Question on linearity of Lorentz transformations

[Al68]

And one of my points is that if you start with a set of assumptions and end up with a contradiction, you have only proved that your theory is inconsistent. You certainly haven't derived a new theory.

That's why I'm saying that the only way to make sense of the "derivation" is to interpret the "postulates" as ill-defined statements, and the "derivation" as finding out which of the corresponding well-defined statements are consistent with the other assumptions we want to make.


I don't have a problem with the fact that the first paper ever written on SR is "not too pedantic". I just don't think that's a good reason for us do the same. It's not even too difficult to talk about SR in a way that makes sense, so we have no excuse. I think it's absurd that professors still give students the impression that SR is defined by Einstein's postulates, and that the rest of the theory can be "derived" from the "postulates". You really can't derive anything from them, and they can't be taken as the definition of SR.

I haven't heard of the postulates being the definition of SR, certainly they aren't. But they marked the historical transition from Newtonian physics to SR.

As far as the postulates being contradictory in Newtonian physics, I think the 1905 paper showed that Newtonian physics was the one of the three assumptions that needed to be modified, not the other two (the postulates).

 

[dx]

These two postulates apply only to free particles. So, you need also a third postulate:

3. Events with interacting particles (e.g., their worldlines) transform by the same formulas as events with free particles.

Then, according to the Currie-Jordan-Sudarshan theorem, your theory must be interaction-free.

Why is this postulate needed? My first two postulates are enough to prove the linearity of transformations between inertial frames.

Also, I said that an inertial frame is a map from events to V, not a map from "free particle worldline events" to V. It doesn't matter what type of event it is. All events transform by the same formulas by definition.

 

[meopemuk]

It doesn't matter what type of event it is. All events transform by the same formulas by definition.

I am not sure about that. This is your postulate (or definition), and I would like to know if you have any evidence to support it?

CJS theorem provides an example in which points (events) on worldlines of interacting and non-interacting particles transform by different formulas.

 

[dx]

I am not sure about that. This is your postulate (or definition), and I would like to know if you have any evidence to support it?

You mean experimental evidence? As far as I know, there are no mainstream theories where different types of events transform differently. The best theory we have about spacetime is general relativity, where spacetime is a manifold, and a coordinate system is a function from some patch of spacetime into R⁴. Given two coordinate systems, i.e. two functions φ, φ' : M → R⁴, it is easy to see that the transformation of coordinates for any event E from φ to φ' is given by φ^-1φ'.

 

[meopemuk]

You mean experimental evidence? As far as I know, there are no mainstream theories where different types of events transform differently. The best theory we have about spacetime is general relativity, where spacetime is a manifold, and a coordinate system is a function from some patch of spacetime into R⁴. Given two coordinate systems, i.e. two functions φ, φ' : M → R⁴, it is easy to see that the transformation of coordinates for any event E from φ to φ' is given by φ^-1φ'.

Yes, I agree that both special and general relativity theories are based on your (rarely mentioned, but important) postulate that time-position transformations of events do not depend on the physical nature of the events and on interactions acting in the observed system. So, you are saying that the validity of your postulate is justified a posteriori by the fact that both SR and GR agree well with experiments? This leaves however the possibility that the postulate is not exactly true, and that there is some dependence of the time-position transformations on interactions between particles. If the effect is small, then it wouldn't contradict existing experiments.

Another important (though not appreciated) point is the logical consistency. Suppose that we accepted your postulate and assumed that time-position transformations between different moving frames do not depend on interactions. Then Lorentz transformations are guaranteed to be linear and universal, and all events can be represented as points in the Minkowski space-time. Suppose also that we constructed a dynamical interacting theory based on this principle. The Maxwell-Lorentz electrodynamics is a good example. Then it would be of interest to verify within our theory whether the initial postulate holds.

For example, we can consider a system of two interacting charges (e.g. an electron and a proton), calculate their trajectories, and find space-time coordinates of some localized event, e.g., when the two particles collide. Next, in our theory, we could repeat the same calculation in a moving reference frame. So, we would have space-time coordinates of the same event (collision of the two particles) in two reference frame. Will they be connected by Lorentz formulas?

If the answer is "yes", then our theory is logically consistent (the initial postulate has been confirmed by a direct dynamical calculation). However, can we be sure that the answer is "yes"? As far as I know, nobody has performed this kind of calculation in Maxwell-Lorentz electrodynamics (if you think I missed some relevant works, I would appreciate the reference). Moreover, I have a strong suspicion that a direct calculation of this sort will *not* yield the expected result.

 

[Fredrik]

It is possible to have well defined postulates from which the linearity of Lorentz transformations follows.

Let V be a four dimensional vector space. An inertial frame is a map ψ from the set of events into V which satisfies the following postulates:

1. The world lines of free particles are straight lines.
2. Clock rates are uniform, i.e. intervals measured by clocks agree with the linear structure of V.

I agree with your opening statement, but I would drop your second postulate and add stuff to the first. I'd choose V=\mathbb R^4, and change #1 to

1. Each transition function* corresponding to two inertial frames is a smooth** bijection that takes straight lines to straight lines.

*) See my posts earlier in this thread for a definition.
**) All its partial derivatives up to arbitrary order exist.

Let T be a transition function. The axiom guarantees that it can be Taylor expanded.

T(x)=T(0)+x^\mu\partial_\mu T(0)+\frac 1 2 x^\mu x^\nu\partial_\mu\partial_\nu T(0)+\cdots

Let's call a transition function with T(0)=0 a "Lorenz transformation". (This will be our definition of a Lorentz transformation for the rest of this post). Note that a Lorentz transformation defined this way takes straight lines through the origin to straight lines through the origin.

Now let T be a Lorentz transformation, and let x and y be two points on a straight line through the origin. We must have y=kx. Postulate #1 and our definition of a Lorentz transformation imply that we also have T(y)=k'T(x), but T(y)=T(kx), so we have

T(kx)=k'T(x)

for all x. Let's Taylor expand both sides.

kx^\mu\partial_\mu T(0)+\frac 1 2 k^2 x^\mu x^\nu\partial_\mu\partial_\nu T(0)+\cdots=k'\Big(x^\mu\partial_\mu T(0)+\frac 1 2 x^\mu x^\nu\partial_\mu\partial_\nu T(0)+\cdots\Big)

These two expressions must mach term by term, and that's only possible if k'=k and all the higher order terms are =0. So any "Lorentz transformation" must be linear.

I don't have any objections to this sort of argument, but one could point out that the axiom is extremely strong. I mean, we're assuming that transition functions take straight lines to straight lines, so it's not exactly a surprise that they turn out to be linear. So one could argue that we might as well have started by requiring linearity. The counter argument to that is that this approach is more intuitive and "natural" than the abstract requirement of linearity. It only expresses the idea that any inertial observer should be able to describe any other inertial observer as moving with constant velocity.

 

[facenian]

I don't have any objections to this sort of argument, but one could point out that the axiom is extremely strong. I mean, we're assuming that transition functions take straight lines to straight lines, so it's not exactly a surprise that they turn out to be linear. So one could argue that we might as well have started by requiring linearity. The counter argument to that is that this approach is more intuitive and "natural" than the abstract requirement of linearity. It only expresses the idea that any inertial observer should be able to describe any other inertial observer as moving with constant velocity.

Fredrik, This line of thought is what I was reffering to from the begining. I think what you call natural and intuitive may be translated as "physical",eg, physical reassons instend of purely mathematical assumption with no conection to physical reality.
However,I do have some concern with your derivation because the transformation I posted in #16 of this thread is not linear and seems to be a counter example for your demostration.
According to other posts of this thread the "homgeneity" hypothesis seems to be neccesary so there must be something wrong in your derivation although I don't know what.

 

[dx]

I agree with your opening statement, but I would drop your second postulate and add stuff to the first. I'd choose V=\mathbb R^4, and change #1 to

1. Each transition function* corresponding to two inertial frames is a smooth** bijection that takes straight lines to straight lines.

Projective transformations are smooth and take straight lines to straight lines, but they are not linear.

The linearity of Lorentz transformations has more to do with our idea of an inertial frame than it does with any specific property of the Lorentz transformation. Like I said before, once we characterize inertial frames by the two simple postulates from my previous post, it follows that any transformation between inertial frames must be linear. The Lorentz behavior of clocks is just one type that is compatible with these postulates, another being the Galilean/Newtonian.

 

[DrGreg]

However,I do have some concern with your derivation because the transformation I posted in #16 of this thread is not linear and seems to be a counter example for your demostration.
According to other posts of this thread the "homgeneity" hypothesis seems to be neccesary so there must be something wrong in your derivation although I don't know what.

The projective transformation you quoted in #16 does not satisfy T(0) = 0, as Fredrik's argument assumes.

 

[facenian]

The projective transformation you quoted in #16 does not satisfy T(0) = 0, as Fredrik's argument assumes.

It does when b_i=0 for i=0,1,2,3

 

[facenian]

2. Clock rates are uniform, i.e. intervals measured by clocks agree with the linear structure of V.

what is the meaning of time intervals agreeing with the linear structure of V.

 

[Fredrik]

It does when b_i=0 for i=0,1,2,3

You also have to set the c_i=0.

 

[facenian]

You also have to set the c_i=0.

I don't see why. Setting b_i=0 for all i seems sufficient for T(0)=0

 

[Fredrik]

I don't see why. Setting b_i=0 for all i seems sufficient for T(0)=0

Your function isn't smooth, or even defined for all x, because of the x in the denominator.

 

[Fredrik]

I just started looking at the articles referenced on page 1. I haven't looked at George's reference yet (because it's not an online article), but all the others look interesting, especially the one atyy posted. It seems that the assumptions in my post #56 are much stronger than they need to be. I think I'm going to have to read that whole article soon.

 

[dx]

what is the meaning of time intervals agreeing with the linear structure of V?

If A - B = C - D, then a clock carried along AB will measure the same interval as it would if it were carried along CD. If A - B = λ(C - D), then a clock carried along AB will measure λ times the interval it would measure along CD.

 

[facenian]

If A - B = C - D, then a clock carried along AB will measure the same interval as it would if it were carried along CD. If A - B = λ(C - D), then a clock carried along AB will measure λ times the interval it would measure along CD.

Thank you dx. The fist condition seems to be consequence of homogeneity, and if A-B is understood only as distance could be consequence of isotropy.