Transformation Vs. Physical Law

[universal_101]

I gave you physical laws for both the decay of unstable particles and the differential aging. Your point is completely refuted. I think that you know it is refuted which is why you have carefully avoided discussing the points I have made.

This thread is heading towards a lock for the same reason as the previous one. You are going around in circles as though you had not already received a complete answer.

I don't understand how did you solved the problem, the radioactivity law that you posted, alone is incapable to explain the time dilation of Muons in the cyclotrons. You must somehow, introduce the gamma factor to calculate the Time Dialtion of moving Muons.

Now, when and where you would introduce this gamma factor in the radioactivity law, will be the point of my question.

And for the locking part, I think this thread has lost it's objective already, so go for it.

 

[universal_101]

Please don't think I'm badgering you, but we need to clear this up.

Muons are heavy unstable leptons that always decay quickly into other particles, whereas radioactive emmission of beta-rays is something else.

We can talk about rates of emission in the beta-ray case, but not decay because beta-particles don't decay.

In the muon case we can talk about decay, but not emission rates.

Did you mean to say

" ... the different number of (beta) particles (counted), or decay (times of muons), depending on the motion of these particles." ?

I apologize for the confusion regarding beta rays, I think I used the term beta at the wrong place.

I was trying to say that recently beta decays(involving weak nuclear forces) are found to depend on the Earth-Sun distance, and the decay of Muons are also mediated by the weak nuclear force.

 

[harrylin]

universal, I'm sorry but I can't make any sense of your last posts. Can you refer to any textbook or publications on which you (think to) base your ideas?

PS I referred to posts 100 and 101 - and with "publications" I don't mean Arxiv (see the rules).

 

[universal_101]

Please post your reference here. The time dilation of muons was found by Bailey to follow the law I posted, which is compatible with special relativity.

Actually, the reference is for the variation in the decay rates of beta decays, which involve the weak nuclear forces, and it is this same force which mediates the decay of Muons. So, it is the extended view that, Muons too could depend on the Earth-Sun distance.

And the dependence of the nuclear decay rates(beta) is well-known nowadays. There are plenty preprints on arxiv.

 

[ghwellsjr]

I didn't say that the Doppler factor is not dependent of the relative speed between the twins, I said it's not dependent on each twin's relative speed in any medium. I'm also saying that the speed is not important to being able to derive the difference in aging. All we need is the knowledge that the two Doppler factors are reciprocals, and that the traveling twin spends the same amount of time traveling away as he does toward the other twin based on his own clock.

For example, with Dopplers of 2 and 1/2, the average of them is 1.25 which means that as the traveling twin kept his eye on the stationary twin's clock through the entire trip, he first saw it ticking at 1/2 the rate of his own, then for the return trip, he watched it tick twice as fast as his own. You can confirm that at a relative speed of 0.6c, the relativistic Doppler factors are 2 and 1/2 and that gamma equals 1.25.

Another example, with Dopplers of 3 and 1/3, the average is 5/3 or 1.667, and this occurs with a relative speed of 0.8c which produces a gamma of 1.667.

The point is that we don't need to know the value of the speed in order to calculate the age difference which happens to be equal to gamma.

You inherently used the relative velocity, when you talk about the reciprocity of the Doppler values, i.e. 2 and 1/2 etc. It is very surprising that you and other people here are claiming that difference in the age is independent of relative velocity.

In the specific example of the Twin Paradox which we are discussing, where one twin remains inertial and the other one travels in both directions at the same speed, I'm not saying that you cannot use that speed to calculate the difference in aging, you can. And you can generalize the question to show that the age difference is a function of just the value of gamma which can be calculated from the value of the relative speed.

But you don't have to analyze it that way. You can also generalize it using just the Doppler for the outbound portion of the trip, whatever that happens to be and the knowledge that it will be the reciprocal on the inbound portion of the trip, and calculate the same answer using the process I described above. It's not that one way is wrong and the other way is right, they're both right. But the first way requires the establishment of a frame with coordinate times defined according to Einstein's synchronization process whereas the second way does not have that requirement. I'm only trying to get you to see that the second way does not require any transformation tools as you claimed in posts #34 and #45.

If you don't understand my argument, please ask specific questions, don't just disregard what I am saying.

 

[universal_101]

universal, I'm sorry but I can't make any sense of your last posts. Can you refer to any textbook or publications on which you (think to) base your ideas?

EDIT : http://www.sciencedirect.com/science/article/pii/S092765050900084X, how about this one.

 

[harrylin]

http://arxiv.org/abs/0808.3283, there you go.

Sorry I posted at about the same time as you and I referred to your earlier posts 100 and 101. But note that Arxiv cannot generally be used as knowledge base for discussions here, see the rules:
https://www.physicsforums.com/showthread.php?t=414380

EDIT : http://www.sciencedirect.com/science/article/pii/S092765050900084X, how about this one.

That looks better! :-)

 

[Dale]

I don't understand how did you solved the problem, the radioactivity law that you posted, alone is incapable to explain the time dilation of Muons in the cyclotrons.

This is false. That law, alone, is in fact capable of explaining the decay of any large bunch of radioactive particles (i.e. n must be large) in an accelerator.

You must somehow, introduce the gamma factor to calculate the Time Dialtion of moving Muons.

Why must I do that when the law works fine as written?

Sure, if you want then you can rewrite the law in terms of coordinate time for some specific frame, and if that specific frame is inertial then the rewritten law will have a gamma factor in it. But if you do that then you are looking at a rewritten special case of the law as applied to a specific frame, not the general coordinate-independent form of the law itself.

And for the locking part, I think this thread has lost it's objective already, so go for it.

I am not a mentor so I cannot lock it.

 

[Dale]

Actually, the reference is for the variation in the decay rates of beta decays, which involve the weak nuclear forces, and it is this same force which mediates the decay of Muons. So, it is the extended view that, Muons too could depend on the Earth-Sun distance.

And the dependence of the nuclear decay rates(beta) is well-known nowadays. There are plenty preprints on arxiv.

The decay rates of muons was found to agree with the forumla I posted above in the following experiments using a highly relativistic storage ring:

Bailey et al., “Measurements of relativistic time dilation for positive and negative muons in a circular orbit,” Nature 268 (July 28, 1977) pg 301.

Bailey et al., Nuclear Physics B 150 pg 1–79 (1979).

 

[universal_101]

In the specific example of the Twin Paradox which we are discussing, where one twin remains inertial and the other one travels in both directions at the same speed, I'm not saying that you cannot use that speed to calculate the difference in aging, you can. And you can generalize the question to show that the age difference is a function of just the value of gamma which can be calculated from the value of the relative speed.

But you don't have to analyze it that way. You can also generalize it using just the Doppler for the outbound portion of the trip, whatever that happens to be and the knowledge that it will be the reciprocal on the inbound portion of the trip, and calculate the same answer using the process I described above. It's not that one way is wrong and the other way is right, they're both right. But the first way requires the establishment of a frame with coordinate times defined according to Einstein's synchronization process whereas the second way does not have that requirement. I'm only trying to get you to see that the second way does not require any transformation tools as you claimed in posts #34 and #45.

If you don't understand my argument, please ask specific questions, don't just disregard what I am saying.

George, I really appreciate your arguments, but even in the second way we end up using the gamma factor.

What I have problem with is, Do we have any way to get the gamma factor from a physical law, instead of the relativistic Doppler which we get from the Lorentz transformation.

That is, the relativistic Doppler is a transformation tool, is it not.

 

[universal_101]

This is false. That law, alone, is in fact capable of explaining the decay of any large bunch of radioactive particles (i.e. n must be large) in an accelerator.

Of-course, the law can explain the decay of Muons in cyclotrons, but to use the same law for lab particles, we must incorporate gamma factor in that law. Because, we are trying to understand the decay rates of both the frames with the same law and then compare them to get the gamma factor.

But if you think, that comparing the decay of Muons in lab and in cyclotrons does not involve the gamma factor. Please post some calculations for the same.

 

[ghwellsjr]

George, I really appreciate your arguments, but even in the second way we end up using the gamma factor.

What I have problem with is, Do we have any way to get the gamma factor from a physical law, instead of the relativistic Doppler which we get from the Lorentz transformation.

That is, the relativistic Doppler is a transformation tool, is it not.

Doppler is something that you observe. You can't avoid it. Just like when you hear the pitch of the siren of an emergency vehicle take a sudden drop as it passes you, the same thing happens with light. It doesn't matter if you have a theory to account for it or not, it's still going to happen. In fact, it's the raw data that we get from making these measurements that a theory has to account for. Special Relativity accounts for it perfectly but the theory doesn't make it happen.

If we could actually travel at relativistic speeds, the difference in clocks would be readily apparent even without any theory and then we would have to develop a theory to explain the facts. Gamma was developed prior to Special Relativity but SR also derives it.

And my point in bringing up gamma was not to show an alternate way of deriving it but merely to show that since we know that gamma is the correct answer based on SR with regard to the simple twin paradox, and the Doppler analysis arrives at the same numerical answer in all cases without resorting to SR, it shows that we don't need transformation tools to solve the age difference in the simple twin paradox scenario. So we are not using the gamma factor, we're just showing that we get the same answer as we would if we did use the gamma factor.

 

[Dale]

Of-course, the law can explain the decay of Muons in cyclotrons, but to use the same law for lab particles, we must incorporate gamma factor in that law.

Nonsense. Where precisely would you put gamma in? Wherever you would add it to the formula given it would give you wrong answers. The formula is correct as written for any reference frame (inertial or non-inertial), in any spacetime (flat or curved), and for any particle worldline (stationary or moving).

But if you think, that comparing the decay of Muons in lab and in cyclotrons does not involve the gamma factor. Please post some calculations for the same.

I can certainly do exactly that, but it will have to be later in the day. In the meantime, please identify precisely where in that formula you think gamma is missing so that I can describe the errors that you would get by inserting it there.

 

[Samshorn]

The radioactivity law that you posted, alone is incapable to explain the time dilation of Muons in the cyclotrons. You must somehow, introduce the gamma factor to calculate the Time Dialtion of moving Muons.

For any standard system of inertial coordinates x,y,z,t, consider a burst of muons moving from the location x1,y1,z1 at the time t1 to the incrementally nearby location x2,y2,z2 at the time t2. If there are n muons at the first location, how many of them will have decayed by the time the burst reaches the second location?

For convenience, let dx denote the value of x2-x1, and so on for the other coordinates, and let dn denote the incremental change in the number of muons. The physical law states that

dn = -lambda*n*sqrt[dt^2 - (dx^2 - dy^2 -dz^2)/c^2]

The value of dn for that incremental segment must be invariant, i.e., independent of the choice of inertial coordinate system, because the number of muons at any given event obviously cannot depend on our choice of coordinate system. However, suppose we divide both sides of that equation by dt. Then the left hand quantity will be dn/dt, which is NOT an invariant quantity, because it DOES depend on our choice of the t coordinate. Therefore, when we divide by dt, both sides of the equation become coordinate dependent quantities. The right side becomes -lambda*n*(1/gamma). Hence the derivative of n with respect to t depends on gamma, which depends on v, which depends on the choice of coordinate systems, but this shouldn't surprise you, because the thing we are computing in that case is dn/dt, which is a coordinate dependent quantity.

This bothers you because you want t and dt to be invariants, as they would be if standard inertial coordinate systems were related by Galilean transformations. But if that were the case, energy would not have inertia, and E would not equal mc^2, and we would be living in a very different universe.

 

[GeorgeDishman]

The Michelson-Morley experiment for example was originally explained using length contraction alone ("FitzGerald–Lorentz contraction") in 1889. Only later was time dilation included (Larmor in 1897 according to Wikipedia) but it didn't remove the need for length contraction.

http://en.wikipedia.org/wiki/Michels...ial_relativity

Every phenomena even those which are not discovered yet can be explained just by considering a supernatural power i.e. a GOD.

I didn't mention "God" or anything like that, I'm atheist. In fact I wasn't even addressing your question, Austin0 asked for an experiment which showed length contraction and I gave him a well-known historical example.

But the problem is it is not falsifiable, the same is the problem with using Length contraction, we don't have any direct confirmation of it,

The above experiment uses light to measure the length of one leg of the interferometer versus the other just like radar, it is as direct a confirmation of the phenomenon as can be obtained.

but it is assumed to be there in order to explain some relativistic effects.

As I said above, length contraction was identified in 1887, relativity wasn't published until 1905 so your suggestion is blatently erroneous.

Whereas, for the electromagnetism it is perfectly fine to use Length contraction ...

If you accept it is "perfectly fine" to use it in one case, it is necessary to use it in all similar cases, either it is a law or it isn't.

While we are talking, let me go back to your original question and ask you for some information so that I can better understand what you mean by "physical law". Suppose two friends walk over a flat salt lake, one goes directly from town A to town C while the other detours via town B on the way. They both have the same steady stride length but the guy who makes the journey directly takes fewer steps that his friend who takes the detour. The question is what "physical law" do you think explains that difference?

 

[russ_watters]

Universal101, you are using an incorrect formulation of the Principle of Relativity based on an incorrect belief about reality to try to create a self-reinforcing challenge to SR. The recursiveness of your argument is both silly and maddening:

You say that because time dilation is frame dependent, the transformation is part of the "physical law", therefore time dilation is a violation of the PoR.

Nevermind the fact that time dilation arises as a necessary consequence of the PoR; there are plenty of frame dependent concepts in physics. Anything having to do with motion has a frame dependency. The fact that (for example) you can transform velocity into another frame and arrive at a different value for kinetic energy of a moving object does not imply a violation of the PoR for kinetic energy. Why? Because the PoR requires you to work in one frame at a time and therefore mandates doing transformations to ensure all calculations are based in that chosen frame. Jumping frames to attempt to create a contradiction (what you are doing) is the violation of the PoR, not the frame dependency of (for example) a velocity calculation.

Put another way: The fact that you have to transform from one frame to another to make accurate calculations of velocity dependent phenomena (energy, momentum, time dilation, air pressure, etc.) isn't a violation of the PoR, it is the whole point of the PoR!

 

[ghwellsjr]

In fact I wasn't even addressing your question, Austin0 asked for an experiment which showed length contraction and I gave him a well-known historical example.

Universal_101 may be confusing his similar request in post #72:

Now, it is this use of the gamma factor to produce difference in the ages of the Twins, make it necessary to have real Length contraction phenomena, to which we don't have any experimental support.

and which I gave him the same answer that you gave Austin0 in post #87:

Don't you consider the Michelson-Morley Experiment to be experimental evidence of length contraction? That's how Lorentz explained it.

 

[GeorgeDishman]

Universal_101 may be confusing his similar request in post #72:

and which I gave him the same answer that you gave Austin0 in post #87:

Ah, indeed, I had missed your mention of it. There seemed to be separate conversations going on as Austin0's question seemed a straightforward request for a pointer.

Anyway, hopefully Universal_101 will answer the question I asked which will make it easier to respond meaningfully to his question.

 

[zonde]

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.

This physical law is quite simple. Zigzag trajectory is longer than forth and back trajectory (with forth and back trajectory being projection of zigzag trajectory). You can use Pythagoras' theorem and derive gamma factor as length difference for two trajectories.

Some additional assumptions are required however.
Let's assume that say muons have some dimensional structure and decay rate of muons depend on their structure evolution at the speed determined by communication speed within that structure. Then decay rate of muons in motion will decrease by gamma factor as compared to muons at rest.

The only question left then is why muon in motion has to have the same dimensions in direction orthogonal to direction of motion as stationary muon. There is no answer to that question but we can simply assume that matter "likes" PoR and PoR is fulfilled when these dimension are equal.

 

[harrylin]

Universal, did I in post #95 - and I see that others gave similar answers - address your point or not? If not, why not?

Here's another way that this physical law about "time" was explained as based on one of the first convincing experiments (and I paraphrase to express the same in a more straightforward way) :

If a light beam is split into two beams that are brought together after traversing paths of different lengths, the resulting interference pattern will not depend on the velocity of the apparatus because the frequency of the light depends on the velocity as required by relativity.
- http://en.wikipedia.org/wiki/Kennedy-Thorndike_experiment

In other words (your words!), one may hold the opinion that this phenomenon is governed by a physical law and not by a transformation between points of view. This physical law is a necessary requirement for the transformation equations to work.

Does that help?

 

[Dale]

But if you think, that comparing the decay of Muons in lab and in cyclotrons does not involve the gamma factor. Please post some calculations for the same.

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}.

 

[harrylin]

[..] 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}.

Very neat (thanks!) - but if I'm not terribly mistaken it's only half true. With the "flat spacetime metric" you obviously mean the space-time interval which is based on the Lorentz group* - and thus on the gamma factor. So implicitly gamma is hiding in your calculation.

However, (and this is for universal): the fact that the gamma factor is used for transformation equations doesn't mean that it's not an implicit or explicit part of natural law as well - as many of us have tried to explain by now.

*http://en.wikisource.org/wiki/On_the_Dynamics_of_the_Electron_%28July%29#.C2.A7_4._.E2.80.94_The_Lorentz_group

 

[Dead Boss]

I would say that the Minkowski metric is more fundamental than the gamma factor.

 

[Dale]

So implicitly it's hiding in your calculation.

Where, exactly?

Certainly you can derive gamma from the metric, but that doesn't mean that every calculation involving the metric involves gamma. (and also not every calculation involving gamma is a transform, but that is a separate discussion)

 

[Dale]

I would say that the Minkowski metric is more fundamental than the gamma factor.

Agreed. A metric can be defined on a Riemannian manifold independently of any coordinate charts and any transforms between different coordinate charts. Gamma is a factor in a specific type of transform between a specific class of coordinate systems on a flat spacetime, which is several steps less general than the metric on the flat spacetime.

 

[harrylin]

Where, exactly?

Exactly in the section to which I referred, about the rotation angle (it's simply Pythagoras):

it is easy to see that this transformation is equivalent to a coordinate change, the axes are rotating a very small angle around the z-axis. [..] Any transformation of this group can always be decomposed into [..] a linear transformation which does not change the quadratic form x²+y²+z²-t²

Certainly you can derive gamma from the metric, but that doesn't mean that every calculation involving the metric involves gamma. (and also not every calculation involving gamma is a transform, but that is a separate discussion)

I think that the fact that not every calculation involving gamma is a transform, is at the heart of the discussion; that it's the sticking point for universal. And he seems to have not responded to a dozen of posts in which this has been stressed.

 

[Dale]

Exactly in the section to which I referred, about the rotation angle (it's simply Pythagoras):

I meant where exactly in my calculation. I agree that gamma is part of the Lorentz transform, but I never did a Lorentz transform in my calculation. I also agree that gamma can be derived from the metric, but that derivation wasn't part of my calculation. So it is quite a stretch, IMO, to say that gamma is in there, even implicitly.

I think that the fact that not every calculation involving gamma is a transform, is at the heart of the discussion; that it's the sticking point for universal. And he seems to have not responded to a dozen of posts in which this has been stressed.

Yes, he does seem to have a strong tendency to simply ignore posts which refute his argument. Otherwise this thread would be much shorter and his previous thread would not have been locked.

 

[universal_101]

I think that the fact that not every calculation involving gamma is a transform, is at the heart of the discussion; that it's the sticking point for universal. And he seems to have not responded to a dozen of posts in which this has been stressed.

I already agree to that fact, that every calculation involving factor gamma is not a transform. Instead this was the one of the points in my original post.

The question was if it does not come from the transformations then where does this gamma factor come from.

But I think, it's pointless, to discuss it here. And as all of you are suggesting, I should stop questioning since everything has been answered.

And it means this thread can be safely locked.

 

[Dale]

The question was if it does not come from the transformations then where does this gamma factor come from.

Do you have a specific non-transform equation with gamma that you are interested in? I think, for a given equation, we could explain the source, but I am not sure that the answer is the same for all such equations.

 

[Mentz114]

I already agree to that fact, that every calculation involving factor gamma is not a transform. Instead this was the one of the points in my original post.

The question was if it does not come from the transformations then where does this gamma factor come from.

But I think, it's pointless, to discuss it here. And as all of you are suggesting, I should stop questioning since everything has been answered.

And it means this thread can be safely locked.

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.

 

[Austin0]

I meant where exactly in my calculation. I agree that gamma is part of the Lorentz transform, but I never did a Lorentz transform in my calculation. I also agree that gamma can be derived from the metric, but that derivation wasn't part of my calculation. So it is quite a stretch, IMO, to say that gamma is in there, even implicitly.


Yes, he does seem to have a strong tendency to simply ignore posts which refute his argument. Otherwise this thread would be much shorter and his previous thread would not have been locked.

Isn't it true that the metric itself is the direct result of the gamma function and its implications regarding the relationship between time,space and relative states of motion.

it seems to me that the coordinate system and derivative body of transformation maths is simply the fundamental gamma function applied to and implemented through the Galilean metric.

This is explicitly manifest in the reverse sign of the time term , which of itself necessarilly implies and expresses time dilation.
The ratio of the temporal and spatial componenets describe a velocity and the resulting proper time value is the gamma corrected, dilated time value related to this velocity.

To me this appears to be a case of math magic, integrating the function so that the gamma calculation is not necessary as a separate operation. As is the case with the Doppler formula.

The relativistic Doppler equation does not have an explicit gamma function either but would you say that the gamma was not implicit in the math?
That to apply the formula was not indirectly applying that function?

Don't you consider the invariant interval equation a Lorentz transformation??

your thoughts?

 

[Austin0]

I already agree to that fact, that every calculation involving factor gamma is not a transform. Instead this was the one of the points in my original post.

The question was if it does not come from the transformations then where does this gamma factor come from.

But I think, it's pointless, to discuss it here. And as all of you are suggesting, I should stop questioning since everything has been answered.

And it means this thread can be safely locked.

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

 

[Dale]

wow Austin0, that is a lot of questions. My answers will be necessarily brief, but if you want to go deeper into one or two, I will be glad to.

Isn't it true that the metric itself is the direct result of the gamma function and its implications regarding the relationship between time,space and relative states of motion

No. The metric is more fundamental than any coordinate system or transform.

This is explicitly manifest in the reverse sign of the time term , which of itself necessarilly implies and expresses time dilation.
The ratio of the temporal and spatial componenets describe a velocity and the resulting proper time value is the gamma corrected, dilated time value related to this velocity.

This is correct when the metric is expressed in terms of an inertial coordinate system in flat spacetime, but not in general.

To me this appears to be a case of math magic, integrating the function so that the gamma calculation is not necessary as a separate operation. As is the case with the Doppler formula.

The relativistic Doppler equation does not have an explicit gamma function either but would you say that the gamma was not implicit in the math?

I have asked several times for people to show me where exactly the gamma was implicit in my formulas. I don't know where it is supposed to be lurking.

That to apply the formula was not indirectly applying that function?

Huh?

Don't you consider the invariant interval equation a Lorentz transformation??

No.

 

[Samshorn]

I have asked several times for people to show me where exactly the gamma was implicit in my formulas. I don't know where it is supposed to be lurking.

This was explained in post #114. Your formula is \tau_P=\int_P d\tau where d\tau^2=dt^2-dr^2-r^2d\theta^2-dz^2, but the latter can be written as d\tau^2=dt^2 / \gamma^2, so you're just integrating a function of gamma. Of course, as explained in #114, written in this form we are essentially computing the ratio of proper time to coordinate time, which is ultimately what we always do. Gamma is a function of the velocity in terms of our chosen coordinate system with the time coordinate t. If we chose a different coordinate system, t would be different and so would gamma. But as explained in #114, the number of muons at any event in invariant. But the elapsed times in two different frames are related by gamma, as confirmed by the ratio of your two answers. Normally we combine your two calculations into just one for the ratio, since that's what we ultimately care about, and hence gamma appears.

This is correct when the metric is expressed in terms of an inertial coordinate system... but not in general.

If both reference systems are accelerating then the comparison of elapsed times would be expressible in terms of the combined effect of two "gammas". Of course, the ratio of differental elapsed times can always be expressed in terms of a single "gamma".

 

[bobc2]

The metric is more fundamental than any coordinate system or transform...

Hi, DaleSpam. Could you show how this is established?

 

[Dale]

This was explained in post #114. Your formula is \tau_P=\int_P d\tau where d\tau^2=dt^2-dr^2-r^2d\theta^2-dz^2, but this can be written as

d\tau^2=\frac{dt^2}{\gamma^2}

No it can't, at least, not without doing a coordinate transform into a standard rectilinear coordinate system.

If you do a transform and differentiate wrt coordinate time as you explained in 114, then you can indeed derive gamma. I have mentioned that before. However, I neither transformed to an appropriate coordinate system nor did I differentiate wrt coordinate time. I didn't use gamma either implicitly nor explicitly, although you certainly can branch off from what I did and derive gamma.

so you're just integrating a function of gamma. Of course, as explained in #114, written in this form we are essentially computing the ratio of proper time to coordinate time, which is ultimately what we always do.

No, in my case it was the ratio of proper time to the parameter a. If I had parameterized the worldline by t then that would be the case. Often, proper time itself is used as the parameter, so you just integrate 1. There is never any need to use t as the parameter.

If both reference systems are accelerating then the comparison of elapsed times would be expressible in terms of the combined effect of two "gammas". Of course, the ratio of differental elapsed times can always be expressed in terms of a single "gamma".

Non inertial reference frames do not, in general, have factors of gamma.

 

[Austin0]

Isn't it true that the metric itself is the direct result of the gamma function and its implications regarding the relationship between time,space and relative states of motion

No. The metric is more fundamental than any coordinate system or transform.

you may be right about the abstract question of general fundamentality
but this is about a specific metric.
I appears to me that this metric could not exist without the basic understanding of the relationship of space and time described by Lorentz in the gamma function. The metric did not appear out of thin air and how could it?

This is explicitly manifest in the reverse sign of the time term , which of itself necessarily implies and expresses time dilation.
The ratio of the temporal and spatial components describe a velocity and the resulting proper time value is the gamma corrected, dilated time value related to this velocity.

This is correct when the metric is expressed in terms of an inertial coordinate system in flat spacetime, but not in general.

Good , so we are agreed that the result does contain the gamma factor.

To me this appears to be a case of math magic, integrating the function so that the gamma calculation is not necessary as a separate operation. As is the case with the Doppler formula.

A The relativistic Doppler equation does not have an explicit gamma function either but would you say that the gamma was not implicit in the math?

I have asked several times for people to show me where exactly the gamma was implicit in my formulas. I don't know where it is supposed to be lurking.

You didn't answer the actual question A

As far as where it might be lurking, I suspect that harrylin might be on the right track.
the Pythagorean operation returns the value of a line interval in Minkowski geometric space. The geometry of that space seems to me to intrinsically require the gamma factor to transform the geometry regarding the moving system into meaningful quantities.

So as the pythagorean operation does perform this transformation, it is in effect a geometric gamma function.

More importantly; as you agree that the result contains the gamma factor and I am sure you would agree that this factor was not implicit in the raw coordinate values, the question becomes ,where else could it possibly be lurking??

Pure logic leads to the inevitable conclusion that it must necessarily be implicit,(hiding) in the operation leading to that result. Unless you think it is just coincidence ;-)

Would you say That to apply the Doppler formula was not (indirectly) applying that function?

Huh?

* Sorry if my phrasing was a little ambiguous. Does this track better??

Don't you consider the invariant interval equation a Lorentz transformation??

No.

Well as it seems clear it is a transformation does this mean you don't think it is part of the general derivative functions stemming from the original math?

 

[Nugatory]

Hi, DaleSpam. Could you show how this is established?

That's not so much something that can be established/proven as it is an argument from aesthetic principles; if the theory is internally consistent I can start at either end and get to the other, so neither end is demonstrably more fundamental.

But I'd expect that this aesthetic judgement is shared by the overwhelming majority of people who have figured out the underlying math and physics. The metric describes properties of space-time that exist independent of any coordinate system; the coordinate systems, reference frames, and transformations between them were invented by us to make it easier to map these properties to our own experience.

 

[ImaLooser]

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.

What is the difference between a physical law and a transform? Not much. Hard to say, really. The boundary is vague. I wouldn't worry about it. If you want to call the Lorentz transform a physical law, that is fine with me.

 

[Nugatory]

I appears to me that this metric could not exist without the basic understanding of the relationship of space and time described by Lorentz in the gamma function. The metric did not appear out of thin air and how could it?

You are right that the gamma factor was described first, and that its discovery set us on the path that led to the discovery of the four-vector formulation of SR, the Minkowski metric, and eventually GR. But that's the history, not the logical structure of the theory that we see when we've made it to the other end of that path.

A perhaps less controversial example: the discovery that dropped objects accelerate towards the surface of the Earth at 10 m/sec^2 preceded Newton's principle of gravitation. But it would be absurd to argue from this history that Newtonian gravity and and G requires 10 m/sec^2 and not the other way around.

 

[Austin0]

You are right that the gamma factor was described first, and that its discovery set us on the path that led to the discovery of the four-vector formulation of SR, the Minkowski metric, and eventually GR. But that's the history, not the logical structure of the theory that we see when we've made it to the other end of that path.

A perhaps less controversial example: the discovery that dropped objects accelerate towards the surface of the Earth at 10 m/sec^2 preceded Newton's principle of gravitation. But it would be absurd to argue from this history that Newtonian gravity and and G requires 10 m/sec^2 and not the other way around.

i think maybe your analogy is a little distorted. The gamma function is more analogous to the G of gravity as being a fundamental aspect of physics than it is to any specific velocity or value. Newton inferred gravity from observation while Lorentz derived gamma from the Maxwell math but they were both discoveries of fundamental principles

 

[Samshorn]

d\tau^2=dt^2-dr^2-r^2d\theta^2-dz^2 can be written as d\tau^2=dt^2 / \gamma^2...

No it can't, at least, not without doing a coordinate transform into a standard rectilinear coordinate system.

I can't imagine why you think that. Those two equations are mathematically equivalent, just two different ways of writing the same thing. Just factor dt^2 out of the right hand side of the first equation. The remaining factor is (1 - v^2), which is 1/gamma^2.

There is never any need to use t as the parameter.

Not true. For example, when trying to determine the elapsed t, we obviously need to use the t coordinate. In your example you just split up the computation into two separate calculations, one for tau and one for t, and declined to note the ratio, but ordinarily the ratio is what we want. In other words, knowing the half-life in the rest frame of the particle, we want to know the half-life in terms of the t of some other coordinate system.

Non inertial reference frames do not, in general, have factors of gamma.

It's always possible to express the ratios of times in terms of gamma factors. I think the first step toward understanding this would be to understand why those two expressions above are mathematically equivalent.

 

[Jorriss]

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.

There is physical law behind it. It's that the speed of light is the same in all reference frames. If the speed of light is the same in all references then a moving clock will appear to run slow which can be quantitatively described by lorentz transformations.

 

[harrylin]

I meant where exactly in my calculation. I agree that gamma is part of the Lorentz transform, but I never did a Lorentz transform in my calculation. I also agree that gamma can be derived from the metric, but that derivation wasn't part of my calculation. So it is quite a stretch, IMO, to say that gamma is in there, even implicitly.[..]

The "metric" that you used for your calculation is effectively the cited "quadratic form" that is equivalent to the Lorentz group (except that you used polar coordinates). The same calculation can be done using gamma explicitly.

[..] Don't you consider the invariant interval equation a Lorentz transformation?? [..]

Yes, as I cited, the invariant interval equation was introduced as an alternative (but equivalent) description of the Lorentz group.

 

[harrylin]

I already agree to that fact, that every calculation involving factor gamma is not a transform. Instead this was the one of the points in my original post.

The question was if it does not come from the transformations then where does this gamma factor come from.

But I think, it's pointless, to discuss it here. [..]

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

 

[zonde]

There is physical law behind it. It's that the speed of light is the same in all reference frames. If the speed of light is the same in all references then a moving clock will appear to run slow which can be quantitatively described by lorentz transformations.

This is not true. Speed of light being the same in all admissible reference frames is a postulate (not a physical law) and it effectively defines "all admissible reference frames".

Physical laws are defined within single reference frame not across many reference frames. So if you try to define speed of light within single reference frame you get physical constant with particular value. Then in a sense we can say that if we treat speed of light as physical constant (kind of trivial physical law) then the statement that "speed of light is the same in all admissible reference frames" becomes redundant as it is already implied by principle of relativity.

 

[A.T.]

Then in a sense we can say that if we treat speed of light as physical constant (kind of trivial physical law) then the statement that "speed of light is the same in all admissible reference frames" becomes redundant as it is already implied by principle of relativity.

No, the 2nd postulate in not redundant and is not implied by the 1st postulate. The ballistic light theory satisfies the 1st but not the 2nd postulate, so there is no implication 1 -> 2.

The 2nd postulate is not just that speed of light is the same in all inertial frames, but also that it is independent of the source velocity in all inertial frames. So the 2nd postulate effectively says that the same light beam, has the same speed across all inertial reference frames. And as you correctly stated "physical laws" in the sense of the 1st postulate are defined within single reference frame, so you cannot lump the 2nd postulate as a "physical law" in there.

 

[Dale]

Hi, DaleSpam. Could you show how this is established?

The place that I learned it was primarily chapter 2 of Sean M. Carroll's lecture notes on GR:
http://arxiv.org/abs/gr-qc/9712019

I list here the fundamental geometric objects of GR, not in the order of presentation in Carroll, but in order of generality:
Topological spaces (continuity)
Manifolds (continuity, dimensionality)
Metric (continuity, dimensionality, distance and angles)
Coordinate chart (continuity, dimensionality, distance and angles, labelling)
Coordinate transforms (continuity, dimensionality, distance and angles, labelling, correspondence)

At each level, everything from all the previous levels is included and something new is added. So, for example, all manifolds are topological spaces, but not all topological spaces are manifolds, making topological spaces more fundamental/general than manifolds.

 

[harrylin]

No, the 2nd postulate in not redundant and is not implied by the 1st postulate. The ballistic light theory satisfies the 1st but not the 2nd postulate, so there is no implication 1 -> 2.

The 2nd postulate is not just that speed of light is the same in all inertial frames, but also that it is independent of the source velocity in all inertial frames. So the 2nd postulate effectively says that the same light beam, has the same speed across all inertial reference frames. And as you correctly stated "physical laws" in the sense of the 1st postulate are defined within single reference frame, so you cannot lump the 2nd postulate as a "physical law" in there.

As that's a bit off-topic, just this for the record: That is a myth[2].
The second postulate is a physical law which adheres to Maxwell's wave model and rejects ballistic emission theory; it is defined for a single reference frame. See:
[1] http://www.fourmilab.ch/etexts/einstein/specrel/www/
[2] http://www.uio.no/studier/emner/matnat/fys/FYS-MEK1110/v06/MythsSpecRelativAJP193.pdf

 

[Dale]

The metric did not appear out of thin air and how could it?

In GR, often times metrics do appear "out of thin air". I.e. it is often easier to find a solution to the EFE by starting with the metric and then solving for the distribution of mass and energy required to make that metric.

You didn't answer the actual question A

The actual question was based on a flawed analogy. So I tried to de-analogize it and answer the underlying question, which I though was a valid question despite the bad analogy.

As far as where it might be lurking, I suspect that harrylin might be on the right track.
the Pythagorean operation returns the value of a line interval in Minkowski geometric space.
... So as the pythagorean operation does perform this transformation, it is in effect a geometric gamma function.

First, I disagree, and second, I never used the Pythagorean theorem nor the Minkowski metric. You are not getting any closer to showing where I used gamma in my calculations.

More importantly; as you agree that the result contains the gamma factor and I am sure you would agree that this factor was not implicit in the raw coordinate values, the question becomes ,where else could it possibly be lurking??

It lurks in differeniating the line element wrt coordinate time in Minkowski coordinates. I neither used Minkowski coordinates nor did I differentiate wrt coordinate time. You cannot get from what I did to gamma without transforming to Minkowski coordinates and differentiating wrt coordinate time, neither of which are required for calculating the decay of the muons.

Well as it seems clear it is a transformation

Nonsense. A transformation requires at least two different coordinate systems and a mapping between them. The metric doesn't even require coordinate systems, let alone a pair of them. So claiming that it a transformation is wrong, and claiming that it is clearly a transformation is absurd.