Transformation Vs. Physical Law

[Samshorn]

The quantity aT is equal to t in units of "laps", not a.

No, for the stationary particle the quantity 'a' does not represent any angular measure of laps or anything like that, because the particle is stationary. Its angular position is not changing. You yourself have admitted that 'a' is nothing but a parameterization of the worldline of the particle, and its only definition is a = t/T where T is just an arbitrary unit of time that you carried over (unwittingly) from a different case. So 'a' is nothing but coordinate time expressed in units of "reference particle lap times". Needless to say, the units of "a" don't matter, because you're multiplying dtau by da/da. So you just need to be sure the limits of integration are specified in consistent units with the variable of integration.

Look, the relevant integrand is (dtau/dt)dt = dt/gamma, but you choose to write this in the more convoluted form (dtau/d[t/T])d[t/T], and you claim with a straight face that, because you've written it this way, replacing t with t/T for some arbitrary constant T, it no longer involves gamma or the coordinate time t, not even implicitly, despite the fact that the T's obviously cancel out, leaving nothing but dt/gamma.

I honestly don't see how you can seriously claim that, simply by multiplying dt/gamma by T/T for some arbitrary constant T, the resulting expression no longer involves dt/gamma, not even implicitly!

 

[Dale]

And I honestly don't see how you can seriously claim that a dimensionless quantity has dimensions of time.

 

[GAsahi]

OK, so I will use units where c=1, and the decay law in post 13: n=n_0 e^{-\lambda \tau}. Assuming that the initial number and the decay constant are known then all that remains is to calculate tau for two cases, one being muons in a cyclotron and the other being muons at rest in the lab. We will neglect gravity and, since cyclotrons are circular it will be convenient to use cylindrical coordinates. The flat spacetime metric in cylindrical coordinates is: d\tau^2=dt^2-dr^2-r^2d\theta^2-dz^2 and, as mentioned in post 37, \tau_P=\int_P d\tau

For the first case, the muons in the cyclotron, we can write their worldline as P=(t,r,\theta,z)=(a T, R, a 2 \pi, 0) where a is the number of "laps" around the cyclotron, R is the radius of the cyclotron, and T is the period for one lap. So we have
\tau_P=\int_P d\tau=\int_0^a \frac{d\tau}{da}da=\int_0^a \sqrt{\frac{dt^2}{da^2}-\frac{dr^2}{da^2}-r^2\frac{d\theta^2}{da^2}-\frac{dz^2}{da^2}} \; da=\int_0^a \sqrt{T^2-0-R^2 4 \pi^2-0} \; da=a\sqrt{T^2-4\pi^2 R^2}

For the second case, the muons at rest in the lab, we can write their worldline as P=(t,r,\theta,z)=(a T, R, 0, 0). So we have
\tau_P=\int_P d\tau=\int_0^a \frac{d\tau}{da}da=\int_0^a \sqrt{\frac{dt^2}{da^2}-\frac{dr^2}{da^2}-r^2\frac{d\theta^2}{da^2}-\frac{dz^2}{da^2}} \; da=\int_0^a \sqrt{T^2-0-R^2 0-0} \; da=aT

So, I have calculated the number of decayed particles for particles at rest in the lab and in a cyclotron without using \gamma = (1-v^2)^{-1/2}.

This is not quite right, since the muons move in a cyclotron, r=R so dr=dz=0 meaning that d\tau^2=dt^2-r^2d\theta^2=dt^2(1-R^2 (\frac{d \theta}{dt})^2) meaning that d\tau^2=dt^2 (1-v^2)=(\frac{dt}{\gamma})^2. There is really no reason to introduce the number of laps, a as a variable of integration, the integration variable is the coordinate time t. This is the standard for problems of this type.

 

[Samshorn]

And I honestly don't see how you can seriously claim that a dimensionless quantity has dimensions of time.

It's quite common in relativity to normalize all variables so that everything (distances, times, masses,...) all have consistent units, such as all having units of meters (so-called geometric units). In these units, the mass of the Sun is 1.475 kilometers. That's right... the mass of the Earth has units of distance. Gasp! And we can just as well divide all these quantities (distances, times, masses,...) by some arbitrary reference distance (such as the diameter of your navel), to make them all dimensionless. These are perfectly fine units. Of course, what we've really done is converted everything to units of "Dalespam's navel diameters".

Again, every numerical value of a physical quantity is a ratio of the variable to some reference value along with arbitrary scale factors - this is the meaning of units. What your high school teacher tried (and evidently failed) to teach you is that we must have consistency of units, not that we are obligated to use any particular units (e.g., metric versus English, or seconds versus Dalespam navels). Of course, in the case under discussion, where the coordinate time appears only as dt/dt, it isn't even necessary for the units of t to be the same as the units of tau. All we need to ensure is that the units are consistent with the limits of integration. But in more general circumstances we are still always free to define whatever units we like, provided only that we are consistent.

So, now that we've cleared up your misunderstanding of elementary units, hopefully you can see that your 'a' is nothing but t in weird units, and your analysis simply consisted of integrating dt/gamma.

 

[Dale]

There is really no reason to introduce the number of laps, a as a variable of integration

There is really no reason not to either. Reparameterizations are common and well accepted.

 

[Dale]

It's quite common in relativity to normalize all variables so that everything (distances, times, masses,...) all have consistent units, such as all having units of meters (so-called geometric units). In these units, the mass of the Sun is 1.475 kilometers. That's right... the mass of the Earth has units of distance.

In geometrized units t has units of length. It is still not dimensionless.

 

[Samshorn]

In geometrized units t has units of length. It is still not dimensionless.

Your misunderstanding of what "dimensionless" means was explained in the very next sentence after the one you quoted:

And we can just as well divide all these quantities (distances, times, masses,...) by some arbitrary reference distance (such as the diameter of your navel), to make them all dimensionless. These are perfectly fine units. Of course, what we've really done is converted everything to units of "Dalespam's navel diameters".

Again, every numerical value of a physical quantity is a ratio of the variable to some reference value along with arbitrary scale factors - this is the meaning of units. What your high school teacher tried (and evidently failed) to teach you is that we must have consistency of units, not that we are obligated to use any particular units (e.g., metric versus English, or seconds versus Dalespam navels). Of course, in the case under discussion, where the coordinate time appears only as dt/dt, it isn't even necessary for the units of t to be the same as the units of tau. All we need to ensure is that the units are consistent with the limits of integration. But in more general circumstances we are still always free to define whatever units we like, provided only that we are consistent.

So, now that we've cleared up your misunderstanding of elementary units, hopefully you can see that your 'a' is nothing but t in weird units, and your analysis simply consisted of integrating dt/gamma.

 

[Dale]

Your misunderstanding of what "dimensionless" means was explained in the very next sentence after the one you quoted:

And we can just as well divide all these quantities (distances, times, masses,...) by some arbitrary reference distance (such as the diameter of your navel), to make them all dimensionless. These are perfectly fine units. Of course, what we've really done is converted everything to units of "Dalespam's navel diameters".

You seem to be missing some basic algebra. If you start with geometrized units t is in e.g. meters and a is dimensionless. If you then "divide all these quantities" by navels then t' will be dimensionless but a' will have units of navels^-1. There is no factor that you can divide out to get t and a into the same units. They are dimensionally inconsistent no matter how much you wish otherwise.

 

[Samshorn]

If you start with geometrized units t is in e.g. meters and a is dimensionless. If you then "divide all these quantities" by navels then t' will be dimensionless but a' will have units of navels^-1. There is no factor that you can divide out to get t and a into the same units.

Of course they have different units. That's what I've been telling you. The variables t and 'a' both represent coordinate time, but in different units. To argue that they can't both represent coordinate time because they have different units is absurd. Hours, seconds, reference lap times... the units don't change what the variable represents, not even if you choose to give the variable different names in different units.

You defined 'a' as coordinate time in units of "reference lap times", which is a pure number and hence dimensionless, but it still represents coordinate time. Just as 'beta' (=v/c) and v both represent velocity, your variable 'a' (=t/T) and t both represent coordinate time. Yes, they have different units. But obviously this does not imply that beta (which is numerically equal to v if we choose units such that c=1) does not represent velocity. And it does not imply that 'a' (which is numerically equal to t if we choose units such that T=1) does not represent coordinate time. It's purely a matter of arbitrary units (and even more arbitrary in your case, since your scale factor T is itself totally arbitrary).

All this merely confirms that you just unwittingly integrated dt/gamma, and confused yourself into thinking you had done something different by using a different symbol for coordinate time in dimensionless units.

 

[Dale]

Of course they have different units. That's what I've been telling you. The variables t and 'a' both represent coordinate time, but in different units. To argue that they can't both represent coordinate time because they have different units is absurd. Hours, seconds, reference lap times... the units don't change what the variable represents, not even if you choose to give the variable different names in different units.

Perhaps you don't understand the difference between units and dimensions. t and a not only have different units, they have different dimensions. That means that there is no conversion factor between them. Here is a good place to start:

http://en.wikipedia.org/wiki/Dimension_of_a_physical_quantity

You defined 'a' as coordinate time in units of "reference lap times", which is a pure number and hence dimensionless, but it still represents coordinate time.

Amazing. You really seem to honestly believe that a dimensionless number nonetheless has dimensions of time.

 

[Samshorn]

Perhaps you don't understand the difference between units and dimensions. t and a not only have different units, they have different dimensions.

No, you're still confused. Take a look at
http://www.owlnet.rice.edu/~labgroup/pdf/Dimensions_units.pdf

In summary, it says:
"It's easy to confuse the physical dimensions of a quantity with the units used to measure the dimension. We usually consider quantities like mass, length, and time as fundamental dimensions, and then we express the dimensions of other quantities like speed, which is length/time, in terms of the basic set. Every quantity which is not explicitly dimensionless, like a pure number, has a characteristic dimension which is not affected by the way we measure it. Units give the magnitude of some dimension relative to an arbitrary standard. For example, when we say that a person is six feet tall, we mean that person is six times as long as an object whose length is defined to be one foot."

Do you understand this? When we say a person is six feet tall, we are giving the ratio of two lengths (the person and the reference object), so the ratio is dimensionless, but we still say the person's height has the dimension of length, and it's magnitude has the dimensionless value 6 in units of feet. The magnitude of the length in any specified units is dimensionless.

This confuses people because they don't understand the degree to which our terminology is conventional, and because they don't clearly distinguish between variables and reference constants. Think about spatial length in a certain direction, represented by the coordinate X (distance from the origin), which has the dimension of length. So far we haven't specified any units. Now we decide to use the length L of King George's foot as our unit of length. This means we divide every value of X by the length L. Letting x denote the numerical value of X in units of feet, we have x = X/L.

Bear in mind that neither X nor L has intrinsic units, and they both have the dimension of length. When we take the ratio of those two lengths, we get a pure dimensionless number, but then we DEFINE this ratio x to be the numerical value of X in units of feet. Students get confused because the ratio of any length to the length of King George's foot is clearly dimensionless, so how can we say the ratio has the dimension of length?

The answer is that we DEFINE the ratio of any length X to the reference standard length L as the magnitude of the length X in units of L. The reason this is confusing is because if we divide X by any other length Y, we usually regard the ratio as just a dimensionless number. However, we COULD regard it as the value of the length X in units of Y, if we were treating Y as simply an arbitrary reference length. The point is that reference lengths defining units are treated (semantically) differently - by convention - from other lengths.

Getting back to the topic of the discussion, when you work with the quantity t/T where T is a reference constant, you are just establishing the units (reference lap times), and the dimensionless ratio gives the numerical value of the coordinate time (which has dimension of time) in units of "reference lap times".

You really seem to honestly believe that a dimensionless number nonetheless has dimensions of time.

To be more precise, the numerical value, in specified units, of any quantity with a certain dimension is always a dimensionless ratio of the magnitude of the quantity to the magnitude of whatever reference quantity defines the units.

Again, this all just confirms that you just integrated dt/gamma.

 

[Austin0]

Not really. Let me give an analogy. Take a piece of paper, and make two dots on it, points A and B. Draw a curve connecting the two points. That curve has a length, and the length has a meaning that is independent of which coordinate system you use to compute the length. We can represent that length in the following way:

Divide the curve into tiny little segments by selecting a bunch of points along the curve P₀, P₁, ... P_N. Then define the vector V_i to be the line connecting P_i to P_i+1. Then the length of the curve is (in the limit as N → ∞) given by L = sum over all i of |V_i|, where |V_i| is the length of vector V_i. This definition doesn't refer to coordinates at all. However, to compute the length of vector V_i, we can pick a coordinate system (x,y) to describe V_i, and compute the length as
√(δx² + δy²).

The spacetime interval is exactly the same sort of thing. You have two events (points in spacetime) A and B. You have a path connecting the two events. You divide the path into segments by picking events along the curve e_i. Then define the spacetime vector V_i to be the vector connecting e_i to e_i+1. Each vector V_i has a "length", |V_i|, and to compute the "length" of the whole path, you just add up the lengths of the V_i. This definition doesn't refer to coordinates at all. However, to compute the length of vector V_i, we can pick a coordinate system (x,t) to describe V_i, and compute the length as
√(δt² - δx²/c²).

Thank you for your explication but as I fully understood the meaning of the interval and the metric I am afraid it completely missed the point.
That point being the meaning of the word invariant. In this context it simply means constant,unchanging, across all inertial coordinate systems
This necessarily implies the existence of other frames.
Would you disagree?

It does not apply to other coordinate systems within a single frame. I.e. changing from orthogonal to polar coordinates for eg.

It does not apply to local measurements as they apply within the frame.

It takes local coordinate measurements and outputs a value that is meaningful and constant in all other frames.

As that output value is related to the input values by the gamma factor it would appear it was a de facto transformation, semantic quibbles notwithstanding.
Yes

 

[Austin0]

DO you think that gamma could have a different equation giving different relationships?
Could the metric have a different form and still correspond to the gamma relationship of space and time?

I don't know what you mean by "the gamma relationship of space and time". Gamma is a particular expression: . It is easy to get this expression from the Minkowski metric. As far as I know, it is not possible to get it from other metrics without doing a transform to a (local) Minkowski coordinate system. Although Samshorn's point that v=dS/dt does expand the class of metrics where you can get it.

"the gamma relationship of space and time". is the inherent relationship that as the spatial part of motion increases relative to the temporal component, the temporal part is decreased relative to other inertial frames. It appears that it was this relationship that determined the metric signiture, the only deviation from a Euclidean metric.
would you agree that this seems to be an intrinsic quality of spacetime geometry??
.

dt (untransformed coordinate time) =====> {BLACK BOX} ======> dtau (transformed time -by the gamma factor)
No matter what may have occurred in the black box, Quija board,quantum computer, whatever, that makes no difference.
It is clear that the final value is related to the initial value by the gamma factor?
do you disagree with this statement?

There is no black box which takes dt as input and gives dτ as output in general. For certain specific problems (constant speed particles) in specific metrics (Minkowski) you can simplify the inputs to the black box as you describe. But in general the inputs to the black box are all of the dxi, not just dt, and in fact in some metrics there is no dt to begin with.

This time I was presenting an analogy ;-) in this case the black box represented whatever math you cared to employ to achieve your result. that it also entailed a dx was overlooked for simplicity.

The point of course being what was inside the box was irrelevant , The outcome making it clear that in some form the gamma factor had to be present in the box and that the operation itself was functionally a gamma (Lorentz) transformation what ever it might be called.

Of course, I do agree that any physical experiment where the outcome is a function of gamma will have the same outcome regardless of the metric used to analyze the experiment. The point I was making is that the correct outcome can be obtained by simple application of the law as stated, without at any point explicitly bringing gamma into the analysis nor doing any transforms.

Yes you did accomplish your goal without explicitly invoking gamma or doing a transformation within a limited semantic loophole regarding what is called a transformation.

For the first case, the muons in the cyclotron, we can write their worldline as P=(t,r,\theta,z)=(a T, R, a 2 \pi, 0) where a is the number of "laps" around the cyclotron, R is the radius of the cyclotron, and T is the period for one lap. So we have
\tau_P=\int_P d\tau=\int_0^a \frac{d\tau}{da}da=\int_0^a \sqrt{\frac{dt^2}{da^2}-\frac{dr^2}{da^2}-r^2\frac{d\theta^2}{da^2}-\frac{dz^2}{da^2}} \; da=\int_0^a \sqrt{T^2-0-R^2 4 \pi^2-0} \; da =a\sqrt{T^2-4\pi^2 R^2}

Looking at your math it appears to me that at this point =a\sqrt{T^2-4\pi^2 R^2}

you have a distance measurement and a time interval. That you are simply applying the Pythagorean operation to these quantities. the operation itself composites vector components and returns a single value of length. in this case a spacetime length expressed in units of proper time.
The fact that the spatial measurement was not linear does not seem relevant as by this point it is simply a value.
Harking back to an earlier proposition of mine that the gamma was hiding in the pythagorean operation as a geometric gamma function , this seems to be supported by your calculations.
As you say, you did not explicitly employ the function or a gamma value , yet the end result was in fact, transformed.
Since what was in the box was the pythagorean operation this would imply that it was the culprit.
This transformation can be easily demonstrated regarding linear motion within the context of Minkowski geometry.
Regarding the other's arguments about integrating with gamma. In this case with motion limited to angular displacement , is integration even necessary?.
wouldn't it be a simple measurement of distance determined by pi?

Perhaps I am not understanding correctly (quite possible) but to me the term itself , invariant interval necessarily implies a multiplicity of frames to have any meaning.

It also didn't seem to apply within a single frame as time intervals are automatically proper time within that frame.

T

his is not true in general. For example, consider a rotating reference frame. For large values of r the t coordinate is spacelike.

Wouldn't it be more correct to say it was true in general but not necessarily true in certain limited cases like rotating frames , which I would think was not even an inertial frame. This is SR

So i understood it to be turning coordinate intervals between events occurring relative to another frame, F ,into a form/value that would agree with all other frames evaluation of that interval relative to F. ------> F's proper time.
Is this incorrect??

I am not certain that I understand what you are saying, but I don't think that is incorrect, just a bit "cumbersome".

Relativity, particularly GR, is a geometric theory. Just think about the spacetime interval as a strange "distance" in spacetime. Distance is a geometric feature, it exists independently of any coordinate system that you might draw on top of the geometric features.

I agree, cumbersome ;-)
yes I understand the spacetime interval,,, the question was; Was determining the interval a transformation.?

It also seems to me that within a frame the metric is still Galilean/Euclidean.
With the assumption of synchronicity of all clocks within the system.
Functionally equivalent to absolute time.

Not in general, particularly for non-inertial reference frames.

We are talking about SR , inertial frames, Not GR or accelerating frames which are outside the topic of this thread and discussion.
No offense intended whatsoever, but you have a general habit of taking questions and statements that have an explicitly limited context and subject, and either answering them with generalities or refuting them for lack of generality.

So the Minkowski metric actually only applies to other frames or is this not so?

This is not so. It applies to every inertial frame, not just "other" inertial frames.

Wouldn't it be truer to say it is applied within every inertial frame regarding every "other" inertial frame.
In any Minkowski diagram the designated rest frame is in a purely Cartesian coordinate structure with a Euclidean metric or do you disagree with this??
If you disagree could you explain what you mean by applying to every frame?

 

[universal_101]

I don't think your refutation holds water, for the reasons explained above. Mind you, we can certainly contrive to avoid writing the greek symbol "gamma", merely by writing out the definition of gamma in full, which basically is what the metric line element represents. (Likewise we can avoid writing "v" by writing dS/dt, but would we really claim we have avoided using v?) But I don't think the OP's fundamental error is in thinking that the results of special relativity are represented by the gamma factor. The gamma factor actually does encode the essential non-positive-definite signature of the Minkowski metric, from which the unique effects of special relativity arise. So although associating everything with "the gamma factor" may be a somewhat dim-witted way of looking at things, it isn't exactly wrong.

Thanks for the support on the basic math magic trick, the same can be criticized by the Dalespam's own analogy of Principle of least action and Newtonian mechanics, or even the simple following basic mathematics principle,

which is A+A = 2A, that is, suggesting A+A is not same as 2A, because he never uses multiplication when he uses addition and vice-versa, is entirely incorrect.

The OP's fundamental problem, as he clarified in his "farewell cruel world" post is that he says he wants to know "where this factor comes from", and yet he in unable to articulate what he means by this question. He began by saying he would be satisfied if someone could give him the physical law, not involving a transformation, but then when you provided that law he shifted his ground, and began asking where that law "comes from".

Physics implies causality principle, i.e. If we have an effect then we must have a cause, atleast in the domain of classical physics.

Therefore, the effect(Time Dilation of unstable particles) must have a cause(where it comes from). Since, atleast you are accepting that we do have to involve gamma factor in order to understand these effects. How do you understand, the use of this gamma factor which is a part of the transformations, to produce the effects is question.

Obviously that question is so vague as to be meaningless, and all efforts to get him to clarify his meaning are doomed to fail, basically becuase he doesn't have any clue what he means, because he has never subjected his own beliefs to any kind of rational scrutiny. My guess is that the only answer about where something "comes from" that would satisfy him is an explanation that conforms to his personal pre-conceptions, prejudices, and misconceptions, none of which he ever intends to give up. Anything else he will simply reject as not satisfactory. Still, it's sometimes of interest to engage someone like that in conversation, if only for the light it sheds on some pathological aspects of human psychology.

As I explained, you simply misunderstood what I was trying to ask, and not for introducing personal war of words, but it does not interests me a bit, how easily you make your perspective of other people, just by reading posts from a forum.

 

[universal_101]

Maybe what you are calling a gamma factor is a property of the Minkowski spacetime, the background ( or theatre) in which the physical laws act.

In Minkowski spacetime, physical laws are unaffected(i.e.produce invariant results), aren't they ?

Then how come we explain, the invariant results of Muons Experiment, by using a property of Minkowski spacetime ?

Whereas, it seems evident(using basic physical concepts) that the explanation must come from a physical law !

 

[universal_101]

The gamma function comes directly from the intrinsic properties of the physical world.
It is simply a description of the fundamental relationship of time, space and motion. If it was not discovered through Maxwell it would have been through particle accelerations or other empirical measurements. So you have it backwards. The function does not come from the transformation; The transformation comes from the function. And that was always there

This is a good description, but what I'm questioning is the simple incompatibility of the two.

That is, the transformation cannot produce a function which can be applied to physical effects, nor can the physical effects be associated with the transformations. This is the basic physics argument.

And Since, we don't observe any other such physical effects(Length contraction, increased density), just as the above argument suggests.

This implies, we must have a physical law to account for the Time Dilation of unstable particles and increase in inertia, whereas, EM phenomena comes validly under Lorentz transformations.

 

[universal_101]

OK. Well, obviously it is part of laws of nature. And the same question can be asked about all laws of nature. So, perhaps that becomes too philosophical indeed!

However, partial answers exist. Special relativity assumes that everything behaves like electromagnetism - and that is only a small stretch from knowing that matter is governed by electromagnetic bonds. From such considerations one can build special relativity "bottom-up" (and in fact this is just how the early development proceeded), for example by analyzing how a light clock would behave in motion.
- http://en.wikipedia.org/wiki/Time_d...nce_of_time_dilation_due_to_relative_velocity

Thanks Herald, I appreciate the insights or answer.

Unfortunately it is incomplete, even if I consider that stretch, Since Special Relativity is all about the transformations of one electromagnetic effect, to be analysed from the other frame so that equations give same result in this another frame too. Special relativity is not about the electromagnetism but it's transformation.

 

[harrylin]

[..]the transformation cannot produce a function which can be applied to physical effects, nor can the physical effects be associated with the transformations. This is the basic physics argument.

Argument of what? Apparently you missed my post #95 (as well as #120). Already in classical mechanics are physical effects associated with the transformations*; thus your denial is simply wrong. Physical laws result in the validity of certain transformation equations between reference systems, so that they are related. If you know the effect (the transformation equations) then you can draw some conclusions about physical laws and causes.

*PS I had in mind Newton's law but did not elaborate, however I see that now there is a fresh thread on that (simple explanation in post #10):
https://www.physicsforums.com/showthread.php?t=610258
Do you really claim that this is wrong?

[..] Special Relativity is all about the transformations of one electromagnetic effect, to be analysed from the other frame so that equations give same result in this another frame too. Special relativity is not about the electromagnetism but it's transformation.

It's about both the physical laws and the resulting transformation equations (emphasis mine):

"Poincaré has objected to the existing theory of electric and optical phenomena in moving bodies that [..] the introduction of a new hypothesis [will be] required [..] each time new facts [are] brought to light. Surely, this course of inventing special hypothesis for each new experimental result is somewhat artificial. It would be more satisfactory, if it were possible to show, by means of certain fundamental assumptions [..] that many electromagnetic actions are entirely independent of the motion of the system. [..] I believe now to be able to treat the subject with a better result. The only restriction as regards the velocity will be that it be smaller than that of light."
- Lorentz 1904

"It is known that Maxwell's electrodynamics—as usually understood at the present time—when applied to moving bodies, leads to asymmetries which do not appear to be inherent in the phenomena. [..] [SR is] a simple and consistent theory of the electrodynamics of moving bodies based on Maxwell's theory for stationary bodies."
- Einstein 1905

 

[stevendaryl]

Thank you for your explication but as I fully understood the meaning of the interval and the metric I am afraid it completely missed the point.
That point being the meaning of the word invariant. In this context it simply means constant, unchanging, across all inertial coordinate systems. This necessarily implies the existence of other frames. Would you disagree?

Yes, I would disagree. The length of the hypotenuse of a triangle is, by Pythagorus, equal to √(A² + B²), where A is the length of one leg, and B is the length of the other leg. Do you think that that definition implies the existence of other frames? The invariant interval in SR, τ = √((ct)² - x²) is a geometric relationship, just like the length of the hypotenuse of a right triangle. It doesn't have anything to do with "frames". The metric doesn't have anything to do with frames. Curved surfaces have associated metrics, and that doesn't have anything to do with frames.

 

[stevendaryl]

It does not apply to other coordinate systems within a single frame. I.e. changing from orthogonal to polar coordinates for eg.

Yes, it certainly does. In cartesian coordinates, the metric tensor is defined by
(in 2-D spacetime):

g_tt = 1
g_xx = -1/c²
g_yy = -1/c²

(all other components are zero)

In polar coordinates, we have:
g_tt = 1
g_rr = -1/c²
g_θθ = -r²/c²

So the invariant interval is

ds² = dt² - dr²/c² - r²dθ²/c²

 

[Austin0]

Each twin appears to age at a certain rate according to physical law, and the difference of these two rates gives you the difference in their ages when they meet again.

Hi I agree completely with this statement and it is the perfect example as it can't get any more physical than that.

In particular, the difference in the ages of the Twins after the trip does not depend on their relative velocity during the trip. For example if the one does not stay at home but the two take off at the same speed in opposite directions, turn around and meet again, then their relative velocity was much more; nevertheless their difference in age can be zero.

But I think this might be a bit ambiguous. yes in this case they have instantaneous relative velocity throughout the trip but the end result depends on total spacetime distance traveled not on the relation during any segment.
I know you know this and that was your point but it could appear you were suggesting that relative velocity wasn't a factor in determining the aging [to someone who didn't understand world lines]
To determine the difference in rates requires relative velocities.wrt some frame.yes??

 

[Nugatory]

or even the simple following basic mathematics principle,
which is A+A = 2A, that is, suggesting A+A is not same as 2A, because he never uses multiplication when he uses addition and vice-versa, is entirely incorrect.

Is this thread a Socratic dialog with us as the students at the knee of a master Sophist? Or have we met the King of Trolls and fallen into his snares? Either way, it's an amazing thing to read from the beginning.

 

[universal_101]

Argument of what? Apparently you missed my post #95 (as well as #120). Already in classical mechanics are physical effects associated with the transformations*; thus your denial is simply wrong. Physical laws result in the validity of certain transformation equations between reference systems, so that they are related. If you know the effect (the transformation equations) then you can draw some conclusions about physical laws and causes.

I never denied the association of the physical effects with the transformations, I'm questioning them.

*PS I had in mind Newton's law but did not elaborate, however I see that now there is a fresh thread on that (simple explanation in post #10):
https://www.physicsforums.com/showthread.php?t=610258
Do you really claim that this is wrong?

No, We all know the Newton's law is invariant under Galelien transformation.

It's about both the physical laws and the resulting transformation equations (emphasis mine):

"Poincaré has objected to the existing theory of electric and optical phenomena in moving bodies that [..] the introduction of a new hypothesis [will be] required [..] each time new facts [are] brought to light. Surely, this course of inventing special hypothesis for each new experimental result is somewhat artificial. It would be more satisfactory, if it were possible to show, by means of certain fundamental assumptions [..] that many electromagnetic actions are entirely independent of the motion of the system. [..] I believe now to be able to treat the subject with a better result. The only restriction as regards the velocity will be that it be smaller than that of light."
- Lorentz 1904

"It is known that Maxwell's electrodynamics—as usually understood at the present time—when applied to moving bodies, leads to asymmetries which do not appear to be inherent in the phenomena. [..] [SR is] a simple and consistent theory of the electrodynamics of moving bodies based on Maxwell's theory for stationary bodies."
- Einstein 1905

Seems like you are suggesting that I'm just as incorrect as the Poincare, well may be I'm incorrect but even then Lorentz and Einstein are only partially verified experimentally, since we don't have any experiment confirming Length contraction as a physical effect. Does that trouble you even the slightest ? or do I need to ignore the Length contraction as a physical effect.

 

[Austin0]

which is A+A = 2A, that is, suggesting A+A is not same as 2A, because he never uses multiplication when he uses addition and vice-versa, is entirely incorrect.

Physics implies causality principle, i.e. If we have an effect then we must have a cause, atleast in the domain of classical physics.

Therefore, the effect(Time Dilation of unstable particles) must have a cause(where it comes from). Since, atleast you are accepting that we do have to involve gamma factor in order to understand these effects. How do you understand, the use of this gamma factor which is a part of the transformations, to produce the effects is question.

As I explained, you simply misunderstood what I was trying to ask, and not for introducing personal war of words, but it does not interests me a bit, how easily you make your perspective of other people, just by reading posts from a forum.

Where does gravity come from? we can describe the effects. Predict accurately how particles react in its presence but neither Newton or GR tells us where it comes from.
Well time dilation comes from the same place.

 

[universal_101]

Where does gravity come from? we can describe the effects. Predict accurately how particles react in its presence but neither Newton or GR tells us where it comes from.
Well time dilation comes from the same place.

You are confusing Time Dilation as an assumption, it is not. Assumption is electrodynamics(interaction of charges) just like gravity(interaction of masses), where we assumed certain things in order to satisfy experiments and observations.

Whereas, Time Dilation falls under the banner of electromagnetic phenomenon. That is, it is in no sense an assumption but an effect of there being an assumption(i.e. electromagnetism).

Your argument would be valid if I would have been questioning the existence of charges or electromagnetism.

So, if you think that Time Dilation is not an effect of relative motion under the laws of electromagnetism(Maxwell's Equations), but an entire theory of it's own, then only your argument is valid.

 

[Austin0]

QUOTE=Austin0;3940936]Thank you for your explication but as I fully understood the meaning of the interval and the metric I am afraid it completely missed the point.
That point being the meaning of the word invariant. In this context it simply means constant,unchanging, across all inertial coordinate systems
This necessarily implies the existence of other frames.
Would you disagree?

It does not apply to other coordinate systems within a single frame. I.e. changing from orthogonal to polar coordinates for eg.

It does not apply to local measurements as they apply within the frame.

It takes local coordinate measurements and outputs a value that is meaningful and constant in all other frames.

As that output value is related to the input values by the gamma factor it would appear it was a de facto transformation, semantic quibbles notwithstanding.
Yes[/QUOTE]

Yes, I would disagree. The length of the hypotenuse of a triangle is, by Pythagorus, equal to √(A² + B²), where A is the length of one leg, and B is the length of the other leg. Do you think that that definition implies the existence of other frames? The invariant interval in SR, τ = √((ct)² - x²) is a geometric relationship, just like the length of the hypotenuse of a right triangle. It doesn't have anything to do with "frames". The metric doesn't have anything to do with frames. Curved surfaces have associated metrics, and that doesn't have anything to do with frames.

Well you completely disregarded the meaning of the word invariant.

τ = √((ct)² - x²) this expression itself implicitly assumes and requires an orthogonal coordinate structure.

You have two points floating in space in front of you.

On this alone how do you define or express a metric? You can not even assign a distance of any kind without a ruler. I.e. a one dimensional coordinate system.

The thing that is fundamental about the metric is the signiture and that falls out of the gamma function. Out of the intrinsic structure of the world.It has nothing to do with historical precedent.
Without this fundamental aspect of reality there would be no Minkowski metric or gamma function and we would be living in a Galilean/Newtonian world.


gamma + Cartesian 4 D coordinates ===> Minkowski metric
gamma + Galilean transforms ====> Lorentz transforms
How do you possibly imagine anyone arriving at the metric without the knowledge of the gamma aspect of reality?

Yes, it certainly does. In cartesian coordinates, the metric tensor is defined by
(in 2-D spacetime):

g_tt = 1
g_xx = -1/c²
g_yy = -1/c²

(all other components are zero)

In polar coordinates, we have:
g_tt = 1
g_rr = -1/c²
g_θθ = -r²/c²

So the invariant interval is

ds² = dt² - dr²/c² - r²dθ²/c²

You have misunderstood.
Of course within a frame every possible coordinate system will have a relevant version of the Minkoiwski metric. We are in the middle of a discussion regarding the -version for cylindrical coordinates right now. So it should be obvious that this is not what I meant.
Within a frame transformation of events between different coordinate systems does not invlove the Minkowski metric.
It only comes into play when regarding things moving relative to the frame. I.e. different coordinate frames.

Within a Minkowski 2 D chart :
Regarding the rest frame the geometry is purely Euclidean with the Euclidean metric in a Cartesian chart. Yes? the normal Pythagorean relationships apply, yes?

It is only regarding the moving frame that the Minkowski metric with it's specific form of the Pythagorean operation applies ,,,,Yes?
Applying that metric returns the proper time value for a segment of the moving particles world line.
That time is related to the time of the rest frame described by the vertical interval of the triangle by the gamma factor Yes?

So it accomplished a geometric gamma transformation or do you disagree?

 

[universal_101]

gamma + Cartesian 4 D coordinates ===> Minkowski metric
gamma + Galilean transforms ====> Lorentz transforms
How do you possibly imagine anyone arriving at the metric without the knowledge of the gamma aspect of reality?

Thanks Austin, And i think this nails it.

That is, we don't need to present the metric or transforms as the physical aspects behind the physical relativistic effects.

In other words, Minkowski metric or spacetime should come under electromagnetism or electromagnetic effects of relative motion. But not as a physical law in themselves. And this is what some people are trying to suggest(by saying we don't need relative velocity to determine differential ageing), that spacetime itself is kind of a physical law.

 

[harrylin]

Yes, I would disagree. The length of the hypotenuse of a triangle is, by Pythagorus, equal to √(A² + B²), where A is the length of one leg, and B is the length of the other leg. Do you think that that definition implies the existence of other frames? The invariant interval in SR, τ = √((ct)² - x²) is a geometric relationship, just like the length of the hypotenuse of a right triangle. It doesn't have anything to do with "frames". The metric doesn't have anything to do with frames. Curved surfaces have associated metrics, and that doesn't have anything to do with frames.

The "frames" of SR are not things that necessarily "exist"; instead they are imaginary entities that represent differing measures of lengths and times or frequencies by means of really existing tools ("rods" and "clocks"). Obviously both the invariant space-time interval and the Lorentz transformation relate to the same "x" and "t"; as a matter of fact, Poincare introduced that invariant interval in the context of the Lorentz group in his 1906 paper (see my citation in post #126).

 

[harrylin]

[..] it could appear you were suggesting that relative velocity wasn't a factor in determining the aging [to someone who didn't understand world lines]
To determine the difference in rates requires relative velocities.wrt some frame.yes??

Exactly. The main point of modern versions of the twin "paradox" is to explain to students that when using SR the relative velocity (in terms of varying distance/time) between particles is not what determines the outcome; instead one has to relate the velocities to inertial reference systems (or to at least one inertial system that may be freely chosen).

 

[harrylin]

I never denied the association of the physical effects with the transformations, I'm questioning them. [..] We all know the Newton's law is invariant under Galelien transformation.

Then the issue is not really in the headline of your thread, "Transformation Vs. Physical Law"?

Seems like you are suggesting that I'm just as incorrect as the Poincare,

?? I see nowhere where I suggested that Poincare was incorrect. What made you think so? Lorentz, Einstein and likely everyone else agreed with his criticism. Some people even consider him to be the real inventor of SR (see for example http://en.wikipedia.org/wiki/Henri_Poincaré#Assessments for an entry point to the debate).

well may be I'm incorrect but even then Lorentz and Einstein are only partially verified experimentally, since we don't have any experiment confirming Length contraction as a physical effect. Does that trouble you even the slightest ? or do I need to ignore the Length contraction as a physical effect.

Maybe you are incorrect about what? You presented a point of view with which half of the people here seem to agree as well as a few erroneous claims that were debunked without comment by you (apparently you don't understand your errors). It doesn't trouble me much if a theoretical effect is too small to be directly measured with current technology (I already referred you to the first indirect experiment). That gives some slack for possible alternative theories, but very little.
Do you think that Newton should have been troubled by the fact that he could not test his theory to very high precision, so that he could not detect its flaws as we can nowadays? Most people are happy to have a better theory than before, and that works rather well.