# Why is the speed-of-light “the upper limit” rather than the speed of “particle type X”?

Basically, I can't stop wondering why light (the photon) is so special, compared to all the other particles known (and unknown) to modern day physics.

Could it be that there exists an upper limit on speed that is instead a property of another "entity" (particle / wave), other than the photon?

A question obviously inspired by the recent very slightly possible superluminal neutrino findings

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c being a limit follows from the assumption that the Michaelson-Morley experiment is correct (which measured the speed of light in orthogonal directions and found no difference). It just so happens that the name given to particles of light is "photon". – Mike Dunlavey Sep 27 '11 at 13:48

The reason is partly historical accident, and partly experimental support, but in the end a certain salient property of light itself.

James Clerk Maxwell inferred, from his laws of electromagnetism, that there exist electromagnetic waves. Computing the speed at which they travel, he found that this speed was in close agreement with the observed speed of light. And so he conjectured that light was an electromagnetic wave. However, his equations make no reference to a medium (beyond bare electric and magnetic fields) in which the electromagnetic waves were to travel. The waves travelled at the fixed speed c in vaccuum — but relative to what, or to whom? There was no answer.

The "luminiferous ether" was conjectured to solve this problem. It was defined simply to be the medium in which EM waves (and light) travelled; and if you moved relative to the ether, then you would notice differences in how light travels relative to you. Michelson and Morely tried to establish how quickly the Earth moved relative to the ether, at different times of year (to account for how the Earth revolved around the Sun) with precision equipment, but could find no detectable difference at all in the speed of light in different directions, at any time of year.

FitzGerald and Lorentz tried to explain this negative result by means of a length contraction of any objects not at rest. There wasn't a good justification for why this should occur, but it did explain the results. And people tried in various ways to figure out what was going on, with ambiguous success.

Once Einstein simply decided to accept as an (experimentally supported!) axiom that the speed of a beam of light in vaccum is the same for all observers, and worked out the consequences, the length contraction postulated by FitzGerald and Lorentz fell out as an immediate corollary. Another corollary was that it would require an unbounded amount of energy to accelerate an object to speeds approaching that of light, from rest in any reference frame other than that of a beam of light. And so unless you think you have access to a very-literally infinite source of energy (and in finite time at that), you cannot accelerate anything up to the speed of light.

In all of this, what made light special was it's property of travelling at the same speed for all observers: predicted by Maxwell, and supported experimentally as well as one can without the use of high-velocity spacecraft, not to mention indirectly by all of the empirical support (including the precession of the perihelion of Mercury's orbit, starlight deflected by the Sun, and the successful operation of GPS as it is designed) for Special and General Relativity themselves.

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An infinite source of energy, or a zero-mass object. – Jerry Schirmer Sep 27 '11 at 14:13
@JerrySchirmer: true; but of course, the way we use the term, no zero-mass object "starts at rest" either. – Niel de Beaudrap Sep 27 '11 at 17:27
Well, the earth is a kind of high velocity spacecraft... – Greg L Sep 27 '11 at 18:36
@GregLoftus: true; and the average speed is 0.0001 times c, which wouldn't be entirely negligible. Still, it would be nice to have other frames of reference to choose from to experiment, particularly given the fact that the modern null hypothesis is that no frame of reference is privileged (or unprivileged). – Niel de Beaudrap Sep 28 '11 at 3:12
Thanks Niel - that's certainly given me plenty of 'food for thought' and different avenues to investigate further. – Will A Sep 28 '11 at 13:35

In addition to Niel's historical answer, I'll try to give a slightly more mathematical one. It will turn out that not only light travels at the speed it does, but as Jerry hinted at in his comment to Niel's answer, any particle with rest-mass zero travels at this speed.

This speed ($\approx3\cdot 10^8m/s$) could have been denoted "speed of gluons" or "speed of neutrinos" (in the Standard Model at least where neutrinos are massless"). The reason why it is called "speed of light" is that photons are by far the easiest massless particles to detect and thus were detected first, leading to the events and advances in physics that Niel described.

To recap, the postulate that the speed of light $c=\Delta x/\Delta t$ is the same in all reference frames is mathematically expressed by means of Lorentz transformations. As you may have heard Lorentz transformations not only change the coordinates of space but also time, which is necessary in order to satisfy the postulate. Lorentz transformations leave the expression $c^2t^2-x^2-y^2-z^2 = c^2t^2-\mathbf{x}^2$ invariant for all reference frames. So if it is zero (which it is for light, as is easy to see when looking at the equation at the beginning of the paragraph) it will stay zero in all frames of reference, just as we postulated. (If it is some other value, it will be this exact value in all other frames as well).

It turns out (I won't derive this here) that for the energy and momentum a similar equation holds: $\frac{E^2}{c^2}-\mathbf{p}^2=m^2c^2$. The energy got the role of time, more or less, while the momentum $p$ took over the role of the position (this is known as canonical conjugation). So written differently $E=\sqrt{\textbf{p}^2c^2+m^2c^4}$. Even if a particle doesn't have any momentum it still has energy (the famous $E=mc^2$). Another way to write the expression for the energy is $E=\gamma mc^2$ where $\gamma=\frac{1}{\sqrt{1-v^2/c^2}}$, a number known as the Lorentz factor.

As you see to have particles with finite energy the expression $\gamma m$ must be finite. But as soon as $v$ approaches $c$ the denominator in the Lorentz factor approaches zero. Assuming the mass $m$ is finite and positive this would result in the infinite Energy Niel mentioned to get the particle to move at velocity $v=c$. One can alleviate this problem by sending $m$ to zero. The energy will be finite if $v=c$ but only if $m=0$ at the same time.

If you $v>c$, $\gamma$ becomes imaginary, so to have non-imaginary energy (which is a reasonable assumption), one would need a imaginary mass, which is what tachyons are.

The takeaway here is that all massless particles move at this speed called $c$, which was just named after the most prominent particle moving at this speed, namely light. No particle with non-imaginary mass can be faster that massless particles, and massive particles can never reach the speed of massless particles (all particles "stay on their mass-shell" as physicists like to say). Remember, as $c^2t^2-\mathbf{x}^2$ has the same value in all reference frames, so does $\frac{E^2}{c^2}-\mathbf{p}^2=m^2c^2$. No matter how you boost this particle, it will always have the same value, because $m$ is a property of the particle independent of its frame of reference.

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Thanks Luksen - I don't even begin to understand the maths involved, but would like to so I'll invest some time in getting to grips with the Lorentz transformations. – Will A Sep 28 '11 at 13:56
@WillA well it's not that hard really, in the end it's a linear transformation (a 4x4 matrix) of time and space with a special property that it leaves something invariant, so if you understand matrix multiplication, it should be possible to follow introductory literature. It just like rotations of objects (3x3 matrices), which also leave something unchanged, namely the length. – luksen Sep 28 '11 at 14:05

Because it's a property of space-time, not a property of the particles. So it applies in a similar way to all of them ("thy speed shall be equal to or less than this").

Of course, if the neutrino thing turns out true, we'll have to revise this. :)

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I offer the following comments based on my experience as an aerospace engineer of 60+ years standing and a major subsystem manager on the Apollo program and also om the Space Station Program, working for the prime contractor. I agree with the physicists who accept that light is the visual evidence of a form of energy released from stars initially as electrons made redundant by the atomic fusion of lighter elements in said star's atomic fusion furnaces. Light is thus just the visual frequencies of such energy and is accompanied by the inseparable constant ratio of mass and gravity.Its upper limit speed is constrained by the longest constituent frequency wave length because greater speed would entail at least one fractional wave length which physics will not allow and is also the basic reason for energy always being transmitted in whole number. A lower limit is constrained only by the constancy of its total energy which is a function of it frequency and amplitude. When such light associated energy approaches a body of energy many times more massive than itself the laws of attraction of gravity will cause it to speed up to the maximum velocity relative to the more massive body or if it approaches at a speed greater than c it will slow down again to eliminate partial waves. Hence the fact that light always moves at speed c relative to its measuring frame (i.e. all frames no matter what their relative velocities) is not surprising or weird and does not act as a limiting value of velocity. It does however cast doubt upon the validity of E=mc^2 and upon the Lorenz equations and other concepts of cosmological physics especially that nothing material can exceed c because all those current concepts absolutely depend upon the assumption that c is a limit. So all those concepts become circular arguments, i.e. are invalid.

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-1: Largely because this is incoherent, not because it disagrees with mainstream physics. But as far as that goes: in those 60+ years of aerospace engineering did you ever work on a high energy particle accelerator where $E=mc^2$, the Lorentz transformations and $c$ being a speed limit are everyday facts of life which you have to engineer around (otherwise your machine doesn't work)? – Michael Brown Mar 21 '13 at 0:42

## protected by Qmechanic♦Mar 21 '13 at 0:42

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