25. K Calculus (#1/6): The twin paradox that can be understood by common sense
Let's take a look at K Calculus from now on six times. K calculus was Hermann Bondi's first proposed relativistic calculation method. The important thing is that we start using basic common sense. If we do this, it will therefore be very easy.
This method starts using simple common sense, and can derive the time dilation, velocity-addition formula, relativistic Doppler effect, and the Lorentz transformation itself. And during 5th and 6th time, I will point out the error of the length contraction and prove the length expansion through K Calculus.
I need a few posts to do that. After explaining that k calculus is a relativistically correct method, we will talk about 'the principle of invariant light speed and length'. I will describe it in the manner described by Bondi. I will let you know and explain in advance what the different is between Bondi's method and my own.
This method only deals with moving in one direction. This is the only drawback. However, expanding the two-dimensional space-time diagram to three dimensions can reveal so many facts. And you can see why a photo-clock is only possible if it is vertical, like kultiroff's method. We will deal later with photo-clocks that are tilted at random angles.
Relativity theory and common sense
People think that there will be something very mysterious about the theory of relativity. However, Bondi starts from common sense and derived all of this. Of course, there are preconditions. There must be a condition that the principle of relativity and the principle of invariant light speed are established.
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| Fig 1. Hermann Bondi |
Most of the content that I'm describing today is from the "Relativity and Commonsense" section of the main body. If you are interested, it would be good to refer to it.
Figure 2 below shows the world-line of particles. (b) is a stationary observer. Even if you stay still, it means that time is passing. (c) shows a case of moving at a low speed, and (d) shows a case of moving at a very high speed. And (e) is the world of light. At this time is 45 degrees. (a) shows the case of moving in the opposite direction.
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| Figure 3. Jones and James regularly signaling |
John and James are far apart. They are stationary relative to each other. If John sends a light signal every four seconds, does James get a signal every few seconds? Naturally, the signal is received every 4 seconds. It is extremely to understand using common sense. Thus K Calculus is common sense.
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| Figure 4. Role of Relativity |
When John sends a signal to James, Alice moves quickly from John to James. In figure 4 above, red is the world-line of Alice. Alice moves fast toward James, so Alice gets a signal at a longer interval than 4 seconds. At certain speeds, Alice will receive a signal every 6 seconds. Of course we can calculate but we will not calculate here. John sends a signal every four seconds, and Alice receives a signal every six seconds.
The fundamental ratio K in relativity
The ratio of this time can be expressed as k. This is not the same as the time dilation. However, this is closely related to the time dilation of the theory of relativity. Suppose that the speed of light is infinite. If the speed of light is infinite, if Bob sends a signal every four seconds, James and Alice will get a signal every four seconds as well. Then the theory of relativity returns to Newtonian mechanics.
When Ellis sends a signal every six seconds, James receives a signal every four seconds. Nothing is wrong with common sense.
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| Figure 5. The fundamental ratio of the theory of relativity K |
When John sends a signal every four seconds, Alice receives it every six seconds. Its ratio is (2: 3) because it is (4 seconds: 6 seconds), or (1: 1.5). If we generalize this now, we can say (1: k). Now, let's apply to Figure 5 above. If 4 seconds is k, 6 seconds corresponds to kT. Their ratio is (T: kT).
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| Figure 6. Return light signal |
Suppose that Alice sent his signal to John, through a mirror, immediately. Then John receives that signal on k2T. So the ratio of these three times is as follows.
T: kT: K2T
So the time that John gets back is 2: 3, so it will be 9 seconds. This alone is enough to understand the twin paradox.
Twin paradox
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Figure 7. The twin paradox that can be understood by common sense |
Alice travels with a clock that sends a signal every six seconds. And after 48 seconds, she delivers this watch to James. Of course, she passed on to James without any time delay. From John's point of view, he receives the signal from Alice every nine seconds and the signal from James every four seconds. Of course, the relative speeds of Alice and James are the same for John.
Of course, someone might argue that Ellis and James were quiet and John traveled. Let's say that she delivered a pack of eggs along with her watch. If she delivered the watch and the egg together, the eggs would have been destroyed at the moment of delivery. Therefore, it is easy to distinguish who moved.
Let's do the calculation now. John experienced 104 seconds in total, but Alice and James experienced 96 seconds. Therefore, it can be confirmed that the time of the moving person has flowed slowly. As a result, the twin paradox was solved.
Relationship between K and v
Now that we know their ratio, let's try to figure out the speed of the spaceship. As you can see in Figure 8 below, the speed is (distance / time).
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| Figure 8. Definition of speed |
Just get the distance (distance / time) from the graph and you will get the speed. Let's look at the speed in the picture below. Hermann Bondi calculated the speed of light c at 1. So we will also set c to 1. This makes calculations very easy. This is called the Planck units or Natural units. It should be calculated as follows.
(1 unit distance) = (speed of light) x (1 unit time)
However, if you use Planck units, you can calculate as follows.
(1 unit distance) = (1 unit time)
That is, it can be calculated as (distance = time). If necessary, you can write down the speed of light again later.
In the figure above, the speed is (OE) / (OD). First, let's get (OE). Since we have set the speed of light to one,
(OE) = (AD) = (DC) = (DB).
(AC) = k2T-T
(AD) = (k2T-T) / 2. Therefore
(OE) = (k2T-T) / 2 since (OE) = (AD).
Let's get the next OD.
(OD) = (OA) + (AD)
(OD) = (k2T-T) / 2 + T. Therefore
(OD) = (k2T + T) / 2
Now that we have both (OE) and (OD), we can get speed.
Since the speed is (distance / time), it is (OE) / (OD).
Now that we know the ratio, we can quickly get the velocity of the object. This is rearranged with respect to k, and k is as follows.
Since we have c set to 1 here, we can return to the general formula below.
Now we have the tools to understand the strange consequences of special relativity in common sense. There are a lot of things you can do with this. Ultimately, we will use K Calculus to see if length contractions are justified. But before that, there are things we should be aware of. So I will introduce them in turn. Thank you.
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