Comments about "Lorentz_transformation" in Wikipedia

This document contains comments about the article Lorentz_transformation in Wikipedia
In the last paragraph I explain my own opinion.

Contents

Reflection

Introduction

The article starts with the following sentence.

1. History

2. Derivation of the group of Lorentz transformations

3. Generalities

4. Physical formulation of Lorentz boosts

The most important is the physical reasoning behind these Lorentz boosts. Its meaning etc.

4.1 Coordinate transformation

A "stationary" observer in frame F defines events with coordinates t, x, y, z. Another frame F′ moves with velocity v relative to F, and an observer in this "moving" frame F′ defines events using the coordinates t′, x′, y′, z′.
How are these parameters t, x, y, and z measured?
The coordinate axes in each frame are parallel (the x and x′ axes are parallel, the y and y′ axes are parallel, and the z and z′ axes are parallel), remain mutually perpendicular, and relative motion is along the coincident xx′ axes. At t = t′ = 0, the origins of both coordinate systems are the same, (x, y, z) = (x′, y′, z′) = (0, 0, 0).
The only realistic way (?) to setup this system is to define at a certain moment t0, the origin O, when the two observers meet. At that same event, flash moment, the equally spaced, clocks in each reference frame should be synchronized.
In other words, the times and positions are coincident at this event.
Okay.
If all these hold, then the coordinate systems are said to be in standard configuration, or synchronized.
Okay. But that does not explain the reason of the transformations.
If an observer in F records an event t, x, y, z, then an observer in F′ records the same event with coordinates:
t' = gamma (t - v*x/c^2) : x' = gamma (x - v*t) : y'= y and z'= z
where v is the relative velocity between frames in the x-direction, c is the speed of light, and gamma = 1/ sqrt(1- v^2/c^2). Gamma is the Lorentz factor.
v is in effect the speed in F of the observer-in-F' .
This requires a continuous monitoring of the positions of the observer-in-F' in F. Even if the observer-in-F' moves with a constant velocity, these measurements are requiered to validate this claim.
The inverse relations (t, x, y, z in terms of t′, x′, y′, z′) can be found by algebraically solving the original set of equations.
A more efficient way is to use physical principles. Here F′ is the "stationary" frame while F is the "moving" frame.
According to the principle of relativity, there is no privileged frame of reference, so the transformations from F′ to F must take exactly the same form as the transformations from F to F′.
The only difference is F moves with velocity −v relative to F′ (i.e., the relative velocity has the same magnitude but is oppositely directed).
The inverse relations are only of mathematical interest. They don't answer the question: what is the purpose of the Lorentz transformations.

4.2 Physical implications

A critical requirement of the Lorentz transformations is the invariance of the speed of light, a fact used in their derivation, and contained in the transformations themselves.
Three counterintuitive, but correct, predictions of the transformations are:
Relativity of simultaneity
Suppose two events occur along the x axis simultaneously (Δt = 0) in F, but separated by a nonzero displacement Δx. Then in F′, we find that dt'= gamma* -v*dx/C^2, so the events are no longer simultaneous according to a moving observer.
But how important is that? Suppose there are two events, and assume event 1 comes first. That means event 1 can be the cause of event 2. But suppose a moving observer sees those events simultaneously. His conclusion is that event 1 can not be be the cause of event 2, which is wrong
The conclusion is that vissible observations are tricky. The scientific method is to establish the time of each event, separately, in one frame.
Time dilation
Suppose there is a clock at rest in F. If a time interval is measured at the same point in that frame, so that dx = 0, then the transformations give this interval in F' by dt' = gamma*dt.
Either way, each observer measures the time interval between ticks of a moving clock to be longer by a factor gamma than the time interval between ticks of his own clock.
The details of these measurements should be explained.
Paragraph https://en.wikipedia.org/wiki/Time_dilation#Simple_inference clearly shows that a moving clock ticks slower than a clock at rest.
My interpretation is that this is a physical issue, because the light path between two ticks of the moving clock is longer than the light path of the clock at rest.
Length contraction
Suppose there is a rod at rest in F aligned along the x axis, with length Δx. In F′, the rod moves with velocity -v, so its length must be measured by taking two simultaneous (Δt′ = 0) measurements at opposite ends.
That means F' is a moving frame with a speed -v in F.
Under these conditions, the inverse Lorentz transform shows that Δx = γΔx′.
In F the two measurements are no longer simultaneous, but this does not matter because the rod is at rest in F.
So each observer measures, the distance between the end points of a moving rod, to be shorter by a factor 1/γ than the end points of an identical rod at rest in his own frame.
Length contraction affects any geometric quantity related to lengths, so from the perspective of a moving observer, areas and volumes will also appear to shrink along the direction of motion.
The length of any material object is something physical. This means you should measure the length i.e. the positions of all objects simultaneous, at the same universal time in the universe. The physical reasoning is that events happening at the same universal time (in the universe) can not influence each other.
Suppose there is a second rod in F. This rod is moving towards the right. To measure it's length you must take two simultaneous measurements of the positions of the front and back of the rod in F. These positions define the length and the position of the rod at that instant. To do that you need clocks in F which run simulataneous.
My guess is that you will not measure any length contraction.

4.3 Vector transformations

4.4 Transformation of velocities

4.5 Transformation of other quantities

5. Mathematical formulation

5.1 Homogeneous Lorentz group

5.2 Proper transformations

5.2.1 The Lie group SO+(3,1)

5.2.2 The Lie algebra so(3,1)

5.3 Improper transformations

5.4 Inhomogeneous Lorentz group

6. Tensor formulation

6.1 Contravariant vectors

6.2 Covariant vectors

6.3 Tensors

6.3.1 Transformation of the electromagnetic field

6.4 Spinors

6.4.1 Transformation of general fields

7. See also

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Created: 3 September 2024

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