03 Invariants of the theory of relativity, space-time intervals





    Today, I’d like to talk about space-time intervals. In the theory of relativity, space-time intervals can be thought of as 'four-dimensional length'. In 2-dimensions, length means the line-segment above the paper. And in 3D, the length is the length of the bar inside the three-dimensional space. The main stage of relativity is the fourth dimension. Here, the fourth dimension has nothing to do with the fourth dimension of mystery novels!

The main tool used to talk about relativity is the Lorentz transformation. This Lorentz transformation is essentially a mix of time and space. The theory of relativity is based on two basic ideas. The second is the constancy of the speed of light.

The invariants that emerge from relativity come mostly from the constancy of the speed of light. This means that the speed of light in a vacuum is always constant regardless of the speed of the light source. This "always constant" is directly related to other invariants of relativity. Here I wrote the four-dimensional length.

      
Two-dimensional length

Three-dimensional length

Four-dimensional length = the space-time interval


Squaring a real number is still a real number. So squaring the above equations does not matter. Squared is shown below.  


Here we call 's' the space-time interval. It is interpreted as 'the length of the object which can not catch up with the light'. It can also be interpreted as a four-dimensional length. 
However, this is always constant regardless of the speed of the observer. The values ​​of s and s^2 should always be the same, even if I observe or not observe them.

They should always be the same, even if the Lorentz transformation is substituted for 's'. The problem arises here. The change in invariant in physics is evidence of a critical defect. This is a very important and urgent event. It is clear evidence that there is a serious flaw in the theory. The relativity theory is in crisis when the velocity of light, which is invariant in relativity, and the space-time interval change.

I am a very enthusiastic advocate of relativity. I will point out the problem and suggest ways to solve it. Let's start with the first problem.





The circle on the first line is the square of the three-dimensional length. It is a real number squared a real number, so if the length is contracted, the square of the length is also contracted. This is undeniable. Now let's set up a situation where the space-time interval is '1'. The number '1' has no meaning. If doesn’t change no matter what number you set.

Now, let's say that the speed of a system increases and its length changes. The invariant 's' can no longer be invariant because the 'length' continues to shrink and the 'time' continues to increase. As the speed increases, the 's' becomes smaller and later becomes negative. If length contraction theory is correct, then an important invariance of relativity will collapse. Serious contradictions occur.


Is there a way to solve this with the existing "length contraction-time dilation theory"?
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Nobody can solve this problem.
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However, if you choose the theory of length expansion by abandoning the theory of length contraction, the problem is easily solved. Now let's look at the scene once again.


The above picture is not a length contraction, but a length expansion. The higher the speed, the longer the length of the object. Then the first circle above will continue to grow. Because the length is bigger and the time is bigger, the invariant is literally invariant. For a more detailed quantitative process, you should draw a hyperbolic graph in a four-dimensional space. I will show you this later.

How about now?

The serious problem of collapse of the invariant is solved very easily by the expansion of the length. If you have any questions about this, please leave a comment.




Reading 'Length Expansion' in Amazon Books


p.s. Now you can answer the following questions correctly.
How do I derive the Lorentz contraction from the invariant interval?

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