# Why is there a universal speed limit?

I am looking for an answer that does not rely on Special or General Relativity -and without recourse to the fact that the speed of light is frame invariant.

Is there another way of showing this universal speed limit to be necessary -one that it would have been possible to find before Einstein made his theories?

For myself I think there should be a universal speed limit because there is a law of diminishing returns when we look for ways to accelerate an object (we have to mine further and further regions of the universe which means that even an infinite universe would only have a finite amount of accessible energy)

However, following my reasoning it does not follow that this limit would be the same as the speed of light in vacuum. (I don't deny that it is identical -just that my "method" does not show this)

• But your method - "the law of diminishing returns" is not a physical law, and we shouldn't expect it to be valid. – levitopher Sep 4 '15 at 16:31
• There is no speed limit but there is a very common misunderstanding about what the speed of light is: it's a constant in all coordinate systems. That action at a distance was not a realistic physical model was already suspected by Newton, but I will leave the more historically inclined folks here elaborate on that. – CuriousOne Sep 4 '15 at 16:34
• Hello, and welcome to Physics SE. Please look around, and take the Tour. I'm trying to figure out exactly what you are asking - special and general relativity were, and are, the answer to the question you posed. Please clarify what physics you are interested in. – Jon Custer Sep 4 '15 at 16:35
• @JonCuster, the Theory of Relativity does not prove that the speed of light is invariant, it assumes it. Maxwell showed that something called "electromagnetic waves", which seemed like the same thing as light, should propagate through vacuum at a constant speed. Einstein's theories are the culmination of work by several physicists and mathematicians who tried to reconcile Maxwell's discovery with the Principle of Relativity---the idea that the laws of physics should be the same in any inertial frame. – Solomon Slow Sep 4 '15 at 17:32
• Thanks for all the replies. It will take me a while to assimilate them (if I can) but @ levitopher my "law of diminishing returns" that you have questioned represents ,as I see it the difficulty of accelerating an object above a particular speed. – geordief Sep 5 '15 at 0:03

Backing up what zeldredge said, what you asked about is known as "relativity without light". According to the intro of this paper (arXiv link) for instance, the original argument was given as early as 1910 by Ignatowski, and has been rediscovered several times. There is a modern version due to David Mermin, in "Relativity without light", Am. J. Phys. 52, 119-124 (1984), but a pretty accessible presentation may also be found in Sec.2 of this paper by Shan Gao: "Relativity without light: A further suggestion" (academia.edu link). The basic idea is that the existence of an invariant speed follows directly from the homogeneity and isotropy of space and time, and the principle of relativity. No reference to a speed limit is needed, but it does follow that the invariant speed acts as a speed limit. The only alternative is a universe without a speed limit (infinite invariant speed), where kinematics is governed by the Galilei transformations. Why it is that our universe has a finite invariant speed, and not an infinite one, remains an open question. Gao's "further suggestion" is that the invariant speed is related to the discreteness of space and time at the Plank scale, which is an intriguing thought in its simplicity, but then it remains just a "thought" so far.

• udrv: "a pretty accessible presentation may also be found in Sec.2 of this paper by Shan Gao: "Relativity without light: A further suggestion" -- After its first, introductory paragraph Sec.2 of the linked paper by Shan Gao starts out with: "Consider two inertial frames $S$ and $S'$, where $S'$ moves with a speed of $\nu$ relative to $S$ ...". What does "speed" mean there?, How is "speed" to be evaluated?? p.s. Note the spelling of the surname of M. Planck. – user12262 Sep 13 '15 at 12:00
• He obviously takes "speed" to mean "velocity". I'd consider it a language slip: some languages do use "velocity" and "magnitude of velocity" or "speed" and "magnitude of speed", so it's easy to slip when using English. As for how it is evaluated, if you are pointing to a loophole, I don't think we necessarily need light signals to measure speed/velocity or synchronize clocks, especially assuming homogeneity& isotropy of space&time, etc. Is that what your Planck reference is about? – udrv Sep 13 '15 at 19:17
• udrv: "He obviously takes "speed" to mean "velocity"." -- Right. But that wasn't my main concern. "if you are pointing to a loophole [...]" -- I meant to point to the "chronometric" definition of distance´(cmp. Synge, GR) and consequently the evaluation of "speed" necessarily in terms of signal front speed $c_0$, together with some real-number factor $\beta$. Or do you consider defensible any other notion of "distance" and consequently of "speed" (or "velocity", as applicable)?? "[...] assuming homogeneity& isotropy of space&time, etc" -- Certainly notions to be questioned ... – user12262 Sep 13 '15 at 20:06
• @user12262:"Or do you consider defensible any other notion of "distance"" -- "any" is stretching it. From an operational point of view I'd say thank goodness Galilei didn't care to question "distance", "speed", "homogeneity", etc in the terms you brought up. Or who knows how long the development of classical mechanics would've been delayed. The archetypical chicken-or-egg dilemma to me. Pick one and stick to it, refine if/when you can. – udrv Sep 14 '15 at 4:59
• @user12262 I just noticed that you provided one of the answers below. Sorry, I didn't quite understand the context of your comment above. Anyway, Synge's chronometric definition is perfectly fine in GR context, but I meant to point out that operational definitions of length and time were (had to be) available before any notion of SR. Therefore I don't have an issue accepting homogeneity & isotropy of space and time without referring to the invariant speed of light. – udrv Sep 14 '15 at 16:18

There is an argument based on deriving an arbitrary coordinate transformation that allows time to change as well as space for different observers. This transformation will turn out to have an undetermined parameter $v$ which corresponds to a maximum speed, and in the limit $v \to \infty$ the transformation becomes the Galilean transformation of classical mechanics. Extending this to $v = c$ the speed of light requires some additional piece of reasoning--Maxwell's equations being historically important here.

There are a few problems with your "diminishing returns" argument. One is that it doesn't explain many phenomena of relativity, for instance, that two observers who recede from each other at $.75 c$ in one frame do not observe each other traveling at $1.5 c$ in their own frames. In addition, it isn't clear to me why you believe an infinite universe would only have a finite amount of energy, especially if you're not assuming such a speed limit. Most importantly, however, you're trying to argue that there is a practical speed limit based on finiteness of energy. But in relativity, energy does not scale with velocity the way you expect, and in fact we do not believe there is a maximum energy-- massive particles can have any kinetic energy, and this kinetic energy goes to infinity as the particle's velocity approaches $c$.

• I am unable to back up my claim regarding the universal speed limit but perhaps I can explain why I think that an infinite universe can have a finite amount of energy. I am talking about accessible energy and that is why it seems to me that it is impossible to assemble a reservoir of energetic material above a certain finite amount in any one neighbourhood of the (hypothetically infinite) universe because beyond a certain point more energy is expended in gathering this resource than is eventually available (what I meant by “diminishing returns”). – geordief Sep 6 '15 at 12:41

Besides the philosophical problems of not having a maximum speed at which interactions propagates: Leibniz was horrified by Newton's theory of gravity, and even Newton himself knew that his theory could not be the complete story. The best shots that comes to my mind right now at guessing that the speed of light has to be finite (ignoring post 1850 evidences) are:

1) The inverse of the Olber's paradox: assuming the universe is finite in space and time if light propagates at infnite speed there would be a very birght night sky (i guess that would hold even for an infinite universe but then the speed of light being infinite or not would be irrelevant)

2) Maxwell equation in vacuum can be writtien in the form $\nabla^2A^\mu-\frac{1}{c^2}\frac{\partial^2}{\partial t^2}A^\mu=0$ then if you put $c\rightarrow \infty$ there would be only stationary solution and light wouldn't propagate, so the fact of seeing things should indicate, knowing Maxwell's equations, that the speed of light must be finite.

3) Philosphical arguments on the local nature of physics, which were just stuff for philosphers until the late 800' when experimental data started to corroborate the notion of a absolute speed limit.

The first two "arguments" just suggest that there is a finite speed of light, they don't shed any light (couldn't resist) on the fact that there should be a speed limit at which interactions and particles can propagate, someone in mid 800' could have just said that a yet unseen particle (or in the case of a fervent newtonian supporter, gravitational interaction itself) could travel at an arbitrary high speed, even infinite. Only with special relativity and modern particle physics we are able to explain in a complete manner what you are asking.

• Olber's paradox is a problem whenever $\operatorname{min}(ct,d) = \infty$, where $c$ is the speed of light, $t$ is the age of the universe, and $d$ is the distance to the "edge" of the universe. So it needs an infinite in space universe (or at least one without an edge), as well as either an infinite speed of light or an infinite age (or both). – user10851 Sep 5 '15 at 20:35

Not sure this is what you are looking for, but let's consider simple energy conservation. An object moving at a speed $v$ has kinetic energy equal to

$$K=\frac{1}{2}mv^2$$

Now, we either assume the universe is finite or infinite. If the universe is infinite, then it follows it contains an infinite amount of energy (saying this "follows" might be a bit strong, but I think it's reasonable). On the other hand, if the universe is finite, it follows that it contains a finite amount of energy.

If you have a finite amount of energy available to make this object move at a velocity $v$, then there is an upper limit to how fast it can go. In this way, each object would have it's own speed limit, given by

$$v_{max}=\sqrt{\frac{2\mathcal{E}}{m}}$$

where $\mathcal{E}$ is the total amount of energy in the universe.

Of course, this answer ignores nearly everything we actually know about physics, but given how you asked it, it made me think like this.

• I'm downvoting because the energy-velocity relation you've posted is not valid relativistically, and I think the implications about finite/infinite universes and their energies are pretty much impossible to actually make (certainly not as one-line dismissals). I realize you're trying to work in the "framework" given by the asker, but I feel like this is the kind of question that calls for education rather than accommodation of the premises of the question. – zeldredge Sep 4 '15 at 16:41
• I think you would be hard-pressed to find a good argument for how an "infinite universe" could possibly have a finite amount of energy. But anyway, physics is nothing if not the study of frameworks, and he clearly indicated the framework he was interested in. Downvoting seems inappropriate to me in this case, but whatever ya feel! – levitopher Sep 4 '15 at 16:46
• Except that even within the theory of special relativity there is no upper bound on the amount of kinetic energy objects can have. Sometimes the best answer to a question like this is "the premise of the question is flawed, and is unanswerable". – Brionius Sep 4 '15 at 19:11
• In case I was misunderstood, I was not suggesting that an infinite universe might have finite energy. I was suggesting that it might have finite accessible energy for the single purposes of accelerating a particular object (because of the "economic" difficulties of assembling the energy (infinitely) at one particular place. So ,yes energy infinite but "usable" energy finite (with a value to be determined) – geordief Sep 13 '15 at 17:25

Why is there a universal speed limit? I am looking for an answer that does not rely on Special or General Relativity -and without recourse to the fact that the speed of light is frame invariant.

Some preliminary arguments can be made without appeal to any particular notion of "frames" (or even the notion of manifolds, coordinates, or somesuch):

(1.) You don't necessarily make all your observations at once (in coincidence);

(2.) To any of your signal indications you don't necessarily receive a response by someone particular right away (in coincidence with your stating the signal indication); not even the very first reception of a response, i.e. the "ping response", referring to the signal front.

(3.) Some participant whose ping response to one particular of your signal indications you didn't receive right away is said to have been "separated" from you, at least in the course from your stating the signal indication until your receiving the corresponding ping response).

In the following a notion of "inertial frame" is required, namely the measurement whether a pair of participants who were separate from each other, but observing each other, had been "at rest" to each other, or not, in the course of an experimental trial. (How to measure this can be defined based on 1. - 3.):

(4.) Any pair of participants who were separate and at rest to each other, throughout a trial, shall be characterized by a finite value of their "distance" with respect to each other.

(5.) By the (coordinate-free) definition of simultaneity which has been explicitly described by Einstein 1917 (though it was know to others, and, arguably to Einstein himself, too, even earlier), a pair of participants who were separate and at rest to each other, and who determined a particular signal indication of one having been simultaneous to a particular signal indication of the other, will necessarily determine the corresponding ping response indication (i.e. the indication of receiving the echo) of one having been simultaneous to the corresponding ping response indication of the other. Thereby these two participants are provided a common measure of "duration", and they determine their mutual ping durations as equal to each other.

(6.) By the defnition of "distance" as "chronometric distance" (which was arguably already implied by Einstein, but explicitly recognized only by J. L. Synge in the 1950s) the mutually equal ping duration between such a pair of participants (say $A$ and $B$) is taken as the value of their characteristic distance between each other: $$\text{distance}_{AB} := \frac{c}{2}~\text{ping_duration}_{AB},$$

where the (nonzero) symbol "$c$" is introduced to express the conceptual distinction of a ping duration, between a pair of participants at rest with respect to each other, from incidental other duration values; and the factor $\frac{1}{2}$ is due to the convention of understanding "distance" as "one-way", rather than "round-trip".

(7.) By the definition of "(average) speed" as ratio between "distance" (between two suitable ends, separated and at rest to each other) and "duration" (of the "course" having been occupied) the value $1~c$ is identified as the value of signal front speed.

As far as these definitions are understood and being adhered to universally, the signal front speed is accordingly considered a universal, non-zero, symbolic value.

And since this value is referring to the signal front (i.e. always the very first receptions of corresponding signal indications) the value $c$ constitutes a limiting value, such that any response to a particular signal indication received after the ping reception is attributed a speed value less than $c$.

I must object by saying, I could go about questioning everything, but one must rather understand how? and why? things are the way they are.

One must understand the Maxwell Equations, the problems and issues that come with it, special relativity, how it solves the problems, how to derive the Lorentz factor and how it breaks down at the speed of light. There is a speed limit cause the math behind all of this says so.

If you still want an answer, the Anthropic principle is waiting. It isn't a scientific principle (or at least i don't consider it as one). In simple it means that things are the way they are because if they weren't, we (humans) wouldn't be here to ask this question. A method to avoid the question rather than answer it.

One example is the Quantum Zeno effect. It shows how measuring the state of a changing system can slow down the rate of change. So, if we look at it more often, it slows down more. If we could see it continuously, it would never change. The universe would freeze. But cause there is a speed limit, light can at max move at that limit, slowing down our observations and stopping the universe from freezing. If this speed limit wasn't there, the universe could never evolve to a point where life on earth could exist and there would be no humans, violating the Anthropic Principle. So, a speed limit becomes necessary.

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. 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, and as intuited by the OP, since acceleration increases the mass of the object and so an exactly quantitative "diminishing returns" applies.

So everything the OP wants 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. Olbers etc. is an observation, as well, but a less reliable one (since interstellar dust might be the explanation). 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 that could never be accepted in physics....)

• 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

Actually light speed is not the universal speed limit. The expansion of space could be faster than light

The speed limit is the limit of us the matter living in space-time medium. Light and other particle is the wave propagate in space-time which is constant limit. And us the matter is product of these wave-particle interaction. So we just can't have speed beyond limit of the wave we created from. And we still never observe anything that is not matter and particle yet

Except the space itself which we see expansion that may faster than light

But as I said. Us the matter, maybe dark matter and antimatter too, is trapped in speed limit. Because all matter is the product of spacetime wave. And any wave in one medium has constant speed

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