Posted: Thu Aug 15, 2002 12:28 am Post subject: Must the Lorentz Transformation Equations be Linear?
Does homogeneity => linearity ?
Mark Q wrote:
Can someone explain to me why spatial homogeneity implies that the Lorentz Transformation equations must be linear?
I’ve seen this mentioned in several LT derivations including one from the man himself (Einstein) in his original paper on the subject, but I’ve never been able to see exactly why the implication holds true.
There is nothing in the assumption of spatial homogeneity that disallows arbitrary coordinate systems and nonlinear transformations between different inertial frames of reference. Linearity in the coordinate transformation equations is not required.
Here is my counterexample.
Let f(x) be any function of x. Let Y(v) = [1-(v/c)^2]^(-1/2).
The set of all transformations on (x,t) of the form
x' = b + Y(v)[(x-b) -v(t-a) + (v/c)f(x)]
t' = a + Y(v){(t - a) - v(x-b)/c^2 -f(x)/c} + (1/c)f{b + Y(v)[(x-b) -v(t-a) + (v/c)f(x)]}
is a group. a = t_0, b = x_0, are constants.
This group is physically indistinguishable from the Lorentz group.
Amazingly, these transformations are easily invertible and the general form is preserved. I will leave the easy math to you to do as an exercise. The group property holds for any nonlinear f(x).
If I remember correctly, the group properties aren’t extremely difficult to verify because there turns out to be a lot of cancellations. For the inverse transform, merely interchange t' with t, x' with x and for parameter v use -v.
The problem is, physicists can’t see why this group is physically indistinguishable from the Lorentz group.
Posted: Sat Oct 25, 2003 10:22 am Post subject: The newsgroup sci.physics.relativity
Do the Laws of Physics Demand that Clocks be Synchronized?
Do the laws of physics demand that clocks be synchronized? Let's take a wild guess and suppose the answer is no. Imagine, then, every observer in every frame of reference deciding to set his clocks in whatever way he wanted. The consequences would be disastrous for relativists. The Lorentz transformation then couldn't possibly be the correct rule to express how events (x, t) get mapped into events (x', t') in moving frames of reference. Have you ever heard that the homogeneity and isotropy of space and time require linear coordinate transformations? I have just now proven that ridiculous myth to be false.
Just for fun I have taken the Lorentz transformation equations (which imply a specific and well-known synchronization scheme) and reset all the clocks in a wild and unsettling manner. The end result is a group of nonlinear transformations that obviously contain the same physical content as the original Lorentz transformations. See exercise 1 and 2 of http://www.everythingimportant.org/relativity/generalized.htm
You can't imagine all the flack that I've received over this. My critics act as if I've violated some cardinal law of physics or something. I piety them for their small-minded delusions. Sometimes I do see a humorous side. Should I laugh more at the implications of their words? What good would it do? In their religious frenzy to condemn me as a heretic, they're not even capable of understanding my words.
Posted: Sat Oct 25, 2003 5:03 pm Post subject: The newsgroup sci.physics.relativity
Eugene Shubert wrote:
My personal predilection for mischief, irreverence for religious relativity and absurdly nonlinear clock synchronizations is no threat to the homogeneity and isotropy of spacetime. If I'm wrong and if my continued use of hazardous and unlicensed clock synchronizations begins to tear at the fabric of spacetime, then, and only then, will I do away with the threat—for a fee.
"Many respectable physicists said that they weren't going to stand for this, partly because it was a debasement of science, but mostly because they didn't get invited to those sorts of parties." —Douglas Adams, The Hitchhiker's Guide to the Galaxy (1979)
Posted: Sun Oct 26, 2003 10:25 am Post subject: The newsgroup sci.physics.relativity
Bilge wrote:
> Do the laws of physics demand that clocks be synchronized? Let's take a wild guess and suppose the answer is no.
Not bad for a wild guess.
Correct; and that conclusion is based on what the fantastic repercussions to the universe might be if the answer was yes. It's obvious that violating the laws of physics would have serious consequences. See the previous post.
Bilge wrote:
> Imagine, then, every observer in every frame of reference deciding to set his clocks in whatever way he wanted.
OK. So that means there exists absolutely no relationship between any two clocks.
Bilge wrote:
> The consequences would be disastrous for relativists.
Why is that? I think you have turned something obvious into something mystical.
Keep reading.
Bilge wrote:
> The Lorentz transformation then couldn't possibly be the correct rule to express how events (x,t) get mapped into events (x',t') in moving frames of reference.
That is completely silly. You've said above that observers set their clocks as they choose. So far you haven't specified how two observers satisfy the first postulate.
We've already agreed that there is no law of physics preventing every observer in the universe from resetting every clock to some time other than what's given by standard Einsteinian synchronization and the Lorentz transformation. Consequently, you're just interrupting a valid argument.
Bilge wrote:
So sure, so long as you don't satisfy the first postulate, you can do anything you want. Once you impose the first postulate as a constraint, your choices are severely limited.
Ah ha. You believe that the laws of physics demand that clocks be synchronized!
Bilge wrote:
> Have you ever heard that the homogeneity and isotropy of space and time require linear coordinate transformations? I have just now proven that ridiculous myth to be false.
No, you haven't.
The point of the Lorentz transformation is to give a rule expressing how events (x,y,z,t) in one frame of reference translate into events (x',y',z',t') for moving frames of reference. Start with Einstein's standard synchronization. If the clock at (x,y,z) in one frame reads time t, reset it to read time t+f(x,y,z) for a nonlinear function f. Do this for all clocks at every point. If you continue this tangling procedure frame by frame, common sense says that you're going to end up with an extraordinarily complicated nonlinear mess. Each frame of reference can have its own synchronization scheme. There is no limit to the complexity. If you dare, index all the frames with a parameter alpha. Let the offset function f(alpha,x,y,z) be nonmeasurable at every point. Physicists claim that the end result must be linear. You, my friend, are in big trouble.
Bilge wrote:
Homogeneity and isotropy require the form of the equations of motion to have no explicit dependence on the coordinates. You have to define things like energy, momentum, angular momentum (or some quantities you want to call "laws of physics") such that any of your inertial observers defines them in the same way in their own frames.
Why is that my problem? It's clear that refusing to synchronize clocks has no adverse affect on the universe. I think you're just backpedaling and trying to argue that the laws of physics demand that clocks be synchronized.
Posted: Mon Oct 27, 2003 11:17 am Post subject: The newsgroup sci.physics.relativity
Bill Hobba wrote:
Wrong Eugene. If you do what you suggested then you have a way to determine where you are by reading of clocks and breaks the homogeneity of space postulate.
The homogeneity of space is a geometrical concept. That's the meaning of my joke about me tampering with the geometry of spacetime. I can see the headlines now. Maniacal scientist poses severe threat to the underlying structure of the universe. I am truly flattered by your testimony that I have the power to affect spacetime geometry.
Bill Hobba wrote:
You are logically allowed to do it but by making that definition you have turned an inertial reference frame into a non inertial one — free particles would not move with constant velocity for instance which is a central requirement of an inertial reference frame.
You can't have it both ways. You say that the laws of physics do not demand that clocks be synchronized but just as soon as I introduce a desynchronization scheme you start whining and complaining that I'm upsetting the laws of physics.
Bill Hobba wrote:
While your choice of synchronization is arbitrary, to conform to the central tenants of an inertial reference frame they must be consistent with it.
What you're saying is that I'm free to perform my desynchronization scheme but if I actually do it then I will be breaking the laws of physics. Let me ask you a question. What are the consequences of breaking the laws of physics? Will the fabric of spacetime begin to unravel? Will there be terrifying interdimensional crossrips of staggering proportions? Do you think that I'll be able to collect my fee? How much should I charge to save the universe?
Bill Hobba wrote:
That is why such a choice is unphysical — you are breaking the natural symmetries of an inertial frame, i.e., by doing that you no longer have an inertial frame.
You say that my choice is unphysical. Does that mean that I claim to be able to do the impossible? Or does it mean that I've fiendishly devised a way to do the impossible thereby being a potential threat to all of reality, personal existence and the known laws of the universe?
Posted: Mon Oct 27, 2003 12:54 pm Post subject: The newsgroup sci.physics.relativity
Dear Bilge,
If you wish to refute Tom Roberts' understanding of clock synchronization and his testimonials in my favor, perhaps you should demonstrate near asymptotic credibility and write up a scientific paper on the threat I pose to the laws of physics and the universe with my very menacing desynchronization scheme.
If you lack courage in refuting recognized experts like Tom Roberts then I don't believe that you've earned the right to pretend, with elaborate charades, that you've refuted my thesis.
Posted: Mon Oct 27, 2003 3:18 pm Post subject: The newsgroup sci.physics.relativity
Jem wrote:
> Have you ever heard that the homogeneity and isotropy of space and time require linear coordinate transformations?
No, but I've heard that the principle of relativity in conjunction with homogeneity and isotropy of space and time leads to linear transformations between inertial frames.
That's nothing. Have you heard that the principle of relativity in conjunction with the homogeneity and isotropy of space and time is a physical law that demands that clocks be synchronized? Do you believe that the White House and State Department should be contacted regarding my very menacing desynchronization scheme?
I have been accused of tampering with the laws of physics. It has already been said that my desynchronization scheme will cause free particles to accelerate uncontrollably and the same effect will be seen in all massive objects. Predictions are being made that the fantastic repercussions from my mathematical design will impact all inertial frames of reference. But what about the threat of possible fractures in the spacetime continuum itself?
Frankly, I am greatly honored, flattered and very surprised that my scientific ideas are being interpreted as the greatest potential source of inexhaustible energy since cold fusion. But my humility and sound state of mind tells me that I'm not deserving of such great recognition.
Oh don't give yourself so much credit Eugene. I've resisted responding to some of the other stuff in this forum, but this one I'm going to respond to. Your "desynchronization scheme" doesn't mean that the laws of physics are breaking down. The laws of physics are only valid in the first place if the clocks are synchronized. If the clocks aren't synchronized, then the laws of physics as presently formulated have nothing to say on the matter. You could, if you wanted to, construct laws that are based on certain desynchronization schemes. But they would have very little predictive power (and thus be useless laws). And if you really took them apart, they would just be the ordinary laws of physics with the proviso that some nut insisted on having the clocks out of synchronization.
Now please don't respond by saying something like 'Your misunderstanding comes from a lack of intuition about how true physics works. I have already desynchronized the clocks and the physics is invariant. See my unsolved exercises on blah blah blah.' because we are currently hammering out that point in the other thread we have going.
Lol. Sorry if I'm giving you more to write about. I'm just a little bored waiting for your other reply.
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