# Do we know why there is a speed limit in our universe?

This question is about why we have a universal speed limit (the speed of light in vacuum). Is there a more fundamental law that tells us why this is?

I'm not asking why the speed limit is equal to $c$ and not something else, but why there is a limit at all.

EDIT: Answers like "if it was not.." and answers explaining the consequences of having or not having a speed limit are not--in my opinion-- giving an answer of whether there is a more fundamental way to derive it or a law to explain this limit.

• @ TheQuantumMan--When you're asking about "our" universe (instead of "the" universe), I'm wondering whether you're assuming that the speed of light might vary between causally-separated regions of a spatial multiverse (like that suggested by field-based inflation) or between different iterations of a spatio-temporal multiverse (like that suggested by some "bouncing" cosmologies}. Could you (and maybe some of those answering) clarify that? (At least one answer refers to one of these two possibilities, and I believe that another refers to the other one.) Thanks. – Edouard Nov 18 '18 at 4:34
• Physics is the study of finding general rules for what we see out in the universe. If you want to prove something that cannot be or is not out there, then using these rules as axioms and giving proof by contradiction is what you are curtailing by your condition of "Answers like "if it was not.." and answers explaining the consequences of having or not having a speed limit are not--in my opinion-- giving an answer of wh – SagarM Jan 1 at 12:19

Imagine that there is a person who prefers to measure the amount of money in his bank account with the value $$V$$. The equation is $$V = C\tanh N$$, where $$N$$ is the actual amount of money in dollars. This person will also be confused:

Why is there a limit ($$C$$) on the amount of money that I can have? Is there any law that says the value of my money, $$V$$, cannot be more than $$C$$?

The answer is that he is just using a "wrong" variable to measure his assets. $$V$$ is not additive — it is a transform of an additive variable, $$N$$, which he has to use in order for everything to make sense. And there is no "law of the universe" that limits the value of $$V$$ — such a limit is just a product of his own stubbornness.

The same thing applies to measure speed — it is the "wrong" variable to describe the rate of motion; speed is not additive. The "correct" variable is called "rapidity" — it is additive, and there is no limit on it.

• I'm not going to lie, that made almost no sense to me. Can we dumb it down a bit? – David Grinberg Jan 22 '16 at 16:26
• But WHY should our measure of speed be determined by an asymptotic function? – user66309 Jan 22 '16 at 18:36
• okay, so the question now is: Why is rapidity = $\arctan(v/c)$? why $c$? – Ant Jan 22 '16 at 18:46
• And why not simply rapidity = $v$? – immibis Jan 24 '16 at 10:17
• I don't quite like that answer. If I measure the time it takes for light to go to the moon and back, I can measure that the distance / time is c. So that's the definition of speed. It is a finite measure. Extracting an inifinite value out of those 2 simple measurements of duration and distance seems artificial at best. – njzk2 Jan 26 '16 at 4:54

There is a wonderful paper I remember reading which uses only basic algebra only to determine the most general form of the formula to add velocities, based only on general principles of symmetry (what works here also works there, etc.).

I can’t find that one, but it’s easy to find Nothing but Relativity. And others that are derived from an initial version by Mermin.

Another paper with the same idea (but different specific axioms) is One more derivation of the Lorentz transformation by Jean-Mark Lévy-Leblond published in 1976 (thanks bdforbs).

In the end, it shows that the familiar special relativity is the inescapable answer. In the paper an arbitrary non-determined value $Z$ came out of it. There are 3 cases: negative did not work (in the paper I’m remembering, the math chokes. In the linked paper, it’s “not self-consistent”, but Greg Egan worked out this case in detail). $0$ gives Galileo’s fixed absolute time, and any positive value gives special relativity with a speed limit.

The speed just is. In natural units it is a value of $1$. The reason the speed of light (or of any massless thing) seems to be something specific is its relationship to other things. In the end, you can find the relationship called the fine-structure constant has a particular value.

Your real question is: why is the fine-structure constant the value that it is? The answer is unknown. It may be determined by a deeper set of rules than we know now, it may come from physics we don’t have an awareness of, or it may be a pure accident like the number of planets in our solar system rather than a law.

So why is $Z$ non-zero? Well, if it’s any random value it has a vanishingly small chance of being exactly zero, and the actual value it has, as long as it’s not exactly zero, just sets a scaling factor and doesn't really mean anything.

The richness of the Universe is due to emergent effects of the basics: given that time is relative combined with quantum mechanics means that antiparticles must exist, along with pair production and annihilation.

If the universe didn't have special relativity but had fixed absolute time, it would be very different and we wouldn’t have the same kind of thing at all. It all fits together, and by and large if you look at one deep feature you find it’s necessarily so based on the other deep features. It's all or nothing: you can’t pick on the finite speed of causality by itself any more than you can ask why one side of a triangle has the specific length that it does.

# Notes

The linked paper and others related (from Mermin?) Use a value K in a slightly different way to the Z in the paper I can't find now. These easy-to-find ones also use calculus and limits, which for this purpose isn’t as satisfying as using algebra alone with four assumed relationships due to symmetry.

The one I recall originally used (IIRC)

1. space is the same here and there
2. if A sees B moving at speed X, B sees A moving at speed X in the opposite direction. equiv. to space is the same in every direction in a one dimensional model.
3. results of experiments now are the same as results of experiments at different times. (same as #1 but for T rather than X)
4. same as #2 but for T rather than X

I think this is more satisfying than the Lévy-Leblond paper, which takes universal agreement of causality as an axiom.

• That is sort of the anthropic approach, but it is no less true. It seems counterintuitive, but only because we see things on a smaller scale where the rules seem to be different. It'd be like asking how people on the underside of the earth can stay attached if the earth is round. That reasoning comes from what we intuitively understand, but that doesn't mean the same applies on a larger scale. – Neil May 31 '16 at 8:57
• Starting with symmetry considerations, you can show that you indeed have only special relativity as a possible answer, but c = infinite is not rulled out at all – agemO May 31 '16 at 12:29
• @Neil True enough, but the principle we intuitively understand in this case is causality: it's a bit hard to make sense of a noncausal universe, which is what would arise if the negative sign were chosen. Moreover, this isn't the only effect, and all the weird effects that a Riemannian universe would show are studied in Greg Egan's "Orthogonal" trilogy. Egan gives a summary of some of the physics here. – WetSavannaAnimal May 31 '16 at 13:37
• Mermin didn't originate this. It goes all the way back to a 1911 paper by Ignatowsky. – Ben Crowell Aug 6 '17 at 16:18
• Is it "One more derivation of the Lorentz transform"? doi.org/10.1119/1.10490 – bdforbes Dec 17 '17 at 13:51

The best answer I can come up with is "because the Universe would be fundamentally unpredictable otherwise."

We can imagine spacetime as a four-dimensional manifold $\mathcal{M}$; the laws of physics then dictate how matter and energy behave on this manifold. (For the sake of argument, you can view this as plain old flat Minkowski space, though the argument generalizes to include curved spacetimes as well.) We can then ask the following question: "Suppose I know how the matter and energy is behaving in some finite portion of the universe at some moment of time $t = 0$. What does this tell me about the behavior of matter and energy in the Universe after that time?"

$\hspace{100px}$

If there is a speed limit to the universe, then there exists a region of spacetime called the domain of dependence, in which one can predict what will happen after our initial moment. It consists of all spacetime events whose past light-cones1, when traced back to $t = 0$, are entirely contained in the region of space we had knowledge of. Viewed as a function of time, the region of space lying in the domain of dependence will gradually shrink away to nothing, as influences from outside our initial region (of which we had no knowledge) propagate inwards. But if the universe has a speed limit at all points, then we are guaranteed that there is some finite volume of spacetime in which we can predict what will happen.

If there is an infinite propagation speed of the Universe, though, then the domain of dependence vanishes. Roughly speaking, there is no way we could predict anything, because causal influences could propagate from outside our region of initial data and mess everything up the instant after $t = 0$. Thus, if there was not a speed limit, then the Universe would basically be unpredictable; without knowledge of everything that was happening in the Universe at a particular instant time, the laws of physics would have no predictive power.

I will freely admit that this isn't so much an answer to "why is there a speed limit?" as "what would the universe be like if there wasn't a speed limit?" Still, a Universe without a speed limit is sufficiently alien and incomprehensible to make me glad that I live in a Universe with one. (This has shades of the anthropic argument—maybe in some parallel Universe, some incomprehensible creature is making an argument about how awful it would be to live in a Universe with a speed limit.)

Finally, note that nothing in this argument relies on special relativity; all that is required is that there be a notion of a "light-cone" at every point in space. The speed limit could vary from point to point, or differ with direction, but so long as it separates the neighborhood of each spacetime point into a causal past, causal future, and causally unconnected regions (as with conventional light-cones in conventional relativity), then the argument still follows.

1 "Light-cone" here doesn't necessarily mean "the path of all light rays traced back in time", but rather "the paths of all rays travelling at the speed limit traced back in time."

• As long as the strength of effects goes to 0 with distance, I see no reason why this "domain of dependence" idea is a necessity. We already can't exactly predict what happens in any finite volume of space, and we're cool with that, so why would it change anything? – immibis Jan 22 '16 at 0:25
• this answer is deep. good one. – Fattie Jan 22 '16 at 3:24
• @R.. Electromagnetic radiation, and gravity, both decay as $\frac{1}{r^2}$ - the quadratic decay cancels out the quadratic amount of stuff that could affect you, so the stuff 500 billion km away can't affect you any more than the stuff 5 cm away. (Total effect is still unbounded in theory, but in practice, the space 500 billion km away is filled to a much smaller proportion) – immibis Jan 22 '16 at 3:53
• Isn't the Universe already fundamentally unpredictable? Radioactive decay: You have no Idea when exactly a single atom will decay. If you have one of these in your observable portion of space, you cannot predict what it will look like at t=1 because it may or may not have been decayed. So with infinitely small chance your whole experiment might just blow up. The same with quantum vacuum fluctuations... – Falco Jan 22 '16 at 12:04
• @immibis At reasonable scales, the universe seems uniform. So the million-light year thick universe-shell at 1 billion light years, 10 billion light years, 100 billion light years, 1000 billion light years, 10000 billion light years, 100000 billion light years all contribute the same amount. The first of these shells generate a single photon hitting your body over your lifetime. Well, there are an infinite number of such shells. And something times infinity is going to ruin your day. – Yakk Jan 27 '16 at 15:07

Physics is a scientific discipline where observations and measurements are fitted with mathematical models which describe existing data and successfully predict new values for new boundary conditions. When this happens one says that the model has been validated.

If new experiments and observations should falsify the model, one will have to re-examine the assumptions and even search for a new model.

At present the validated model we have for elementary particles is the Standard Model which uses relativistic quantum mechanics and has been tested innumerable times with laboratory and observational experiments. This mathematical model , because it incorporates special relativity, agrees with the observation that the speed of light is a constant c in vacuum. True, the value of c is serendipitous for this discussion. It is the existence of the limit that is questioned, and the only possible answer is : because the theoretical model agrees with experiment and is very predictive.

If new data falsify the standard model to the point of a new theoretical model being necessary, this new model will have to incorporate the existing structure for the cases that it has been validated, including the velocity of light limit. The standard model would become a limiting case for the new theory, for the energies and boundary conditions that were validated, in a similar way that Newtonian physics emerges from special relativity at the limit of low energies.

• @annav so, the bottom line is that the most fundamental explanation that we have for it right now is that it "just is", right? – TheQuantumMan Jan 21 '16 at 13:47
• @LandosAdam Yes. The theory that demands a limited and fixed velocity for zero mass particles derives from the data. Physics does not answer ultimately "why" questions. It says "how" with the accepted model one can arrive at an observation. If we had observed a variable velocity of light we would have developed a different theory. – anna v Jan 21 '16 at 14:53
• @annav i know that, that is the beauty of its objectivity in a sense. But, sometimes, a theory can explain some things that were previously considered to be fundamental. That is why i made this question. And, the way i look at it, the fact that there is a chance that it could be derived from more fundamental laws that we are not aware of yet makes me more enthusiastic. But, the key word here is "MIGHT" because it could as well just be as you said. – TheQuantumMan Jan 21 '16 at 14:57
• @LandosAdam: exactly. If we ask, "why do objects move around/past the Sun in conic sections?", then physics can say "why" that is in terms of other properties of the model that can be seen in some sense as more fundamental: an inverse-square acceleration results in certain geometrical features. Of course it's still not an "ultimate" answer to "why", but it is an answer. But when it comes to why light doesn't arrive instantaneously, anna is stating here that there's only one possible answer, that is to say the model doesn't provide any "deeper" foundation for the phenomenon. – Steve Jessop Jan 21 '16 at 15:13
• It's also worth bearing in mind that whether or not something answers a "why" question is somewhat subjective, since we're talking about simplifying explanations ("multiples of 10 end in a 0 because we're writing them in base 10"), not physical causes ("My foot hurts because it's softer than the rock I just kicked"). So others have concluded that there is an answer to "why", and in fact they can talk about features of the model that for them partially explain the existence in the model of a limit. – Steve Jessop Jan 21 '16 at 15:15

In physics, you cannot ask/answer why without ambiguity. Now, we observe that the speed of light is finite and that it seems to be the highest speed for the energy.

Effective theories have been built around this limitation and they are consistent since they depend on measuring devices which are based on technology/sciences that all have c built in. In modern sciences, one doesn't care about what is happening, but of what the devices measure.

The validation of these theories lets says easily that there is a universal maximum speed. In fact, there is a maximum speed for a moving energy-made object when it is measured in a static space. It's not exact in an expanding universe or in other critical relativistic contexts. It's not exact if it is not energy, ie with the presumed influence which would be exchanged by entangled particles. Even so, one cannot travel faster than light unless in dreams.

Modern physics is new. It extends in all directions. It is still difficult to make a synthesis of all that is known, what is useful and what is not consistent and relevant. Perhaps the finite value of c ( and not only c ) will come from quantum deep fields analysis , in the same way that relativity followed the very rich Maxwell theory. Meanwhile, the scientific community is not aware of such a new analysis, even if it has been published already in some obscure repository.

• That's "Why do we know there is a speed limit in our universe?" which is a very different question to "Do we know why there is a speed limit in our universe?" – immibis Jan 22 '16 at 0:20
• @immibis I tried to show that there is no answer to this why because it is an experimental fact which is assumed like a postulate, even if the concept of speed may be discussed. Perhaps its status will evolve with more knowledge and more synthesis, but today who knows how to deduce the speed limit from other assumptions ? – user46925 Jan 22 '16 at 0:43
• There is a problem with asking 'why' in the sciences. In everyday English, 'why' and 'how' are often synonymous: The answer to "Why is the sky blue" is the same as "How is the sky blue". But for deeper questions, 'why' becomes meaningless, because it is a teleological question (not theo_logical)-- it's ultimately inquiring about _purpose or end goal. But the laws of nature are not designed for a particular goal; they just are. Thus, 'how' is really the only question science can answer. In this case, the how results from the equations. – user151841 Jan 22 '16 at 16:58
• @user151841 yes, because why must be answered by a logical demonstration in the current theory. But we cannot demonstrate a postulate. This one stays on robust observations and experienced theories – user46925 Jan 22 '16 at 17:03
• @user151841 Although I partly agree with you and tend to place those questions in some part of metaphysics, I disagree with you reducing it to a teleological argument. It is not the same as asking why the value of $\pi$ is what it is, which would be meaningless. Rather, it may be the case here that the alternative is not even consistent, in a mathematical sense, which would provide an answer not based in teleological arguments. – G. Bergeron Jun 9 '18 at 2:14

Maxwell made a rigorous, mathematical study of the properties of electricity and magnetism, and he proved that there must be a phenomenon that he called electromagnetic waves. According to Maxwell's theory, an electromagnetic wave must propagate at a constant speed that he called $c$, and which could be calculated from other physical constants that were known and measurable at the time.

Around that time, scientists were actively debating the nature of light. After Hertz demonstrated the existence of electromagnetic waves in a laboratory experiment, Maxwell's theory suddenly became the most favored explanation for light.

The funny thing about Maxwell's theory was, that the speed was relative to whoever measured it. If you and I both measured the speed of waves emanating from the same source, we should both get the same result, regardless of our motion relative to the source or, to each other.

Some physicists found that to be troubling, and they tried different ways to justify it. Einstein's big achievement was to combine their various ideas into a single consistent, rigorous, mathematical theory. One of the consequences of his theory---proven in the math---is that if anything has a characteristic speed that must be the same for every observer, then nobody can ever observe anything moving faster than that speed.

So, the universal speed limit is a mathematical consequence of certain measurable, and as far as anybody knows, fundamental physical constants; e.g., the permeability of free space.

Your question then comes down to, "why does the universe have those properties?"

Every time physicists answer a "why" question, the answer always rests on deeper levels of "why?"

• Actually, the equations show the speed as a function of the magnetic and electric constants. It's a further observation that these don't behave like a normal menium that the observer moves relative to, but stubbornly read the same value no matter your motion. – JDługosz Jan 21 '16 at 19:15
• "The funny thing about Maxwell's theory was, that the speed was relative to whoever measured it." Shouldn't that be: "wasn't relative to whom ever measured it", i.e. independent of the reference frame? – Ziezi Jan 22 '16 at 21:56
• the speed was relative: He means that the calculated speed, c, is taken to be the same value for any observer. It's the same sentence ambiguity that gave relativity its name, as Einstein notes the same issue with it in English. The speed of the wave is c relative to Alice, and the speed is c relative to Bob. – JDługosz Jan 23 '16 at 3:18
• This is pointing in exactly the right direction, and deserves to be far more highly upvoted than the current "top" answer. I would love it if you expanded this to include the basic points of the derivations (both Maxwell's, and Einstein's). – Floris Feb 12 '16 at 14:08
• @Floris, I'm only a tourist here. I would love to learn some physics some day. I spent more time that I like to admit working my way through "The Road To Reality" by Roger Penrose. My biggest takeaway was that physics is a big subject. That book basically is eleven hundred pages of brief introductions to some of the mathematical techniques and forms that one would have to learn in order to understand physics. Maybe when (if!) I retire... – Solomon Slow Feb 12 '16 at 15:25

This question has sparked some interesting answers, and I'd like to throw a contribution in as well. It should be perfectly clear that we are living in a world with a finite upper speed, and many answers have touched upon the consequences of and reasons for this.

I would, however, like to point out an aspect that seems to have been forgotten altogether in the other answers. If the speed of light were infinite, we would not have light at all.

To see this, take a look at Maxwell's equations again. Note that in them $c=\frac{1}{\sqrt{\mu_0\epsilon_0}}$, so if you set $c\to\infty$ then either (or both) of $\mu_0$ and $\epsilon_0$ would have to be zero. This will effectively kill the existence of dynamic magnetic fields.

Especially, for light, it means that $\nabla\times B=\frac1{c^2}\frac{\partial E}{\partial t}\to 0$, so magnetic fields would be static (and of zero intensity, remember no magnetic monopoles). Thus the only thing left of electromagnetism would be simply electrostatics.

Physically this also makes sense, if $c\to\infty$ then the electric field response to any rearrangement of charges would be instantaneous, so there is no place (time?) for a magnetic field response.

Also, thinking of magnetic fields (and especially the Lorentz force), it also makes sense that magnetic fields should vanish. If $c\to\infty$ there is no length contraction, and so there will be no Lorentz forces on any particles.

Thus when we speak of signals propagating infinitely fast, it's dubious what signals we are referring to.

• I think, your answer is the first, that really shows, what breaks down when $c$ tends to infinity, without actually invoking the circular argument of "causality because SR, and with infinite c no causality" etc. Thank you and +1 – LLlAMnYP Jan 25 '16 at 17:25
• To be honest, i don't really by the argument that $c\to\infty$ means no causality. This is just the Newtonian limit, where time is absolute, and this works fine as far as I'm aware. – Mikael Fremling Jan 27 '16 at 9:24
• Infinity is far greater than light speed. Why is light speed the same everywhere and all the time? Because space changes density, which is observed by redshift and such, but doesn't time (light speed) change as well? – Cees Timmerman Jan 28 '16 at 10:12
• @CeesTimmerman: I would say we are not sure why $c$ has the same value in all of spacetime. The strongest argument would be that of the general principle of relativity. However, I'm sure that if the speed of light was (spacetime)position-dependent, that you would get some weird effects, that would hopefully be detectable from earth. – Mikael Fremling Jan 28 '16 at 10:31
• Saying "speed of light cannot be infinite" is not at all the same as saying "nothing can go faster". – Floris Feb 12 '16 at 14:09

The existence of the speed limit is related to the existence of time [UPDATE: time is a measurement which is only available when $c$ is limited. If you do not agree, provide a way to measure time when $c$ is infinite before down voting]. If there'd be no speed limit, everything would happen instantly. Also, any waves in any matter would not be affected and spread momentarily. Time would disappear (as well as distance and, consequently, space, btw).

So, it is the same as "why is there time?". Instant energy transfer which is currently limited would change the world as we know it and it would not be this world that we know any more. The Newtonian physics would disappear as a concept, since the matter itself would not work like that any more. As well as the concept of form. The consequences would dawn on everything. However, we are not observing this, we observe the limit.

There is some inherent separation present in the Universal matter, which allows it to exist the way we know/perceive it. If there exists a world without the limit, we did not emerge in it, we appeared here.

Not exactly an answer, but there's nothing else to say

UPDATE

In response to @Davors comment:

It is hard to picture what would exactly happen because we can't be sure what is the actual underlying structure of the reality that makes up for the speed of light and how is it intertwined with the rest of things. That is - how would the other 3 forces form the matter if EM was instant. But lets explore some options that support the notion:

1. There are 4 forces, and if the EM force would transfer all energy instantly, then even if the 3 other forces still would hold, it would invalidate most structures bigger than atoms of the matter we see now. Since on macro level only gravity and EM do matter, and gravity will have no big sense in this scenario, then all processes that are conducted via EM forces will be instant.

2. No macro structure could exist, and with all EM processes going infinitely fast, there would be no possibility to know anything about electron states in the atoms. They will become infinitely every possible states. All possible absorption and emission will happen at once. Not sure even if atoms would hold up.

3. Try putting infinity instead of $c$ in all the relations and see what happens. Also, as all speeds can effectively be measured as a fraction of $c$, then if $c = \infty$, all other speeds will also be infinite no matter the fraction coefficient.

4. See the reply by @Nikos M.

• Comments are not for extended discussion; this conversation has been moved to chat. – David Z Jan 22 '16 at 10:20
• You're saying that if causality had no speed limit, then all processes would occur at infinite speed? – JDługosz Jan 23 '16 at 2:41

Why do we have a universal speed limit? Is there a more fundamental law that tells us why this is?

The more fundamental laws are causality and locality. Causality expresses the fact (or assumption) that effects cannot precede causes, and locality expresses the fact (or assumption) that fundamental causal relations are described by differential equations.

Given these two fundamental principles, the logic of mathematics dictates that the differential equations are either parabolic (heat equation like) or symmetric hyperbolic (wave equation like).

If they are parabolic, there is no speed limit. For example, according to the heat equation, heat propagates instantaneously to arbitrarily far places, though suppressed exponentially with distance.

If they are symmetric hyperbolic, mathematical theory implies a finite propagation speed. For example, this is the case for Maxwell's equations, which limits the speed of electromagnetic signals to a number called the speed of light.

It is an experimental fact that Nature behaves according to the second possibility - even independent of considerations of the speed of light. There is overwhelming evidence that all fundamental processes in Nature are of the symmetric hyperbolic kind. Even heat - the heat equation is just the simplest approximation, in which the speed limit is lost. But more sophisticated derivations from nonequilibrium statistical mechanics produce symmetric hyperbolic equations, which become parabolic only upon further approximation.

That the limiting speed is the speed of light is very likely but not necessarily the case. It is linked to the assumption that photons are massless. If photons were massive but gravitons are massless, the speed of light would be smaller than the theoretical limit of signal speeds in the universe - which would then be the speed of gravity.

However, according to the particle review of the Particle Data Group, the upper bounds on the mass of a photon are extremely tiny, and observations are currently in full agreement with the assumption of massless photons.

• Thank you! I was looking for this answer. I wanted to ask if causality and locality (hyperbolicity) are different in any way. If hyperbolicity provides an upper speed limit say the speed of light, it also implies the prohibition of superluminal transfer of information, and hence it implies causality. So, they are the same thing, no? – Nanashi No Gombe Mar 13 '18 at 11:42
• @NanashiNoGombe: Locality means describable by a differential equation. A parabolic equation is local but not causal. – Arnold Neumaier Mar 13 '18 at 12:12
• Thank you for your quick response. I was asking if hyperbolicity is the same thing as relativistic causality? I think it is. Please correct me if I am wrong. – Nanashi No Gombe Mar 13 '18 at 13:36
• @NanashiNoGombe: It is the same in classical relativistic physics. In quantum physics, the relation is more complicated. – Arnold Neumaier Mar 13 '18 at 14:48

$\hspace{50px}$
The above picture I drew to expand on Kostya's wonderful answer.

Basically, imagine people who measure height of buildings in degrees of angle of the buildings' visibility from the certain fixed distance. This is not at all unreasonable if you fix the distance C large enough compared to the building heights'. However, for taller buildings you'd notice that their angular height is not additive. Also maximal possible angular height is fixed at absolute value of 90 degrees.

This is very similar to the way humans measure speed: we picked a certain measure "distance/time" that makes sense for smaller speeds, but for higher speeds it's not additive. Also, there is the unreachable "maximal" speed, the speed of light.

However, the above problem is solely due to the wrong choice of measuring speed. The "right" choice measuring speed is "rapidity", as explained by Kostya. And rapidity is both additive and unlimited.

Do we know WHY there is a speed limit in our universe?

"Do we know WHY there is a length limit?"

The same way we need finite lengths to measure size or interval between two points in 3D - Euclidean space, we need a finite speed of light to measure the interval between events in 4D - Minkowski space. Minkowski developed his theory to expand Maxwell's equation in four dimensions. In order for $$s^2$$ in Minkowski space, (2), to preserve its invariance, as an extension of the Pythagoras' Theorem, (1), which in three dimensions is: $$s^2 = x^2 + y^2 + z^2 \tag{1} \,,$$

and in four dimensions becomes: $$s^2 = x^2 + y^2 + z^2 - {\left(ct\right)}^2 \tag{2} \,,$$ $$c$$ needs to be not only finite but same for all reference frames, which is supported by the observed evidence that light (EM waves) speed was independent of observes's frame of reference.

The speed of light needs to have a limit, i.e. to be finite for the Special Relativity to work:

If you go back to the famous A. Einstein's thought experiments, specifically the one where there are two observers, one not moving, $$A$$, standing on a train station and another, $$B$$, moving standing in a train, that passes through the train station.

Now, when the train passes and $$A$$ and $$B$$ are right opposite each other, lightning strikes on both sides of $$A$$, at the same distance. $$A$$ sees them simultaneously:

However, because $$B$$ is moving relative to them, i.e. away from one and towards the other it, sees them successively:

...right?

Well, NO, this would have resulted in the light being measured differently in different reference frames, something that was refuted by Michelson–Morley experiment that used Earth as the train:
$$\hspace{100px}$$,

and measured light speed in two perpendicular directions:

assuming that the light moving in the direction matching Earth's direction of movement would have to be smaller (similar to the person, $$B$$, who was on the train) than the other, assumption that was scientifically proved to be wrong.

Consequently, the speed of light is constant and anyone measuring it will find the same value, regardless of his speed or in other words speed of light is invariant. Invariance could be contrasted with relativity, for example, relativity of time, which by the way was used to describe why the two people, $$A$$ and $$B$$, observe the same speed of light, namely because the time of the moving person $$B$$ is ticking slower, in general, the larger the relative velocity between the two observers, the greater the difference of the ticking rate of their watches, i.e. time dilation.

Finally, time dilation could be observed in the presence of an object with mass that generates a gravitational field, or in terms of General Relativity in stretched space-time, that will cause the time of the observer located closer to the mass object to tick at a slower rate, i.e. time dilation and respectively, the observer located at a greater distance will observe his watch to tick with faster rate.

As you see time is relative, space stretches and speed of light is the constant, with a finite value that "holds them together" and "synchronizes them", defining event simultaneity. Moreover, with its help, we can define an invariant interval between two points in space-time, i.e. between two events. Space-time intervals depend on the temporal and spatial separations of the two points and they could be: time-like, light-like(time distance = space distance) or space-like(time distance < space distance). Thanks to that Special Relativity is a successful theory, with a long list of supporting experiment evidence.

## Edit:

As a response to the first comment, which argues that time dilation is a result of the constancy of light speed, I will present you an example of the opposite, i.e. constancy of light speed can be shown as a direct consequence of velocity time dilation:

Let us consider a hypothetical clock called photon clock. In it, light is reflected back and forth between its mirrors and whenever the light strikes a given mirror, the clock ticks once. If this clock is in inertial motion with respect to an observer, then velocity time dilation will cause it, like all other kinds of clocks, to tick slower. However, because the clock is moving, the light pulse will trace out a longer, angled path between the mirrors. The net result of velocity time dilation and increase in the path length is that the speed of light in the moving photon clock remains equal to the speed of light in the rest photon clock. In other words, the speed of light remains constant.

Moreover, Lorentz transformation (LT), which were derived by Joseph Larmor [1] in 1897, and Lorentz (1899, 1904) [2], directly predicted time dilation. In fact, time dilation by the Lorentz factor was correctly predicted by Joseph Larmor (1897)[3] long before Einstein published his paper in 1905.

Your question tends to be a bit Philosophical, for all we know the value of $$c$$ could be related with a property of space-time itself, determined along with the other Fundamental Physical Constants during the Big Bang, which is difficult to observe and realize similar to why fish is not aware of all the water around it.

[1] Larmor, J. (1897), “Upon a dynamical theory of the electric and luminiferous medium”, Philosophical Transactions of the Royal Society 190: 205-300.
[2] Lorentz, Hendrik Antoon (1899), “Simplified theory of electrical and optical phenomena in moving systems”, Proc. Acad. Science Amsterdam I: 427-443; and Lorentz, Hendrik Antoon (1904), “Electromagnetic phenomena in a system moving with any velocity less than that of light”, Proc. Acad. Science Amsterdam IV: 669-678.
[3] Larmor, J. (1897), “On a Dynamical Theory of the Electric and Luminiferous Medium, Part 3, Relations with material media”, Phil. Trans.Roy. Soc. 190: 205-300

I can sense from your question that you are looking for a simple and basic explanation without jargons. I will give it an honest shot and will keep it really simple and classical. I am a classical thinker, so, I do not even have any more complex explanation. Hope more qualified and accreditted users will not frown upon the answer.

Let me break the question into two parts -

(1) Why there is a certain speed of electromagnetic waves (which happens to be c)

Speed of any wave is property of the medium through which it travels. So, it is property of empty space that electromagnetic waves travel at a certain speed (no more, no less). It is a property, not a limit. If it was a limit, then light (or EM) could travel at < c through empty space. But it travels exactly at c, in empty space. So, it is a property. If you have problem with this explanation, then you should also have problem with - sound having a certain speed through air, and need to go to more basic level. If you accept a certain speed of sound, I will expect you to accept this property in case of EM waves as well. The value of this property happens to be c. james large has also indicated this, in his answer to your question on Jan 21.

(2) Why any material body can not move faster than c

This is a direct consequence of (1) and so, turns out to be a limit. (It is limit because bodies can move at any speed as long as it does not exceed c)

Let us consider how do we increase speed of a mass - we apply a force on it. For example, we can make a standing car move by pushing it with our hands. The electrons in our hands, and the ones in the car (where we touch it) repel each other and that repulsive force causes increase in the speed of the car. Suppose you were running at your maximum speed and a car passes by you at 300 miles/hour. Can you increase speed of the car by pushing with your hands when it passes by you? Answer is no (a normal human being can not move the hands faster than 300 miles/hr). To increase the speed of a moving body, the force has to act on it faster than the speed of the moving body.

Any force we apply to accelerate a body, the force is ultimately exerted on the body as one of the fundamental forces. All the fundamental forces themselves travel at c. Just as example, electromagnetic force propagate at the same speed as EM waves i.e. c. For simplicity let us agree that all fundamental forces propagate at c as a property per (1) Therefore they can not increase speed of any material body that is already moving at a speed of c to > c.

Note that the forces have to travel faster than c through space in order to cause a speed greater than c. But we know they move at c. So even the fundamental forces become ineffective for a body that is already moving at c.

The forces become ineffective in the direction of movement of the body at c. They are still effective in other directions and , so body can be slowed down etc.

Therefore, speed of propagation of fundamental forces is a property (not a limit). This property has highest value in empty space which happens to be c. And, nothing can propagate faster than the forces themselves - what will cause anything to do so? Speed of forces (a property just like speed of sound) turns out to be a speed limit for material bodies. Which is not a mystery at all.

May I also comment that in particle accelerators, they use the electric/magnetic fields to accelerate the particles and obviously can not accelerate them faster than c.

Please LMK if this explanation works for you.

• So the answer is "speed of forces" are not bigger than c? Then why is there an increase in mass of the particles when there is an increase in velocity? – Koray Jul 9 at 8:28
• How would you explain the force of gravity on a photon falling straight into a black hole? If the photon doesn't go faster than c then is that because gravity is not applying any force to it (e.g. it doesn't gain mass, energy, isn't blue shifted)? – Quantumplate Jul 25 at 10:13

A lot of the answers here are focusing on the wrong half of the problem, I think. They're telling you how we know there's a limit, rather than explaining why it has to be that way.

For the most part, there's nothing preventing the creation of a universe with infinite light speed that's otherwise similar to ours1. However, there is one important property such a universe would have to have: it has to be finite and/or non-homogeneous.

This is related to Olber's Paradox. Basically, it goes like this:

1. The light from a star at a given distance is inversely proportional to the square of distance. $$L\propto{1\over D^2}$$.
2. The number of stars at a given distance is directly proportional to the square of distance. $$N\propto D^2$$.
3. The total light from stars at a given distance is equal to the light per star times the number of stars. $$T\text{light}$$ $$=LN\text{light}$$ $$=L{\text{light}\over \text{star}}N\text{stars}$$. ("light" and "star(s)" are units here.)
4. Therefore the total light from one distance is the same as the light from any other distance. $$T\propto LN$$ $$\propto D^2{1\over D^2}$$ $$\propto 1$$.
5. If stars are homogeneously distributed through the universe, and the universe is infinite, we can separate the stars into an infinite number of shells, with each shell having a finite, constant brightness. The sum of all this light is infinite. $$\sum_{D=0}^{\infty}T=\infty$$.

In the real universe, this isn't a problem because of two effects: first, the expansion of the universe means every star in the universe is receding from us (on average), and the rate of recession is directly proportional to the star's distance; and second, because the finite speed of light means light from distant stars takes longer to reach us than it should, due to said expansion.

Combined, this means the light that reaches us per second from any given shell of stars goes down linearly with distance. $$L\propto{1\over D}$$. Also, this means there's a finite distance where all objects at that distance are receding from us faster than the speed of light, so we'll never see light from beyond that distance (effectively making the universe finite). $$\sum_{D=0}^{N,N<\infty}{T\over D}<\infty$$.

But if light traveled at infinite speed, expansion wouldn't help. It would mean the brightness of the universe is slowly decreasing (stars are more spread out, so the brightness per shell is lower), but a decrease in infinite brightness doesn't help much. So the universe would have to be finite in size and/or the stars would have to be less dense the further you get from the center.

Alternately, we could posit some universe where light attenuates more quickly. $$L\propto{1\over D^3}$$ or something. But that no longer makes geometric sense and requires extra fiddling to work. Perhaps the light is somehow absorbed by the expansion mechanism, with the light being absorbed more readily as its power is near zero. Infinite space means proportionally-infinite light absorption results in a finite expansion rate, and increases light attenuation. But that's kind of hand-wavium crap I made up and belongs in worldbuilding more than current physics.

Even if the universe were finite, note that the brightness of all the stars in our own galaxy would far outshine the sun. So we really can't have a universe almost identical to ours unless light has a finite speed.

Also, note that I don't know anything about quantum mechanics or how light-speed affects that branch of science. It's possible current quantum mechanics couldn't exist with infinite light-speed, which you might declare means any universe with infinite light-speed is totally different from ours. 1 However, I consider "similar to our universe" to mean any universe with particle physics that allow the formation of planets, stars, neural pathways etc., at a macro level that a typical human would recognize as similar. I don't care if "gold" has more protons than "hydrogen", etc., much less quantum effects.

• Check out the physics of Greg Egan's orthogonal series. Your arguments presume a fixed global time like Galileo, which is not the only solution. Egan has 4 dimensions with time being a relative direction: infinite speeds (for orthogonal observers' recerence frame) and no Olber's catastrophe. – JDługosz Jan 23 '16 at 3:11
• @JDługosz no Olbers' paradox, but some pretty weird stuff besides! That's a great text - I'm not usually a fan of SciFi but Orthogonal did it for me. – WetSavannaAnimal May 31 '16 at 1:33

This question has a very short answer but uses the assumption that all relativity uses. i.e. The speed of light is constant for all observers.

Based on this assumption it is trivial to show that an event horrison is observed at a velocity of c when trying to accelerate infinitely.

To answer why this assumption is valid you have to look at the derivation of Maxwell's equations which show the propogation speed of electromagnetic waves to be independent of reference frame. The derivation of these equations is founded in concepts such as "Conservation of charge" and Faraday's law. I don't believe the question can be answered any deeper than that.

tldr: Through conservation of energy => charge and emperical observations of electromagnetic interactions one can show that the speed of light is independant of inertial frame. This can be used to proove that there is a universal speed limit of c.

As you have read from other answers, it's not an easy thing to explain. It seems so counter-intuitive. "If I want to go faster, why not just accelerate more?" or "If my speed is very near the speed of light, then I shoot a bullet, won't it be going faster than light? Why doesn't it?"

## Relativity

Let's start by refining what we mean by "speed limit". Let's say you're in a space ship with an infinite fuel source and an arbitrary acceleration potential. You are at rest (docked at a space station), and you want to travel to a star system 10 light-years away. How long will it take you? In this hypothetical space ship, you could get there in 10 MINUTES, not years (assuming the acceleration doesn't scramble your delicate human body).

But isn't this a violation of the cosmic speed limit? Nope! Technically, you're not traveling faster than light. From your point of view, it appears as though space is flattening and your destination is getting closer to you. If someone at the space station was watching you embark, from their point of view you would be traveling very near the speed of light, but they would only see you arrive at your destination shy of 10 years into the future.

Now let's take this a step further, and imagine that you are a beam of light traveling through space. From your point of view, how long does it take before you interact with something? No time at all. A beam of light will instantly teleport from the source to a destination without a passage of time. But of course the same relativity principals apply here - an outside observer will not experience this instant teleportation.

## Non-Relativity

Now imagine an alternate universe where there is no cosmic speed limit. First, light would travel instantaneously. So when we look up at the sky, we would see other stars and planets exactly how they are right now. We could travel to and from any location in the universe in an arbitrarily small amount of time. Seems realistic right?

The problem is what happens at a smaller scale. Imagine an atomic process - like the one in our Sun - in this hypothetical universe. The core of the sun is about 15 million degrees Celsius (remember - temperature is related to Kinetic energy). The sun is about 4.6 light-seconds across.

A relativistic speed of light acts as a throttle, preventing these atomic chain reactions from happening too quickly. It helps limit how hot something can be (by increasing the mass of particles that move very fast, to prevent them from moving too quickly), and how fast reactions can happen (fractions of a second for energy to travel vs. instantaneous, which is a HUGE difference). This could mean that stars explode too quickly to even form. It could also mean that the energy that fuels the reactions escapes the sun too quickly and doesn't give it time to react. I'm not sure which would happen, but either way the results are catastrophic.

## Conclusion

The "cosmic speed limit" is an important aspect of our universe. While it is conceivable to design a universe with no relative speed limit, the results wouldn't be very interesting. Our intuition tells us this should be possible, but sometimes our intuition about things we don't fully understand isn't very good.

The question of "why is there a cosmic speed limit?" is as fundamental as "why does the universe contain more matter than antimatter?" or "why does magnetism exist?". The question should be restated, "Why do we live in a universe with these characteristics?" Or "Could we live in a universe with different characteristics?" Because it is possible that other universes with different characteristics DO exist, and only a small portion of them can actually sustain life. If human life exists, naturally it will do so in a universe that can sustain it.

• This post examines some of the consequences of having a speed limit, but does not really touch the why, except for the "classical" it has to be this way, otherwise we wouldn't exist. In a sense, that is as good an answer as any I suppose. – Mikael Fremling Jan 21 '16 at 16:36
• It's like asking "Why is the Earth 93 million miles from the Sun?" The answer is, nobody knows exactly. It just kind of happened that way. The question itself is flawed - it's completely irrelevant. The real question is, "Could we even exist to ask the question if Earth was some other distance?" – James Watkins Jan 21 '16 at 18:20
• You said "If someone at the space station was watching you embark, from their point of view you would be traveling very near the speed of light" and they would not "see" you embark until 10 min before you arrived, right? Otherwise it is absurd. I don't see light "embark" until the moment it arrives. – user95006 Jan 21 '16 at 19:20
• @nocomprende Not sure what you mean. I'm not talking about the light embarking, I'm talking about the ship. Any observer would witness the ship taking about 10 years to reach the destination, but the person on the ship would witness only 10 minutes passing. – James Watkins Jan 21 '16 at 19:29
• "we would see other stars and planets exactly how they are right now. We could travel to and from any location in the universe without any passage of time." - wtf? how does the former imply the latter? – immibis Jan 22 '16 at 0:28

In contrast to the other answers I'll try to give a simple answer.

First, be aware that "Why" is a poor question for modern science as modern science prefers to predict "what" will happen as accurately as possible using "models" of what they guess reality is doing.

Speed and Time are heavily inter-related and are effectively under the same "speed limit". When you are going as slow as possible through space then you are going as fast as possible through time and visa-versa.

Our observable universe appears to have a universal "space-time" limit. This limit is part of the interconnected "fundamental constants" of our universe.

The interesting thing is that if any of them were altered in any significant way our current models predict very different versions of reality in which it is very unlikely sentient life would come into existence (please excuse the wild speculation here) to be able to ask this question.

This is called the Anthropic principle

Do we know WHY there is a speed limit in our universe?

is "yes, because we're lucky enough to live in a universe those apparently random speed limit allowed sentient life to appear"

There may be many such universes, possibly with different fundamental constants to ours, possibly with radically different forms of sentient life.

The real PHYSICS questions behind all this philosophy include

• "Are there other universes ?"
• "What can we know about them ?"
• "What are the probability distributions of fundamental constants across those universes ?"
• etc etc etc

Sadly I don't think any practical scientific experiments have been proposed to test these models yet ?

Thus, this topic is more philosophy rather than physics, so your question should probably be closed as off-topic ?

Wording updated in response to comment.

• FYI the anthropic principle is speculation. It is used as extra-factor in order to reduce the zoo, string theory has found itself into, with no good solution. If the universe was different then life would be different, simple as that, no reason to feel luckier than necessary. Note the difference between different forms of life vs no life – Nikos M. Jan 21 '16 at 22:01
• Some say the anthropic principle is simply put, the following: "The universe is such and such so cosmologists can observe it". Not good for my standards – Nikos M. Jan 21 '16 at 22:04
• The anthropic principle doesn't say "this is the only possible universe life could evolve in". It simply says "if this universe couldn't evolve life, nobody would be here to ask why not". It's an argument against "the universe was specially designed for us", not against the existence of alternate universes. – MichaelS Jan 22 '16 at 2:52
• "When you are going as slow as possible through space then you are going as fast as possible through time and visa-versa." Is there any well-known equation which models this relationship? – James Watkins Jun 3 '16 at 16:42
• @JamesWatkins Yes, the Lorentz equation decribes Time Dilation in the context of Special Relativity. see simple.wikipedia.org/wiki/Time_dilation – John McNamara Jun 8 '16 at 18:12

One can say that this is just an experimental result. That light (and other signals / interactions) do not travel with infinite velocity / speed.

One can leave it there and say that is how it is.

One can also say, look if you take that variable and do that transformation (e.g rapidity) it can be added ad-infinitum, so the question is around the correct variable to use. Although this just bypasses the actual question instead of addressing it.

i will take another approach and addres the question directly to the heart.

Finite velocity of signal transmission (or interaction) is a basic requirement for causality to hold.

Else if signal transmission can be infinite, an effect may outrun its own cause and the resulting causal loops make causality cease to be as causality. Something that is also an experimental fact but of even more basic level. In this sense this is the answer to the question.

One may take this even further and derive a (upper-limit) finite velocity transmission directly from thermodynamics considerations sth that is outside the scope of this question, but mention it for further study.

However, finite velocity of signal transmission is not necesarily equivalent to the Special Relativity postulate that the speed of light is this maximum (and constant) available velocity.

One can have many different finite velocities of transmission less than or even greater than the speed of light depending on the process under study.

In fact, there is some research into faster-than-light signal transmission through quantum entanglement. But i will just leave this at this point

• Suppose the universe followed Galilean relativity. Then in what way would causality be violated? It already follows Galilean relativity approximately at low speeds, yet we don't violate causality at low speeds. – immibis Jan 22 '16 at 0:27
• If information could be transmitted instantaneously between objects at arbitrary distance, how would that abolish causality? If we were unable to determine the order of cause and effect due to not having a nice duration gap between the two, would that stop causality or only interfere with our perception of causality? – Corey Jan 22 '16 at 0:27
• @Corey: Causality isn't violated. People are just too hung up on SR equations that predict such violations based on what we currently know about our universe. Information still wouldn't be instantaneous. Physical structures would be made of massive particles that would have finite speeds, so there would always be some lag between transmission, reception, comprehension, response, etc. Also, the effect could never reach the cause at any moment prior to the cause happening, so causality is still not violated. – MichaelS Jan 22 '16 at 3:03
• @MichaelS It just confuses me when people make claims about causality breaking down under some set of conditions that don't appear to actually lead to any conflict with causality. Maybe it's how they think about time? – Corey Jan 22 '16 at 3:09
• @immibis, the argument is moot since galilean reltivity or not does not exclude finite signal transmission. You confuse finite signal transmisiion with maximum and constant velocity of special relativity. In the sense of this post special relativity is just that, a special case. No more no less. In fact through thermodynamics finite signal velocities (or characteristic velocities can be derived with no assumtion of special relativity). – Nikos M. Jan 22 '16 at 15:48

Well, it's possible to prove (theoretically, and I advice you for Feigenbaum, 2008) that the homogeneity and isotropy of the space and the homogeneity of time lead necessarily to the existence of a speed limit.

Let's do that: imagine about taking the Universe and delete (remove) every kind od object. You remain only with the spacetime itself. In this spacetime there is nothing, no matter no energy.

Now: is it reasonable to think that, in an empty spacetime, there is a single point which is privileged than the other? No, so the empty spacetimes is homogeneous.

Is it reasonable to think that, in an empty spacetime, a single direction is privileged than the others? Nope, so the spacetime is also isotrope.

From those two assumptions, Feigenbaum shows the existence of a limit velocity. On the other way, it's also interesting to notice that in Einstein's theory, the existence of a limit velocity is an axiom. However, this is, in a certain sense, unnecessary. Indeed assuming less things (like only homogeneity and isotropy) it can be shown that there has to exist a limit velocity.

Now, the fact that THAT limit velocity is the speed of light's one it's a question which Feigenbaum's theory does not either show or prove. This fact has to be fixed by an experiment!

Last Caveat

What does really remain true in a spacetime full of matter? All I wrote above still runs and it's valid, but only locally, namely in really small areas of the spacetime, and area by area.

Globally it has no meaning to state/say that there exists a limit velocity (because the concept itself of a global velocity is ill-defined) and it may happen that sometimes moves with a superluminal velocity. For example: far galaxies move far away from us with velocities ways greater than the speed of light.

• Thanks for your answer, but how did you go from isotropic and homogeneous universe to a speed limit?(i did not completely follow your argument) – TheQuantumMan Jan 23 '16 at 10:28
• @LandosAdam this kind of follows from Galilean relativity. Once you demand that physics are the same in all reference frames, you quickly come to the demand that, say, lightspeed is the same in all reference frames, and that leads to Lorenz invariance and SR. But what if there was indeed a priviliged reference frame? Not a lot of stuff is moving at relativistic speeds relative to us, so I don't see, why a prviliged frame of reference should break much stuff. – LLlAMnYP Jan 23 '16 at 13:45
• would be good to provide the actual reference to the Feigenbaum article. Note that homogeneity and isotropy may be used to derive a speed limit but are not basic in the sense that space-time with absolutely no matter and energy maybe isotropic and homogeneous but does not mean much. On the other hand WITH matter and energy isotropy and homogeneity do not necesarily hold.One can also derive speed limit from the principle of relativity (note, not the theories of relativity), which itself can also be derived from causality – Nikos M. Jan 24 '16 at 12:02
• I have added the link to Feigenbaum's paper on arxiv. However the answer should also contain the crucial part of the mathematical derivation from the paper, otherwise it is a link-only answer. – mpv Jan 25 '16 at 16:26
• You might want to take a look at arxiv.org/abs/1209.0563 and at the Einstein digital papers here and here. Space is not homogeneous, and the speed of light varies. – John Duffield Mar 7 '16 at 22:47

It is not particularly unusual for physical systems to have speed limits.

Consider the classic vibrating elastic string, defined by the equation

$$\frac{\partial ^2}{\partial t^2}\,y(x, t) = -a\frac{\partial ^2}{\partial x^2}\,y(x, t)$$

Using that equation, you can see that a small disturbance in one part of the string will propagate outwards at a particular speed. In fact you will see that the rate at which any disturbance can travel along the string is limited by that speed.

You can make intuitive sense of this speed by imagining that the string is made up of little beads joined by elastic threads, and the signal has to propagate by going from bead to bead, which limits its speed.

You will see a similar phenomenon in the differential equations for a 3D elastic solid (like a cube of jello). Also the differential equations for an electric signal in a wire, or an electromagnetic wave in space, or a sound wave in the air.

Pretty much any system that can be described by a differential equation that relates the rate of change over time to a local property like the derivative or density will end up having a speed limit. And it is pretty common for physical systems to behave that way, because most things in the world are made up of smaller parts, and the macroscopic behavior of the system can be analyzed in terms of the behavior of the smaller parts.

I realize that this does not at all answer why. I just want to point out that having a speed limit isn't such an unusual or surprising thing.

To put it simply, it is Nature's way to preserve causality. From Wikipedia:

"On the other hand, if signals could move faster than the speed of light, this would violate causality because it would allow a signal to be sent across spacelike intervals, which means that at least to some inertial observers the signal would travel backward in time. For this reason, special relativity does not allow communication faster than the speed of light."

If there wasn't a speed limit, every sort of paradox involving violation of causality would be possible (maybe you've heard about the Grandfather paradox?).

• But, this answer, like most others, has to do with the effect of the limitation and not the reason behind it.. – TheQuantumMan May 12 '16 at 1:38
• Then I don't think an answer is possible. This is like asking "why is force equal mass times acceleration?", or "why is quantum mechanics governed by Schroedinger's equation?"... – valerio May 12 '16 at 13:27
• Well, i did not really think that there would be an answer. I just asked just in case there is an answer because i am just in my first steps as a physicist, so there are a lot of thing i do not know and not able to derive(if they are derivable)! :) – TheQuantumMan May 12 '16 at 21:31

When an electrically charged body moves relative an observer an induced magnetical field could be measured by the observer. The energy stored in this magnetic field tends to infinity while the velocity of the moving body come closer to c.

• This is not entirely intuitive with general relativity... Are you also saying that two charged bodies heading at each other at .9c will have a larger induced magnetic field for a stationary observer present at the point they meet each other than each traveler observes? (Is it just 2x for the external observer, or does it go crazy because the delta speed between the two approaching objects is 1.8c?) – BenPen Sep 20 '16 at 20:05
• @BenPen, all depends on the movement of the observer. And the magnetic field is different in different systems of inertia, although all systems are "housed in the same universe". – Lehs Sep 21 '16 at 5:25

In GR, the speed of light is not constant, it varies with the curvature of space--time. So the constancy of this universal speed depends on space--time's having constant curvature. Which it doesn't, but this is locally a useful approximation, and in order to address the OP's intention, we will from now on assume that the Universe is a space of constant curvature. In any case, we know the speed of light is slower when the curvature is greater, so if we are looking for the limit, we have to consider the case of constant zero curvature, since everywhere else it will be slower.

Now, for simplicity, assume this curvature is zero.

It is experimentally observed that mass is equivalent to energy, so they have the same units. But the additional mass produced by the kinetic energy of a veolcity v is ${1 \over 2} m v^2$ so v must be dimensionless. Therefore, there is a coordinate system for space--time in which the x coordinates have the same units as the t coordinates. Since the manifold is completely flat, practically Euclidean (except for the -1 in the signature of the metric), we can choose a coordinate system which is, naively speaking, the same everywhere. So a space-like direction can be rotated to a time-like direction in the same way, uniformly, everywhere. (This may sound like SR, but it's not yet SR. This is simply a dimensional analysis plus simple geometry plus that one experimental fact of the equivalence of mass with energy). But then we have a universal speed, this same conversion between the x-coordinate and the t-coordinate.

So far, this does not say the speed is a speed limit, nor that it has to do with light. But it is canonical and intrinsic and "physical" since it depends on the conversion ratio between mass and energy.

The next step is to deduce that this is a universal speed limit. That is done as usual, since acceleration increases the mass of the object and so an exactly quantitative "diminishing returns" applies.

So everything we want follows from Newton's relation between mass, kinetic energy, and velocity, plus the one experimental fact of the mass-energy equivalence.

Note: William Davidon somewhere published a note showing how all of SR followed from the mass--energy equivalence. I didn't read it, but just hearing the fact that he did it has, obviously, clued me in to this. So one must acknowledge that "priority".

There cannot be a very basic philosophical reason why mass has to be equivalent to energy, since theoretical physics is possible in a Galilean way, where it is not true. On the other hand, philosophically, one could always consider the Galilean case as included in this framework in the sense that $\infty$ is a universal constant and a universal speed limit, too, with equal rights as 1. (It is zero as a universal speed limit that could never be accepted in physics....not even in theory.)

• Dear joseph f. johnson. It is often frown upon to post nearly identical answers to similar posts. In such cases, it is often better to just flag/comment about duplicate questions, so they can get closed. – Qmechanic Feb 13 '16 at 15:50

The question in the heading was

Do we know why there is a speed limit in our universe?”

Then there was an amplification

This question is about why we have a universal speed limit (the speed of light in vacuum). Is there a more fundamental law that tells us why this is? I'm not asking why the speed limit is equal to c and not something else, but why there is a limit at all.

I think that of all the answers @Anna_v has come closest to answering the question.

At present the answer to the question is “No”.

The idea that there is a universal speed limit comes from observations of the Universe.
These observations result in some theories which can be used to make predictions about the Universe.
A number of these theories which are good at making predictions have the idea that there is a universal speed limit.
Put another way, the universal speed limit is a useful postulate because it makes some of the theories “work”.
Present theories are not able to predict everything that has happened and will happen and so scientist look for better theories.

Whether there is a theory which will explain why there is a universal speed limit is unknown.

At present such a theory does not exist.

Some constant c with dimensions of a velocity is necessary because boosts do not commute, and therefore boosts must be done by dimensionless (mathematical) radians. The constant c converts velocities to radians. The constant c can not be infinite because that would make boosts commute.

Giving an object a velocity (boosting) in the x-direction does not commute with boosting in the y-direction. Boosts and rotations empirically obey the definition of a group. Special relativity discovered that boosts are members of the non-abelian Lorentz group. For $\frac{v}{c} \ll 1$ it is true that

$$\operatorname{Boost}{\left(\frac{v_x}{c}\right)} \, \operatorname{Boost}{\left(\frac{v_y}{c}\right)} - \operatorname{Boost}{\left(\frac{v_y}{c}\right)} \, \operatorname{Boost}{\left(\frac{v_x}{c}\right)} = \operatorname{Rotation}_{z}{\left(\frac{v_x}{c}\frac{v_y}{c}\right)} \,.$$

There must be a constant $c$ with dimensions of velocity to make the boost parameters dimensionless radians so that their product (the angle of rotation about the $z$-axis) can also be in dimensionless radians. It is okay that $radians^2=radians$ as evidenced by the terms in the power series expansion of $\sin{\left(\theta\right)}$. It is nonsense to do a rotation about the $z$-axis by ${\left(\frac{\mathrm{m}}{\mathrm{s}}\right)}^{2}$.

If $c \to \infty$, then the boosts would commute, and they would no longer be part of the Lorentz group. The constant c is similar to the constant $a={\left(\frac{180}{\pi}\right)}\,\mathrm{degrees}$ which is used to convert angles $\Theta$ from degrees to radians. The rotation group (which is a subgroup of the Lorentz group) is not abelian

$$\small{ \operatorname{Rotation}{\left(\frac{\Theta_x}{a}\right)} \, \operatorname{Rotation}{\left(\frac{\Theta_y}{a}\right)} - \operatorname{Rotation}{\left(\frac{\Theta_y}{a}\right)} \, \operatorname{Rotation}{\left(\frac{\Theta_x}{a}\right)} = \operatorname{Rotation}_z{\left(\frac{\Theta_x}{a}\frac{\Theta_y}{a}\right)} } \,.$$

The "$a$" is necessary. It would be nonsense to do a rotation about the $z$-axis by $\left[\text{degrees}\right]^2$. If $a \to \infty ,$ then the rotations would commute, and they would no longer be part of the Lorentz group. If rotations commuted, our world would be very different. There would be no such thing as rotating an object by a nonzero angle and having it come back to its original orientation. Also, angular momentum would not be quantized and particles would have no spin.

In summary, c (and a) are necessary and must be finite because boosts (and rotations) are part of the non-abelian Lorentz group. This group is where the boost parameter $\left[\text{rapidity}\right]= \tanh^{-1}{\left(\frac{v}{c}\right)}$ comes from in Kostya's answer.

• Nice approach to the topic, but it does posit the conclusion before reaching it, implicitly. Assuming spacetime symmetries to satisfy the Lorentz group amounts to the contancy of the speed of light... – G. Bergeron Jun 9 '18 at 1:40
• We are used to the historical path to SR in which the spacetime symmetry invariance of the Minkowski metric was discovered first and a c was needed to give ct the same dimension as x. That these symmetry transformations form the Lorentz Group we learned next as deriving from that symmetry. – Gary Godfrey Jun 9 '18 at 4:15
• ...However, suppose we first found empirically that the product of boosts was not just adding velocities but caused some rotation and that boosts don’t commute. By seeing how products behave we find that we can identify the group elements of O(3,1) with rotations and boosts and successfully make our symbols on a piece of paper copy the real world observations. Only then we see the line element $ds^2=dx^2+dy^2+dz^2-(cdt)^2$ is left invariant by the group. Now we might say the group is the fundamental reason a c is needed. The symmetry invariance of the metric is then the secondary result. – Gary Godfrey Jun 9 '18 at 4:17
• The group of what? ... Of spatial and/or temporal transformations that maps physical systems to physical systems isomorphically (with respect to the physics). This is what I meant by spacetime symmetries. Now, my initial point is that assuming (or observing) this structure directly implies the constancy of the speed of light. The question on the speed limit now becomes why is this particular group structure exhibited by reality. – G. Bergeron Jun 13 '18 at 18:39

I've noticed that the closer one gets to fundamental physical theories, the ones that describe the most basic interactions in our universe, the more the equations all start to look like coordinate transformations. Sometimes these coordinates are in abstract spaces – the groups of the Standard Model of particle physics and the Hilbert spaces of quantum mechanics – but, ultimately, physics is a description of the motion of things.

In order to locate something in the universe, you need both position and time. Now, even if you are sitting still while reading this, you are moving through time. The rate at which you move through time is one second per second according to your own watch, but not everyone will agree with that. Let's figure out how fast you're going according to any observer.

Lets send you on a trip to the Alpha Centauri star system at a large fraction of the speed of light. A resident at your destination watches your journey and sees that you traveled a distance of $d$ (about 4 light-years). According to the watch you've been wearing, you've aged by a time $t$, which is less than the time the Alpha Centaurian measured your trip to be due to time dilation. To find your total travel through spacetime, we can combine your trip in two dimensions with the Pythagorean theorem: $$x = \sqrt{d^2 + t^2}.$$ The total distance you traveled in space and time is $x$; the distance you traveled in space is $d$; and the distance you traveled in time is $t$ (which is equivalent to saying how much you aged). The problem with this equation is that $d$ and $t$ are in different units: meters and seconds. Luckily, Einstein's relativity provides a conversion factor: the speed of light. So, the equation should read: $$x = \sqrt{d^2 + {\left(ct\right)}^2}.$$

Now, the distance you traveled is equal to the spaceship's velocity times the time of the trip as measured by the Alpha Centurion (Distance measured in the rest frame of two points is called the proper distance. Proper time is measured by a clock at rest with the entity being timed, namely your watch.). Let's call the time elapsed in Alpha Centauri $t_\alpha$. $$x = \sqrt{{\left(vt_\alpha\right)}^2 + {\left(ct\right)}^2}.$$ We can relate $t$ and $t_\alpha$ with the time dilation equation: $$t = \frac{t_\alpha}{\gamma} = t_\alpha\sqrt{1-{\left(\frac{v}{c}\right)}^2}$$ where $\gamma$ is the relativistic factor that appears in nearly all relativistic equations. Notice that $t_\alpha$ is smaller than $t$ to reflect the slower aging that fast-moving objects (you) undergo.

So, now we have $$x = \sqrt{{\left(vt_\alpha\right)}^2 + \left(ct_\alpha\sqrt{1-\left(\frac{v}{c}\right)^2}\right)^2}.$$ Simplifying: \begin{align} x &= \sqrt{v^2t_\alpha^2 + c^2t_\alpha^2 \left(1-\left( \frac{v}{c} \right)^2\right)} \\[5px] &= t_\alpha\sqrt{v^2 + c^2 \left(1-\left(\frac{v}{c}\right)^2\right)} \\[5px] &= t_\alpha\sqrt{v^2 + c^2 - v^2} \\[5px] &= t_\alpha\sqrt{c^2} \\[5px] &= ct_\alpha\,. \end{align} The total distance you traveled through space and time is equal to the speed of light times your travel time. This is true no matter what your speed is. Thus, when you take your motion through space and time together, you are always moving at the speed of light! Thinking that different object travel at different velocities ignores their motion through time. So, the speed of light isn't only a maximum speed. It's also a minimum speed. You could say it's the only speed.

A consequence of this is that the faster you move through space, the slower you move through time and vice versa. You can picture this situation as if you are driving in a car with no accelerator pedal and no brake pedal – just a steering wheel. It always travels at the same speed. If you want to drive east, you have to sacrifice some speed in the northerly direction. In the same way, if you want to move through space, you have to sacrifice some speed through time. In fact, the math works out the same for relativity if you picture one axis as space and the other as time as I did in the derivation above.

Now, is there a fundamental reason to answer why this is so? The best I can come up with is to observe that we have no control over the rate at which we age. One year for you is exactly the same year for me (unless some advanced spacecraft are invented soon). If time is not so different from space, as it seems to be in our universe, then, like time, travel through space would also be limited to a certain speed. The base fact about our universe that sets the speed of light limit is the interdependence of motion through space and time.

OK, we are talking about speeds, and the fact that there is a limit to speed. Thus we are talking about motion. Motion contains two variables. One is speed, and the other is distance. Variables range from zero to infinity. Thus to look at the biggest picture possible concerning motion, one would naturally push both of these variables to infinity.

To travel at an infinite "speed", means to travel across any distance, in zero time. Meaning, one could travel from point A to point B in let's say 1 minute, but that also means that you could travel faster and complete the trip in less than say just 1 second. The faster you go, the less time is required. These are finite speeds. But if you travel from point A to point B in no time at all, then there is no way that you can beat this particular speed. This is the infinite speed.

Next. To travel across an infinite "distance", means that you will go on forever since there is no end to an infinite distance. Thus if you combine the two and travel across an infinite distance at an infinite speed, this means that you will go on forever, in no time at all. Holistically, this is simply not possible. However, relativistically, it is possible, and it is possible since the two extremes become separate.

In one possible extreme, you may move across space but will not be moving across time. In the other possible extreme, you will be moving across time but not across space. To make this transition possible, if you gain in one, you must lose in the other. Thus you can't have both extremes at the same time. Thus they in turn must be encompassed within a finite to cause this gain and loss phenomena.

In turn, you may be moving across space but not across time, while for those observing you, time for them is still ticking, and thus even though time is at a standstill for you, time may go on forever, elsewhere. Thus you may go on forever, in no time at all.

Thus finite motion across space-time is a requirement to make motion possible. Thus there is a finite limit to speed of motion across space.

If you then analyze the outcome of this phenomena, you independently discover Special Relativity, and independently derive all of its mathematical equations. See this video for verification.

• Please, if down voting, provide an explanation. After all, my logical thinking concerning a simple analysis of "Motion" had led me to my independent discovery of the SR phenomena, and to my independent deriving of the SR equations, and I did this in a manner that no one else has thought of as of yet. If my logic is wrong, then there is no speed limit, and the SR equations are also wrong. Granted, I have received no prior physics education at all. But I have learnt that if you discover SR on your own, you are considered to be a backward person. This may relate to any negative votes appearing. – Sean Jun 9 '18 at 0:17
• I'm not the downvoter, but I guess the general idea is that your explanation is fuzzy (not rigorous enough to derive any claim). Here's why: saying that ''[If you] travel across an infinite distance at an infinite speed, this means that you will go on forever, in no time at all.'' is false. You're canceling infinities without specifying the asymptotic behavior of your quantities. It amounts to saying $0 \cdot \infty = \infty$, which is undefined. Second, you implicitly assume the conclusion when stating your two extremes, by already taking time to be akin to a spatial dimension. – G. Bergeron Jun 9 '18 at 2:01
• G. Bergeron - Well you could always watch my YT videos and see what the outcome is. They are accessible via my network profile. – Sean Jun 16 '18 at 4:34

The amount of mass an object has = how resistant to change in momentum that object is. The more mass, the more energy required to change momentum. Since photons are massless, they have no resistance to changes in momentum. Photons are essentially the speed that energy propagates through the universe. As for why it's c, that's just how the universe happens to work out. There's no law that something can't move faster, but if we consider the physical world to be the duality of energy/mass, then nothing physical can go faster than light.

The reason there is a speed limit in our Universe, is that we have nothing (no energy/force) that can help us move something faster than the highest speed available to us $\left(c\right)$.

In other words, if the speed of $c$ were $2 {\cdot} {10}^{8} \frac{\mathrm{m}}{\mathrm{s}}$ (or $4 {\cdot} {10}^{8} \frac{\mathrm{m}}{\mathrm{s}}$), then this would be the Universe's speed limit. The best that we could do, would be to match the speed of $c$.

The speed of light applies to more than just light, nothing can propagate faster, including forces that act on matter. There's a non-traditional answer for why, though. The speed of light is a maximum speed because we are all traveling in a wave carrying medium, spacetime. Light, as a wave in the medium, travels at a constant speed in it; and like everything else in smooth travel in a "fluid", we move slower than a wave traveling in the fluid of space time, as observed with objects in water or air or waves in solids. (Solids are a special case, since only waves travel through them.)

The analogy hints that there may not be an absolute speed limit, but everything we have observed travels through the "fluid" of spacetime and can't be used to accelerate past the speed of light, so we're stuck. There are classically two cases for Acceleration.

1) The thing that's accelerating has mass:

We have observed that the mass, as observed by an external observer, varies as a ratio of $$\frac{1}{\sqrt{ 1 - v^2 / c^2}}$$ So, our observation says that mass goes to infinity as v approaches c, but we don't know "why," exactly, but, it acts like increasing resistance to travel through a "medium" or diminishing returns trying to add momentum to the traveling object. As a function of the energy required increase velocity goes, your yield in velocity increase gets less and less because the resistance to movement, "mass," gets larger and larger and eventually, because of "mass," energy (which is waves or matter on the same medium as the object we're accelerating, so it must travel slower than the speed of light) eventually gets so ineffective as to not really cause an effect for any known source of energy.

2) The thing that's accelerating has no mass:

The only thing we know of currently that has no mass, but is measurable in some way is a wave/particle thingie, and it can't accelerate, it just travels through the fluid of spacetime. Gravity, EM like light, fundamental forces all propagate as a wave on a media. And again, that media is "Spacetime" but we don't know what it's made of. Other waves travel through matter, and there's a wave speed that's measurable in all of them. BUT, it is travel in matter, within spacetime.

3) The third case of accelerating matter past the speed of light is interesting speculation, but ultimately futile, since is not possible according to known physics as there's no way to add more momentum after you get to the speed of light. If someone were to add a force that pushed on something other than "space..." there's a thought experiment for faster than wave travel in the medium of spacetime.

If you look at faster than wake travel in other medium, you can see it's possible, but things change. For a while, we thought that we couldn't travel faster than sound, but because there's no acceleration limit, something special happens at the boundary, a plane can exceed it and it causes a conical shock wave behind it. A boat can travel faster than the waves in the water, and again, a breaking wave, a "shock wave" appears on the surface behind the boat. What it would mean to accelerate to travel faster than the waves traveling on space time? We don't think it can be done, but it seems like we sort of know how it might look from slower speed FTW relativistic travel. There is a name for the shock wave created when matter travels faster than the local speed of light (a fission reactor is the usual example), called Cherenkov radiation, which tracks with normal faster than wave travel in a fluid. (We see visible light, a pretty shade of blue.) For travel in normal fluids, traveling faster than a wave causes a new wake regime. With the speed of light, the speed of a wave in spacetime, could it be the same? Maybe, if there were some other force, some other way to add momentum to the system. Something traveling faster than c would potentially form a shock wave and maybe deform spacetime in a new way. There are hints that spacetime acts like a fluid compressible by mass; If you excluded gravity as a force that could act on light directly, light curving around a large enough mass looks like what happens when the speed of an object changes in the direction perpendicular from its travel, which we call refraction. It's probably a mistaken comparision, though, since there's no evidence that light CAN speed up; otherwise the event horizon around a black hole would not swallow light. But there are people looking at density of space So, there is research around the edges, but breaking that limit of c Not so likely, but interesting to consider.

Tachyon theory might bring some insights to this question. According to tachyon theory it is possible that a particle can travel faster than the speed of light as long as the particle was created with a higher velocity than the light speed. And the speed of light boundary, according to tachyon theory, can be approached from below or above, but not be crossed. A tachyon with infinite speed will have zero relativistic energy, but will gain energy as it decelerates towards the speed of light boundary.

$\hspace{100px}$

But such faster than light speeds has strange time properties, particles travels so fast that they constantly jump back in time, and will always come from the future and go to the past. Seen from our reality this might looked upon as particles constantly traveling wormholes to the past, and we have never observed such particles, and it might not even be possible to observe such particles.

The quantum link has by using lasers and quantum entanglement been measured to be at least 10 000 times faster than light, so it has either infinite speed or close to infinite speed. Einstein wasn't fond of quantum entanglement, as it seems to break his light speed limit. We don't know what the quantum link exactly are, some suggest wormholes, some suggest superposition, some say it is linked on a fundamental plane, particle free waves, etc., but if the quantum link involves traveling particles these particles must be tachyon particles with infinite or close to infinite speed.

And it is strange that in quantum mechanics the observer might influence the result of an experiment. The result of the experiment happened back in time as it takes a few nano-seconds before the light reaches us so we can observe the result of the experiment. As observation influences a result, is observation like picking past quantum potentials from the future? And relatively speaking is the experiment really happening in the past? These things often confuse me.

Richard Feynman and Wheeler had a strange hypothesis, what if light is emitted by an advanced photon which traveled back in time, and emits the light. We might argue that this is not breaking causality as for this particle linear time goes the other way. This answer by Terry Bollinger explains this hypothesis beautifully: Can a photon get emitted without a receiver?

Inspired by Feynman I wonder if there is a relation here, if a tachyon with close to infinite speed and close to zero energy is emitted, from our eye in the same moment as a light is received, could the tachyon travel back in time to the emitter and influence the quantum jump which emitted the light? The relation is then: Does infinite speed match the speed of light in reverse direction? Anyone know how to calculate this?

If there is such a relation, then we might wonder if infinite speed plays a part in setting the speed of light boundary.

## protected by Qmechanic♦Jan 21 '16 at 10:12

Thank you for your interest in this question. Because it has attracted low-quality or spam answers that had to be removed, posting an answer now requires 10 reputation on this site (the association bonus does not count).