Transformation Vs. Physical Law

DaleSpam

That is indeed reasonable, but the correct adjective is "invariant", not "independent".

The muon ratios you mention are invariant and are given by the law I cited at the beginning of the thread, without using any transform. So your question seems to have been resolved more than a hundred posts ago.

DaleSpam

See post 154 where I derive gamma directly from the metric without a Lorentz transform.

harrylin

Checking for it, I see that you already qualitatively clarified the whole matter in post #7 of this thread - that's more than 200 posts ago - and you stated that you already did so in an earlier thread.

And concerning the time tau, I thought that I gave that but if not, here it is for universal as given in 1905:

"the time marked by the [moving] clock (viewed in the stationary system) is slow [on the reference clock] by 1-√(1-v²/c²) seconds per second, or—neglecting magnitudes of fourth and higher order—by 1/2 v²/c²."

This effect must be valid in general: not only mechanical clocks slow down by a factor γ but any natural process, including the half-life of radioactive "clocks" such as muons.

Note also that the expression "viewed in the stationary system" is the reference for "laws of physics"; it's not a transformation equation, and universal can use that law for arriving muons with the earth as "stationary system".

GeorgeDishman

A transformation changes numbers in one coordinate system into the equivalent numbers in another system. If you have first calculated the number in the muon frame, you can use a transform to convert any frame-dependent numbers (e.g. lengths and times) to the Earth frame. The number of muons reaching the Earth is an invariant so applying a transform will not change it.

Physics is nothing but mathematics applied to measurable quantities. The basic physics we are using has been posted several times, it is the exponential equation which predicts the number of particles that decay in a given period based on the half-life of the muon. That equation is obviously just mathematics.

Using a transform simply converts that half-life to a different coordinate system.

No, I doubt that you have grasped that the question you are asking is the same as querying the "physical cause" for taking more strides to get from A to B via C than going directly from A to B. Most people think of that as a consequence of geometry rather than a "physical" effect but your definition of what you mean by "physical" is different to what I have seen before.

universal_101

See my post #207, Which shows very clearly, that you are just juggling mathematics..... and by NO means it can be said a derivation.

Mentz114

No, you're wrong. The gamma factor emerges from the Minkowski metric with a simple rearrangement of the differentials. You cannot deny this - it is elementary mathematics.

DaleSpam

Post 207 doesn't show anything "very clearly". And yes, as Mentz114 mentioned, my post 154 derives gamma without a coordinate transformation.

If you choose to call a correct derivation "just juggling mathematics" then so be it. It is still a correct derivation and completely refutes your claim.

GAsahi

You are totally missing (or willfully ignoring) the point: your derivation is not quite right, since the muons move in a cyclotron, [itex]r=R[/itex] so [itex]dr=dz=0[/itex] meaning that [itex]d\tau^2=dt^2-r^2d\theta^2=dt^2(1-R^2 (\frac{d \theta}{dt})^2)[/itex] meaning that [itex]d\tau^2=dt^2 (1-v^2)=(\frac{dt}{\gamma})^2[/itex]. So, the correct result is [itex]d\tau = \frac{dt}{\gamma}[/itex].

DaleSpam

That isn't the only correct way to work the problem.

universal_101

Let me put it this way, consider the Newtonian mechanics,

Just because we can understand a particular event by using energy conservation(Least action principle), does not mean that there is NO Force involved in the event.

Whereas, you are suggesting, because we can calculate the correct results also by using spacetime metrics, it means that Time Dilation is independent of relative motion, i.e. independent of gamma factor.

DaleSpam

What I have proven is that the mere fact that gamma shows up does not imply that there was a coordinate transform. I didn't make any claims about the independence of time dilation and relative motion.

Your entire complaint from the OP is that physical effects like muon decay should come from physical laws and not coordinate transforms. I gave the physical law for muon decay. You then complained that simply because gamma pops out of the law that it must be a transform. In fact, you said that the "only way I know of producing gamma factor is using Transformations(Lorentz)". So I showed you that it can be produced directly from the metric without any transform.

So now that you know that gamma can come from other sources than a coordinate transform it should be clear that your complaint is completely resolved.

Do you now accept that the decay of muons can be explained by a law of physics that is not a coordinate transform? If not, then what possible logical reason can you have for not accepting it?

GeorgeDishman

[QUOTE=universal_101;3950519]OK, then you should note that the null result of the MMX is invariant hence length contraction is a "physical effect" by your definition.

That is incorrect, LET predicts a null result for the MMX, if a non-null result was found, LET would be falsified. However, you miss the point, the null result of the MMX is an invariant hence valid by your definition. It is indirect in that it uses a radar-like technique but you say that's not what you meant and such inference is an unavoidable part of most measurements anyway.

Physics is purely mathematics in which the variables are identified with measurable quantities.

Well "gamma" is just a mathematical shorthand for a commonly ocurring term, it is part of many "physical" effects. I'm not at all sure what it is that you are arguing about.

universal_101

Do you mean, that Minkowski spacetime or metrics are the physical laws that govern the decay of Muons in motion ?

But then I'm wondering where does this concept of spacetime and the metrics comes from if not from the transformations. Since there must be a physical concept behind any physical law.

Why were these physical laws not discovered on their own !!

And how come this physical law can be used to derive the transformations, which it could not if it were an actual physical law !

Mentz114

In order to do physics with equations we need coordinates and time derivatives of coordinates. This means we implicitly assume some space or spacetime where the coordinates make sense. The physical laws must be written in some such space, although coordinates are themselves not physical.

Relativity demands that we examine the laws in moving frames, and this where transformations of coordinates happen. We demand that physical laws do not change under a relabelling of spacetime points.

Your remarks show a deep lack of understanding of how physics is actually done and makes me wonder why you think you are qualified to make any meaningful remarks about any aspect of physics.

Your last two posts make no sense to me at all.

Muphrid

I admit, I've not read through the entire thread, but can I give this a shot?

A muon travels on an inertial path from point [itex]A[/itex] to point [itex]B[/itex], separated by some proper time [itex]\tau[/itex], at which point it decays. This trajectory is characterized by four-velocity [itex]v[/itex].

An observer with four-velocity [itex]u[/itex] will measure the elapsed time between events [itex]A[/itex] and [itex]B[/itex] as [itex]\tau v \cdot (-u)[/itex]. If it happens that the observer has the same four-velocity as the muon, then this reduces to [itex]\tau[/itex] (in [itex]c=1[/itex] units), but any arbitrary observer will come up with a different result.

There is no transformation of coordinate systems going on here. There is only each observer taking the dot product of their own four-velocity vector and the four-velocity of the muon to find one component of the muon's four-velocity in a particular basis. That, in itself, makes a physical statement about the universe: that we are free to choose any basis we wish because any direction is equivalent to any other. We choose Minkowski spacetime to model the universe because it shares that property of isotropy with what we observe in the real world.

harrylin

Such things are usually not called laws, as some of us explained much earlier in this thread.
Not sure which laws you mean, but physical laws with gamma were largely discovered on their own - see post #236. Physicists solve puzzles of nature by contemplating different types of information.
??? The classical physical laws can be used to derive the "Galilean transformations" as I tried to explain many times incl. in post #236. Similarly the Lorentz transformations were first derived from physical laws, deduced from observations. And note that physical laws are always generalisations of ideas that emerged from observations.

DaleSpam

I noticed that in your response you carefully avoided answering the questions:

Do you now accept that the decay of muons can be explained by a law of physics that is not a coordinate transform? If not, then what possible logical reason can you have for not accepting it?

The metric is the mathematical structure which defines geometry. I.e. it describes distances, durations, angles, and the causal structure. The metric will appear in any law where geometry is important (which is most).

The law that governs the decay of muons (in motion or at rest) is the one I provided >240 posts ago.

The various geometric concepts don't need coordinates nor coordinate transforms, no matter how much you wish they did. They are physical concepts in and of themselves. Or are you trying to claim that my table leg is perpendicular to my table top only if I perform a coordinate transform?

Nonsense. What possible chain of reasoning could lead you to this absurd comment.