Boy of Age 16 Asks Me about Relativity, etc.16. What Is the Paradox about Time Travel?

Hand colored etching Mr. Fezziwig’s Ball by John Leech from A Christmas Carol by Charles Dickens. [Public domain], via Wikimedia Commons.
A Christmas Carol is considered to be one of the first depictions of time travel in both directions, as the main character, Ebenezer Scrooge, is transported to Christmases past, present and yet to come.

A friend of mine on Twitter, Aaron (a pseudonym), is an overseas, 16-year old boy, who seriously admires Albert Einstein and wants to become a physicist. He continually writes me (Ted, also a pseudonym) questions about the theory of relativity and related topics, and I am sending answers. In this series of blog posts, those questions and answers are reproduced with modifications. I am not an expert in the fields of physics related to relativity. So, my answers might contain errors. If you find any error, please do not hesitate to write a comment for the benefit, not only of the boy and me, but also of other readers.

Aaron: I think they say that there is a paradox about time travel. What is it?

Ted: Any theory that would allow time travel would require that problems of causality (the relationship between the cause and effect that the former should come before the latter) be resolved. From this viewpoint, the concept of time travel seems to give contradictions, examples of which are stated as paradoxes. One of the best examples is the grandfather paradox.

The grandfather paradox is a hypothetical situation in which a time traveler goes back in time and attempts to kill his grandfather at a time before his grandfather met his grandmother. If he did so, then his mother or father never would have been born, and neither would the time traveler himself. In that case, the time traveler never would have gone back in time to kill his grandfather. This is in contradiction to the assumption at the start.

This paradox has been used to argue that backwards time travel must be impossible. A number of hypotheses have been postulated to avoid the paradox, such as the idea that the past is unchangeable. However, any of those hypotheses has not become an accepted theory because the theoretical possibility of time travel itself is unknown.

To write the above explanations, I referenced the Wikipedia pages of "Time travel" (Ref. 1) and "Grandfather paradox" (Ref. 2). So, if you want to learn in more details, you can consult those pages.

Aaron: It's amazing. Now, I have an idea about how to go backwards in time. Time is like a line and flows in one direction, like a river, and we can go in both directions, in a river. This means that we can also control ourselves in time. If we can go back in time, we can kill Hitler and make the future without stupid World War II. But, I have to find how this is possible in a theoretical way. What do you think? Is it funny?

Ted: Your idea is appealing. However, it does not seem to be a physical idea about how to go backwards in time, but I'm afraid that it is an idea about what you would like to do if you could go backwards in time. Further, only killing Adolf Hitler would not prevent the World War II totally. You may need to kill also Benito Mussolini in Italy and Hirohito in Japan and to change all the factors related to nationalism or imperialism and international tensions of those days.

[Next day, Ted again wrote to Aaron, writing as follows:]

However, your idea also included a good point. If you go backwards in time not to kill your grandfather but to kill Hitler, you can escape the paradox of your not being born. Thus, your idea is a good step toward the solution of the paradox.

I compared your idea with Novikov self-consistency principle. This principle was proposed by a Russian (and former Soviet) theoretical astrophysicist and cosmologist, Igor Dmitriyevich Novikov, in the mid-1980s and have been regarded as an important contribution to the theory of time travel (Ref. 3). I have just learned it from Wikipedia.

According to this hypothetical principle, the only possible time lines are those entirely self-consistent. So, anything a time traveler does in the past must have been "part of history all along." Your idea is partly similar to this principle, in the successful removal of the inconsistency about the time traveler's birth, though killing Hitler is inconsistent with the real history. You can have confidence in your ability of thinking about physics.

References

1. Time travel, Wikipedia, The Free Encyclopedia (December14, 2012 at 23:29).
2. Grandfather paradox, Wikipedia, The Free Encyclopedia (December17, 2012 at 08:09).
3. Novikov self-consistency principle, Wikipedia, The Free Encyclopedia (November 26, 2012 at 03:04).

(Originally written from July 29 to 31, 2011)

Boy of Age 16 Asks Me about Relativity, etc.15. What is String Theory?

Different levels of magnification of matter, ending with the string level: 1. Macroscopic level – Matter. 2. Molecular level. 3. Atomic level – Protons, neutrons, and electrons. 4. Subatomic level – Electron. 5. Subatomic level – Quarks. 6. String level. [By MissMJ (CC-BY-3.0), via Wikimedia Commons.]

A friend of mine on Twitter, Aaron (a pseudonym), is an overseas, 16-year old boy, who seriously admires Albert Einstein and wants to become a physicist. He continually writes me (Ted, also a pseudonym) questions about the theory of relativity and related topics, and I am sending answers. In this series of blog posts, those questions and answers are reproduced with modifications. I am not an expert in the fields of physics related to relativity. So, my answers might contain errors. If you find any error, please do not hesitate to write a comment for the benefit, not only of the boy and me, but also of other readers.

Aaron: What is string theory? I have read a little about it. It seems to be the theory of everything that Einstein was working on. Dr. Michio Kaku is probably working on how to find it. However, there are many equations in this theory. How can I understand it?

Ted: "What is string theory?" is a difficult question for me. In my student days, this theory was not yet born. So, some years ago I wanted to learn a little bit of it and bought a graduate level text book on this theory written by the physicist you just mentioned, i.e., Michio Kaku. However, it was pretty difficult for me to learn it by myself, and I have not read the book yet.

The essential idea of string theory is that all of the different "fundamental" particles are different manifestations of one basic object, a string (see the figure above). I hear that the equations of this theory gives a lot of solutions, and presently it is difficult to determine which of those solutions reflect the laws of physics in the real world. In this situation, there is the supposition that there may be many worlds, in each of which one of many solutions is applicable. (However, it is a vexing problem how we can verify the applicability of solutions in other worlds). A number of gifted physicists are studying this theory, but some famous physicists do not think that this is the right direction to advance the study of theoretical physics. Further, it is said that we humans don't yet have enough mathematical methods fully to explore this theory.

String theory is such a complex and difficult thing. You had better learn it after enough mastering of quantum mechanics and relativity. Taking such a step is indispensable also considering the fact that string theory aims at the unification of quantum mechanics and general relativity. However, there are a number of Web pages explaining string theory for non-scientists. See, for example, Ref. 1 and links given in it.

Reference

1. Alberto Güijosa, What is String Theory?

(Originally written on July 18, 2011)

Leo Tolstoy and Mathematics

Leo Tolstoy. By Scan by User: Gabor [Public domain], via Wikimedia Commons.

These days I am reading Leo Tolstoy's autobiographical novels Childhood, Boyhood and Youth by Japanese translations in Shinchō World Literature Volume 16 (1972). In the first half of Youth, there is an episode that the hero, "I", passes the entrance examination to the mathematics department of a university.

The examination is made in the following manner: Each applicant takes a piece from the problem cards held by a professor and answers to the problem chosen, in front of the professor. In the mathematics examination, there were two professors. The hero was called at the same time as another applicant, and they secretly exchanged the cards they chose. The card the hero initially took was of the problem about "combinatorics," which he thought difficult. However, the question he got by exchanging the cards was about "Newton's binomial theorem," which he had tried to solve just before the examination. Thus, the hero was able to answer it perfectly.

Reading this, you may wonder if Tolstoy studied in the mathematics department. In fact, Tolstoy began studying law and oriental languages at Kazan University and left university in the middle of his studies (Ref. 1). Therefore, the story of the entrance to the mathematics department is fictional.

I learned "combinatorics" and "the binomial theorem" in senior high school. However, the latter was not Newton's but the basic one. The basic "binomial theorem" describes a formula for the algebraic expansion of nth power of a binomial x + y, where n is a positive integer. Isaac Newton generalized the formula to allow real exponents, and the formula can be generalized further, to complex exponents (Ref. 2).

It was probably at university that I learned about generalized binomial theorem. Therefore, Newton's binomial theorem seems to be too difficult for the entrance examination of university. It is also strange that the hero thought Newton's binomial theorem easier than combinatorics. This is because we learn combinatorics as preparation for studying the basic binomial theorem. Thus, I guess that problems of mathematics were also invented by Tolstoy on the basis of his superficial knowledge of these terms of mathematics. What do you think?

References

1. "Leo Tolstoy," Wikipedia: The Free Encyclopedia (December 8, 2012 at 19:10).
2. "Binomial theorem," Wikipedia: The Free Encyclopedia (November 25, 2012 at 04:39).

Classifications of Theoretical Physicists, Especially of Yukawa and Tomonaga

The theoretical physicist Susumu Kamefuchi published an essay entitled "Gramsci's words, Yukawa, Tomonaga, and Sakata" [1]. At the beginning of the essay, Kamefuchi quotes the following words:

Passage from knowing to understanding and to feeling and vice versa from feeling to understanding and to knowing —Antonio Gramsci, "Prison Notebooks" [2]

Kamefuchi likens the three elements in the above quotes, i.e., feeling, understanding and knowing to three stages of research in theoretical physics, i.e., (I) practitioner's stage, (II) theorist's stage and (III) natural philosopher's stage. Then, he thinks about the question in which stage each of Hideki Yukawa, Sin-Itiro Tomonaga, and Shoichi Sakata was good at working or liked to work, in order to classify them into corresponding three types, I, II and III, of physicists.

Sakata was called a person of methods and his successful studies, i.e., the two-meson theory and the Sakata model of elementary particles were phenomenological. From these facts, Kamefuchi classifies Sakata into type I.

Tomonaga had an excellent mastery of mathematics and expertise in constructing theories based on different physical requirements, producing the super‐many‐time theory, which lead him to the finding of the renormalization method and to the winning of Nobel Prize. Thus, Kamefuchi classifies him into type II.

Yukawa's work to create a comprehensive theory of particles starting from "nonlocal fields" or "elementary domains" corresponded to the process of going from knowing to understanding and to feeling, but was not completed. However, Yukawa said in his later year, "Such a fundamental theory was my ultimate purpose, and the meson theory was a byproduct on my way." Yukawa often presented his opinion about various cultural problems (creativity, genius, learning, peace, etc.), displaying his characteristic of being an excellent thinker in culture as well as in physics. From these facts, Kamefuchi classifies Yukawa into type III.

Kamefuchi's essay concludes as follows:

The fact that Yukawa, Tomonaga and Sakata belonged to the three different types was rather lucky to the development of particle theory in Japan. The three leaders played the role of antithesis against each other so that the study of particle physics in our country made a balanced progress. […] I believe that this was the basis of the Nobel-prize winning studies by the physicists of the next generation, Yoichiro Nambu, Masatoshi Koshiba, Toshihide Maskawa and Makoto Kobayashi.

Kamefuchi's classification scheme of physicists reminds me of a similar classification proposed by Yoichiro Nambu. His classification as summarized by himself is as follows [3]:

Once I classified theoretical physicists into three types according to their different styles of approach, and called them Heisenberg (H), Einstein (E) and Dirac (D) modes, referring to their most characteristic contributions respectively, i.e., quantum mechanics, theory of gravitation and the Dirac equation. Heisenberg’s is heuristic, bottom-up and inductive. Einstein’s is axiomatic, top-down and deductive. Dirac’s is abstract, revolutionary and esthetic.

As for the modes to which Yukawa and Tomonaga belongs, Nambu writes as follows [3]:

It would be safe to say that Yukawa belonged to H when he proposed the meson. He failed in E when he tried his hand at nonlocal theory. I have a bit of difficulty applying this to Tomonaga, but I will assign him to E. Most theorists belong to H or E. But, when it comes to contrasting Yukawa and Tomonaga, it may be appropriate to use the analogy to designer vs. craftsman.

Kamefuchi's type II and type III seem to correspond to Nambu's H mode and E mode, respectively. However, when we consider the corresponding categories identical, it causes inconsistency between Kamefuchi's and Nambu's classification of Yukawa and Tomonaga. The inconsistency comes from the difference in the viewpoint between Kamefuchi and Nambu. Namely, Kamefuchi attach importance on the physicist's preference of a method, especially for the classification of Yukawa, and Nambu, on the physicist's successful work.

On the other hand, Kamefuchi's type II and type III seem to correspond to Nambu's category of craftsman and that of designer, respectively. In this case, we can regard the corresponding categories as nearly equal without causing inconsistency between Kamefuchi's and Nambu's classification of the two physicists. The consistency in this case arises because Nambu's classification here is based on methodology of the physicists, similarly to Kamefuchi's.

References

1. S. Kamefuchi, Tosho No. 766, p. 2 (December, 2012) in Japanese.
2. English translation has been obtained from: Antonio Gramsci, Selections from the Prison Notebooks, edited and translated by Q. Hoare and G. N. Smith, p. 767 (ElecBook, London, 1999).
3. Y. Nambu, The Legacies of Yukawa and Tomonaga, AAPPS Bulletin Vol. 18, No. 6, p. 7 (2008)

Boy of Age 16 Asks Me about Relativity, etc.14. Relations among the Expansion of the Universe, Gravity, Relativity Theory and Dark Energy

George Gamow's book My World Line, in which Einstein's words "the biggest blunder
I had ever made in my life" were first written.

A friend of mine on Twitter, Aaron (a pseudonym), is an overseas, 16-year old boy, who seriously admires Albert Einstein and wants to become a physicist. He continually writes me (Ted, also a pseudonym) questions about the theory of relativity and related topics, and I am sending answers. In this series of blog posts, those questions and answers are reproduced with modifications. I am not an expert in the fields of physics related to relativity. So, my answers might contain errors. If you find any error, please do not hesitate to write a comment for the benefit, not only of the boy and me, but also of other readers.

Aaron: Why is the universe expanding? Where is gravity? What about Einstein's relativity? Some scientists say that the universe is expanding because of dark energy, don't they?

Ted: The expansion of the universe is considered possibly due to the initial condition of the Big Bang, with which our universe started. In 1998, two teams of astronomers suggested on the basis of their observations of Type Ia supernovae that the expansion of the universe had been accelerating. (Saul Perlmutter and Adam Riess of the U.S. and Brian Schmidt of Australia contributed to this finding and won Nobel Prize in Physics in 2011.) Until this discovery, physicists were convinced that gravity should be causing the expansion rate of the universe to slow. To explain the accelerated expansion, dark energy, which produces the mysterious force to repel gravity, was proposed and has been constituting the most accepted theory. Scientists are still trying to find what dark energy exactly is (Ref. 1).

As explained above, dark energy is the notion that appeared after the discovery of the accelerated expansion. With regard to the relation between the expansion of the universe found earlier and the relativity theory, there is a fascinating history. After formulating the equation of general relativity, Einstein tried to find the distribution of masses that would lead to a stable universe unchangeable with time (a static universe was the prevailing hypothesis those days). He found that the equation was incorrect to produce such a universe. Therefore, he added a term to the equation, which became known as the "cosmological term" or the "cosmological constant."

The Russian mathematician Alexander Friedmann found that Einstein's treatment had been wrong and that the original equation of general relativity was correct to predict time-dependent universes as well including an expanding one, which became the observational fact by Edwin Hubble's work, in the late 1920s, of measuring the redshifts of light from galaxies. Thus, changing the original equation was a mistake, and Einstein once told Gamow that the introduction of the cosmological term was the biggest blunder he had ever made in his life (Ref. 2).

One possible source of dark energy, supposed to explain the accelerated expansion, is the "cosmological constant," a constant energy density filling space homogeneously, and the other is scalar fields (Ref. 3). Therefore, Einstein's biggest blunder has become a central concept of the present cosmology.

Note: Earlier, Aaron asked what would happen to the relativity theory if dark energy were true (see here). I took this as the question about a possible failure of general relativity under the presence of the accelerated expansion. So, I quoted from Ref. 4 the description of some theorists' thought that a failure might happen on scales larger than superclusters. However, the equation of the general relativity with the cosmological constant might prove to be an excellent theory except for such an extreme case.

References

1. Physics Nobel Explainer: Why Is Expanding Universe Accelerating? National Geographic, Daily News (October 2011).
2. George Gamow, My World Line: An Informal Autobiography (Viking, New York, 1970) p. 44.
3. Dark energy, Wikipedia, the free encyclopedia (November7, 2012 at 00:22).
4. 3 Alternative Ideas, ibid.

(Originally written on June 27 and July 5; modified to a large extent.)