Transformation Vs. Physical Law

Mentz114

Good, because it is invariant.

The transformation does not predict how many many particles should reach a particular destination. The transformation changes the observers coordinates.

There is a physical law that decides the number, which law happens to be invariant under transformation .

Not necessarily. We can use the rest frame of the particle. We only use the LT when we want to see what happens in a different frame.

Nugatory

No contradiction that I see... The transform describes certain aspects of physical law, namely how observations of time and space differ between observers in relative motion. It's very convenient to describe these differences in terms of coordinate transforms because we generally state our observations of time and space in terms of coordinate systems.

PeterDonis

The transformation tells how different observers view the invariant event--the death of a character in the play, or the arrival of a given number of unstable particles at a given detector. So just as we expect the "transformation" from one audience viewpoint to another to keep invariant the death of the character in the play (while changing observer-dependent details such as the exact angle at which the character's face is viewed), we expect the transformation from one observer's viewpoint to another to keep invariant the number of unstable particles arriving at the detector (while changing observer-dependent details such as the time, according to that observer, that the particles take to travel from source to detector, or the distance between the two). As of course it does.

So your own analogy perfectly supports the facts of the Lorentz transformation; yet you talk as if you are somehow pointing out a problem. What problem?

universal_101

If something is there for decades and so many people admire it, does not make that something correct or does it. I don't want to include history, which says otherwise.

But what I would really like to mention is that, to refute a theory we need just one experiment where as to give a theory the stature of fundamental fact there is NO limit on the Experiments.

And No, I'm not looking for paradoxes, instead I'm looking for solutions.

Thanks

universal_101

I don't think even using the rest frame of the particle, you can calculate the number of particles reaching Earth without using any kind of transformation of any property what so ever.

Mentz114

Which something are you talking about ? Special relativity ?

Sure. What experimental evidence have got ?

In the first post of this thread you asked for an explanation of a paradox

Solutions to what problem ? The fact that physical laws must be Lorentz invariant is not a problem.

PeterDonis

If you know the time in the rest frame of the particle, what could you possibly need to transform? That's the only variable in the formula DaleSpam posted (everything else is a physical constant or a known initial condition).

universal_101

The problem is, it is the audience viewpoint of one special position in audience, which is utilized in determining how many characters will die in a certain play.

That is, the number of particles reaching Earth are determined by the tools of transformation. This is a big problem, at-least to my understanding.

DaleSpam

Yes.

Since it is a law of physics and since all laws of physics are diffeomorphism invariant, we know that it is invariant under the Lorentz transform. But no transform is required in order to use it.

universal_101

I think you left the Length part, since in order to calculate how many particles reached, we must know how much they traveled.

And this length is to be transformed.

Mentz114

If you do the calculation in the rest frame of the particle, only the coordinates of that frame are used. No transformation is used. As others have said above.

Yes, measured in rest frame coordinates.

PeterDonis

No, it isn't. Remember we're talking about the rest frame of the particle: in that frame, the particles are at rest. So there is no "length" involved--only the particles' travel time (or, if you want to be really precise, since the particles are not moving but the source and detector are, in this frame: the time between when the source is co-located with the particles and when the detector is co-located with the particles, by the particles' clock).

Look at the formula DaleSpam posted, which explicitly uses the time in the particles' rest frame. Do you see any length in there?

Nugatory

Yes.
I was tempted to add "of course", but obviously it's not obvious or you wouldn't be asking.

So here goes.... You find yourself riding a relativistic particle down from the top of the atmosphere. You see the surface of the earth rushing towards you at speed v=.999c, from a distance of 1 light-usec away. Note that neither this distance nor the speed came from any sort of transformation - you measured them directly.

Now, what is the probability that your relativistic but unstable steed will hit (be hit by) the surface of the earth before it decays? Calculate the time the particle needs to live, by dividing the distance by the velocity, and plug it and lambda (the half-life of the particle expressed in terms of the particle's proper time, which is the time that you are measuring - see, still no transforms) into the formula... And out pops your answer.

universal_101

I thought it would be simple to explain the necessity of the use of the transformation even in the rest frame of the particles.

In order to pinpoint, how does one calculate when was the particle at source and when at the detector? The simple equation would have been, contracting the distance between the source and detector and dividing it by the relative velocity.

But you never mentioned how are you going to calculate when the particle was at source and how much time it took to reach the detector.

universal_101

You are using the increased half-life time of the particle, which is a transformation tool. Remember you would either use Time Dilation of half-life or the length contraction of the distance which the particle needs to travel. And these are not transformations but tools of it.

DaleSpam

Where, exactly? [itex]\lambda[/itex] is the usual non-increased half life. It is not a function of speed.

universal_101

So that everyone can see it, does the differential ageing of the twins after the trip can be explained without using transformation tools ?

No it cannot, the same applies to the number of particles reaching Earth. That is, No matter what, in the end the ratio of the number of particles reaching Earth to the number of particles survived in the lab, is always a function of the speed of the particles which reach in higher quantity.

What you guys are missing is the point that, we need to use that same law for lab particles also.

Thanks