Mystery of Dimension(PART 1)
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Kato. . . , what kind of mystery is that?
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Diane, have you ever heard of a four-dimensional world?
Of course, I have. . . We're living in the four-dimensional world or spacetime world, aren't we?
Yes, most of us believe we are, but actually we're living in a 10-dimensional world.
You gotta be kidding!
I'm dead serious!
But what about other six dimensions? What the heck are they? . . . And where in the world do we find those six dimensions?
Good question! . . . Tell me, Diane, what the dimension is.
Well. . . the dimension is a measurable extent of some kind, such as length, breadth, depth, or height. . . So usually we can move along the vertical, horizontal, height and time axis. . . Therefore, it is said that we're living in a four-dimensional world.
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You're darn right on that. . . But according to some mathematicians, our world has ten dimensions, which is mathematically proven.
Are you pulling my leg?
Oh no. . . I'm profoundly dead serious. . . If you're in doubt, you should view the following clip.
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In other words, the 10-dimensional world has something to do with the birth of the universe, huh?
That's right. . . If you look carefully at the clip above, you can understand it. . .
Kato, can you understand it by only watching the above clip?
I think I can. . .
You can understand because you've got an engineering background, but I'm a student of liberal arts. . . Please explain it easily so that I can understand. . .
Well, first of all, I'll talk about two theoritical physicists. . .
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The above two men are said to be the creators of the so-called superstring theory, but Joël Scherk, French physicist, died at the age of 34. . .
Was he killed by a traffic accident?
Oh no. . . He is said to have died because he suffered from severe diabetes. They say, he died of overdose. . .
So, that superstring theory is related to the 10-dimensional world, isn't it?
That's right. . . Currently, in his 70s John Schwarz teaches theoretical physics at the California Institute of Technology. . .
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This photo shows that Professor Schwarz teaches students in the lecture room, huh?
Oh no, he doesn't. . . He's writing mathematical formulas on the blackboard in his own office. . .
Is he doing research on a blackboard without using a computer? It’s much faster to do research using a personal computer. . . He could get better research, I suppose.
No, not really because no matter how great software he uses on his personal computer, he cannot get an original idea.
Why is that?
It is because every software and programs were originally created by humans. . . As long as you use such software, you will not come up with original ideas. . . So, in order to derive original formulas, it is best to do research while writing formulas by hand using a blackboard or whiteboard. . .
I see. . .
By the way, the big bang that occurred 13.8 billion years ago was predicted by a mathematical formula. . .
What is that formula?
That is Einstein's general theory of relativity. . .
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Is Einstein's general theory of relativity flawed compared to the superstring theory?
Yes, it is. . . The general relativity cannot unravel the bottom of the black hole, that is, the profoundly microscopic world. . .
why. . . ?
It turns out that the birth of the universe has a deep connection with black holes. . .
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The big bang and the black hole are very similar in structure. . . The big bang and the bottom of the black hole are mathematically the only keys to the birth of the universe. . . In other words, in order to explore the birth of the universe, it is necessary to elucidate the black holes mathematically. . . However, the general theory of relativity cannot unravel the bottom of the black hole. . .
Why can the general relativity not unravel it?
If the depth of the black hole is squeezed by general relativity, the denominator will be zero at the depth of the black hole. . . This means infinity!
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What's wrong when the denominator becomes zero and becomes infinite?
When the denominator beconmes zero, it cannot be calculated in the mathematical formula. . . So, if you try to unravel the depth of a black hole with a mathematical formula, you cannot apply the general relativity, which is not valid. . .
In short, in order to unravel the depth of a black hole with a mathematical formula, you must understand the birth of the universe, huh?
That's right. . .
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■"Zoom In"
In other words, unless the infinite problem is solved, the birth of the universe cannot be solved, huh?
You're telling me. . .
So what happened?
Somebody came up with an idea. . . How about combining quantum mechanics with general relativity? . . .
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This idea came to the mind of Matvei Bronstein (1906-1938) in the Soviet era. . .
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He was a genius of theoretical physics. . . It is said that at the age of 19 he had understood the mathematical formulas of quantum mechanics and general relativity completely.
Wow! . . . So why did he die so young?
Just when he was immersed in his research, that is, in 1937, he was arrested and killed by Stalin. . . The intellectuals who were doing suspicious research seemed to Stalin an enemy who must be purged. . .
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Stalin did a terrible thing, didn't he? . . . So what happened to the study of Matvei Bronstein?
Unfortunately, combining the two mathematical formulas did not provide an infinite solution. . . As the research progressed, the theoretical physicists found a great deal of infinity in the profoundly microscopic world. . .
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So the problem of infinity had deepened, huh?
That's right. . . So, most researchers gave up and threw away the infinite mystery.
What happened then?
About 40 years later, precisely in 1974, two unknown theoretical physicists published a paper that solved the mystery. . . They are Joël Scherk and John Schwarz introduced above. . .
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They were studying string theory that no one looked at. . . In this string theory, elementary particles are not dots but strings like threads.
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John Schwarz went further and discovered the superstring theory that solves an infinite problem. . . If an elementary particle is a point, it becomes infinite when two elementary particles collide.
So . . . ?
In the superstring theory, the distance does not become zero even if it collides because elementary particles are strings---not points, so there is no infinity. . .
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However, there was a problem here. . . The condition that supports the superstring theory seems impossible in reality!
What is that condition?
The superstring theory is valid only in a 10-dimensionnal world.
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That means that the universe was born as a 10-dimensional universe, huh?
That's right. . . However, from a common-sense point of view, our world is not a 10-dimensional world. . .
According to Einstein, we're living in a four-dimensional world or spacetime world. . . What are the other six dimensions?
So, many researchers came to conclusion that this theory was absurd and they gave up.
How about the followers of the superstring theory?
They combined two formulas, namely those of general relativity and quantum mechanics.
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When they verified whether the two formulas were included in the superstring theory formula, they were surprised to see the superstring formula include two formulas that seem to be completely unrelated. . . And they found the perfect number (496) appearing one after another.
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Perfect number
In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself.
For instance, 6 has divisors 1, 2 and 3 (excluding itself), and 1 + 2 + 3 = 6, so 6 is a perfect number.
The sum of divisors of a number, excluding the number itself, is called its aliquot sum, so a perfect number is one that is equal to its aliquot sum.
Equivalently, a perfect number is a number that is half the sum of all of its positive divisors including itself i.e. σ1(n) = 2n.
For instance, 28 is perfect as 1 + 2 + 4 + 7 + 14 + 28 = 56 = 2 × 28.
This definition is ancient, appearing as early as Euclid's Elements (VII.22) where it is called τέλειος ἀριθμός (perfect, ideal, or complete number).
Euclid also proved a formation rule (IX.36) whereby q ( q + 1 ) / 2 {\displaystyle q(q+1)/2} {\displaystyle q(q+1)/2} is an even perfect number whenever q {\displaystyle q} q is a prime of the form 2 p − 1 {\displaystyle 2^{p}-1} 2^{p}-1 for prime p {\displaystyle p} p—what is now called a Mersenne prime.
Two millennia later, Euler proved that all even perfect numbers are of this form.
This is known as the Euclid–Euler theorem.
It is not known whether there are any odd perfect numbers, nor whether infinitely many perfect numbers exist.
The first few perfect numbers are 6, 28, 496 and 8128 (sequence A000396 in the OEIS).
Source: “Perfect number”
Free encyclopedia "Wikipedia"
In other words, it was thought for ages that perfect numbers symbolize completeness. . .
That's right. . . Anyway, the news that the two formulas are included in the formula of the superstring theory became known to researchers all over the world. . . It was a sensation!
That's how the superstring theory attracted attention, huh?
You're telling me, Diane.
But what are the other six dimensions?
Well, for example, the one-dimensional world for humans are actually two-dimensional or three-dimensional world for ladybugs. . .
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In other words, as shown above, the person on a tightrope is one-dimensional on the cable. . . The person can only move forward and backward. . . However, there are dimensions invisible to the human eye because, for the ladybug, it is two-dimensional or three-dimensional world. . .
I see. . . But what exactly do you mean by a 10-dimensional world?
The remaining 6 dimensions are hidden in one ultra-micro world of one trillionth of a trillion of an atom!
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This solved the 10-dimensional problem, but there were more problems. . . The genius of that wheelchair proposed the “Hawking Paradox”.
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“Why is heat generated at the bottom
of a black hole where even elementary particles
cannot move?”
So who solved it?
Joseph Polchinski, who appeared in the photo above, succeeded in calculating the heat of the black hole by adding a group of strings called "D-brane".
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The film-like elementary particles move around in six dimensions, generating heat. . .
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In other words, Joseph Polchinski solved all the problems?
No, not really. . . In the latest research, some scientists insist that this universe is not 10-dimensional but 11-dimentional, and there is a hypothesis that there are a great number of universes---10 to the 500th power. . . The problem is deepening. . .
I wonder if scientists try to experimentally confirme that there are actually the remaining 6 dimensions.
Yes, they do. . . At the famous European Organization for Nuclear Research (CERN) in France, scientists are experimenting to actually verify the remaining six dimensions using a giant accelerator. . .
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Talking about the European Organization for Nuclear Research (CERN), the World Wide Web (WWW) was born here, huh?
That's right. . . You're telling me. . .
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(To be continued)
