every other discipline has an implicit “As far as we know.” Do you go around saying that someone’s height is 5’10” “as far as we know”? Salt is sodium chloride “as far as we know”? Water is wet and fire is hot “as far as we know”?

Why is this finding, reaffirmed by millions of experiments and experiences by millions of people in all societies over hundreds of years, subject to a qualifier by you?

[Lumpy]

@Exapno_Mapcase

everything in science and every other discipline has an implicit “As far as we know.”

Perhaps because Relativity was founded upon the very notion of our conceptual frameworks requiring radical revising– in this case, replacing Newtonian mechanics. So while Relativity has been the best description of reality at large scales we have for over a century, to accept it at all requires that the lesson that scientific knowledge is ultimately provisional be ground in our faces. Especially given that at the limits- singularities, reconciliation with Quantum Physics- we have good reason to think that the current theory is NOT the ultimate description of reality.

[Pasta]

@CC, rewinding to your original question in the OP –

@CC

I understand that as an object nears the speed of light its mass increases and approaches infinite mass so it’s near its maximum speed.

Not true unless you define mass in a silly way (which people did use to try to do, so you aren’t making up the words here.)

@CC

But my question is why the speed of light? What is so magical about it that it presents such a limit to the entire universe.

I hope to show here that it’s not so crazy. I’m happy to engage if you stop me at any point for follow-up.

3D space has three directions. You could label these as forward/backward and up/down and left/right. Lengths of objects (like sticks or people) in 3D space do not change when you rotate your point of view (e.g., look at it upside down or something). What you call “up” or “left” might change, but when the three directions are taken together coherently, the ultimate measured length of the object is the same regardless of any rotation of you (the observer) or the object. This “law of nature” is not terribly upsetting, and it stems from the very geometrical behavior of space.

Instead of saying “length of object”, let’s say “distance between the two ends of an object”. This means the same thing, but will help in a moment.

4D spacetime has the same three regular directions and a fourth direction called “time”. Movement through time feels different than movement through space, and it is, but it helps greatly to think of it as “just another direction” at first, and then later add on any complications that make it different.

In 4D spacetime, a point is both a location and a time, not just a location. Such points are often called “events” (place + time, like an event would be).

The distance between two events (points in 4D spacetime) can be calculated or measured, too, but time enters the distance with a negative sign. You can’t slap high school geometry on this 4D space. You need this slightly different geometrical rule about how distances are calculated. This is one way “time” is different from “space”.

Additionally, we would like these distances calculated in 4D spacetime to be unchanged when we change our point of view. Rotating our point of view is more subtle than in 3D, and I’ll leave it aside for the present purposes, but know that there is a concept of rotation in spacetime, and we want distances to stay unchanged. Consider this unchanging of distances another law of nature (just like in 3D, up above) that is a reflection of how spacetime geometry works. Not a crazy law to have, again.

Additionally, we would like events that are related by cause-and-effect to always stay in the same time order when we rotate our point of view. Keep in mind that rotating our point of view in 4D can and will jumble up “time” and “space” directions, so care is needed. But requiring that cause-and-effect remains unchanged under rotations seems like a sensible law to impose.

Putting this all together, we have the following consequence: Any object that moves from one event (4D point) to another event must necessarily be moving a total 4D distance (squared) less than zero. Remember that distance is calculated using different geometrical rules, hence negative squared distances being possible.

One can recast that consequence as a limit on how much spatial change you can have given some amount of temporal change. “Change in space” has to be less than “change in time”. This sounds like a speed limit, but it’s can be thought of as a consequence of the geometry and a couple of other not-terribly-upsetting laws imposed, as walked through above. This geometrical limit on motion is (space change) / (time change) < 1.

Completely separately, humans have historically measured spatial directions in 4D spacetime using one set of units (meters, miles, etc.) and the time direction using another set of units (seconds, hours, etc.) In terms of these historical units, the geometrical limit picks up an arbitrary numerical appearance that reflects our choice of units. My other posts above continue the story from here.

@CC

Is this a matter of “that’s just the way the universe works”?

To summarize:
If we accept that spacetime is a coherent 4D thing; that distances can be calculated in this space using a slightly different geometrical rule (negative sign for time direction); that there is a particular concept of rotating our point of view, thereby mixing space and time up with each other; that we want 4D distances to be unchanged when rotating; and that we want cause-and-effect to be preserved when rotating, then there is a geometrical limit that requires steps in space to be smaller than than steps in time. That’s the story. Additionally, we can apply historical units to the distance and time parts of that limit, and it looks like a speed. That’s the speed of light.

[Hakushon]

Let me just say this about that: it’s pretty easy to measure someone’s height, or to show that sodium chloride is a type of salt. It is also easy to hypothesize that there might be a particle that would travel faster than light, and to name it a “tachyon.”

However, to actually detect a tachyon would be a lot more difficult. “Ah,” you might say, “it’s especially difficult because they do not exist!”

So let’s look at a different particle: the Higgs boson. Here is a particle which the standard model of physics predicted, ought to exist. It was as if a jigsaw puzzle had been pieced together, and there was a hole in it with a very particular shape, and everyone was saying, “Yeah, there’s a missing piece, and it should go right there.”

And yet, at first, no one could detect it–despite that there should have been many, many of these particles, which were supposed to be ubiquitous throughout the universe. What it actually took to detect just one of these particles, was a particle accelerator (the Large Hadron Collider) that measures over five miles across, which required international cooperation and great expense to construct, maintain, and operate. Even after completion, the search for the Higgs boson required running the collider at different energy levels over the course of years, and shutting it down between searches–a time consuming process.

Years later, when a particle finally was detected, which appeared to be the Higgs boson, the data had to be analyzed and verified before anyone was willing to announce, “Yes, we have found it.” (I seem to recall that the grand announcement was actually something to the effect of, “We have detected the shadow of a particle which we believe to be the Higgs boson.”)

This is a lot more difficult than, for instance, slapping a measuring tape up against a wall. So while the saying “as far as we know” may seem arbitrary and unnecessary, it is actually more meaningful, and carries with it more scientific labor and history, than most of us would naturally, commonly assume.

[wolfpup]

@Hakushon

So let’s look at a different particle: the Higgs boson. Here is a particle which the standard model of physics predicted, ought to exist. It was as if a jigsaw puzzle had been pieced together, and there was a hole in it with a very particular shape, and everyone was saying, “Yeah, there’s a missing piece, and it should go right there.”

And yet, at first, no one could detect it–despite that there should have been many, many of these particles, which were supposed to be ubiquitous throughout the universe.

Just to clarify a bit, Higgs bosons are not particles that exist all over the universe, and what the Standard Model predicted was not the Higgs boson itself, but more specifically, the Higgs field. The Higgs field permeates the universe and is responsible for giving particles their mass. What the experiments with the LHC accomplished was to instantiate the Higgs particle from the Higgs field, which required enormous energy because the particle is so massive, and it was brought into existence for a mere 1.6 x 10^-22 seconds, or around one ten-thousandth of a billionth of a billionth of a second. It was an incredibly important discovery nonetheless, which confirmed the existence of the Higgs field and added more support for the Standard Model, as well as winning François Englert and Peter Higgs the Nobel Prize in Physics the year following the Higgs discovery.

[Mijin]

@RitterSport

I think @Chronos explained it in some thread somewhere that you should think of c as the same for everything moving through space-time, and with more useful (for this conversation) units, would have the value of 1.

This. Think of the universe having a grid where each cell is 1 second high and 300,000km wide. These seem arbitrary units to us because we’re coming from a position where we’ve devised our own units, but to the universe this is the fundamental mapping.

(I know it’s more complex than this because this grid would have to change based on velocity of the observer…analogies like this always break down anyway, as I said in another thread)