Lorentz Transform Derivation questions

In summary: Therefore, at any given moment, the x' coordinate of this object will be 0. This is why the author states that "for the origin of k' we have permanently x' = 0." This does not mean that x' lingers at the K' origin, but rather that the definition of x' is always tied to the position of this object. In summary, the concept of x' being permanently 0 at the origin of the K' frame is simply a matter of definition and does not have any implications for the movement of the frame.
  • #1
kwestion
63
0
I'm trying to follow along with Simple Derivation of the Lorentz Transformation, but am having some hurdles.

I'll be referring to step (5) which states:
x'=ax-bct
ct'=act-bx​
In paragraph marked 6, I see that the author tries to get eqn (5) to describe motion of the K' frame. This is an important move, but not understood. Up until that point, I believe x' has been a description of the position of light on the frame K' with x' having rules of motion that include x'=ct'. x'=ct' suggests to me that x' is at the K' origin for only a moment when t'=0, but the author states that:
For the origin of k' we have permanently x' = 0[...]​
I don't understand the "permanence" here. Does x' linger at the K' origin? Did x' change meaning? Is it poor notation? Is it that since t'=0 is the only valid moment* for (5) that the state of that moment constitutes a permanent state for (5)? Is there a better description of why (5) begins to be used to track the motion of the frame? I don't see how the position of x' helps understand the movement of K' here. * "The only valid moment" is an unconfirmed assumption on my part. (5) was derived from equations like x-ct=0 and x+ct=0 (inferred) and x'-ct'=0 and x'+ct'=0 (inferred). Upon combining equations in (5), I think all former conditions need to be satisfied by any x, t, x', or t' used with (5). That is, valid x,t,x',t' must not contradict any of: x-ct=0, x+ct=0, x'-ct'=0, or x'+ct'=0, which implies that x=0, t=0, x'=0, and t'=0. What perspective am I missing?
 
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  • #2
This is just a matter of definition. The x' coordinate in the frame K' is defined by measuring distances from some inertially moving object that is at rest in the K' frame.
 

Related to Lorentz Transform Derivation questions

1. What is the Lorentz Transformation?

The Lorentz Transformation is a mathematical equation that describes how space and time coordinates change when an observer moves at a constant velocity relative to another observer. It is a fundamental concept in Einstein's theory of special relativity.

2. Why is the Lorentz Transformation important?

The Lorentz Transformation is important because it allows us to understand how physical laws and measurements change for observers in different reference frames. It also helps to reconcile the apparent discrepancy between the constant speed of light and the principle of relativity.

3. How is the Lorentz Transformation derived?

The Lorentz Transformation is derived using the principles of special relativity, which state that the laws of physics should be the same for all observers moving at a constant velocity. By applying these principles to the equations for time and space measurements, the Lorentz Transformation can be derived.

4. What are some applications of the Lorentz Transformation?

The Lorentz Transformation has many applications in modern physics, including in particle accelerators, GPS systems, and the development of nuclear energy. It also helps to explain phenomena such as time dilation and length contraction.

5. Are there any limitations to the Lorentz Transformation?

While the Lorentz Transformation accurately describes the relationship between space and time for observers in relative motion, it does not account for the effects of gravity. In these cases, the more complex equations of general relativity are needed. Additionally, the Lorentz Transformation only applies to objects moving at constant velocities, not those undergoing acceleration.

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