28. K Calculus (#4/6): The Lorentz transformations derived from common sense




  Hello. This time we will look at how we can derive Lorentz transformations using K Calculus. If you follow only the principle of invariant light speed and principle of relativity, anyone can derive the Lorentz transformation equation.

Hermann Bondi


This method, created by Hermann Bondi (K Calculus), can derive to most conclusions of special relativity theories such as twin paradox, velocity-addition law, Lorentz transformation, time dilation.

 The k in this article is not particularly mysterious, but rather the ratio of John and Alice' time. When john sends a signal every 4 seconds, James receives it every 4 seconds, but Alice receives a signal every 6 seconds. So it becomes (4: 6), which can be expressed as (1: 1.5). At this time, k is 1.5.
Figure 1. The nature of relativistic effects

And for two observers with different relative movements, K is transformed as follows.

Figure 2. Signal transfer and k value.
When the John sends a signal at 4 second intervals,
Alice receives it every 6 seconds.


In Figure 2 above, if 6 seconds is passed again, the relationship is as shown below.

Figure 3. Returned signal and k value.
When Alice sends a signal at 6-second intervals,
the John receives at 9-second intervals.


Every time we convert, we multiply by k.

Derivation of Lorentz transformation

   Hermann Bondi has one thing in his book that derives to various expressions. It is calculated by setting the speed of light c to 1. Planck unit system is used. In the picture below, James shoots the light at t-x and receives the light again at t+x after the light is reflected at A.
Do not have a question, 'Why is the x axis in the vertical axis?' In Planck unit system, unit length and unit time are the same. For example, 3×108m is equivalent to 1 sec. Hermann Bondi used Plank unit system, and set c to 1. So if you force x to interpret it, you can interpret it as the time it takes for the light to go to x.
Figure 4. Light signal returned to John

In any case, in Figure 4, the light was fired at t-x and reflected back to t+x. I think there is no doubt here.

Figure 5. A round-trip signal recognized by James

   At the origin O in Figure 5, however, the two are separated from each other. The signal that John fired passes through James twice and returns to John again. Figure 5 illustrates this situation. We talked a few things about the ratios in K Calculus. According to it, the following relationship holds.

(2) Let's change the left and right terms of equation. Then it looks like this:


Now, (1) × (3) will be as follows.


Here, k on both sides is cleared and summarized as follows.


 And, (2) summarizing the equation for x ', it becomes as follows. 

(5) is substituted into equation (4) and summarized for t '.


 If we summarize this expression (6), it becomes as follows.
 

 Since the value of k is as follows, put this in (7) and organize it.
If you put it together, it will look like (8).




  In the same way, x' can be summarized as (9). You might think it's a bit different from what you see in plain text, but Bondi has just set c to 1. This is an expression of Hermann Bondi, so it would be nice to respect it. In this way, there are advantages. You should look closely at (8) and (9). Time and space are perfectly equal and perfectly mixed. In special relativity, time and space are always symmetrical.


Space-time symmetry and time-space equivalence



In special relativity theory, time and space are symmetrical to each other. Equation (8) and (9) are perfectly equivalent. Please remember this correctly. Typically, length is the product of time and speed.



If you set the speed of light to 1, you can write as below. Since the speed is set to 1, it is very natural that the time and the length are the same.



If you agree with this, I will ask you one question.
Is time and space really equal?
Is it true?

 Of course it is.

 In the special relativity theory, it is true and so on.
If yes, what is below?



This length and time are no longer the same when the current theory of relativity is applied. Distance traveled by light and travel time traveled by light are not the same. If time and distance are equal 'length contraction = time dilation' must be established. But this never happens. One side becomes smaller while the other becomes larger. How can it be so?

 In the space-time diagram in Figure 6 below, the length of time(red) and the length of space(blue) are the same. Because it is exactly symmetrical around the world of light. Unit time = unit length, and the speed of light is set to 1, which is a natural result.




Let '300,000km/sec' be the speed of light. Then, in the space-time diagram in Figure 6 above, the unit time of red in the time axis corresponds to 1 second, and the unit distance in the space axis corresponds to 300,000 km. (Precisely the unit of time axis is ict). At this time, the red unit length and the blue unit length are the same. So if you follow Hermann Bondi's method, the length in the space-time diagram tells you everything, even if you do not show the unit.
This method does not change this way, even when in motion.


The unit time shown on the time axis became unit time ', and the unit length displayed on the space axis became unit length'. Since the speed of light is still c, if we omit the speed c of light, we can still see that the following equation holds true.

However, if the theory of length contraction is correct, this relation is never formed.



This can be seen clearly from the equation below. If the length contraction is correct even if the speed of light is not set to 1, the principle of invariant light speed is no longer established.



However, if you abandon the theory of length contraction and choose the theory of length expansion, you can eliminate all contradictions. When the speed of light is set to 1, the following relationship holds.


If you select the length expansion, even if you observe the moving partner, there is no problem because the following relationship is established.



The equation is as shown below, and the principle of invariant light speed is established naturally.



















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