How Gödel broke the taboo of time travel

When you are talking about Kurt Gödel with me, you'd better be careful as you are probably speaking with the one of the biggest admirers of him in the universe. It's unarguable that he is one of the greatest minds in the science history however for me he is rather an inspiration to follow an academic career. Once the famous mathematician Fermat wrote for his mysterious claim, later known as "Fermat's Last Theorem" in Arithmetica as follows : "I have discovered a truly marvelous proof of this, which this margin is too narrow to contain". I can confıdently repeat this statement for Gödel. This margin is too narrow to contain him.

Until several decades, time travel has only been a topic of sci-fi movies, books and articles and merely an imagination of human mind, along with other supposed superstitions. Probably , people were having a "thoughtless" fun while reading Memoirs of the Twentieth Century, in 1733, in which a guardian angel was traveling to 1997-98 from the present. But I do not think people were just "enjoying" themselves on the discussions about the TV series Lost, while brainstorming on conspiracy theories based on time travel and Einstein's relativity theory. (Get over it, the series did not have a scientific point at all :) )

But , if time travel suddenly started being taken seriously by physicists and scientists in general, who broke this long-lasting taboo?

Gödel's Universe

In 1949, Gödel published a paper called "An example of a new type of cosmological solutions of Einstein's field equations of gravitation" to honor his best friend and Princeton neighbor Einstein. In his paper, he constructed a hypothetical universe, derived from the equations of the general theory of relativity, that admits time travel into the past; it is infinite, static (not expanding), rotating, with non-zero cosmological constant. It was an exact solution to Einstein's field equations and it had many, literally many strange properties.

I am not sure if Einstein felt quite honored though. Gödel must have been quite an annoying person for the scientists of his time.First, with his "Incompleteness theorem", he probably ruined everyone’s day in math, which basically pointed out that every system of math capable of computing addition and multiplication produces some true conclusions that the system itself can’t even prove it is true.

His new universe was also annoying for physicists, especially for Einstein. He tried without success to find an error in Gödel's physics or a missing element in relativity itself that would rule out the applicability of Gödel's results-but he failed.

The results were shocking. In Gödel's hypothetical universe, you could not only travel "anywhere" but "anywhen" also.


How is time travel possible in Gödel's universe?

Schematic view of Gödel's universe

 Above, the two vertical lines are a world line of a matter particle (which may be regarded as a galaxy). These particles are always at the same distant. However, the center vertical line is a center of rotational symmetry.  The rotation of the universe  (which Gödel introduced into his model) produces a peculiar effect on the light-cones, as is shown in the figure above. The center light-cone is up-right, but as we go farther from the center, the light-cone tilts and widens. For the universe is rotating around this center , according to special relativity, the velocity of a particle changes the hypersurface of simultaneity and also the direction of the time axis. 

In the Gödel's universe, there is a critical distance from the center; at which the light-cone becomes tangent to the plane (hypersurface) of simultaneity of the central light-cone. And beyond this critical circle,( see the circles in the first figure of this topic) the light-cone straddle the plane. This means that a light ray can go below the plane , as shown above.

The shocking implication of the mentioned phenomena is that, this makes time travel possible! Imagine that , in the above figure, your galaxy was once at point p. But now you are at q. In order to visit p again in the timeline , you first accelerate and aim at the outside of the critical circle for the light rays to be able to go below the plane. Then when you passed the critical circle, you go downward for a while, enough below the original hypersurface at p. Next, you aim at the critical circle (below p, of course), and if you passed it, you aim at p, from the past region of p. Thus you came back to your past by also travelling through your future! This makes defining the cosmological time  impossible! (This section is a summary of Hawking and Ellis,1973) 

 Time travel is possible.....theoretically

This seems all fine, but if this is a hypothetical universe, how did that help breaking the taboo of time travel in science communities? Well, because Gödel showed that you can "theoretically" make time travel possible, without having to deal with practical issues. Since then there have been lots of hypesthesia for time travel, which requires another article to explain.(along with the paradoxes they brought. See: Grandfather paradox)

Who knows? Maybe one day the humanity will be able to travel through the time. One thing that is certain to me is that, we will need more geniuses to chit chat with each other as below.

Gödel and Einstein

Graham's Number and Infinity

Ronald Graham and his wife Fan Chung

What is the biggest number that you know of? Centillion, googol,googolplex? A googol is 10¹⁰⁰  and  a googolplex is  10^Googol  , which is already an unbelievably large number. Of course you can go on forever, by taking the powers of these numbers until you reach the infinity. But what is the largest number that is "useful" for mathematics? The answer is the mysterious Graham's number. Graham's number is so ridiculously huge that  it trumps googolplex by a long shot.

"The Mathemagician"

Ronald Graham is not one of those "high IQ, low EQ" mathematicians that you are familiar with. He is the only one of its kind, and the people tend to call him " The Mathemagician". Literally... Believe it or not, he is "a highly skilled trampolinist and juggler", and the past president of the International Jugglers' Association. In fact, he was even on stage with Cirque du Soleil (recently visited Istanbul for a set of shows) and in an issue of Discover magazine about the Science of the Circus.

Ronald Graham while juggling

Recently a professor in  the Computer Science Department of UCSD(University of California at San Diego), Prof. Graham considered a problem in Ramsey theory, and gave a "large number " as an upperbound for its solution. Since then, this number, known as Graham's number, became well known as the largest number ever used in a mathematical proof. It was so large when compared with the largest numbers previously used in mathematics that, people had tough times in perceiving it.Well actually, they still do.Perhaps the most ironic thing about this number was, it was an upper bound solution to a problem to which most of people would give the trivial answer: 6. Isn't it quite hilarious that we cannot even calculate how many times Graham's number is bigger than 6 today?


Graham's Number

Graham's number is connected to the following problem in the branch of mathematics known as Ramsey Theory:

Consider an n-dimensional hypercube, and connect each pair of vertices to obtain a complete graph on 2ⁿ vertices. Then colour each of the edges of this graph either red or blue.

What is the smallest value of n for which every such colouring contains at least one single-coloured 4-vertex planar complete subgraph?

Well , this may sound quite complicated and I do not want to bore you with details, so let's proceed with what Graham's number really is.

where the number of arrows in each layer, starting at the top layer, is specified by the value of the next layer below it; that is,

So to explain better we can say that g(1)= 3^^^^3 meaning basically 3^27. For g(2), there will be g(1) number of arrors between 3s, that is to say g(2)= 3^^^^.....^^^3, where the number of ^'s is g(1) and so on until g(64) , which is the Graham's number itself.

It comes as a no surprise to say that this is an unbelievably huge number, so huge that if we could write each number of the Graham's number on every atom in the observable universe, it would not be enough.  As a result:

1.We will never learn how many digits it has.

2. We will never learn the first digit of the Graham's number

3.We will never learn if there are more 1s than 0s in Graham's number

Well, we will pretty much never learn anything.  Why am I so confident? Can't a modern computer in the future store the Graham's number? The entire number is far too big to be stored in perfect precision by any computer that has ever existed or ever will exist. How can I say "ever will exist"? Because, even written in scientific notation, i.e. with only one digit of precision, the number of digits in the exponent would exceed the number of atoms in the observable universe. The total number is easily larger than the number of Planck volumes into which the observable universe can be divided. If the whole observable universe were a computer, and every tiny quark and neutrino represented a bit of data, it could not store the entire number in absolute precision.

The only thing we can do about the Graham's number is that we are able to calculate the last digits of it by using the "modular exponentiation" technique. In fact, I implemented this technique in a Matlab code ( you can download it HERE), and I was able to calculate the last 11 digits of the Graham's number in , well, 6 hours. Either I need a better computer or I should quit writing programs so inefficient :)

It is obvious that the size of Graham's number is beyond our perception. But the ironic thing is, you, me and Ronald Graham is  exactly at the same distance with infinity: Infinity

Multiverse: The place where everything started off?

Science has come to a point in which one started questioning things more than ever. I do not know of any time in history where science had caused so many philosophical questions arise. "How did I get here?" , "Why am I here","What will happen after I die". Well , science has always been pretty good at answering "How" and "What" questions and it certainly has never liked "Why" questions and always seemed to dismiss them. Because scientifical theories are constructed to "explain" the way things happen in the best and most accurate way possible. For example, Newton's theory of gravity is good enough to explain the "macroscopic" events around us , however , it fails at the "quantum" level where "general relativity" lends a helping hand.

Big Bang is probably where science fails at a lot of questions. Frankly, "Big Bang Theory" tells  almost nothing about the "Big Bang" itself. For a start, it doesn't say "what" banged, "how" it banged and "why" it banged. It is only concerned with how the universe came to being "after" 10^-37 seconds from the  very moment Big Bang happens. From this point backwards, science & physics fail. What I mean by "failling" doesn't mean that they need improvements, they "utterly" fail. Because for years scientists accepted that time started at the very moment of Big Bang, and there was an "absolute nothing" before that. That is why all the equations etc go to infinity and blow up at the moment of Big Bang.

With the discoveries of spring theory, 11th dimension and finally M-theory, scientists now strongly believe that Big Bang is not a beginning. Actually, with the new way of thinking, it is absolutely nothing. They claim that our universe is actually not alone and is floating in the "multiverse" in the 11th dimension along with the other "infinite number" of universes. A big bang is nothing more than a collision of two universes, which, they believe ,happens very frequently. That way, they are able to explain the moment Big Bang happens, because it is not a "singularity" any more but rather it is one of the infinitesimal "instants" in the multiverse history.


This may seem like sci-fi , but as long as it is not "falsified" and it is enough to explain the history of the universe, it will definitely keep on being taken seriously. Actually there are extensive research projects now to find out any proof in our universe about the collisons with other universes . From a sci-fi point of view, it raises a lot of questions:

1. Is it possible to travel between different universes? 
2.If there are infinite number of universes in the multiverse, are there infinite number of civilizations since the beginning of the multiverse( if that makes any sense)?
3.How advanced are they(other civilizations) in technology? Why could they not find the way to travel between universes? Is it impossible?
4. Can our universe collapse in a matter of seconds due to a  severe collision with another universe? (As some scientists argue that our universe is actually colliding with others even now)
5.What caused the multiverse to emerge? How did the first universe came to being?


It might quite be true that we are only some "things" with  some complex electrical signals going through our nervous systems and being interpret by our brains, living in a "planet" located in a universe,which is merely one of the infinite number of universes floating in the multiverse, whose size our brains cannot perceive. Humiliating for the human beings? I think so. Crazy? Well, like Bohr said, is it crazy enough to be true?

Mehmet Kurt