Transformation Vs. Physical Law

Samshorn

Still not true. Your premise is that we can experimentally determine that the Doppler shift when receding at a certain speed is the reciprocal of the Doppler shift when approaching at the same speed. The problem is that you haven't thought about how they would deterime that they are approaching each other at the same speed that they were formerly receding from each other. They obviously can't use the Doppler shift, because that would be circular and devoid of physical content. In other words, they can't simply define their approach speed to be equal to their receed speed when the Doppler shifts are reciprocal. For that proposition to have physical meaning, they need some independent measure of speed, which comes from the systems of coordinates in which the homogeneous and isotropic equations of mechanics hold good. There is simply no way of getting the effects of special relativity without establishing the correlation (implicitly or explicitly) with inertia.

Well, it obviously doesn't require any transforming of coordinates, but it does imply Lorentz invariance, which entails the covariance of the physical parameters under a certain class of transformations.

The answer to the OP is that the physical law describing the half-life of a sub-atomic particle moving in the x, y, and z directions by the amounts dx, dy, dz in the time dt is purely a function of the quantity sqrt[dt^2 - dx^2 - dy^2 - dz^2] where x,y,z,t are any single system of inertial coordinates. No transformation is involved. (But of course x,y,z,t do have to be coordinates in terms of which the laws of mechanics hold good.)

In fact, we find that every physical process and phenomenon (not just the half-lives of sub-atomic particles) has this same form, in the sense that the physical laws don't depend on the absolute values of x,y,z,t, nor even on the absolute values of dx,dy,dz,dt or their ratios, but only on the quantity dt^2 - dx^2 - dy^2 - dz^2. The fact that these physical laws work equally well in terms of any standard system of inertial spacetime coordinates implies that this quadratic quantity is the same in all of them. After noticing this, and then seeing it confirmed over and over again for all known physical laws, we begin to expect it to be true, even when trying to formulate the laws governing previously unknown phenomena. This property, called Lorentz invariance, is not itself a physical law, it is an attribute of all known physical laws.

It's useful to know about Lorentz invariance because it enables us to compute things very easily by taking a short cut. If we already know that a certain physical law (such as the law for the half-life of a particle) is Lorentz invariant, we know that we can compute things in any convenient system of standard inertial coordinates, and then very simply express the results in terms of any other system of coordinates using the Lorentz transformation (which happens to be the transformation that preserves that quadratic quantity appearing in the physical laws). But this is just a computational shortcut, used by people who know what they're doing. If it confuses the OP, he can just go ahead and do things the more laborious (and less insightful) way.

Austin0

Actually, no, I don't see any contradiction.

Do you doubt the validity of time dilation as a function of relative motion as it is described in the Lorentz math??

If you don't then I don't understand why you think there is a problem. Is it the semantic question of whether time dilation is called a law or a transformation? You seemed to agree that it could be both so I am confused as to your point here.

Austin0

Hi regarding relative velocities in these scenarios. Isn't that always problematic if we are considering hypothetical real world situations?? But normally in a case like ghwellsjr's it is assumed we extended a virtual frame all the way to the destination to measure velocity exactly , no? We do make the assumption that approach is equivalent to recession.

Regarding the invariant interval, I understood it was a direct derivation from the Lorentz math. Is this incorrect?

Austin0

Hi Firstly I don't agree or in fact even understand the OP's point, but I have to mention that the Doppler shift equation is itself derived from and expressing the fundamental transformation isn't it? With classical Doppler it seems to me there would be no age differential, no?

DaleSpam

If you start with the spacetime interval you can derive the Lorentz transform as a class of transforms that leaves the interval invariant. If you start with the transform you can derive the interval as a quantity that is invariant. It just depends what you want to consider a postulate and what you want to consider a derived result. The math doesn't care which direction you go.

I find a certain appeal to starting with the interval. After all, to me, the notion of distance seems more basic than the notion of coordinates.

ghwellsjr

I said if two inertial observers start off approaching each other (from far apart) and then pass each other so that they are then receding, they will continue at the same speed, won't they? I wasn't talking yet about the twin scenario.

But beyond that, I have thought about how we can demonstrate that the two Doppler factors (coming and going at the same speed) are reciprocals and I found the answer in Hermann Bondi's book, Relativity and Common Sense, pages 77 to 80. So we can figure it out either by experiment or by analysis.
I wasn't addressing the initial issue (which was already thoroughly addressed and discarded by universal_101 in his other thread that got locked because it was going around in circles) but only his contention that transformation tools are required to explain the twin paradox, which I did in a way that I thought might make sense to him.

ghwellsjr

I am not using the Doppler shift equation, if by that you mean the one that calculates the Doppler factor as a function of relative speed. I'm only saying that the approaching and receding Doppler factors are reciprocals for the same relative speed but we aren't concerned with what that relative speed is or how it relates to the Doppler factor. Of course, you can also confirm that this is true based on that Doppler shift equation, but that is immaterial for the analysis that I have given.

There are many ways to derive the equation but that is irrelevant to what I am saying. And yes, the classical Doppler shift equation won't work because it is not relativistic.

Austin0

Hi
could you point me to the experimental tests revealing length contraction?
I have looked without coming across anything. Thanks

Austin0

yes I understand your point regarding reciprocity and the relative length of time in each phase. And certainly agree.
But to suggest you can apply this principle to the twins question to explain the difference in final age, without invoking the gamma factor inherent in the relativistic Doppler equation, is a different story. Wouldn't you agree?

ghwellsjr

The story I am discussing now does not look at the relative length of time in each phase for both twins but only for the one that turns around. His two times are equal and knowing the Doppler factors are reciprocal allows him to derive the value of gamma without invoking any other considerations.

Samshorn

But an inertial observer doesn't constitute a basis for defining a velocity. For that we need an extended system of space and time coordinates. And if the velocities are going to correlate with the Doppler shift in the expected way we need them to be defined in terms of a standard inertial coordinate system. Of course, we can simply decline to consider any actual numerical velocities, but then we forfeit the ability to provide any quantitative answers to real world questions, and we don't have a physical theory at all. At some point we need to connect numerical velocities with the predicted quantitative effects.

Moreover, the assertion that every pair of inertial observers will each see the (presumed) standard frequency shifted by reciprocal factors when approaching and receding is tantamount to the assertion of not only source independence, but also directional independence and frame independence, meaning we are asserting the complete invariance of light speed in terms of any and every system of standard inertial coordinates.

Naturally we aren't required to explicitly construct such coordinates, but they are implicit in those premises. If two twins are directly approaching a central transmitter from opposite directions (all unaccelerated) and they see equal frequencies, we must say they have equal speeds relative to the rest frame coordinates of the transmitter. They pass the transmitter simultaneously and again see equal frequencies and therefore have equal speeds, so they implicitly define a system of space and time coordinates based on light synchronization. (We say they are at equal distances when they have received equal numbers of pulses.)

Bondi doesn't provide an analytical derivation of reciprocal Doppler factors, he simply assumes it (or rather, he assumes relativity and, tacitly, lightspeed invariance, from which it trivially follows, along with all the rest of special relativity), and spends a few pages trying to disguise the fact that he's simply assuming these things. Also, you can't on your terms "figure it out by experiment" either, because the thing to be figured out involves quantitative velocities (if it is to have any physical significance), and you can't even define velocities without some system of space and time coordinates.

universal_101

Exactly, since to account for the differential ageing of unstable particles in different frames, we must use a physical law and not a part of a transformation.

This is the center point of the debate, in special relativity it is the Lorentz transformations which are used to explain the differential ageing. But instead we should have a physical law explaining these differences, which then can be validly transformed for any other inertial observing frame using Lorentz transformation.

Austin0

Perhaps you could explain this trick?
Reciprocity of Doppler by itself ,without the gamma factor , does not imply aging differential.

So you are assuming that factor behind the scene , applying that to Speedo's hypothetical
observations and then asserting that Speedo, if he were mathematically inclined, could derive the Lorentz transformation directly from these observations.

Are you really claiming that the gamma is not involved or necessary to the explanation?

universal_101

This must be a new physics, since what you are suggesting is that, difference in the ages of the Twins after the trip, is independent of their relative velocity during the trip.

I mean, its alright to disagree with me or anyone for that matter, but rejecting everything that I post is gravely unscientific.

atyy

Exactly. However, if one is able to specify a coordinate system, then one can use the coordinates to describe events. In special relativity as in geometry, both the coordinate-system invariant and the coordinate-system descriptions are useful, with the proviso that when using the latter the coordinate system must be specified.

universal_101

This is incorrect, 2 and 1/2, 3 and 1/3, or any other form like x and 1/x are inversely symmetrical, but saying that these values, for example, 2,3 and x is independent of the relative velocity makes them arbitrary. I mean if they does not depend on the relative velocity, then how come you choose one over the other and say they are different, since 2 and 3 are obviously different.

Austin0

I don't know what you think physical law means. As far as I can see they don't really explain much. They simply describe phenomena in exact terms and provide a basis for predicting certain aspects of those phenomena.
So GR predicts certain cases of time dilation but no particular explanation of the mechanism. The Lorentz math predicts certain other cases of time dilation also with no explanation of mechanism. If you want, you can say GR is a law and the Lorentz math a transform but in this case that is a distinction without a difference.
A semantic quibble not worth pursuing. The function and utility are exactly the same.
I would say that the Lorentz math was fundamentally a physical law and only secondarily a transformation but that is also a semantic question not worth any effort.
So i think you might be better served directing your intelligence towards more interesting questions and subjects, just mHO