A Note To The Linearity Proof Of Lorentz Transformation

[Shubee]

This abstract by Günther, H. (2003), A note to the linearity proof of Lorentz transformation. PAMM, 3:464–465. doi:10.1002/pamm.200310502 sounds interesting to me. It states,

 

The traditional proof of the linearity of Lorentz transformation makes use of a definition of simultaneity and a physical law for the propagation of light. For unravelling these two elements we use a reverse axiomatic approach to special relativity. In this case we are free, in principle, to introduce an arbitrary simultaneity. Linear coordinate transformations require a linear synchronisation. Lorentz transformation results from a special kind of linear synchronisation.

Does anyone here have access to Günther’s note? I’d like to know if Günther actually derived the nonlinear Lorentz transformation based on first principles and arbitrary simultaneity. I’m not a subscriber to that journal and would hate to pay to read an incomplete note. How thoroughly does he develop his comment?

[Rade]

This abstract by Günther, H. (2003), A note to the linearity proof of Lorentz transformation. PAMM, 3:464–465. doi:10.1002/pamm.200310502 sounds interesting to me. It states,

 

 

Does anyone here have access to Günther’s note? I’d like to know if Günther actually derived the nonlinear Lorentz transformation based on first principles and arbitrary simultaneity. I’m not a subscriber to that journal and would hate to pay to read an incomplete note. How thoroughly does he develop his comment?

 

Here is his address:

Prof. Dr. Helmut G¨unther, FH Bielefeld FB2, Wilhelm-Bertelsmann-Str.10, D-33602 Bielefeld, Germany

 

Sorry, copyright laws do not allow me to post the link I have to the paper. But, I can maybe answer your two questions:

 

So, he starts with equations (1) and (2) then:

 

Using (2) the traditional proof of the linearity of the transformations (1) can be given, cf. e.g. V. Fock [1]. Formulating Einstein’s principle of relativity according to, ”there is a definition of simultaneity in all inertial frames so that we always measure one and the same value for the speed of light”, we se that we call for two different elements in order to perform the linearity proof of the coordinate transformations, 1. a definition of simultaneity and 2. a physical law concerning the propagation of light. In order to unravel these two elements we will use a reverse axiomatic approach to special relativity.

Then..it sure looks arbitrary approach to me:

 

2. An arbitrary fixed inertial frame

 

Let Σo be an arbitrary and fixed inertial frame. For Σo we demand homogeneity and isotropy: The quantities lv/lo

and Tv/To , i.e. the length of a moving rod devided by the length of the rod at rest as well as the period of a moving

clock devided by the period at rest should be velocity dependent parameters, which depend neither on space and time

coordinates nor on the length or the period. Eq. (1) expresses two different things:

1. For t = 0 in Σo function f4 defines the synchronisation for the clocks in Σ,

t = f4(x, 0, v) := Ω(x, v) . (3)

The synchronisation function Ω(x, v) can be chosen arbitrarily in principle.

[Shubee]

Thank you Rade, but I’m particularly interested to know if Günther derived the nonlinear Lorentz transformation between inertial frames of reference in terms of arbitrary synchronization functions. That would be a Lorentz-like transformation equation for space and time coordinates that is logically equivalent to equation numbers (44) and (45) or (54) and (55) of http://www.everythingimportant.org/relativity/

[Rade]

Thank you Rade, but I’m particularly interested to know if Günther derived the nonlinear Lorentz transformation between inertial frames of reference in terms of arbitrary synchronization functions. That would be a Lorentz-like transformation equation for space and time coordinates that is logically equivalent to equation numbers (44) and (45) or (54) and (55) of http://www.everythingimportant.org/relativity/

 

===

 

Hope this helps--

 

Applying the homogeneity principle to (4)b we find To/Tv = t/t = f4(x + vt, t, v) − Ω(x, v) /t = Q(v), hence

f4(x + vt, v) = Ω(x, v) + Q(v)t , (6)

 

which generally is a non-linear coordinate transformation, since Ω(x, v) can be chosen arbitrarily, in principle.

(PAMM · Proc. Appl. Math. Mech. 3, 464–465 (2003) / DOI 10.1002/pamm.200310502)

 

Restricting however to linear synchronisation according to Ω(x, v) = Θ(v) x , (7)

we get f4(x + vt, v) = Θ(v)x + Q(v)t . Changing the variables and writing Q(v) = Θ(v) v + q we arrive at

f4(x, t, v) = θ(v) x + q(v) t . (8) According to (5) and (8) a linear synchronisation (7) results in linear coordinate transformations.

 

--

 

3. Lorentz transformation

 

Up to this point we have an apparently preferred frame Σo. On the other hand, we have the freedom of choosing the linear synchronisation parameter Θ(v). Let us choose it according to what we will call the elementary principle of relativity: = After an observer resting in Σo has measured that Σ possesses the velocity v with respect to Σo, all clocks in Σ should be set such that an observer resting in Σ,measures that the reference system Σo possesses the velocity −v with respect to Σ.

 

A consequence of the linear transformations (5) and (8) is the general form for the composition of velocities. For an

object with velocity u = dx/dt in Σo we measure velocity u = dx/dt in Σ,u = ku − vθ u + qwith u =−k vq

in Σ for u = 0 in Σo . (9)

For u = 0 in Σo an observer in Σ measures the velocity u for the frame Σo. Applying the elementary principle of

relativity it must be u = −v, hence

q = k . Demanding elementary relativity for the synchronisation of the clocks in Σ (10)

With Ω = Θv + q we get from (4)b To/Tv = Θv + q. With lo/lv = k and k = q according to (10) we find

θ(v) = To/Tv − lo/lv (all over v). Synchronisation in case of elementary relativity (11)

 

Note that we measure Lorentz contraction and time dilatation in our apparently preferred frame Σo according to

Σo : lvlo=1k=1 − v2c2 ,TvTo= k = 11 − v2/c2. (12)

From (10)-(12) follows

k = q =1 1 − v2/c2 θ=−v/c21 − v2/c2. (13)

Introducing (13) into the linear transformations (5) and (8) we arrive at Lorentz transformation,

x = x − v t1 − v2/c2, y =y z =z t = t − xv/c21 − v2/c2.

[Shubee]

Rade,

 

I don’t see that you’ve cited any equation that looks like a generalization of the Lorentz transformation.

[Rade]

Rade,I don’t see that you’ve cited any equation that looks like a generalization of the Lorentz transformation.

 

2. A simple calculation shows that (1) expresses the length of a moving rod lv and the period of a moving clock Tv

in terms of the corresponding quantities at rest lo resp. To by the following equations,

a) lolv= f(x, v)x,) ToTv= tt= f4(x + v t, t, v) − Ω(x, v)t. (4)

Note that the periods Tv and To are reciprocical to the indicated times t and t, cf. H. G¨unther [2].

Applying the homogeneity principle to (4)a we find lo/lv = f(x, v)/x = k(v). The argument x is arbitrary, hence

f(x − vt, v) = k(v)(x − vt) . (5)

 

Applying the homogeneity principle to (4)b we find To/Tv = t/t =f4(x + vt, t, v) − Ω(x, v)

/t = Q(v), hencef4(x + vt, v) = Ω(x, v) + Q(v)t , (6)

 

which generally is a non-linear coordinate transformation, since Ω(x, v) can be chosen arbitrarily, in principle.

 

==

 

It would be best you contact Dr. Helmut--seems like the two of you would have much of interest to each other to discuss. I find I am at the limit of what I can post here.

[Shubee]

Rade,

 

From the snippets you shared, it seems to me that Günther didn’t derive an explicit form of the nonlinear Lorentz transformation, as is done in the paper at http://www.everythingimportant.org/relativity/

 

Thanks for your help.