Talk:Invariant speed

Invariant Speed vs. Speed of Light

"Special relativity is crucially based on the concept of inertial frames (IFs)...The crucial mathematical discovery that made SR possible was that, if one is willing to give up the idea of absolute time...then a whole new family of transformation groups becomes possible...These are the various Lorentz transformation (LT) groups, each characterized by exactly one finite invariant speed; that is, a speed that transforms into the same speed in all IFs. The Gallilean transformation turns out to be that limiting LT which transforms infinite speeds...Of course, Einstein picked that LT group which leaves the speed of light invariant--exactly the content of his second axiom."--Rindler, Relativity: Special, General and Cosmological"

Relativity, insofar as the invariant speed is concerned, is purely mathematical, so it cannot be disproven by having things travel faster than the speed of light. If anything did travel faster than light, then it would just mean that Einstein picked the wrong LT group. DonQuixote 00:02, 20 November 2005 (UTC)

Okay, but how is this relevant to the article? ‣ᓛᖁᑐ 03:46, 20 November 2005 (UTC)

It's just stating that the invariant speed is independant of the speed of light—it just that they seem to coincide.DonQuixote 06:05, 20 November 2005 (UTC)

Er, excuse me, but I don't think that this hits the point. In the mathematical formulation of special relativity, there is exactly one invariant speed, fixed by the representation of the Lorentz group that you choose. It is then predicted by Maxwell's theory, and verified by experimental evidence, that this speed is the speed of light. So these concepts coincide (and that's not coincidence, that's a prediction of the theory). It doesn't warrant a separate article. --B. Wolterding 17:34, 1 June 2007 (UTC)

Applying Velocity Transformation to $v$ and $c$

Suppose we are in on inertial reference frame. There is a car with a velocity $v$. If $u$ is the speed of any object in our IRF and $u^{\prime }$ is the speed of any object in the car's IRF, then the relationship between the two is
$u^{\prime }={\frac {u-v}{1-uv/c^{2}}}$
Utilizing this, the speed of the car itself is $u=v$ and $u^{\prime }=0$ and the speed of light is $u=c$ and $u^{\prime }=c$.

Now, let's say we accelerate into the the IRF of the car. In which case our IRF corresponds to the car's IRF and $u=u^{\prime }=0$ for the car and $u=u^{\prime }=c$ for light. Suppose, then, another car travelling at a speed $v^{\prime }$, with $u^{\prime \prime }$ being any speed within the second car's IRF. Following the same steps we get, $u=v^{\prime }$ and $u^{\prime \prime }=0$ for the second car's speed and $u=c$ and $u^{\prime \prime }=c$ for the speed of light. This can be continued on for a series of faster moving cars (provided that their speeds are less than $c$). DonQuixote 00:02, 20 November 2005 (UTC)

This doesn't look right. Speed is meaningless without a referent. The tautology that one is always stationary relative to oneself is unhelpful and does not prove objects cannot reach c; we should consider our speed to be undefined, not zero, if there are no other objects to compare our speed to. If there are external objects, we can determine our speed relative to a rest frame. At speeds close to c, relativistic effects become significant and it becomes apparent we are not stationary.

You're mixing relative speed and absolute speed. You state: "At speeds close to c, relativistic effects become significant..." That's not true. At relative speeds close to c relativistic effects become significant to the observer. The fact is that the thing travelling close to speed relative to us doesn't experience any relativistic effects, rather how it appears to us is affected relativistically (in fact, to that thing it is we that are experiencing relativistic effects). The crux of the matter is that all inertial reference frames are the same. Each observer inhabits his or her own inertial reference frame. That observer's speed relative to that reference frame is zero. Different observers have different relative speeds and therefore occupy different inertial reference frames. We cannot say which observer occupies a frame that is of a higher speed than the other. We can only say that one observer appears to be in motion while the second is at rest from the frame of the second observer and conversely the second observer appears to be in motion while the first is at rest from the frame of the first observer. DonQuixote 06:33, 20 November 2005 (UTC)

If the car's speed relative to the rest frame is in fact c (as with photons in a vacuum), all other observers will measure its speed to be c. It would be peculiar for the observer in the car to be the only one to think the car is not travelling at the speed of light.

Yes, well, the caveat is that the speeds are less than c (no one knows what exactly happens in the reference frame of light).DonQuixote 06:33, 20 November 2005 (UTC)

Properly, the reason objects cannot reach the speed of light is because they have mass. As an object approaches c, the acceleration produced by force becomes inconsequential. ‣ᓛᖁᑐ 04:33, 20 November 2005 (UTC)

Again, this is only what appears to happen as we look at an object. From our perspective, the object's length contracts, the object's mass increases exponentially, etc. But from the perspective of the object none of these things occur. Suppose that the force accelerating the object was attached to the object: i.e. the object is a rocket. From our perspective the force of the rocket's thrust becomes inconsequential, but from the perspective of the rocket the thrust is constant and constantly accelerates the rocket with the same amount of acceleration. The connection between the two is that, although the rocket accelerates at a constant rate with respect to the rocket (that is, it "jumps" inertial reference frames at a constant rate), to us the rocket appears to decrease in acceleration (that is, the jumps appear to become smaller and smaller). DonQuixote 06:33, 20 November 2005 (UTC)

So in effect, we always accelerate into an inertial frame where our speed is always zero and the speed of light is always $c$—which is the Red Queen's race: no matter how much you move, you're always where you started from. DonQuixote 00:02, 20 November 2005 (UTC)

This approach to the analogy is okay, but the analogy is awkward here because the Red Queen has a nonzero speed. The race as described by Lewis Carroll is essentially the opposite of the car's situation: all observers will consider the queen's speed to be zero, whereas the queen considers herself to be accelerating. ‣ᓛᖁᑐ 04:33, 20 November 2005 (UTC)

You're thinking too literally. The idea behind the Red Queen's race is that you're not getting anywhere even though you're trying your hardest. In regards to the speed of light, you're trying to accelerate to light speed but you're not actually getting anywhere in terms of speed. See the article on Red Queen's race, particularly the biological Red Queen (it has nothing to do with speed). DonQuixote 06:33, 20 November 2005 (UTC)

Rewrite

The article has been rewritten per the AFD. Much of the information in the old version was unecessary, confusing, and/or not entirely correct. I intended for the new version to do nothing more than answer the question, "What is invariant speed?" Readers can follow the links to find out more, or hunt down a copy of Mermin's paper, which I cited, and which contains a very nice discourse on invariant speeds. I chose not to summarize his conclusions on this page as that is already done in postulates of special relativity. Someguy1221 05:57, 7 June 2007 (UTC)