# B Is the case for a Universal Speed Limit experimental or theoretical?

#### geordief

You need some premises to derive conclusions using logic.
Yes I appreciate your point.There is a temptation to expect logic to work,as it were in a vacuum.

It reminds me of Relativity which needs two FORs to apply **and where the origin of radiation may be inconsequential ( in measuring its speed for example).

Logical reasoning ,on the other hand is relentlessly attached to its original premise even though we have the illusion that it exists in its own right.

I am just making an analogy -not drawing any consequences.

Your point has undercut one of the bases of my question and I would need to clarify what premise I had in mind.

Interesting that Einstein claimed not to rely on the MM result (very interesting really)

** perhaps a mischarecterization but it is just an analogy.Relativity requires to know one's relationship with what is being observed,I would say.

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#### x-vision

Is it possible to show without recourse to experimental observation (not ,of course rejecting those observations) that it would lead to contradictions if there was not an upper speed limit?
The upper speed limit follows from the Lorentz transformation. The Lorentz factor becomes an imaginary number for speeds higher than the speed of light.

#### Nugatory

Mentor
The upper speed limit follows from the Lorentz transformation. The Lorentz factor becomes an imaginary number for speeds higher than the speed of light.
It is true that the Lorentz transformations don't make sense for $v$ greater than $c$, but that doesn't mean that the upper speed limit follows from them.

In fact, it's the other way around: the Lorentz transformations are derived from the assumption of the invariance of light speed, and this assumption is equivalent to assuming the upper speed limit. Thus, the imaginary numbers that appear in the Lorentz transformations when plug in values of $v$ greater than speed of light aren't proving anything. They're just reminding us that we started with the assumption that faster than light travel is impossible.

To derive the speed limit logically, it's most natural to follow the same path that Einstein did in 1905, starting with the assumption of lightspeed invariance.

#### bob012345

Gold Member
Actually, a real bad day for both. Stopping a warp bubble near a planet would vaporize the planet.

From the conclusion:

"any people at the destination would be gamma ray and high energy particle blasted into oblivion"

Well, we don't typically stand on the tracks when a train comes. I'm sure a sufficiently advanced civilization could deal with it. Thanks.

#### bob012345

Gold Member
It is true that the Lorentz transformations don't make sense for $v$ greater than $c$, but that doesn't mean that the upper speed limit follows from them.

In fact, it's the other way around: the Lorentz transformations are derived from the assumption of the invariance of light speed, and this assumption is equivalent to assuming the upper speed limit. Thus, the imaginary numbers that appear in the Lorentz transformations when plug in values of $v$ greater than speed of light aren't proving anything. They're just reminding us that we started with the assumption that faster than light travel is impossible.

To derive the speed limit logically, it's most natural to follow the same path that Einstein did in 1905, starting with the assumption of lightspeed invariance.
Did Einstein actually derive what value that the speed of light had to have independent of measured data? Didn't it come out of the Maxwell equations first as a consequence of the properties of space?

#### Nugatory

Mentor
Did Einstein actually derive what value that the speed of light had to have independent of measured data?
No, but he did assume that whatever it was, it was invariant; and that assumption was motivated by Maxwell's equations of electricity and magnetism. At a somewhat handwavy level, we can say that two observers moving relative to one another in a vacuum shouldn't see different laws of E&M, and if they're using the same laws of E&M then the electromagnetic waves predicted by these laws should behave the same for both of them.

As an aside.... it turns out (although this was not recognized until many decades later) that if there is an invariant speed it doesn't make sense to try measuring or calculating it. Instead we can take advantage of its invariance to define our units, and that's we now do. The meter is defined to be 1/299792458 of the distance that light travels in one second so $c$ necessarily is exactly 299792458 meters per second. If I tried measuring and got any other result that would just tell me that one or both of my clock and meter stick need to be recalibrated.

#### bob012345

Gold Member
No, but he did assume that whatever it was, it was invariant; and that assumption was motivated by Maxwell's equations of electricity and magnetism. At a somewhat handwavy level, we can say that two observers moving relative to one another in a vacuum shouldn't see different laws of E&M, and if they're using the same laws of E&M then the electromagnetic waves predicted by these laws should behave the same for both of them.

As an aside.... it turns out (although this was not recognized until many decades later) that if there is an invariant speed it doesn't make sense to try measuring or calculating it. Instead we can take advantage of its invariance to define our units, and that's we now do. The meter is defined to be 1/299792458 of the distance that light travels in one second so $c$ necessarily is exactly 299792458 meters per second. If I tried measuring and got any other result that would just tell me that one or both of my clock and meter stick need to be recalibrated.
Thanks. Since you need both a clock and a stick to measure it, don't you need some other independent metric? Like the radius of a hydrogen atom for instance?

#### jbriggs444

Homework Helper
Thanks. Since you need both a clock and a stick to measure it, don't you need some other independent metric? Like the radius of a hydrogen atom for instance?
You can define the meter using the distance between two scratches on an artifact and then measure the speed of light in meters per second.

Or you can define the meter in terms of the speed of light, measure the distance between the two scratches in light-seconds and convert to meters. The point is that the latter is our current and preferred practice.

#### bob012345

Gold Member
You can define the meter using the distance between two scratches on an artifact and then measure the speed of light in meters per second.

Or you can define the meter in terms of the speed of light, measure the distance between the two scratches in light-seconds and convert to meters. The point is that the latter is our current and preferred practice.
In either case you have one invariant quantity, a speed, so to convert it to a distance you need an independent definition for a second. You can't define both a meter and a second from c.

#### jbriggs444

Homework Helper
In either case you have one invariant quantity, a speed, so to convert it to a distance you need an independent definition for a second. You can't define both a meter and a second from c.
Right. And we have an independent definition for the second.

#### FactChecker

Gold Member
2018 Award
You might be interested in this. It describes how some very natural assumptions lead to a velocity addition law that implies a single maximum velocity. And the fact that the velocity of light is constant in any inertially moving reference frame forces its speed to be the maximum speed.

#### geordief

You might be interested in this. It describes how some very natural assumptions lead to a velocity addition law that implies a single maximum velocity. And the fact that the velocity of light is constant in any inertially moving reference frame forces its speed to be the maximum speed.
Thanks,yes I am. It is one of the links on the page PAllen suggested I look at on post#5 of this thread.

I am working my way very slowly through it and am hopeful about it .

Actually I was also interested in how the velocity addition law might play into this so I am pleased that it is discussed there in the way you are suggesting.

#### Ibix

Actually I was also interested in how the velocity addition law might play into this
That's just another consequence of the Lorentz transforms. Since they are linear, for some pair of events separated by coordinate difference $(\Delta t,\Delta x)$ we can write$$\begin{eqnarray*} \Delta x'&=&\gamma\left(\Delta x-v\Delta t\right)\\ \Delta t'&=&\gamma\left(\Delta t-\frac v{c^2}\Delta x\right) \end{eqnarray*}$$Assuming that there is a timelike inertial worldline that connects the two events then $u=\Delta x/\Delta t$ is its velocity in the original frame and $u'=\Delta x'/\Delta t'$ is its velocity in the primed frame. Thus$$\frac{\Delta x'}{\Delta t'}=\frac{\Delta x-v\Delta t}{\Delta t- v\Delta x/c^2}$$The left hand side is $u'$, and dividing top and bottom of the right hand side by $\Delta t$ gives you the velocity transformation law.

#### geordief

Please Check out the link I provided. It has links to papers showing from pure symmetry considerations, there are only two possibilities: an infinite invariant speed (Newtonian physics) or a finite invariant speed (SR). Experiment has shown the latter.
Thanks for that link .It was very much what I was looking for.

Sadly the maths may be a little beyond me for now and so I will not attempt to finish this admittedly short and digestible piece. Still I wonder if you could just explain a little bit this next section to me ?

"
Suppose we now displace the rod such that its end which used to be at x1 is now at
the point x1 + h. Its length in the frame S should not be affected by its position on
the x-axis by virtue of the principle of homogeneity of space, so that its other end
should now be at the point x2 + h. In the frame S′, its ends will be at the points
X(x2 + h, t, v) and X(x1 + h, t, v). However, homogeneity of space implies that the
length of the rod should not be affected in the frame S′ as well, so that

l′ = X(x2 + h, t, v) − X(x1 + h, t, v)
.

The problem I have with this is it seems to be saying that the length of the rod in the primed frame is the same as that in the unprimed frame .I was under the impression the the lengths of bodies in moving frames were indeed shortened in the direction of motion .

I am obviously wrong . What is my mistake?

(I won't be pestering you any more about this as it is clearly for me to bone up a bit on the partial differential equation section and the notation used etc before (if ever) I can make it through to the end.

#### Ibix

The problem I have with this is it seems to be saying that the length of the rod in the primed frame is the same as that in the unprimed frame .
It's saying just that moving the rod doesn't change its length - not that a moving rod doesn't change length. So if I measure the length of my bedroom by repeatedly placing a meter rule along one wall, the ruler has the same length each time I put it down (moved along the wall), although ultra-precise measurements might show that its length varied while I was in the process of lifting it up and putting it down again.

He then goes on to point out that this applies just as much in a moving frame. So if one end of a rod is at $X(x_1,t,v)$ (knowing the answer, we will eventually see that $X(x,t,v)=\gamma(x-vt)$) and the other is at $X(x_2,t,v)$ then the difference (which isn't necessarily the length, but is related to it) ought to be the same even if you displace the rod by some distance $h$ (or use another identical rod offset by a distance $h$) and repeat the process.

#### pervect

Staff Emeritus
True, but there are so many reasons to expect isotropy of physical laws, it is in practice, absurd to consider anisotropy of the speed of light. Allowing it while matching all experiment is indeed possible, but almost every equation in physics then becomes more complex. In modern symmetry focused physics, isotropy is taken as an assumption. This would be questioned only if it led to complexity rather than simplification.
I think the one of the easiest to understand arguments for isotropy is to note that if light is non-isotorpic then so are other particles such as electron beams. The argument is that there is a limiting speed for particles that we observer experimentally in particle accelerators that is equal to "c". So if "c" is anisotropic and the speed of particles approaches the limit "c" for high enough energies (which we observe experimentally), we need to also conclude that the speed of high energy particle beams is also anisotropic.

) demonstrates the principle of a limting speed with a linear particle accelerators. There is also a paper to go along with the youtube video, for those who wish to look iup the peer reviewed source, though the video is more accessible IMO.

I think the argument becomes even more compelling when one tries to puzzle out how relativistic particle beams can be stored in circular storage rings, yet (if we assume non-isotropy) have different speeds depending on which direction we release said beams.

Basically, if light is anisotropic, so must everything else be, with particle beams that move essentially at "c" being the obvious example, one which we have a lot of experimental evidence for.

#### bobob

...Or even based on logic?
Actually, even thinking in these terms is rather archaic. Start with the idea that you have a standard euclidean space, x,y,z. Now draw 2 (or more) lines in different directions and find the distances using the pythagorean theorem. How well does that work if you measure the x direction in feet, the y direction in meters and the z direction in fathoms? Not very well, so you have to have a conversion factor. Furthermore, to use the pythagorean theorm you have to define your space as a metric space, so that you know what a right angle is. So, just to do what seems like common sense, you need a lot more than common sense to understand what you are measuring. Euclidean geometry is a space equipped with a metric: $ds^2 = dx^2 + dy^2 +dz^2$. You could extend that to as many dimensions as you wish. In a Gailean universe, time also plays a role but somewhat different. I'll get to that later.

Also, a rotation in the x-y plane for example is given by:

$$dx' = dx \cos\theta - dy \sin\theta$$
$$dy' = dx \sin\theta + dy \cos\theta$$

The angle of rotation is just $\theta$ and $\theta$ can be any value between 0 and $2\pi$.

Relativity is just the same, except the "pythagorean theorm" (i.e., the metric) contains a minus sign:
$$ds^2 = dt^2 - (dx^2 + dy^2 =dz^2)$$

There is speed of light constant 'c' because like the previous example, we want to measure all quantities using the same units, so here, time is measured in meters. The constant 'c' just converts seconds to meters in the same way that the constant 2.54 converts centimeters to inches. What is really different the minus sign in the "pythagorean theorm" (i.e. metric). If we now rotate in the x-t plane (similarly to the example above for the x-y plane, because of the minus sign we get:

$$dx = dx\cosh\phi - dt\sinh\phi$$
$$dt' = dt\cosh\phi - dx\sinh\phi$$

Due to the minus sign, the trransformation uses hyperbolic functions, not circular functions. However, this means we really aren't talking about a "finite velocity," because the hyperbolic angle $\phi$ corresponding to a hyperbolic rotation in the x-t plane ranges from $-\infty\ to\ \infty$. The hyperbolic tangent of $\phi$, however ranges from $-1\ to\ 1$ and if we are using minutes or seconds instead, then we have the more familiar form:
$$(v/c) = tanh\phi$$
So, to say there is an "ultimate speed" and wonder why, is sort of like asking why there is a "maximum rotation" of $2\pi$ to make a complete circle. The answer to both is that it's just the choice of geometry you use to describe physics. Either choice is valid philosophically and logically, but physics discriminates between them by which one agrees with experiments and experiments favor einstein's version of relativity.
I understand that it is expected that there might be a Universal Speed Limit and that this seems with extremely high probability to coincide with the speed of em transmission in a vacuum.
Btw does the existence of this universal speed limit necessitate the invariance of the speed of light (and massless objects)?
All massless objects must travel at 'c'. This is not necessarily true for light. The "phrase" speed of light to mean the velocity 'c' is more of an historical artifact due to what einstein was trying to explain and the general lack of thinking in geometric terms. Einstein was trying to reconcile maxwell's equations with classical mechanics and so he made use of the fact that in maxwell's equations, the speed of light is a constant, independent of frame. Hence the idea that the speed of light has something to do with relativity seems to be pervasive, even though it need not be.

Around 1914, Proca demonstrated a perfectly relativistic theory of electromagnetim in which light propagated just like any other massive particle, the consequence being that light was no longer massless. But, it was perfectly consistent with relativity. The upshot of this is that relativity is the geometry of thr universe and whether or not light is massless and propagets at 'c' or if it has a mass and propagates like anything else with mass, is more properly reserved for theories of electromagnetism. If the photon has a rest mass, it is known from experiment to be less than around $10^{-17} eV$

Finally, the choices that have potential to describe the universe have been Galilean relativity and special relativity (neglecting gravity). From those two possibilities, you can deduce the phyical laws of mechanics. A priori, thee is no reason to choose one over the other, so the reason for choosing the geometry of the universe to be special (or general relativity) over Galilean relativity is a matter that only experiment can distinguish and so it is experiment that tells us that of the two choices we have, (einstein) relativity is the correct choice.

It's also possible to write galilean relativity using time as a coordinate, but it becomes somewhat less transparent on how you end up with classical mechanics, so, I will skip over that.

"Is the case for a Universal Speed Limit experimental or theoretical?"

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