Very basic astrophysics question

[Sela]

Why is lightspeed the fastest speed anything can go at?

I can understand light being the fastest thing we've ever observed; but I can't wrap my head around how we can know that it's impossible to go faster. Let's ignore wormholes and such for the purpose of this discussion. This isn't something I need to know in my life, career, or for any real practical purpose. But its fascinating and completely mystifying to me. I've sat through two basic-level physics classes in undergrad and solved questions regarding time dilation, mass-lightening (or something like that), and comparing newtonian to einsteinian results for near-light-speed travel. That said, I don't think I've grasped the underlying concept at all.

Something Destructionator XIII said in a different topic kinda triggered this curiosity. Anyway - if this is one of those things that's wayyyy too complicated to answer in a simple post and I'd be better off reading up on it, I'd really appreciate a link to a *layman's, SIMPLE* explanation that I might actually understand.

Wikipedia soared right over my head on this one. I'm just greatful that I'm into computers and medicine career-wise.

[Terralthra]

We don't "know," it, per se. We have observed that nothing goes faster, and observed that as we approach it, various factors (lorenz transformation, time dilation) prevent us from attaining it, and even as we do approach it, light still appears to travel at c (rather than it appearing to travel at c - our velocity). From these and others, we create a model of the universe (SR) under which nothing can travel as fast as light (or transmit information faster than light, to be a stickler), and calculate the effects of that model. We observe these same effects in the universe, consistently, and conclude that our assumption is accurate, as far as we can so far observe.

[starslayer]

What he said. In addition to that, what it really boils down to is that the fact that c is the universal speed limit is a central assumption of our models of the universe; we really have no idea why this is so, only that it is. In fact, SR can be entirely derived from just two assumptions resting on top of Newtonian dynamics:

1. The laws of physics are the same in all inertial reference frames.
2. The speed of light in a vacuum, c, is the same in all reference frames.

These are obviously two very big assumptions, but they are just that - assumptions. As Terralthra said, we believe these to be correct because we have been able to confirm that the predictions arising from them agree with our observations. As an aside, however, we did not observe time dilation and lorentz contraction until well after the theory was proposed by Einstein; in fact, he himself started from the electrodynamical point of view based off of magnetic induction, and only later was it realized that at least mechanically, SR can be derived from the two assumptions I provided above.

[Kuroneko]

Note: since you say you worked relativistic problems before, I'm going to assume that you're comfortable with coordinate geometry.

I think that it would be best to explicitly separate this question into two:
-- Whether or not there is a finite upper bound to the speed between different inertial frames.
-- If there is, whether the speed of light is that speed.
Doing so at least partly demystifies the role of light and helps explain why the special theory of relativity made Einstein so famous. It wasn't anything about electromagnetism per se that was new, but rather the connection to relativistic principles. If for the moment we take the first as a given, then the latter is just a matter of experiment: is light massless? To a very tight bound, it is.

As for the first property, as was already said, there is not particular reason why the universe 'should' behave this way, although it's possible to at least motivate it. Say you have a clock moving about, and you're tracking when and where it is, making a path in (t,x)-space, if it's moving in only one spatial dimension. Say it's at some event (t,x) and some short time lapse dt later, it has moved by dx, i.e., to (t+dt,x+dx). In the meantime, it measured some amount of time ds.

Question: how much time does the clock register while moving between those two events?

Possible answer: Same time you do! In other words, ds = dt, which for various reasons I'm going to put as ds² = dt² + 0dx². This way lies Galilean relativity.

Note that since my velocity is v = dx/dt, I can take a "vector" (dt,dx) and think of accelerating the clock as making the dx part larger, covering more space in the same amount of time dt. So if I start with a velocity vector (1,0) represing the clock at rest, 1 being an arbitrary unit of time the clock registers, "accelerating" traces out a horizontal line in the (t,x) plane.

Another possible answer: ds² = dt² - (1/c²)dx², where c is some scale factor between space and time. Now taking a clock at rest, with vector (1,0), and "accelerating" it while keeping the time registered on the clock, ds, constant traces out a hyperbola 1² = t² - x²/c² in the (t,x) plane, and the hyperbola's asymptote prevents reaching c. This gives special (Lorentzian) relativity.

The third possibility, ds² = dt² + (1/c²)dx², gives Euclidean geometry, with "acceleration" tracing out a circle in units of c = 1. Since time is different from space, we know right away that this can't be right. You should recognize ds² = dt² + dx² as the Euclidean metric/distance formula/Pythagorean theorem in the (t,x)-plane.

None of this is reason why things 'must' be that way, but I think it goes a long to motivating relativity as in some sense natural. Euclidean geometry is simple. If time is different from space, then the simplest way of modifying Euclidean metric is to either
(1) Disconnect time and space entirely, as in Galilean relativity, putting a zero in the metric.
(2) Flip the signs in the metric so that the time and space coordinates are opposite.
Those are the only two "minor tweaks" of Euclidean geometry that distinguish time from space. Any other possibility would be significantly more complicated, and it just turns out that the universe prefers the latter.

Interestingly, if gravity is formulated geometrically in spacetime, the Lorentzian case (i.e., general relativity) becomes a lot simpler than the Galilean one (i.e., Newton-Cartan theory).

As an aside, however, we did not observe time dilation and lorentz contraction until well after the theory was proposed by Einstein...

That's not quite true. There were very impressive theoretical derivations that predate relativity--including length contraction by Lorentz and FitzGerald, time dilation to second order by Larmor, and at least two different derivations of an E = mc² contribution to mass for an electron at rest--which should not be discounted entirely because some of them explained the result of the Michelson-Morley experiment that was done before Einstein. But arguably more important was the Fresnel drag measured by Fizaeu. Say the velocity of light in a certain fluid at rest is v = c/n (n the index of refraction), and the fluid is now piped to have some velocity w, what would now be the measured speed of light in the fluid? Naive expectation would be v+w, but the predicted and measured result was v+w-w/n², which is exactly the second-order expansion of the relativistic velocity addition formula in v. And that was even a full decade before Maxwell's equations, never-mind relativity!

[Eternal_Freedom]

I feel I should point out that when we talk about c or "light speed" we mean "speed of light in a vacuum. It is possible to travel faster than light in other mediums, water or air for example, and this causes the phenomenon called Cerenkov (sp?) radiation, which is a blue glow commonly seen in nuclear reactors, and from cosmic rays hitting the atmosphere.

So, to summarise, nothing can go faster than light in a vacuum, and that value is c. But you can go faster than light in other mediums.

[Simon_Jester]

Or, put another way, light is massless, therefore it travels at the speed limit. If light had mass it would travel slower than c, and nothing else about the universe would be different- all relativistic equations would look pretty much the same, and the universal speed limit would still be the same; the difference is that nothing in the universe would be capable of going that fast.

[Bottlestein]

You can derive the speed of light from Maxwell's equations without any fixed reference frame.
We have experimentally verified all of Maxwell's equations that are required for the derivation.

Special relativity uses this fact - it isn't a "proof" of it. General Relativity and Classical Electrodynamics both follow a frame-independent formulation because of this fact.

[someone_else]

Since you are asking for "*layman's, SIMPLE* explanation that I might actually understand." I'll link something that may help.

Atomic rockets, Faster Than Light page

[someone_else]

I feel I should point out that when we talk about c or "light speed" we mean "speed of light in a vacuum. It is possible to travel faster than light in other mediums, water or air for example, and this causes the phenomenon called Cerenkov (sp?) radiation, which is a blue glow commonly seen in nuclear reactors, and from cosmic rays hitting the atmosphere.

Sorry to be spoil-sport, but this is incorrect.
Cerenikov radiation is electromagnetic radiation caused by particle radiation (electrons, or whatever) travelling faster than the speed of light in a medium.

Speed of light in vacuum is ALWAYS more than the speed of light in a medium (since speed of light in a medium depends from the light bouncing around within the medium, if there is no medium, it cannot bounce around to go slower, thus it has max speed in a vacuum).

So, there can be particles, actual matter, going faster than such light slowed down by bouncing in a medium. If they do, they are slowed down by the medium and the kinetic energy is finally emitted as the blue glow.
Still, those aren't true FTL particles, since the light speed in the medium is ALWAYS lower than the speed of light in a vacuum. The denser the more significantly slower light goes.
Other wikipedia wisdom about it

[Eternal_Freedom]

I feel I should point out that when we talk about c or "light speed" we mean "speed of light in a vacuum. It is possible to travel faster than light in other mediums, water or air for example, and this causes the phenomenon called Cerenkov (sp?) radiation, which is a blue glow commonly seen in nuclear reactors, and from cosmic rays hitting the atmosphere.

Sorry to be spoil-sport, but this is incorrect.
Cerenikov radiation is electromagnetic radiation caused by particle radiation (electrons, or whatever) travelling faster than the speed of light in a medium.

Uh, how is that different to what I said? I said it was caused by going faster than light in that medium.

[starslayer]

That's not quite true. There were very impressive theoretical derivations that predate relativity--including length contraction by Lorentz and FitzGerald, time dilation to second order by Larmor, and at least two different derivations of an E = mc² contribution to mass for an electron at rest--which should not be discounted entirely because some of them explained the result of the Michelson-Morley experiment that was done before Einstein. But arguably more important was the Fresnel drag measured by Fizaeu. Say the velocity of light in a certain fluid at rest is v = c/n (n the index of refraction), and the fluid is now piped to have some velocity w, what would now be the measured speed of light in the fluid? Naive expectation would be v+w, but the predicted and measured result was v+w-w/n², which is exactly the second-order expansion of the relativistic velocity addition formula in v. And that was even a full decade before Maxwell's equations, never-mind relativity!

I knew about Lorentz and his derivations (We do still call them Lorentz transformations after all), which IIRC also started from Maxwell's equations, but didn't know about Fizaeu and the others (except Larmor for his work on the power emitted by a moving charged particle). Very interesting stuff, thank you.

[Kuroneko]

I can definitely recommend the book:
Einstein, A. Relativity: The Special and the General Theory
which is definitely addressed to laymen, yet still has interesting insights for those that have already studied relativity from more formal sources.

That said, questions of why are always tricky because it is rather unclear what one is allowed to assume. Here in particular since the question was why there is a special speed, whereas relativity is typically derived from the (experimentally confirmed) postulate that there is such a speed. That doesn't mean it's a bad question; on the contrary, it's a very good one. But it does mean that re-treading the usual ground of Lorentz transformation derivations will not do.

So what could possibly be in some sense more fundamental than the already exceptionally simple postulate like
"The speed of light in a vacuum, c, is the same in all reference frames."
that could be used to explain why that postulate holds? Besides the purely experimental (it works!), I think the best one can do is exhibit a triple of geometries:

Code: Select all

    A   Pythagoras:ds² = dt² + dx² (Euclidean geometry) 1=t²+x² circle
   /|   Galileo:   ds² = dt²       (Newtonian physics)  1=t²    line (x2)
ds/ |dt Lorentz:   ds² = dt² - dx² (special relativity) 1=t²-x² hyperbola
./  |
O---B  AOB is a right triangle
  dx     the hypotenuse, ds, is normally given by Pythagorean theorem
        Recall tangent as the ratio of opposite to adjacent legs.

Euclidean geometry should be simple and intuitive in anyone's book, and from this point of view, both Galilean and Lorentzian relativity are simply its two closest cousins. The Lorentzian case might even be seen as 'more natural' because Galilean geometry is degenerate (dx never contributes to the interval). And all of the results of relativity are derivable from that relationship, e.g., time dilation as dt/ds = 1/sqrt[1 - (dx/dt)²], which, since v = dx/dt, is exactly the Lorentz gamma, in units of c = 1.

Since you are asking for "*layman's, SIMPLE* explanation that I might actually understand." I'll link something that may help.

I'm not sure that's all that helpful by itself, since the geometric perspective is necessary to understand half of what Atomic Rockets says. For example, why is hyperbolic tangent even there? Because rotation in the Lorentzian case traces out a hyperbola, as discussed in the previous post. So we can define the angle:
(Euclid) tan A = dx/dt, though here "velocity" dx/dt is actually a slope
(Galileo) tanp A = dx/dt
(Lorentz) tanh A = dx/dt
And in all three cases velocity addition adds angles, though for Euclid dx/dt is not a physical velocity but a slope and the Galilean case is trivial, as cosp A = dt/ds = 1 for any angle.

In this formulation, there is also a nice identification between Euclidean rotation, the Galilean transformation between reference frames in Newtonian physics, and the Lorentz transformation in relativistic physics. In trigonometric form, they're just about identical. There's also a very fascinating connection to Cayley algebras, but it's probably best not to go there.

I knew about Lorentz and his derivations (We do still call them Lorentz transformations after all), which IIRC also started from Maxwell's equations, but didn't know about Fizaeu and the others (except Larmor for his work on the power emitted by a moving charged particle).

I first heard of Fizaeu in Ch. 13 of Einstein's book, actually, and was even more surprised to later find out that he did it this in 1851. It was particularly impressive because it was both a theoretical prediction strongly foreshadowing relativity before Maxwell's equations were around, and a very direct experimental verification of such.

As to the rest, length contraction was of course fairly unambiguously predicted, not because of spacetime considerations, but because it was the understanding of the time that matter is composed of electric charges, and such systems were predicted to be shortened by motion through the aether. As for time dilation, that's a bit tricky. Lorentz certainly had his notion of local time, which was exactly the time component of what is now called the Lorentz transformation, but he seems to have treated it as a mathematical trick of no particular physical importance. Larmor's involvement I first encountered as an side comment in
Bell, J.S. Speakable and Unspeakable in Quantum Mechanics
in Ch. 9, How to teach special relativity, a chapter that seems oddly out of place in the book though has a very interesting perspective on some relativistic issues. I'll quote:

... we can assert from the correlation, when this pair [of orbiting oppositely-charged electrons] is moving from the aether with velocity v in a direction lying in the plane of the orbits, these orbits will be flattened along the direction of v to ellipticity 1 - 1/2v²/C², while there will be a first-order retardation of phase in each orbital motion when the electron is in front of the mean position combined with acceleration when behind it so that the whole of the period will be changed only in the second-order ratio 1+1/2v²/C². The specification of the orbital modification produced by the translatory motion, for the general case when the direction of that motion is inclined to the plane of the orbit, may be made similarly: it can also be extended to an ideal molecule of any orbital system of electrons however complex.

So Larmor calculated that the period of an orbiting pair of charges will be dilated in a way that agrees to second order with the relativistic prediction, and claims that this conclusion is extendable to nigh-arbitrary system of charges. Although Larmor does not say this explicitly, with the then-current conception of matter as an electromagnetic interaction of charges, this would be suggestive of the claim that material clocks run slow while in motion. So should this count as time dilation? I think so, but not everyone does (notably, Rindler doesn't).

[Bottlestein]

^ Out of curiosity - what is the connection to Cayley Algebras?

[Sela]

Thanks to everyone for the outpouring of explanations! To be 100% honest, it's very confusing even still; but I had a few key misconceptions that I'm really glad I got cleared up. Most notably the idea that somehow the lightspeed-limit was: 1.) Intrinsically light-related and 2.) Somehow 'proved/certain' beyond the usual measure of the entire body of science (ie: reproducible, empirical proof).

For the real meat of it - understanding how relativity works and thus *why* lightspeed *must* be a limit; I'm only partially following even some of the more simple explanations given. For now I'll just accept that even the simple answer is a tad complicated and put off figuring it out till I've got a solid amount of break time.

PS: Thanks for the book recommendation too, Kuroneko!

[someone_else]

I feel I should point out that when we talk about c or "light speed" we mean "speed of light in a vacuum. It is possible to travel faster than light in other mediums, water or air for example, and this causes the phenomenon called Cerenkov (sp?) radiation, which is a blue glow commonly seen in nuclear reactors, and from cosmic rays hitting the atmosphere.

Sorry to be spoil-sport, but this is incorrect.
Cerenikov radiation is electromagnetic radiation caused by particle radiation (electrons, or whatever) travelling faster than the speed of light in a medium.

Uh, how is that different to what I said? I said it was caused by going faster than light in that medium.

Yes, but you forgot to mention the pretty important fact that light speed in a medium is lower than light speed in vacuum. And that those particles still go slower than light speed in vacuum.

Those particles start at the same speed in any medium (assuming the same radiation source), it's just the light that is slowed down.

[Eternal_Freedom]

Ah, I see. Oops. My apologies for not being clear enough.

[Kuroneko]

... 1.) Intrinsically light-related and 2.) Somehow 'proved/certain' beyond the usual measure of the entire body of science (ie: reproducible, empirical proof).

To be fair to your past self, although it's certainly not beyond the usual measure of science--reproducible, empirical evidence--it's still difficult to understate the sheer amount of said evidence, in part because, as starslayer said, it's built-in in every successful fundamental theory of physics we have. Which reinforces the first point, as the fact that STR was derived from the behavior of light was simply because electromagnetism was the only well-explored force known at the time. Every other force is Lorentz-invariant as well, although generally much more complicated than light.

Good luck.

^ Out of curiosity - what is the connection to Cayley Algebras?

The usual construction of the complex numbers from the reals can be generalized to work for an arbitrary algebra with an involution operator * to produce a square root of any particular number λ. This is the usual way the complex numbers are constructed from the reals, with λ = -1 and the involution doing nothing. With the usual exponential function ex = 1 + x + ... + xn/n! + ..., define the trigonometric functions via eiθ = cos θ + i sin θ. Then:
[1] (x',y') = e-iθ(x,y) = (x cos θ + y sin θ, x sin θ - y cos θ),
which is the Euclidean rotation. Repeat for quaternions, although one needs to be a bit more careful for three-dimensional rotations because the algebra will be noncommutative. On the other hand, taking λ = 0 gives a root of zero instead, ε² = 0, and eεv = cosp v + ε sinp v = 1+εv, as higher-order terms simply drop, so that
[2] (t',x') = e-εv(t,x) = (t,x-vt)
is the Galilean transformation between inertial reference frames in Newtonian mechanics. Mixing this construction with the previous λ = -1 case produces an algebra that can naturally represent any rigid motion (rotations and translations). Finally, a root of unity (λ = +1), j² = +1 yields ejα = cosh α + j sinh α and
[3] (t',x') = e-jα(t,x) = (t cosh α - x sinh α, x cosh α - t sinh α)
is the Lorentz transformation between inertial frames in special relativity. This algebra behaves very similarly to the complex numbers.