30. K Calculus (#6/6): Three proofs of length expansion
Hello.
Today, let's take a look at proof of length expansion using K Calculus. The first is to use the K value, the second is to modify the method of Hermann Bondi, and the third is to follow David Bohm's notation.
The first method and the second method are the ones we've already seen. I will show you again in the intention to refine your thinking, and I will look at the third method. Let's take a quick look at the meaning of the K value before that.
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| Figure 1. Meaning of k |
When John sends a light signal every four seconds, James receives a signal every four seconds. By the way, when Alice is moving from Johnson to James, Alice receives this signal at six seconds instead of four seconds. The ratio of John to Alice's signal interval is (4 seconds: 6 seconds). This can be expressed as (1: k), where the ratio is the meaning of k. It's simple. The following relation holds between the speed of Alice and k.
Where v is the relative speed when you see the speed of light as 1.
1. Proof of length expansion through K
There is a circle that is relatively stationary. The radius of this circle is r.
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| Figure 2. Length and radius of a relatively stationary object |
However, if this circle moves at a relativistic speed, its shape will change. An object with a diameter of 2r in the above circle becomes γ(2r) as shown in the figure below. But this is what you see when you go across the front of the observer.
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| Figure 3. Length of object due to transverse Doppler effect |
If the observer is looking from the left, it will look like the picture below. Whether you are moving across my front, observing from the left, or observing from the right, the overall shape of the object is always constantly 2L. This is the overall outline.
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| Figure 4. Length due to longitudinal Doppler effect |
From now on I will start the proof. In the previous post, we saw that the wavelength appears as Kλo when the light goes away, and k-1λo when the light comes back. In Figure 5 below, when John observes Alice's light clock moving relatively, it is observed as Kλo when it goes away, and as k-1λo when it comes back. Here we will set the speed of light to 1 in all of the ways of Hermann Bondi.
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| Figure 5. length of parallel moving object and path of light |
If you are moving parallel to the direction of the light clock, the light clock will have the path shown in Figure 5 above. Here, the proper length of the light-clock is cto.
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Figure 6. Relationship between path of parallel moving light and k |
Now John is measuring the length of Alice's light clock. It is kLo when it goes away, and k-1Lo when it is reflected. This is 2L because it is round-trip length. So if you average these two, John will measure Alice's length. If you want to know more details, please read below.
2. Proof of length expansion according to the method of Hermann Bondi
In order to obtain the length of (5) in Fig. 7, Hermann Bondi expresses the equation as (17).
After going through several steps, he concluded that equation (17) is (19).
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| Figure 7. Meaning of an equation that Hermann Bondi wanted |
(17) represents the inherent length, the final equation is as follows.
If so, then this equation must be concluded as follows.
If this is true, the expression that Hermann Bondi derives is the length of the expansion. Click on the links below for more information.
3. Proof by David Bohm's notation
The third method is a proof by David Bohm's notation. David Bohm did not prove length contraction in his book. To be precise, he could not prove length contraction. According to the theory of relativity and the clear definition of K Calculus, length contraction is never derived. Rather, the expansion of the length is derived.
In other words, David Bohm did not prove the expansion of the length, but according to his symbol system, the expansion of the length is induced. Now, let's derive a legitimate relativistic length to his method.
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| Figure 8. Length of bar in static state according to David Bohm's notation |
Figure 8 above shows a measure of the length of a relatively stationary rod. Round-trip time is t2.
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| Figure 9. Spacetime diagram showing the length of the moving rod. |
The right-leaning rod is moving at relativistic speed. The rods moving here are (ON ') = (ON) + (NN'). This equation can be summarized as follows. It means ct1 = cto + vt1. Expression (1) can be obtained by summarizing the equation.
ct1-vt1=cto
(c-v)t1=cto
In Figure 9, we have the following relationship:
(OD) k = (OS)
c (OD) k = c (OS) ---------- (2)
ct2 = 2L = c (OD)
(OD) k = (OS)
c (OD) k = c (OS)
2Lk = 2ct1 ------------ (3)
Here k and t1 are as follows.
Substituting this into equation (3) results in:
Guided through David Bohm's notation, length expansion is induced, not length contraction. Derived by any method, length-expansion is derived.
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