## Transformation Vs. Physical Law

 Quote by ghwellsjr I'm saying that since two inertial observers can experimentally determine that the Doppler based on light is the same for both of them as they approach each other and that it is the same for both of them as they recede away from each other and that these two Doppler factors are reciprocals of each other, then that is all they need to know to predict ... their accumulated age ratio ... from the Doppler factor. I'm also saying that this analysis does not require ... the establishment or definition of any frame of reference or coordinate system...
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.

 Quote by ghwellsjr I'm also saying that this analysis does not require any ... theory about transforming coordinates between different coordinate systems, which is what universal_101 is contending.
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.

 Quote by universal_101 Thanks for the view, I agree that Lorentz transformation is more than just a transformation in modern physics. It is exactly what I'm questioning. It seems as if the transformation is multipurpose, it can be a physical law at times and also can be a transformation at other. Do you see this contradiction of basic physics concept.
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.

 Quote by 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.
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?

 Quote by ghwellsjr I'm not trying to derive relativistic Doppler. I'm saying that since two inertial observers can experimentally determine that the Doppler based on light is the same for both of them as they approach each other and that it is the same for both of them as they recede away from each other and that these two Doppler factors are reciprocals of each other, then that is all they need to know to predict that if they enact the twin scenario where they depart from each other and remain inertial for a while and then one of them accelerates back toward the other one with the recprocal Doppler, their accumulated age ratio can be calculated from the Doppler factor. I'm also saying that this analysis does not require any synchronization of remote clocks by any method or the establishment or definition of any frame of reference or coordinate system or any theory about transforming coordinates between different coordinate systems, which is what universal_101 is contending.
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?

Mentor
 Quote by Austin0 Regarding the invariant interval, I understood it was a direct derivation from the Lorentz math. Is this incorrect?
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.

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Quote by Samshorn
 Quote by ghwellsjr I'm saying that since two inertial observers can experimentally determine that the Doppler based on light is the same for both of them as they approach each other and that it is the same for both of them as they recede away from each other and that these two Doppler factors are reciprocals of each other, then that is all they need to know to predict ... their accumulated age ratio ... from the Doppler factor. I'm also saying that this analysis does not require ... the establishment or definition of any frame of reference or coordinate system...
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.
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.
Quote by Samshorn
 Quote by ghwellsjr I'm also saying that this analysis does not require any ... theory about transforming coordinates between different coordinate systems, which is what universal_101 is contending.
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...
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.

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 Quote by 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?
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.

 Quote by GeorgeDishman Yes there is, you even say it is well known and DaleSpam gave you it mathematically. No we don't, my point was that the law applies equally well in both the Earth frame and the particle frame. The value of the half-life obtained in the lab is in the particle's rest frame while we usually measure the thickness of the atmosphere in the Earth frame. It is inherent in the question you asked that that those are not the same hence applying the transform is one way to get both to the same frame. However, that isn't the only way. If you want to know the value of the particle half-life in the Earth frame, you must apply the time dilation factor but that can be obtained from many experiments, that of Ives and Stilwell for example, you don't need to use the Lorentz Transforms. The Lorentz Transforms can be used to convert between the frames to check for consistency but they aren't needed to predict the particle numbers, both length contraction and time dilation can be obtained empirically from experiment as independent laws without using the transforms.
Hi
could you point me to the experimental tests revealing length contraction?
I have looked without coming across anything. Thanks

 Quote by 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.
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?

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 Quote by 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?
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.

 Quote by 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?
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.)

 Quote by ghwellsjr 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.
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.

 Quote by 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??
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.

 Quote by 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.
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?

 Quote by Mentz114 No, I don't agree. The clock time of a twin depends only on their *own* worldline. It is completely irrelevant what the other twin is doing. Relative velocity does not come into it, except implicitly when we choose a frame in which to do the calculation. This does not have to be one of the twins frames. The difference in age is the only time both twins come into the calculation.
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.

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 Quote by universal_101 Whereas, a transformation, let's consider a co-ordinate transform in geometry first, then we can simply extend the concept for the Lorentz Transformation. In geometry the shape of any object(circle, parabola, line) does not depend on the position of the origin of the co-ordinate system, even though the co-ordinates(x,y,z) of these objects can change.
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.

 Quote by ghwellsjr If we assume the Principle of Relativity for light, we are assuming that what each twin sees of the other one is symmetrical and not dependent on their relative speed in any medium.
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.

 Quote by 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.
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