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The simplest derivation of the Lorentz transformation simplified: J.M.Levy "A simple derivation of the Lorentz transformation and of the accompanying velocity and acceleration changes," Am.J.Phys 35,615 (2007) arXiv:physics/0603103revisited.[1]

Levy [1} presents a derivation of the Lorentz transformation (LT) conisdering that it is the simplest one. It is based on the knowledge of the lenght contraction formula

L=L(0)sqrt(1-VxV/cxc) (1)

obtained by considering the light clock from its rest frame and from a reference frame relative to which it moves with constant speed V (L(0) represents the proper lenght of a rod measured measured by an observer relative to whom it is in a state of rest. L representing its distorted length measured by an observer relative to whom it moves with speed V in the direction of the rod.) The derivation is based on the fact that expressing the length of the rod as the sum of the lengths of its components then all the lengths should be measured in the same inertial reference frame.

The single change we make in Levy's approach is to consider that proper length and distorted length are related by

L=f(V)L(0) (2)

where f(V) is an unknown function of the speed of the rod and not of the proper length (to a given rod of proper length L(0) corresponds a single distorted length L.

As in [1] the involved inertial reference frames are K and K' in the standard arrangement. K' moves with speed V in the positive direction of the overlapped OX(O'X') axes. Consider the events M(x,t) detected from K and M'(x',t') detected from K'. The two events take place at the same point in space when the clocks C(x) and C'(x',0) located at that read t and t' respectively. By definition M(x,t) and M'(x',t') represent the same event. All the clocks at rest in K and all the clocks in K' are synchronized following a synchronization procedure proposed by Einstein, respectively.

Consider the relative positions of K and K' as detected from K when the clocks of that frame read t. At that very time the distance between the origins of the frames is Vt, the distance Dx=x-0 being a proper length. The proper length Dx'=x'-0, measured by observers from K is f(V)Dx'=f(V)(x'-0). Adding only lengths measured by observers from K we obtain

Dx=Vt+v(V)dx'. (3)

Considering the relative positions of K and K' as detected from K' when the clocks of that frame read t'. then the distance between the origins O and O' is VDt', Dx'=x-0 is a proper length whereas the length Dx=x-0 measured by observers of K' is f(V)Dx. Adding only lengths measured by observers from K' we obtain

Dx'=f(V)Dx-VDt' (4)

Consider that observers from K and K' measure the speed of a light signal that propagates in the positive direction of the overlapped axes. In accordance with the second postulate they obtain

Dx/Dt=Dx'/Dt'=c (5)

and so we can present (3) as

f(V)cDt'=(c-V)Dt (6)

and (4) as

f(V)cDt=(c+V)Dt'. (7)

Combining (6) and (7) we obtain

f(V)=sqrt(1-VxV/cc) (8)

We have derived so far the Lorentz transformations for the space coordinates of the same event (3)

x'-0=[(x-0)-V(t-0)]/f(V) (9)

and (4)

x-0=[(x'-0)+V(t'-0)]/f(V) (10)

Dividing both sides of (9) and (10) by c and taking into account (5) we obtain the LT for the time coordinates of the same event

t-0=[t'-0)+V(x'-0)/cc]/f(V) (11)

and

t'-0=[(t-0)-V(x-0)/cc]/f(V). (12)

I invite all the participants on the forum to express oppinions about the facts presented above helping to elucidate if the LT could be derived without length contraction or time dilation.

Thanks

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# Help: Lorentz transformations with and without thought experiments

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