SIZE DOESN'T MATTER – OR DOES IT ? - THINKING LONGER AND HARDER

(- Copyright © 2016 by Mike Stewart -)

Albert Einstein was upset about the quantum concept of entanglement or as he called it “spooky action at a distance.”.

If you really think about it, this entanglement or "spooky connection" might also mean that we have a fundamental lack of knowledge and understanding of time and distance.

I remember a Simpson's episode that began, like most, with Homer and his family sitting on their couch. We had a close up of Homer's head. We then moved up and away, seeing first Homer's house, then clouds, then the entire earth. This moving away process continued until we saw our solar system, our entire galaxy, and then all the galaxies in the universe. As we continued to expand our view, structures appeared that looked like atoms, then we saw molecules and DNA. Finally, we could make out living cells and then we ended our journey back where we had started, looking closely at Homer's head.

The Simpson writers were trying to give us a unique perspective on reality. They may have been closer to the truth than they realized. I wonder if they recognized what they were saying about the nature of size.

If you really think about it, this entanglement or "spooky connection" might also imply we have a fundamental lack of knowledge and understanding of time and distance.

Not understanding what distance means is the same as saying we don't understand what size means. To illustrate, if we walk for one hour at 3 miles per hour, we will travel three miles. If we then turn right 90 degrees and travel for another hour at the same speed, turn right 90 degrees again, continue walking at the same speed for another hour, and then, finally, turn and, at the same speed, return to our starting point, we would have walked for four hours and traveled 12 miles to return to the same place. If we traced our steps, we would have laid out a square, each side being 3 miles long. The area of this square would be nine square miles (3 miles X 3 miles = 9 square miles).

Now, if we board a helicopter and slowly, at a rate of 3 miles per hour, rise vertically for one hour, we would arrive at a point three miles above our starting point. If, traveling at the same speed, the helicopter now moves horizontally and follows our earlier walking journey, it will, four hours later, arrive back at the point three miles above our starting point. Again, a square with an area of nine square miles would have been laid out, this one identical to the first, but separated by three miles. In fact, if we connect each point on this square where we changed directions to the corresponding point on the ground with three mile long lines, we will have constructed a cube. The volume of this cube would be twenty seven cubic miles (3 miles X 3 miles X 3 miles = 27 cubic miles). In this case, size and volume are the same thing. The size of the cube is 27 cubic miles.

What can you say about the size of this cube? To us, it seems pretty big. To one of the living cells we have been discussing, this cube appears to be almost infinitely large. On the other hand, compared to the known universe, this cube is infinitesimally small.

Can we say any more about this cube? Maybe Albert Einstein, who hated "spooky action at a distance", should have thought a little longer and harder about one of his thought experiments.

When I was trying to understand relativity, I read about what is called the "Twin Paradox" which was an Einstein thought experiment that involved one person traveling in a rocket traveling at a high rate of speed (almost the speed of light). His twin, who remained on earth, would, according to Einstein, age faster than the space traveler. I think, after some effort, I understand what Einstein was saying. If I do understand correctly (quite honestly, it makes my head swim to think about it), I can modify this space journey to address our current subject, size.

If you would like to google "Twin Paradox" you can try to figure out what is being said, but I would like to set up another space journey to make a comment on size. Imagine a rocket leaves earth traveling to a planet circling one of the three stars in the star system, Alpha Centauri. This planet is four light years away.

On this far away planet, we have an older observer who spent half his life traveling to this distant point. He watches carefully through a powerful telescope as our large rocket, piloted by a brave astronaut, blasts off from earth. A second observer, who will stay on earth, also watches the proceedings.

As we have said before, time seems to be important. The astronaut and the two observers are habitual clock watchers and each demands his own personal, extremely accurate clock. Before the astronaut or the first observer leave earth, all three clocks display identical times.

Some of our fellow earthlings have previously placed twelve, three mile long, straight steel pipes in earth orbit. Using the cube we traced out above as a model, they have constructed a real, orbiting cube that has a volume (size) of twenty seven cubic miles. Our astronaut, using several cables, attaches the cube to the rocket, and then heads for Alpha Centauri.

Our rocket is capable of accelerating continuously so that the astronaut will feel as if he weighs the same as he would on earth (the astronaut would feel he was being pulled toward the earth from which he was rapidly retreating).

If you asked the earth observer, two years later, about the rocket, he would say it is moving away from him at almost the speed of light. The rocket, the astronaut , and the cube all display a "red shift" which you remember from Hubble's observations means the rocket is rapidly receding. The astronaut's clock is running slow. The four steel pipes of the cube that are pointed toward earth are compressed, with a length of perhaps 1.5 miles rather than the 3.0 miles you would expect. If you calculate the size of the cube, it has shrunk from the original 27 cubic miles to 3 X 3 X 1.5 = 13.5 cubic miles.

Now let us turn to the second, remote observer. Since when I start thinking about the "Twin Paradox" I start getting confused, I will not ask what the second observer sees "at the same time". Instead I will note that the earth observer is looking at the rocket when it is half way to Alpha Centauri and ask what does the second observer see when he and the observer on earth believe the rocket, cube, and astronaut are at the half way point of the journey?

The remote observer would see that the rocket, astronaut , and cube all are displaying a "blue shift" which means these objects are approaching rapidly. The astronaut's clock is running fast. The four steel pipes of the cube that are pointed toward Alpha Centauri are expanded, with a length of perhaps 4.5 miles rather than the 3.0 miles you would expect. If you calculate the size of the cube, it has grown from the original 27 cubic miles to 3 X 3 X 4.5 = 40.5 cubic miles.

Let us stop at this point and make a few observations. To the astronaut, time is passing normally, and the cube, the rocket, and he, himself, are all unchanged. To the observer on earth, the astronaut's clock is running slow, and everything observed is slightly smaller. To the remote observer, the astronaut's clock is running fast, and everything observed is slightly larger.

If the astronaut makes the cube rotate (the connections between the cables and the cube could be modified so that the cables wouldn't get tangled up), the size of the cube will change as its orientation to the remote observers change. To the observer on earth, the cube would rotate slowly and its size would change slowly. To the other observer, it would rotate rapidly and there would be a rapid, periodic, size change.

What happens if the astronaut, at this point, shuts down the rocket engine? The astronaut, rocket, and cube will be suddenly weightless. All three would stop accelerating, but their velocity and speed do not change. The speed, from the standpoint of the remote observers, is approaching the speed of light. The red shift or blue shift observed would not be affected. I may be wrong, but I suspect that any size changes observed by the remote observers would also not go away.

There is one thing that might be useful to realize. On the other hand, it may, if you are like me, just contribute to a feeling that you are not sure you understand relativity. Both remote observers will not know anything about the astronaut (like he shut down the rocket engine) until light reaches them. The astronaut is at a point where he has traveled one light year (the distance light travels in a year) and has three light years left to travel. The earth observer will not know that the rocket engine is off until a year has past. The Alpha Centauri observer will have to wait three years. And don't forget, the two observers did not start counting at the same time - there is no same time.

If the astronaut does not shut down the engine, the rocket will be traveling at nearly the speed of light when it reaches the remote observer. In fact, we could get the rocket to come as close as we would like to light speed by assuming the rocket had had longer to accelerate. Instead of starting from earth, the rocket could have started its journey on the other side of the galaxy.

In any case, you could imagine the cube being the size of a galaxy as it passes the remote observer. It might be that only one atom from the astronaut's left thumb would pass close enough to be detected. Yet, as soon as the rocket passes, the whole configuration (cube, rocket, astronaut) becomes microscopic.

There is one major problem with the mental exercise I have presented. There may, in fact, be other problems. With relativity, as with other things, the devil is in the details. Nevertheless, I hope that much of what I say is valid and has value.

The problem I recognize is we have been thinking in three dimensions. As the rocket approaches the remote observer, two of the three dimensions of the cube remain a constant three miles. The third dimension has expanded, and with it, the volume or size of the cube. The cube has grown, but in only one direction. It has been transformed into a very, very long rectangular prism.

Let us revisit our astronaut, observers, cube, and rocket and think about String Theory.

I have said that my biggest problem with particle physics is that the mathematics involved is so advanced that few people can understand the equations that support the conclusions the experts have reached. The intelligent layman has to, unfortunately, rely on unsupported faith when he listens to these experts.

String Theory math makes Particle Physics math look like third grade arithmetic. String Theory mathematics is useless to the vast majority of people who would like to learn about this subject. I will, therefore, only use the inaccurate mental images and analogies that are commonly used to discuss String Theory. Maybe then, while working on this foundation of sand, I can ask a question or two that is intelligent, useful, and relevant.

Even the definition of String Theory brings up more questions than it answers. From a Google Snippet: "string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. It describes how these strings propagate through space and interact with each other.".

What is a one-dimensional object? Is it something with length, but no height or width? Is it a particle of time, like Einstein considered in his four dimensional space time world? Or, is it a small particle of time, or timelet, like discussed in the intelligent ant story? Is it like a scalar value having magnitude, but no direction (for example, to an outside observer, the astronaut may have a certain absolute speed. His velocity can include this speed, but must also specify a direction - in this case, toward Alpha Centauri)? Or, is it like something else?

I wanted to discuss String Theory because it is concerned with, among many other things, how size might be affected by velocity. Before we get to this, however, let us try to understand some of these many other things.

A one-dimensional string can be viewed as a real life guitar string under tension. When plucked, the string will vibrate and make a sound. Increasing the tension or length of the string will change the sound. String Theory says that we see a string as an elementary particle. If you change the sound, you get a different elementary particle. For example, an electron is a string vibrating one way, a photon is a string vibrating another way. We could also say that a particle is an excitation of a string.

Quantum Mechanics or Particle Physics is concerned with the interaction of sub-atomic particles with each other. It accurately describes the observed behaviors and properties of these particles.

Particle Physics, however, has logical problems. Buried in the supporting equations are the assertions that the interacting particles are point-like objects and the distance between the objects is zero (you could say the objects occupy the same space). For Particle Physics to work, you have to assume that gravity does not exist. To understand why, consider that the force of attraction (gravity) between two objects increases when the their total mass increases or when the distance between them decreases. If the distance between the two objects becomes zero, the force of attraction becomes infinitely large. This is impossible, but Particle Physics still accurately describes the behaviors and properties of sub-atomic particles.

Note that we were discussing distance which implies size which is what this chapter is about.

Experts, including scientists, often change their terminology to sound more hip or modern. Information technology (IT) is defined by Google as the study or use of systems (especially computers and telecommunications) for storing, retrieving, and sending information. With the rise of the internet, experts in the field wanted to sound more modern so they invented the term Information and Communications Technology (ICT) and made IT a subset of this "new" discipline. This modernization of words, without fundamental change in meaning, continues on a more detailed level. Years ago, we had computer programmers writing and testing computer programs. Today, we have application developers creating apps. When you consider that an app runs on a smart phone and a smart phone is a computer, you realize "app" is just another word for "computer program".

Scientists, as well as their closely allied Science Writers, are influenced by the desire to sound modern. Scientists, however, may also be greatly affected, in a real sense, by another phenomenon, new theories. Other scientists may occasionally develop a new theory that they believe is a better way to describe reality.

A new theory can make a scientist's life work irrelevant. Funding for new research, which in the modern world seems to be a lot of "thinking longer and harder", can dry up overnight. Science writers no longer pester the now irrelevant scientist for interviews. They quickly turn their attention to the new stars, the expert developers of the new theory.

Has String Theory suffered such a fate - Is it now irrelevant? Bryan Green, a well known physicist, in a recent article, asks "... Is String Theory revealing deep laws? Or, as some detractors have claimed, is it a mathematical mirage that has sidetracked a generation of physicists?".

I don't know the answer, but unless some newer theory has drastically simpler mathematics associated with it, I'd like to postpone discussing this until we've looked more closely at String Theory.

String Theory says that when we observe a specific elementary or sub-atomic particle (for example, a photon), we are really seeing a string vibrating in a particular way. String Theorists, therefore, want to determine how many different ways a string can vibrate, and to show that for each way, there is a corresponding sub-atomic particle. The things that make particles unique are their properties, for example, an electron has different properties than an photon. Looking at it another way, String Theorists seek to relate how strings vibrate to the observed properties of different elementary particles. Observations show that particles can be divided into two groups, fermions and bosons. Fermions are the particles that make up matter. Fermions cannot occupy the same space at the same time. Bosons are the particles that transmit force. Most bosons can occupy the same space at the same time.

One of many strange things about strings, based on the mathematics that support and define string theory, is that they do not vibrate just in the normal dimensions we are use to, namely, length, height, width, and time; or what Albert Einstein called space-time. If strings vibrated only in this space-time, we could visualize them as small strands of spaghetti slowly twisting and turning as time passed. Now is a good time to note that strings can be either closed or open (the ends of the spaghetti strand are connected or not). Either way, we can visualize them easily in "normal" space-time.

Unfortunately, string theory mathematics, which normal people have to take on faith, dictates that there must be, not the four dimensions of space-time, but either ten dimensions or twenty six dimensions.

Ten dimensions or twenty six dimensions naturally brings up a lot of questions.

Many of these questions occurred to string theory experts, who I must say, have made a profession out of proposing "out of the box", somewhat outlandish, questions and then developing answers which also might seem absurd. You could say that string theorists, like me, are trying to find unique thoughts. They are the experts. I am the amateur.

An amateur like me who only has an undergraduate degree in mathematics and who readily admits long ago forgetting most advanced mathematical knowledge has only one advantage over these experts. They tend to ask questions and propose very strange answers to explain the mathematical models they have constructed. I, on the other hand, can ask any crazy question and propose any silly answer. I can then ask an expert "somewhere" to modify or create a math model that supports me.

String Theory tells us that we can have multiple dimensions if most of these dimensions are too small to be seen. I need to explain the word "seen" in the last sentence. I feel like I can see three of the four dimensions of space-time, namely, length, width, and height. Time, on the other hand, is experienced rather than seen. String Theory is saying that if we have ten dimensions, six are too small for us to realize they exist. If we have twenty six dimensions, twenty two are too small. String Theory also tells us that these small dimensions are curled up and intertwined together and each has its own shape.

How small are these dimensions? Our most powerful accelerators can detect many sub-atomic particles. Strings are small enough to float through and vibrate in not just space-time, but also in these small dimensions. The shape of each dimension affects how the string vibrates. We can view a string as floating in 10 or 26 dimensions and its vibration defines a subatomic particle. These dimensions are not much larger than strings and strings are a million billion times smaller than the realm explored by the above mentioned accelerators.

String Theory is sometime broken down into multiple theories. A feature of one theory is that there are twenty six dimensions. There are five other theories that feature ten dimensions. As time has passed, however, we should note that String Theorists have tended to both create more string theories and decrease the number of string theories. When String Theorists looked at (mathematically) how to make unseen dimensions very small so that they get the everyday four dimensional world we see, they discover that they can do this with many different mathematical models. Each new model can become a new string theory. On the other hand, Theorist may sometimes discover that two mathematical structures supporting two theories are actually equivalent. In this case, they realize they don't have two theories, but only one that they had been looking at in two different ways. Looking more closely, however, at these six theories is a good starting point.

The first of these six is called bosonic and sports twenty six dimensions. To a layman, this theory seems really stupid. Or, since I have said all of quantum mechanics is gibberish, maybe I should just say this theory seems to be more gibberish-like than the others. I am critical because this theory starts with the assumption that matter does not exist.

To be fair, the mathematical model that supports the theory could be useful in creating better, hybrid string theories. Under the bosonic model, only bosons exist. There are no fermions. This means there can be forces, but no matter. To my simple mind, it doesn't seem you could have a force unless it had some matter to act on.

Probably the most famous boson is the Higgs Boson, the so called "God Particle", recently found by the Large Hadron Collider. Bosons have integer (values of 0, 1, 2, 3, etc.) spin. Spin is a quantum mechanical term that is very important in the mathematics behind the bosonic theory (as well as in much of other math involved with quantum physics). Spin is often defined using analogies to the world we see around us, but these don't really apply and underlines that we don't know what spin is. The best explanation I have seen was by Jon Butterworth in an online blog hosted by The Guardian, a British Newspaper.

Jon Butterworth is a physics professor at University College London. He is a member of the UCL High Energy Physics group and works on the Atlas experiment at Cern's Large Hadron Collider. His book Smashing Physics: The Inside Story of the Hunt for the Higgs was published in May 2014.

To quote Jon Butterworth:

"Bosons have, by definition, integer spin. The Higgs has zero, the gluon, photon, W and Z all have one, and the graviton is postulated to have two units of spin. Quarks, electrons and neutrinos are fermions, and all have a half unit of spin.

This causes a huge difference in their behaviour.

The best way we have of understanding fundamental particles is quantum field theory. In quantum field theory a "state" is a configuration describing all the particles in a system (say a hydrogen atom). The maths is such that if you swap over two identical fermions with identical energies (say, two electrons) then you introduce a negative sign in the state. If you swap two bosons, there is no negative sign.

Since swapping two identical particles of the same energy makes no physical difference to the overall state, you have to add up the two different cases (swapped and unswapped) when calculating the actual real probability of a physical state occurring. Adding the plus and the minus in the fermion case gives zero, but in the boson case they really do add up. This means any state containing two identical fermions of the same energy has zero probability of occurring*. Whereas a state with two identical bosons of the same energy has an enhanced probability.

This fairly simple bit of maths is responsible for the periodic table and the behaviour of all the elements.

** This is known as the Pauli exclusion principle”.*

Now that we understand spin, there are a couple of other noteworthy points to mention regarding bosonic theory. The theory allows both open and closed strings (loops). The theory also, in what some call its greatest flaw, postulates a new particle. I would prefer, in discussing this particle, to quote Spock of Star Trek and say "fascinating".

Bosonic String Theory says that there exists a particle that we have named the tachyon. A tachyon has two unique properties. It has imaginary mass and it can travel faster than the speed of light.

Reluctantly, in order to even hint at what this means, I need to bring in a little math. You may or may not remember from the long ago days of school what a square root is. To me, the definition is kind of verbally awkward, but giving several examples should make the meaning clear.

Consider the following seven multiplications: (1) 0 X 0 = 0; (2) 1 X 1 = 1; (3) -1 X -1 = 1; (4) 2 X 2 = 4; (5) -2 X -2 = 4; (6) 3 X 3 = 9; and (7) -3 X -3 = 9.

Now what is a square root of a number? For convenience, I am going to replace the phrase "square root of", not with its mathematical symbol, but with "SQROOTof". In the first case above, the SQROOTof zero is zero. In the other cases, the SQROOTof a number has two answers, a positive value and a negative value. The SQROOTof four is both +2 and -2. The SQROOTof nine is both +3 and -3.

When you consider these examples or read a more formal definition of square roots, you realize there is no such thing as the square root of a negative number. The SQROOTof -1 does not exist. The SQROOTof any negative number does not exist, they are completely imaginary. They are called Imaginary Numbers.

Scientists, especially Quantum Physicists, seek to create sophisticated mathematical models to describe and support their views of reality. Mathematicians want to build similar models just to prove they can. You would think that both would have no use for Imaginary Numbers. You would be wrong.

Speaking of being wrong, please bear with me while I introduce a little more simple math which I hope I haven't made so simple that I am wrong. I want to do this in the context of our traveling astronaut and eventually show a relationship to imaginary numbers.

Before our astronaut blasts off from earth, we will assume he is carrying with him a one pound ball. He is an American astronaut so he doesn't feel compelled to carry a heavier one kilogram ball. On earth this ball weighs one pound and also has a mass of one pound. The important thing to know, however, is the weight of the ball can change. On the moon, with its lower gravity, the ball would weigh about a sixth of a pound. The mass would still be one pound and does not change (except for the effects of relativity). From now on, we will talk about mass rather than weight.

When the rocket configuration (astronaut, rocket, cube, and now ball) is moving away from the earth, the earth bound observer may notice that the mass of the ball has increased. When we ask how much the mass of the ball has increased (I want to ask does it now "weigh" 2 pounds, 10 pounds, or 5000 pounds, but sensitive scientists would get upset).

Another way of asking this question is to ask what is the mass of the ball compared to what the mass would be if the ball (or a duplicate ball) had remained on earth (which we know is one pound)? Or better still, what is the ratio of the mass of the ball on the rocket to the mass of the ball on earth? And this is where our SQROOTof comes in.

In the early twentieth century, a dutch physicist named Hendrik Lorentz figured out how to find the ratio of the mass of the ball on the rocket to the mass of the ball on earth. Hendrik developed a lot of complicated math to address other related subjects, but one relatively simple equation answered this question about masses. This equation states that the MASSonrocket divided by MASSonearth (their ratio) is equal to the Lorentz Transformation Factor - which we will call the LTR.

Now here is where the SQROOTof comes in. The LTR is equal to the number one divided by the SQROOTof the number one minus another number. This other number is the ratio of the velocity of the rocket multiplied by itself to the speed of light multiplied by itself.

I have written this description in English in hopes of being clear, but we may have reached the point where something more equation-like is better. Using the letter V for velocity and the letter C for the speed of light ("C" is the first letter of celeritas, the Latin word meaning speed and is typically used to denote the speed of light), we can write the following:

MASSonrocket / MASSonearth = LTR = 1 / SQROOTof ( 1 - ( V times V ) / ( C times C ) ).

This equation can also be written as:

MASSonrocket = MASSonearth times LTR.

With this, if we know the velocity of the rocket and the mass of the ball on earth, we can calculate the mass of the ball on the rocket.

If we examine more closely the SQROOTof part of LTR, that is, the denominator, we can gain a lot of insight.

The velocities we see in everyday life (the V in the SQROOTof) are very small compared to the speed of light. A satellite circling the earth may be traveling at five miles per second. If you multiple this by itself, you get twenty five. On the other hand, the speed of light is 186,282 miles per second. If you multiply this by itself, you get 34,700,983,524.

You can use these numbers to see how much the mass of our one pound ball would increase if it was on a satellite, traveling a five miles per second:

MASSonsatellite = MASSonearth times 1 / SQROOTof ( 1 - ( V times V ) / ( C times C ) ).

MASSonsatellite = 1.0 pounds times 1 / SQROOTof ( 1 - ( 25 ) / ( 34,700,983,524 ) ).

MASSonsatellite = 1.0 pounds times 1 / SQROOTof ( 1 - .00000000007204068).

MASSonsatellite = 1.0 pounds times 1 / SQROOTof ( 0.99999999992 ).

MASSonsatellite = 1.0 pounds times 1 / 0.99999999996.

MASSonsatellite = 1.0 pounds times 1.00000000004 = 1.00000000004 pounds.

We can see from these calculations that the mass of the ball is increasing more and more rapidly as the velocity approaches the speed of light. The equation, in fact, predicts that if its velocity could reach light speed, the ball would have infinite mass. Another way of saying this is that it would require infinite energy to push the ball to light speed. This is why conventional wisdom says that the ball can never travel at the speed of light.

And you know how I hate conventional wisdom.

The LTR equation invites additional conjectures and Einstein-like thought experiments. I can't resist this temptation before returning to String Theory. We are at a point, however, where we can say what Bosonic String Theory means by imaginary mass.

Bosonic String Theory postulates that particles exist called tachyons. They can travel faster than light. They cannot travel at the speed of light or less than the speed of light. Suppose our rocket configuration, which includes our one pound ball, is made up of tachyons, traveling faster than light speed.

If we look at part of the LTR equation, namely, ( V times V ) / ( C times C ) where V is greater than C, we realize that the numerator (V time V) is greater than the denominator (C times C).

When you divide a larger number by a smaller number, you get a number greater than one: ((V times V) / (C times C)) equals a number greater than one.

At this point, we should observe that when we try to determine the mass of the ball on the rocket, this is the third of three distinct situations. These three are:

(1) A "normal universe" where light speed is constant for all observers. Note that this universe would not have been considered normal a couple of centuries ago:

LTR = 1 / SQROOTof ( 1 - ( a number less than one ).

LTR = 1 / SQROOTof (a positive number between one and zero). In this case, since MASSonrocket = MASSonearth times LTR, the mass of the ball on the rocket can vary from one pound to a very large, but still imaginable, value.

(2) One brief shining moment where there is a universe that supposedly cannot exist:

LTR = 1 / SQROOTof ( 1 - ( the number one ) = 1 / SQROOTof ( Zero ).

LTR = 1 / 0 = INFINITY (whatever that means). In this case, the ball on the rocket would have infinite mass.

(3) A very strange universe that is hinted at by Bosonic String Theory:

LTR = 1 / SQROOTof ( 1 - ( a number greater than one )).

LTR = 1 / SQROOTof ( a negative number )) - and, of course, we remember that the SQROOTof a negative number is an IMAGINARY NUMBER! And so we have

LTR = 1 / (an imaginary number).

Before we continue, I would like to do a bit of rationalizing. People like to believe that they are very rational creatures (to be more exact, I want to believe that I am rational, you may be, or you may not be, rational). The truth may be that we are creatures that are very good at rationalizing - we find plenty of proof to support the view we want to hold, while we ignore, or even angrily reject, information that contradicts us. This does not mean that rationalizing never leads to valid conclusions (is this a rationalization?).

Now I would like to discuss infinity and imaginary numbers. Both of these concepts are strange to our way of thinking, but they are very important to lots of scientists and mathematicians. They are always using them in their icky math equations.

When you look at these equations, you will find that the math guys write infinity as the number 8 turned on its side and our SQROOTof a negative one as the lower case "i". Mathematicians have written rules to manipulate these concepts. I learned these rules in college so I could solve equations and get my math degree.

Since I have forgotten a lot of these rules and don't want to spend the time needed to relearn them, I am going to ignore them - but I am going to say why any assertion I am making is still true (as you will see, in some situations, I believe the rules themselves are wrong).

Looking first at Infinity.

Normalization is a process or rule used to remove infinities from equations and get meaningful results. When a physicist does this, he will say the equation has been renormalised. Before I explain what this is and why I doubt that it is valid, let me quote from a Google snippet for Renormalization, apparently a similar process, that is for a specific field (quantum mechanics) - remember that I said that scientists are verbose.

Renormalization - In quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, renormalization is a collection of techniques used to treat infinities arising in calculated quantities by altering values of quantities to compensate for effects of their self-interactions.

As is usually the case, the ordinary person has to have faith in the validity of these undefined "collections of techniques", so let us take a closer look at what it takes to get to just a simple renormalised equation.

I read somewhere this example:

In physics, Renormalization is a great tool, or cheat, to get rid of infinities in equations and give meaningful results:

Unrenormalized Equation - (infinity +3) x (infinity +3) = infinity

Renormalized Equation - (infinity +3) x (infinity +3) = 9

This renormalization process is really a combination of the normal math we are use to and "something else". In normal math, we could have the equation (4 + 1) X (4 +1) = 25 which is so self evident that we can skip the intermediate step. With this step included, we have (4 + 1) X (4 + 1) = (4 X 4) + (4 X 1) + (4 X 1) + (1 X 1) = 16 + 4 + 4 +1 = 25.

If we follow the same process with our infinity equation, we get:

(infinity +3) x (infinity +3) = (infinity X infinity) + (3 X infinity) + (3 X infinity) + ( 3 X 3) = (infinity X infinity) + (6 X infinity) + 9 = infinity

If you continued with normal math, you could get:

(infinity X infinity) + (5 X infinity) + 9 = 0 or

minus (infinity times infinity) minus (five times infinity) = nine

To tell you the truth, I don't know what this means. Apparently, physicists and mathematicians don't either so they invented normalization to get rid of infinities when they didn't want them. Normalization may be what I call a "weird rule" that Reality may sometime follow, but, maybe, will more often ignore. Anyway, rather than figure out how to use a "weird rule", I had just as soon invent my own.

Before I leave infinities, let me mention the simplest weird rule.

infinity + 1 = infinity. The non-weird version of this rule would be "you cannot add one to infinity". I seem to remember that once I was told that you couldn't divide one by zero, but later I learned that one divided by zero was (the or a) definition of infinity. Saying you cannot add one to infinity makes some sense - infinity is so big there are no ones left to be found to add.

infinity + 1 = infinity says that some infinities are bigger than others. Renormalization treats all infinities the same. Mathematicians seem to want to put infinities into a weird, Orwellian Animal Farm universe where all infinities are equal, just some are more equal than others.

It is interesting that for infinity + 1 = infinity,

we can subtract infinity from both sides of the equation giving infinity - infinity + 1 = infinity - infinity or 0 + 1 = 0.

We have just proved that one equals zero. Now on to the Imaginary.

As I said earlier, Hendrik Lorentz figured out an equation that could be used to calculate the ratio of the mass of our one pound ball on a rocket to the mass of the same ball if it had remained on earth. This equation is:

MASSonrocket / MASSonearth = 1 / SQROOTof ( 1 - ( V times V ) / ( C times C ) ).

Or, as Lorentz said, LTR = MASSonrocket / MASSonearth.

When V (velocity) exceeds the speed of light, SQROOTof becomes, as we have said before, an imaginary number. And it is at this point that I want to rebel against relearning the rules that typically govern calculations involving imaginary numbers.

One of these rules is that if we multiply the imaginary number "i", which is our SQROOTof negative one, by itself, the results is minus one ( i squared = -1 ). I ask where have they (math or science people) proven this "fact" relates to reality.

i times i = i squared = -1.

(i times i times i) or (i squared times i) or (i cubed) = - i.

(i times i times i times i) or (i raised to the fourth power) = i.

Other rules assert that you can break up imaginary numbers into real number parts plus imaginary parts. Mathematicians can then use regular math on the real numbers.

I know this is hard to follow so let me give a couple of examples where these rules might be applied, one where V is slightly greater than light speed and the other where V is almost twice light speed.

For slightly greater:

MASSonrocket / MASSonearth = 1 / SQROOTof ( 1 - ( V times V ) / ( C times C ) ).

LTR = 1 / SQROOTof ( 1 - (a number slightly greater than 1) ).

LTR = 1 / SQROOTof ( a very small negative number ).

When V almost reaches twice light speed, ( V times V ) / ( C times C ) becomes almost ( 2C times 2C ) / (C times C) = ( 4C squared) / (C squared) = 4 and:

MASSonrocket / MASSonearth = 1 / SQROOTof ( 1 - 4 ) and

LTR = 1 / SQROOTof ( almost negative 3 ).

In both these cases (small negative number, almost negative 3), the math rules say LTR has a real number component and an imaginary number component. I do not believe reality follows these rules. All we can say is that both LTRs are imaginary. Are they equal? Maybe yes, maybe no. It may be analagous to asking if you add one to infinity, is the new infinity equal to the old infinity.

Remember the alternate universe of Schrodinger's cute kitten and old cat? In this story, alternate universes could be easily created. Maybe imaginary numbers in our equations are implying there are also many alternate realities. To see how, let us embark on more thought experiments.

If we think about our rocket configuration which includes a one pound ball, we can calculate the mass of the ball when the velocity is less than the speed of light. If the velocity reaches 99.99 percent of the speed of light, the mass of the ball will be a little over seventy pounds. LTR is still a real number and we could conclude that the ball still exists in what we call our reality.

Light in a vacuum travels at (for this discussion, we will say) exactly 186,282 miles per second. If the rocket / ball was traveling one mile per second slower, or 186,281 miles per second, the observer on earth would say light is outrunning the ball at a rate of one mile (or 5,280 feet) per second. The mass of the ball has increased significantly since it left earth, but can still be calculated and the ball is still in our reality.

We can say the same thing about the ball if its speed increases again to (186,281 miles plus 5,279 feet) per second or even to (186,281 miles plus 5,279 feet plus 11 inches) per second. In the latter case, the observer on earth would say light is moving one inch per second faster that the ball and LTR is a very large, but real, number, which is still equal to the mass of the ball. The ball is still in our reality.

We can continue this process of imagining the ball is getting closer and closer to the speed of light. We, in this case, might say light is moving one inch per second faster than the ball, then light is moving one-tenth of an inch per second faster than the ball, then light is moving one one hundredth of an inch per second faster than the ball, and so on. We cannot, however, continue this process forever.

To understand why, consider the following: One Angstrom is defined as 10 billion meters which means that one inch is equal to 254 million angstroms. When we say, as we did above, that light is moving one inch per second faster than the ball and light is traveling at, well, the speed of light, we would think we could ignore the difference. When we say the same thing, but express it as light is moving 254 million angstroms per second faster than the ball, we may want to take a closer look. If we do, we will find that a lot happens when we try to increase the speed of the ball a mere one inch per second.

Until now, we haven't cared what the ball on the rocket is made of. We were just concerned that when it was on earth, it had a mass equal to one pound. Now we realize that it is a steel ball made up primarily of an alloy of iron and carbon atoms. Our scientist friends could probably tell us about how many atoms are in the ball, but I just want to say to there are less than an infinite number of atoms in the ball - but, to our way of thinking, not much less.

We can, of course, continue the process of increasing the speed of the rocket ball so that it is only one inch or 254 million angstroms per second slower than light, then 127 million angstroms slower, then 64 million angstroms slower, and so on. At some point, maybe when we are down to a few thousand or a few hundred angstroms, interesting things begin to happen.

At some point in the latter stages of this process, when the ball is almost traveling at light speed, we need to stop thinking of the ball as a ball and start thinking of it as a vast collection of atoms, or, since we are looking at String Theory, vibrating strings.

Let us take atoms first, looking at an individual atom speeding away from the same atom in the ball if it had stayed on earth. When we look at an atom on earth, it will display quantum properties. Heisenberg's Uncertainty Principle asserts, more or less, that you can never know exactly where an atom or its components are. You can only assign a probability to the position of the atom's nucleus or, for example, one of its electrons.

We have discussed an unbound electron (not part of an atom) being fired at a steel pole. Heisenberg said we could never be sure the electron would hit the pole - it might miss by various amounts. In fact, there was an extremely remote possibility that you might miss by a mile, or by a hundred million miles. I am not sure if this is true for a bound electron, but it would be very difficult to say the electron was closer to or further away to the observer on earth than the associated neutron.

Now what happens if we put the ball on the rocket and it eventually reaches a speed a few angstroms per second short of light speed?

By the way, I am pretty sure we cannot measure the speed of light in angstroms. Calculations show it is equal to 2.99792458 quintillion angstroms per second, which can also be written as 2,997,924,580,000,000,000 angstroms per second. The "true" speed of light in a vacuum could be, for example, 2,997,924,580,123,456,789.123456789 angstroms per second.

According to quantum mechanics, the ball and by extension a particular atom in the ball doesn't exist until an observer looks at it. If the atom is moving close enough to the speed of light, the observer on earth may decide that the atom is a few angstroms closer or a few angstroms further away. In the first case, the observer will note that in the last second the atom has traveled a few angstroms per second less than light speed and the atom will still be in our reality. On the other hand, the distance the atom has traveled in the last second will be greater than light speed. To the observer on earth, the atom would have evaporated. It would have entered a new reality. This would be analogous to, or even an explanation of, quantum tunneling (refers to the quantum mechanical phenomenon where a particle tunnels through a barrier (in this case, the speed of light) that it classically could not surmount).

As the ball came closer and closer to light speed, the probability that atoms would land "on the other side" would increase until the ball faded from our reality.

We might get another effect if we looked at the situation from the standpoint of an electron. As we get closer and closer to light speed, the probability increases that an electron that is furthest from its neutron will cross over into the land of Oz. The atom left in our reality would display a positive electrical charge. Again, as above, as speed increased, more and more atoms would lose an electron, and the ball would become more and more highly charged.

Wikipedia states that the conservation of energy, a bedrock of our physics, means that the total energy of an isolated system remains constant—it is said to be conserved over time. Energy can neither be created nor destroyed; rather, it transforms from one form to another. For instance, chemical energy can be converted to kinetic energy in the explosion of a stick of dynamite.

It seems likely to me that electrons dropping out of existence might violate the law of the conservation of energy.

Now is a good time to remind the reader we are still talking about Bosonic String Theory under which our ball that is made up of fermions could not exist. I don't know why this is true, either because I don't have the time to research it, or more likely, the mathematical model that supports this "fact" is beyond my comprehension - so I am turning to faith. Nevertheless, my arguments about disappearing atoms or electrons could apply under other theories.

Our next question should be "What is happening on the String level when our subatomic particles are dropping out of existence?". After all, the vibration of strings define these particles. One obvious problem, however, crops up when we try to do this. When we look at light in a classical sense, we think it takes a few quintillionths of a second to cross atomic distances. The scale we are dealing with when discussing strings is a million billion times smaller and thus light must cross string related distances a million billion times faster.

I want to use the phrase "hard to imagine" in the next paragraph, but first I want to say what I mean. If I asked you to describe something imaginary, you might describe a ghost, looking like a sheet floating through the air. This is not what I mean. The ghost is imaginary, but you have described it using common terms like floating, air, and sheet. By "hard to describe", I mean you can't imagine what it is you are talking about or how to describe it.

To the human mind, which can't really even visualize a billion, considering a quintillion seems hopeless. Now we are talking about a million billion quintillion. In fact, I don't even know or care if a million billion quintillion is equal to a quintillion million billion. Such big number make even visualizing what form a thought experiment should take hard to imagine.

But let's try anyway.

In our reality, when we are discussing subatomic particles, whether they are stationary or moving away from us at nearly the speed of light, we calculate their mass using the familiar Lorentz equation:

MASSonrocket/MASSonearth = LTR = 1/SQROOTof (1-(V times V)/(C times C)).

In this equation, we can ask what is the velocity V, and for that matter, what is the velocity C. The simple answers are V is the distance the particle travels in one second while C is the distance light travels in one second. For the significant part of the above equation, namely:

(V times V) / ( C times C ) we have

((the distance the particle travels in one second) times (the distance the particle travels in one second)) divided by ((the distance light travels in one second) times (the distance light travels in one second)).

As long as the distance that the particle travels is less than the distance light travels, LTR may be arbitrarily large, but it remains a real number and we stay in the Reality that we, more or less, recognize. If, on the other hand, the particle travels the same distance or a greater distance than light, LTR becomes infinite, or even worse, imaginary. And we have said goodbye to the Reality we love so much.

We are now faced with a question that is rarely asked: What is distance?

We build great mathematical models to describe our universe. The problem is these mathematical models are made up of building blocks that are all approximations.

A good example - one that involves distance - is the definition of PI, a ubiquitous symbol in many mathematical equations and is also a symbol intimately related to the circle.

When we see a circle, what we usually think is "that is a circle", but what we are really seeing is a line, called the circumference of the circle, which runs around the center of the circle.

When the mathematician imagines a perfect circle, he sees a point in the center of the circle, called a focal point, and an infinite number of points that make up the circumference. To be a perfect circle, every point on the circumference must be the same distance from the focal point. This distance is called the radius of the circle.

The distance across a circle is called the circle's diameter and is equal to twice the radius. By "across", we mean we start drawing a straight line anywhere on the circumference and then pass that line through the focal point before stopping when we again intercept the circumference. By the way, this line will divide the circle into two equal halves.

PI is defined as the ratio of the circumference of a circle to the circle's diameter. PI has a value slightly greater than three and this value is not affected by the size of the circle.

To clarify this, imagine the value of PI is exactly three. You could draw two circles side by side on the ground, one with a diameter of two hundred feet, the other with a diameter of four hundred feet. You could walk across the smaller circle by traveling 200 feet or you could get to the same point by following the circumference for 300 feet (if you continued following the circumference for another 300 feet, you would be back where you started) . By the same token, You could walk across the larger circle by traveling 400 feet or you could get to the same point by following the circumference for 600 feet.

In our reality, PI is not equal to 3, but is more accurately listed as 3.14159. PI, however, is said to be an irrational number, that is, a number that cannot be expressed as a ratio of two whole numbers. If PI were exactly 3.14159, this could be expressed as the ratio of two whole numbers 314,159 and 100,000. It is said that PI is irrational and the digits to the right of the decimal point never repeat themselves no matter how accurate you try to calculate the value.

PI has been said to have been calculated to a least a million digit, but I will just show a few. PI =

3.141592653589793238462643383279502884197169399375105820974944592307.

I would now like to assert that in the reality we live in, which includes the gibberish of quantum mechanics, any more exact definition of PI than the above is meaningless.

To start to show this, let me first quote a Google snippet defining the wave particle duality of matter.

Wave-particle duality is the concept that every elementary particle or quantic entity may be partly described in terms not only of particles, but also of waves. It expresses the inability of the classical concepts "particle" or "wave" to fully describe the behavior of quantum-scale objects.

To a non-physicist like me, this definition means, and I hope I am accurate, that even a particle, like an atom of iron, has certain wave-like properties, and by extension, so does our one pound steel ball.

I also wanted to bring up one more definition to help prove my assertion about PI, but my research indicates that I should really talk about three related definitions. At the same time, since I like to think of myself as someone willing to take things to extremes, I found a new hero. This hero is the German theoretical physicist who won the Nobel Prize in 1918 for his work on quantum theory. My new hero is Max Karl Ernst Ludwig Planck and let us look at three definitions: Planck's Constant, Planck's Length, and Planck's Time.

We can look to Google for a definition of Planck's Constant:

Planck's constant, symbolized h, relates the energy in one quantum (photon) of electromagnetic radiation to the frequency of that radiation. In the International System of units (SI), the constant is equal to approximately 6.626176 x 10-34 joule-seconds.

Just for the fun of it, let's write this as

0.000000000000000000000000000000000626176 joule-seconds and let me sum up by saying this is a very, very, very, very small amount of energy.

Before going on to Planck's Length and Planck's Time, I would like to point out that Werner Heisenberg used Planck's constant in his equations that correctly predict the probability that a particular electron would be found at a particular place - that is, there was a possibility an electron shot at a pole would hit the pole, a possibility it would miss by a foot, and another possibility it would miss by ten miles. Heisenberg was saying you couldn't know in advance where an electron would be at a particular time. It wasn't a matter of measuring more precisely, it was the result of the wave-like nature of the electron. And now we are saying that iron atoms and even entire one pound steel balls also have a wave-like property. Although the probability is exceedingly small, there is the possibility that the ball we think is here is really over there.

Planck's length is the smallest unit of distance that can have any meaning in the "real" world. Any length less than Planck's length cannot exist because quantum effects dominate. The diameter of a proton is roughly 100,000,000,000,000,000,000 greater than Planck's length. Planck's length is 1.61622837 E-35 meters, which we can write as:

0.0000000000000000000000000000000000161622837 meters.

The word quantum means the smallest possible discrete unit of any physical property. Once you have stated that there is a smallest possible unit of length, Planck's length, there must also be the smallest possible unit of time, Planck's time. This is the time it takes for light to travel across one Planck's length. It might be better to visualize light as making a "quantum leap" across Planck's length - if light did not leap, it would travel a distance less than Planck's length which is impossible because no distance can be less than Planck's length.

Planck's time is equal to 5.39116 E-44 seconds.

Time makes 539,116,000,000,000,000,000,000,000,000,000,000,000,000,000 quantum leaps every second.

Now, finally, we are ready to make an argument about the meaninglessness of calculating PI to a million digits, or even a thousand digits, to the right of the decimal point.

The diameter of any circle is determined by starting at a point on the circumference and drawing a straight line through the center of the circle and continuing until we again intercept the circumference. If, however, we are looking at this process at a super microscopic level, namely, measuring in Planck lengths, we only have probabilities that the line will start exactly on the circumference, pass exactly over the center of the circle, and then stop exactly when the circumference is again reached.

If we imagine that we have created the circumference by laying down carbon (graphite) atoms with a very sharp pencil, we would be trying to connect one atom to the corresponding atom that is on the other side of the circle. Quantum effects plus the wave-like property of the carbon atoms would make determining each atoms precise location impossible. Every time we measured the distance, we could get a slightly different answer. When we calculated the circumference, we would also get different answers and it would not help to increase the precision (number of decimals to the right of the decimal point).

There is another way to get an exact value for PI.

We start by visualizing the smallest possible circle - one with a diameter of one Planck's length. What is the circumference of this circle? The answer is three Planck's lengths. On this scale, the value of PI is exactly 3.00 (If you multiple the diameter by a value of PI equal to 3.14159, you get 3.14159 Planck's lengths for the circumference, but this is clearly impossible - all lengths have to be whole number multiples of Planck's length).

The value of PI seems to remains 3.00 for Planck's length diameters of 2 and 3. On the other hand, if we have a diameter of 4, the impossible results (that is, using PI = 3.14159) is 12.56636. The question then becomes is the "true" circumference equal to 12 or is it 13. And is the "true" value of PI 3.00 or 3.14159?

It seems to me that, when we are dealing with subatomic scales, we must always remember that everything is probabilities. For a diameter of 4, the circumference is slightly more likely to be 13, but there is still a significant chance it is 12. All that extending the precision of PI is doing is letting us look at the probabilities more closely - instead of saying there is a 30% chance of a particular outcome, we can say there is a 32% chance, or even a 32.648% chance.

What happens if we try to calculate, in Planck's lengths, the circumference of a slightly larger circle. Believe it or not, we know the approximate diameter of the observable universe. It is many, many light years, or 2.7199622 E+61 Planck's lengths. This number is not usually written as:

27,199,622,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000.

When I see this number written out, it makes me, reluctantly, think that professionals, with their "scientific notation" might have a point. I can't really visualize E+61 or E-35, but who wants to spend all day writing zeroes.

The thing that makes 2.7199622 E+61 so long when written out is what I want to discuss, namely the 54 zeroes.

I said earlier that I don't know or care if a million billion quintillion is equal to a quintillion million billion. In my student days, when I thought in scientific notation, I knew they were equal. A million is E+6, a billion is E+9, and a quintillion is E+18. A million billion quintillion is equal to 1.0 E+(6+9+18) or 1.0 E+33.

Our 54 zeroes can be expressed as 1.0 E+54 or 1.0 E+(6+6+12+12+18) or, as we like to say, a million million trillion trillion quintillion. We can therefore say that the diameter of the observable universe, in Planck's lengths, is 27,199,622 million million trillion trillion quintillion.

No reasonable person, if they think about it, would ever say this is the exact number of Planck's lengths in the diameter of the universe. The 54 zeroes actually show how uncertain we are of the true value. A better way to state the diameter of the universe is to say it is:

27,199,622 million million trillion trillion quintillion plus or minus one million million trillion trillion quintillion.

If we want to calculate the circumference of the universe, we multiply by PI = 3.1415926 and get:

85,450,131.198 million million trillion trillion quintillion plus or minus 3.1415926 million million trillion trillion quintillion.

Note that we could make this calculation other ways, by varying the precision of the value of PI, multiplying first by 3.14, then, as we just did, by 3.1415926, and then by 3.141592653 (and so on). In each case, we are getting a more exact value, but our uncertainty is also rising from (leaving out million million trillion trillion quintillion) 3.14 to 3.1415926 to 3.141592653.

We can easily calculate PI to the necessary precision, but until we can also determine the diameter of the universe to the same precision, we cannot get a "true" value of the circumference.

The value we need for the diameter of the universe is a number that has 61 significant digits and begins 27,199,622, and continues for 53 more digits with zeroes appearing only when there should be zeroes. We, of course, have no idea how to get such a number.

If we then used a PI with a precision of 62, this should be adequate for any calculations involving distances less than ten times the width of the observable universe. We might treat this PI like we do the speed of light - in some calculations, PI might be less precise than this Super PI, but it could never be greater.

(Super) PI =

3.1415926535897932384626433832795028841971693993751058209749445

Return to INTRODUCTION TO THINKING LONGER AND HARDER. Or, if you liked this, send an email to Mike Stewart. - mike@esearchfor.com

SIZE DOESN'T MATTER – OR DOES IT ? - THINKING LONGER AND HARDER

(- Copyright © 2016 by Mike Stewart -)

Albert Einstein was upset about the quantum concept of entanglement or as he called it “spooky action at a distance.”.

If you really think about it, this entanglement or "spooky connection" might also mean that we have a fundamental lack of knowledge and understanding of time and distance.

I remember a Simpson's episode that began, like most, with Homer and his family sitting on their couch. We had a close up of Homer's head. We then moved up and away, seeing first Homer's house, then clouds, then the entire earth. This moving away process continued until we saw our solar system, our entire galaxy, and then all the galaxies in the universe. As we continued to expand our view, structures appeared that looked like atoms, then we saw molecules and DNA. Finally, we could make out living cells and then we ended our journey back where we had started, looking closely at Homer's head.

The Simpson writers were trying to give us a unique perspective on reality. They may have been closer to the truth than they realized. I wonder if they recognized what they were saying about the nature of size.

If you really think about it, this entanglement or "spooky connection" might also imply we have a fundamental lack of knowledge and understanding of time and distance.

Not understanding what distance means is the same as saying we don't understand what size means. To illustrate, if we walk for one hour at 3 miles per hour, we will travel three miles. If we then turn right 90 degrees and travel for another hour at the same speed, turn right 90 degrees again, continue walking at the same speed for another hour, and then, finally, turn and, at the same speed, return to our starting point, we would have walked for four hours and traveled 12 miles to return to the same place. If we traced our steps, we would have laid out a square, each side being 3 miles long. The area of this square would be nine square miles (3 miles X 3 miles = 9 square miles).

Now, if we board a helicopter and slowly, at a rate of 3 miles per hour, rise vertically for one hour, we would arrive at a point three miles above our starting point. If, traveling at the same speed, the helicopter now moves horizontally and follows our earlier walking journey, it will, four hours later, arrive back at the point three miles above our starting point. Again, a square with an area of nine square miles would have been laid out, this one identical to the first, but separated by three miles. In fact, if we connect each point on this square where we changed directions to the corresponding point on the ground with three mile long lines, we will have constructed a cube. The volume of this cube would be twenty seven cubic miles (3 miles X 3 miles X 3 miles = 27 cubic miles). In this case, size and volume are the same thing. The size of the cube is 27 cubic miles.

What can you say about the size of this cube? To us, it seems pretty big. To one of the living cells we have been discussing, this cube appears to be almost infinitely large. On the other hand, compared to the known universe, this cube is infinitesimally small.

Can we say any more about this cube? Maybe Albert Einstein, who hated "spooky action at a distance", should have thought a little longer and harder about one of his thought experiments.

When I was trying to understand relativity, I read about what is called the "Twin Paradox" which was an Einstein thought experiment that involved one person traveling in a rocket traveling at a high rate of speed (almost the speed of light). His twin, who remained on earth, would, according to Einstein, age faster than the space traveler. I think, after some effort, I understand what Einstein was saying. If I do understand correctly (quite honestly, it makes my head swim to think about it), I can modify this space journey to address our current subject, size.

If you would like to google "Twin Paradox" you can try to figure out what is being said, but I would like to set up another space journey to make a comment on size. Imagine a rocket leaves earth traveling to a planet circling one of the three stars in the star system, Alpha Centauri. This planet is four light years away.

On this far away planet, we have an older observer who spent half his life traveling to this distant point. He watches carefully through a powerful telescope as our large rocket, piloted by a brave astronaut, blasts off from earth. A second observer, who will stay on earth, also watches the proceedings.

As we have said before, time seems to be important. The astronaut and the two observers are habitual clock watchers and each demands his own personal, extremely accurate clock. Before the astronaut or the first observer leave earth, all three clocks display identical times.

Some of our fellow earthlings have previously placed twelve, three mile long, straight steel pipes in earth orbit. Using the cube we traced out above as a model, they have constructed a real, orbiting cube that has a volume (size) of twenty seven cubic miles. Our astronaut, using several cables, attaches the cube to the rocket, and then heads for Alpha Centauri.

Our rocket is capable of accelerating continuously so that the astronaut will feel as if he weighs the same as he would on earth (the astronaut would feel he was being pulled toward the earth from which he was rapidly retreating).

If you asked the earth observer, two years later, about the rocket, he would say it is moving away from him at almost the speed of light. The rocket, the astronaut , and the cube all display a "red shift" which you remember from Hubble's observations means the rocket is rapidly receding. The astronaut's clock is running slow. The four steel pipes of the cube that are pointed toward earth are compressed, with a length of perhaps 1.5 miles rather than the 3.0 miles you would expect. If you calculate the size of the cube, it has shrunk from the original 27 cubic miles to 3 X 3 X 1.5 = 13.5 cubic miles.

Now let us turn to the second, remote observer. Since when I start thinking about the "Twin Paradox" I start getting confused, I will not ask what the second observer sees "at the same time". Instead I will note that the earth observer is looking at the rocket when it is half way to Alpha Centauri and ask what does the second observer see when he and the observer on earth believe the rocket, cube, and astronaut are at the half way point of the journey?

The remote observer would see that the rocket, astronaut , and cube all are displaying a "blue shift" which means these objects are approaching rapidly. The astronaut's clock is running fast. The four steel pipes of the cube that are pointed toward Alpha Centauri are expanded, with a length of perhaps 4.5 miles rather than the 3.0 miles you would expect. If you calculate the size of the cube, it has grown from the original 27 cubic miles to 3 X 3 X 4.5 = 40.5 cubic miles.

Let us stop at this point and make a few observations. To the astronaut, time is passing normally, and the cube, the rocket, and he, himself, are all unchanged. To the observer on earth, the astronaut's clock is running slow, and everything observed is slightly smaller. To the remote observer, the astronaut's clock is running fast, and everything observed is slightly larger.

If the astronaut makes the cube rotate (the connections between the cables and the cube could be modified so that the cables wouldn't get tangled up), the size of the cube will change as its orientation to the remote observers change. To the observer on earth, the cube would rotate slowly and its size would change slowly. To the other observer, it would rotate rapidly and there would be a rapid, periodic, size change.

What happens if the astronaut, at this point, shuts down the rocket engine? The astronaut, rocket, and cube will be suddenly weightless. All three would stop accelerating, but their velocity and speed do not change. The speed, from the standpoint of the remote observers, is approaching the speed of light. The red shift or blue shift observed would not be affected. I may be wrong, but I suspect that any size changes observed by the remote observers would also not go away.

There is one thing that might be useful to realize. On the other hand, it may, if you are like me, just contribute to a feeling that you are not sure you understand relativity. Both remote observers will not know anything about the astronaut (like he shut down the rocket engine) until light reaches them. The astronaut is at a point where he has traveled one light year (the distance light travels in a year) and has three light years left to travel. The earth observer will not know that the rocket engine is off until a year has past. The Alpha Centauri observer will have to wait three years. And don't forget, the two observers did not start counting at the same time - there is no same time.

If the astronaut does not shut down the engine, the rocket will be traveling at nearly the speed of light when it reaches the remote observer. In fact, we could get the rocket to come as close as we would like to light speed by assuming the rocket had had longer to accelerate. Instead of starting from earth, the rocket could have started its journey on the other side of the galaxy.

In any case, you could imagine the cube being the size of a galaxy as it passes the remote observer. It might be that only one atom from the astronaut's left thumb would pass close enough to be detected. Yet, as soon as the rocket passes, the whole configuration (cube, rocket, astronaut) becomes microscopic.

There is one major problem with the mental exercise I have presented. There may,in fact, be other problems. With relativity, as with other things, the devil is in the details. Nevertheless, I hope that much of what I say is valid and has value.

The problem I recognize is we have been thinking in three dimensions. As the rocket approaches the remote observer, two of the three dimensions of the cube remain a constant three miles. The third dimension has expanded, and with it, the volume or size of the cube. The cube has grown, but in only one direction. It has been transformed into a very, very long rectangular prism.

Let us revisit our astronaut, observers, cube, and rocket and think about String Theory.

I have said that my biggest problem with particle physics is that the mathematics involved is so advanced that few people can understand the equations that support the conclusions the experts have reached. The intelligent layman has to, unfortunately, rely on unsupported faith when he listens to these experts.

String Theory math makes Particle Physics math look like third grade arithmetic. String Theory mathematics is useless to the vast majority of people who would like to learn about this subject. I will, therefore, only use the inaccurate mental images and analogies that are commonly used to discuss String Theory. Maybe then, while working on this foundation of sand, I can ask a question or two that is intelligent, useful, and relevant.

Even the definition of String Theory brings up more questions than it answers. From a Google Snippet: "string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. It describes how these strings propagate through space and interact with each other.".

What is a one-dimensional object? Is it something with length, but no height or width? Is it a particle of time, like Einstein considered in his four dimensional space time world? Or, is it a small particle of time, or timelet, like discussed in the intelligent ant story? Is it like a scalar value having magnitude, but no direction (for example, to an outside observer, the astronaut may have a certain absolute speed. His velocity can include this speed, but must also specify a direction - in this case, toward Alpha Centauri)? Or, is it like something else?

I wanted to discuss String Theory because it is concerned with, among many other things, how size might be affected by velocity. Before we get to this, however, let us try to understand some of these many other things.

A one-dimensional string can be viewed as a real life guitar string under tension. When plucked, the string will vibrate and make a sound. Increasing the tension or length of the string will change the sound. String Theory says that we see a string as an elementary particle. If you change the sound, you get a different elementary particle. For example, an electron is a string vibrating one way, a photon is a string vibrating another way. We could also say that a particle is an excitation of a string.

Quantum Mechanics or Particle Physics is concerned with the interaction of sub-atomic particles with each other. It accurately describes the observed behaviors and properties of these particles.

Particle Physics, however, has logical problems. Buried in the supporting equations are the assertions that the interacting particles are point-like objects and the distance between the objects is zero (you could say the objects occupy the same space). For Particle Physics to work, you have to assume that gravity does not exist. To understand why, consider that the force of attraction (gravity) between two objects increases when the their total mass increases or when the distance between them decreases. If the distance between the two objects becomes zero, the force of attraction becomes infinitely large. This is impossible, but Particle Physics still accurately describes the behaviors and properties of sub-atomic particles.

Note that we were discussing distance which implies size which is what this chapter is about.

Experts, including scientists, often change their terminology to sound more hip or modern. Information technology (IT) is defined by Google as the study or use of systems (especially computers and telecommunications) for storing, retrieving, and sending information. With the rise of the internet, experts in the field wanted to sound more modern so they invented the term Information and Communications Technology (ICT) and made IT a subset of this "new" discipline. This modernization of words, without fundamental change in meaning, continues on a more detailed level. Years ago, we had computer programmers writing and testing computer programs. Today, we have application developers creating apps. When you consider that an app runs on a smart phone and a smart phone is a computer, you realize "app" is just another word for "computer program".

Scientists, as well as their closely allied Science Writers, are influenced by the desire to sound modern. Scientists, however, may also be greatly affected, in a real sense, by another phenomenon, new theories. Other scientists may occasionally develop a new theory that they believe is a better way to describe reality.

A new theory can make a scientist's life work irrelevant. Funding for new research, which in the modern world seems to be a lot of "thinking longer and harder", can dry up overnight. Science writers no longer pester the now irrelevant scientist for interviews. They quickly turn their attention to the new stars, the expert developers of the new theory.

Has String Theory suffered such a fate - Is it now irrelevant? Bryan Green, a well known physicist, in a recent article, asks "... Is String Theory revealing deep laws? Or, as some detractors have claimed, is it a mathematical mirage that has sidetracked a generation of physicists?".

I don't know the answer, but unless some newer theory has drastically simpler mathematics associated with it, I'd like to postpone discussing this until we've looked more closely at String Theory.

String Theory says that when we observe a specific elementary or sub-atomic particle (for example, a photon), we are really seeing a string vibrating in a particular way. String Theorists, therefore, want to determine how many different ways a string can vibrate, and to show that for each way, there is a corresponding sub-atomic particle. The things that make particles unique are their properties, for example, an electron has different properties than an photon. Looking at it another way, String Theorists seek to relate how strings vibrate to the observed properties of different elementary particles. Observations show that particles can be divided into two groups, fermions and bosons. Fermions are the particles that make up matter. Fermions cannot occupy the same space at the same time. Bosons are the particles that transmit force. Most bosons can occupy the same space at the same time.

One of many strange things about strings, based on the mathematics that support and define string theory, is that they do not vibrate just in the normal dimensions we are use to, namely, length, height, width, and time; or what Albert Einstein called space-time. If strings vibrated only in this space-time, we could visualize them as small strands of spaghetti slowly twisting and turning as time passed. Now is a good time to note that strings can be either closed or open (the ends of the spaghetti strand are connected or not). Either way, we can visualize them easily in "normal" space-time.

Unfortunately, string theory mathematics, which normal people have to take on faith, dictates that there must be, not the four dimensions of space-time, but either ten dimensions or twenty six dimensions.

Ten dimensions or twenty six dimensions naturally brings up a lot of questions.

Many of these questions occurred to string theory experts, who I must say, have made a profession out of proposing "out of the box", somewhat outlandish, questions and then developing answers which also might seem absurd. You could say that string theorists, like me, are trying to find unique thoughts. They are the experts. I am the amateur.

An amateur like me who only has an undergraduate degree in mathematics and who readily admits long ago forgetting most advanced mathematical knowledge has only one advantage over these experts. They tend to ask questions and propose very strange answers to explain the mathematical models they have constructed. I, on the other hand, can ask any crazy question and propose any silly answer. I can then ask an expert "somewhere" to modify or create a math model that supports me.

String Theory tells us that we can have multiple dimensions if most of these dimensions are too small to be seen. I need to explain the word "seen" in the last sentence. I feel like I can see three of the four dimensions of space-time, namely, length, width, and height. Time, on the other hand, is experienced rather than seen. String Theory is saying that if we have ten dimensions, six are too small for us to realize they exist. If we have twenty six dimensions, twenty two are too small. String Theory also tells us that these small dimensions are curled up and intertwined together and each has its own shape.

How small are these dimensions? Our most powerful accelerators can detect many sub-atomic particles. Strings are small enough to float through and vibrate in not just space-time, but also in these small dimensions. The shape of each dimension affects how the string vibrates. We can view a string as floating in 10 or 26 dimensions and its vibration defines a subatomic particle. These dimensions are not much larger than strings and strings are a million billion times smaller than the realm explored by the above mentioned accelerators.

String Theory is sometime broken down into multiple theories. A feature of one theory is that there are twenty six dimensions. There are five other theories that feature ten dimensions. As time has passed, however, we should note that String Theorists have tended to both create more string theories and decrease the number of string theories. When String Theorists looked at (mathematically) how to make unseen dimensions very small so that they get the everyday four dimensional world we see, they discover that they can do this with many different mathematical models. Each new model can become a new string theory. On the other hand, Theorist may sometimes discover that two mathematical structures supporting two theories are actually equivalent. In this case, they realize they don't have two theories, but only one that they had been looking at in two different ways. Looking more closely, however, at these six theories is a good starting point.

The first of these six is called bosonic and sports twenty six dimensions. To a layman, this theory seems really stupid. Or, since I have said all of quantum mechanics is gibberish, maybe I should just say this theory seems to be more gibberish-like than the others. I am critical because this theory starts with the assumption that matter does not exist.

To be fair, the mathematical model that supports the theory could be useful in creating better, hybrid string theories. Under the bosonic model, only bosons exist. There are no fermions. This means there can be forces, but no matter. To my simple mind, it doesn't seem you could have a force unless it had some matter to act on.

Probably the most famous boson is the Higgs Boson, the so called "God Particle", recently found by the Large Hadron Collider. Bosons have integer (values of 0, 1, 2, 3, etc.) spin. Spin is a quantum mechanical term that is very important in the mathematics behind the bosonic theory (as well as in much of other math involved with quantum physics). Spin is often defined using analogies to the world we see around us, but these don't really apply and underlines that we don't know what spin is. The best explanation I have seen was by Jon Butterworth in an online blog hosted by The Guardian, a British Newspaper.

Jon Butterworth is a physics professor at University College London. He is a member of the UCL High Energy Physics group and works on the Atlas experiment at Cern's Large Hadron Collider. His book Smashing Physics: The Inside Story of the Hunt for the Higgs was published in May 2014.

To quote Jon Butterworth:

"Bosons have, by definition, integer spin. The Higgs has zero, the gluon, photon, W and Z all have one, and the graviton is postulated to have two units of spin. Quarks, electrons and neutrinos are fermions, and all have a half unit of spin.

This causes a huge difference in their behaviour.

The best way we have of understanding fundamental particles is quantum field theory. In quantum field theory a "state" is a configuration describing all the particles in a system (say a hydrogen atom). The maths is such that if you swap over two identical fermions with identical energies (say, two electrons) then you introduce a negative sign in the state. If you swap two bosons, there is no negative sign.

Since swapping two identical particles of the same energy makes no physical difference to the overall state, you have to add up the two different cases (swapped and unswapped) when calculating the actual real probability of a physical state occurring. Adding the plus and the minus in the fermion case gives zero, but in the boson case they really do add up. This means any state containing two identical fermions of the same energy has zero probability of occurring*. Whereas a state with two identical bosons of the same energy has an enhanced probability.

This fairly simple bit of maths is responsible for the periodic table and the behaviour of all the elements.

** This is known as the Pauli exclusion principle”.*

Now that we understand spin, there are a couple of other noteworthy points to mention regarding bosonic theory. The theory allows both open and closed strings (loops). The theory also, in what some call its greatest flaw, postulates a new particle. I would prefer, in discussing this particle, to quote Spock of Star Trek and say "fascinating".

Bosonic String Theory says that there exists a particle that we have named the tachyon. A tachyon has two unique properties. It has imaginary mass and it can travel faster than the speed of light.

Reluctantly, in order to even hint at what this means, I need to bring in a little math. You may or may not remember from the long ago days of school what a square root is. To me, the definition is kind of verbally awkward, but giving several examples should make the meaning clear.

Consider the following seven multiplications: (1) 0 X 0 = 0; (2) 1 X 1 = 1; (3) -1 X -1 = 1; (4) 2 X 2 = 4; (5) -2 X -2 = 4; (6) 3 X 3 = 9; and (7) -3 X -3 = 9.

Now what is a square root of a number? For convenience, I am going to replace the phrase "square root of", not with its mathematical symbol, but with "SQROOTof". In the first case above, the SQROOTof zero is zero. In the other cases, the SQROOTof a number has two answers, a positive value and a negative value. The SQROOTof four is both +2 and -2. The SQROOTof nine is both +3 and -3.

When you consider these examples or read a more formal definition of square roots, you realize there is no such thing as the square root of a negative number. The SQROOTof -1 does not exist. The SQROOTof any negative number does not exist, they are completely imaginary. They are called Imaginary Numbers.

Scientists, especially Quantum Physicists, seek to create sophisticated mathematical models to describe and support their views of reality. Mathematicians want to build similar models just to prove they can. You would think that both would have no use for Imaginary Numbers. You would be wrong.

Speaking of being wrong, please bear with me while I introduce a little more simple math which I hope I haven't made so simple that I am wrong. I want to do this in the context of our traveling astronaut and eventually show a relationship to imaginary numbers.

Before our astronaut blasts off from earth, we will assume he is carrying with him a one pound ball. He is an American astronaut so he doesn't feel compelled to carry a heavier one kilogram ball. On earth this ball weighs one pound and also has a mass of one pound. The important thing to know, however, is the weight of the ball can change. On the moon, with its lower gravity, the ball would weigh about a sixth of a pound. The mass would still be one pound and does not change (except for the effects of relativity). From now on, we will talk about mass rather than weight.

When the rocket configuration (astronaut, rocket, cube, and now ball) is moving away from the earth, the earth bound observer may notice that the mass of the ball has increased. When we ask how much the mass of the ball has increased (I want to ask does it now "weigh" 2 pounds, 10 pounds, or 5000 pounds, but sensitive scientists would get upset).

Another way of asking this question is to ask what is the mass of the ball compared to what the mass would be if the ball (or a duplicate ball) had remained on earth (which we know is one pound)? Or better still, what is the ratio of the mass of the ball on the rocket to the mass of the ball on earth? And this is where our SQROOTof comes in.

In the early twentieth century, a dutch physicist named Hendrik Lorentz figured out how to find the ratio of the mass of the ball on the rocket to the mass of the ball on earth. Hendrik developed a lot of complicated math to address other related subjects, but one relatively simple equation answered this question about masses. This equation states that the MASSonrocket divided by MASSonearth (their ratio) is equal to the Lorentz Transformation Factor - which we will call the LTR.

Now here is where the SQROOTof comes in. The LTR is equal to the number one divided by the SQROOTof the number one minus another number. This other number is the ratio of the velocity of the rocket multiplied by itself to the speed of light multiplied by itself.

I have written this description in English in hopes of being clear, but we may have reached the point where something more equation-like is better. Using the letter V for velocity and the letter C for the speed of light ("C" is the first letter of celeritas, the Latin word meaning speed and is typically used to denote the speed of light), we can write the following:

MASSonrocket / MASSonearth = LTR = 1 / SQROOTof ( 1 - ( V times V ) / ( C times C ) ).

This equation can also be written as:

MASSonrocket = MASSonearth times LTR.

With this, if we know the velocity of the rocket and the mass of the ball on earth, we can calculate the mass of the ball on the rocket.

If we examine more closely the SQROOTof part of LTR, that is, the denominator, we can gain a lot of insight.

The velocities we see in everyday life (the V in the SQROOTof) are very small compared to the speed of light. A satellite circling the earth may be traveling at five miles per second. If you multiple this by itself, you get twenty five. On the other hand, the speed of light is 186,282 miles per second. If you multiply this by itself, you get 34,700,983,524.

You can use these numbers to see how much the mass of our one pound ball would increase if it was on a satellite, traveling a five miles per second:

MASSonsatellite = MASSonearth times 1 / SQROOTof ( 1 - ( V times V ) / ( C times C ) ).

MASSonsatellite = 1.0 pounds times 1 / SQROOTof ( 1 - ( 25 ) / ( 34,700,983,524 ) ).

MASSonsatellite = 1.0 pounds times 1 / SQROOTof ( 1 - .00000000007204068).

MASSonsatellite = 1.0 pounds times 1 / SQROOTof ( 0.99999999992 ).

MASSonsatellite = 1.0 pounds times 1 / 0.99999999996.

MASSonsatellite = 1.0 pounds times 1.00000000004 = 1.00000000004 pounds.

We can see from these calculations that the mass of the ball is increasing more and more rapidly as the velocity approaches the speed of light. The equation, in fact, predicts that if its velocity could reach light speed, the ball would have infinite mass. Another way of saying this is that it would require infinite energy to push the ball to light speed. This is why conventional wisdom says that the ball can never travel at the speed of light.

And you know how I hate conventional wisdom.

The LTR equation invites additional conjectures and Einstein-like thought experiments. I can't resist this temptation before returning to String Theory. We are at a point, however, where we can say what Bosonic String Theory means by imaginary mass.

Bosonic String Theory postulates that particles exist called tachyons. They can travel faster than light. They cannot travel at the speed of light or less than the speed of light. Suppose our rocket configuration, which includes our one pound ball, is made up of tachyons, traveling faster than light speed.

If we look at part of the LTR equation, namely, ( V times V ) / ( C times C ) where V is greater than C, we realize that the numerator (V time V) is greater than the denominator (C times C).

When you divide a larger number by a smaller number, you get a number greater than one: ((V times V) / (C times C)) equals a number greater than one.

At this point, we should observe that when we try to determine the mass of the ball on the rocket, this is the third of three distinct situations. These three are:

(1) A "normal universe" where light speed is constant for all observers. Note that this universe would not have been considered normal a couple of centuries ago:

LTR = 1 / SQROOTof ( 1 - ( a number less than one ).

LTR = 1 / SQROOTof (a positive number between one and zero). In this case, since MASSonrocket = MASSonearth times LTR, the mass of the ball on the rocket can vary from one pound to a very large, but still imaginable, value.

(2) One brief shining moment where there is a universe that supposedly cannot exist:

LTR = 1 / SQROOTof ( 1 - ( the number one ) = 1 / SQROOTof ( Zero ).

LTR = 1 / 0 = INFINITY (whatever that means). In this case, the ball on the rocket would have infinite mass.

(3) A very strange universe that is hinted at by Bosonic String Theory:

LTR = 1 / SQROOTof ( 1 - ( a number greater than one )).

LTR = 1 / SQROOTof ( a negative number )) - and, of course, we remember that the SQROOTof a negative number is an IMAGINARY NUMBER! And so we have

LTR = 1 / (an imaginary number).

Before we continue, I would like to do a bit of rationalizing. People like to believe that they are very rational creatures (to be more exact, I want to believe that I am rational, you may be, or you may not be, rational). The truth may be that we are creatures that are very good at rationalizing - we find plenty of proof to support the view we want to hold, while we ignore, or even angrily reject, information that contradicts us. This does not mean that rationalizing never leads to valid conclusions (is this a rationalization?).

Now I would like to discuss infinity and imaginary numbers. Both of these concepts are strange to our way of thinking, but they are very important to lots of scientists and mathematicians. They are always using them in their icky math equations.

When you look at these equations, you will find that the math guys write infinity as the number 8 turned on its side and our SQROOTof a negative one as the lower case "i". Mathematicians have written rules to manipulate these concepts. I learned these rules in college so I could solve equations and get my math degree.

Since I have forgotten a lot of these rules and don't want to spend the time needed to relearn them, I am going to ignore them - but I am going to say why any assertion I am making is still true (as you will see, in some situations, I believe the rules themselves are wrong).

Looking first at Infinity.

Normalization is a process or rule used to remove infinities from equations and get meaningful results. When a physicist does this, he will say the equation has been renormalised. Before I explain what this is and why I doubt that it is valid, let me quote from a Google snippet for Renormalization, apparently a similar process, that is for a specific field (quantum mechanics) - remember that I said that scientists are verbose.

Renormalization - In quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, renormalization is a collection of techniques used to treat infinities arising in calculated quantities by altering values of quantities to compensate for effects of their self-interactions.

As is usually the case, the ordinary person has to have faith in the validity of these undefined "collections of techniques", so let us take a closer look at what it takes to get to just a simple renormalised equation.

I read somewhere this example:

In physics, Renormalization is a great tool, or cheat, to get rid of infinities in equations and give meaningful results:

Unrenormalized Equation - (infinity +3) x (infinity +3) = infinity

Renormalized Equation - (infinity +3) x (infinity +3) = 9

This renormalization process is really a combination of the normal math we are use to and "something else". In normal math, we could have the equation (4 + 1) X (4 +1) = 25 which is so self evident that we can skip the intermediate step. With this step included, we have (4 + 1) X (4 + 1) = (4 X 4) + (4 X 1) + (4 X 1) + (1 X 1) = 16 + 4 + 4 +1 = 25.

If we follow the same process with our infinity equation, we get:

(infinity +3) x (infinity +3) = (infinity X infinity) + (3 X infinity) + (3 X infinity) + ( 3 X 3) = (infinity X infinity) + (6 X infinity) + 9 = infinity

If you continued with normal math, you could get:

(infinity X infinity) + (5 X infinity) + 9 = 0 or

minus (infinity times infinity) minus (five times infinity) = nine

To tell you the truth, I don't know what this means. Apparently, physicists and mathematicians don't either so they invented normalization to get rid of infinities when they didn't want them. Normalization may be what I call a "weird rule" that Reality may sometime follow, but, maybe, will more often ignore. Anyway, rather than figure out how to use a "weird rule", I had just as soon invent my own.

Before I leave infinities, let me mention the simplest weird rule.

infinity + 1 = infinity. The non-weird version of this rule would be "you cannot add one to infinity". I seem to remember that once I was told that you couldn't divide one by zero, but later I learned that one divided by zero was (the or a) definition of infinity. Saying you cannot add one to infinity makes some sense - infinity is so big there are no ones left to be found to add.

infinity + 1 = infinity says that some infinities are bigger than others. Renormalization treats all infinities the same. Mathematicians seem to want to put infinities into a weird, Orwellian Animal Farm universe where all infinities are equal, just some are more equal than others.

It is interesting that for infinity + 1 = infinity,

we can subtract infinity from both sides of the equation giving infinity - infinity + 1 = infinity - infinity or 0 + 1 = 0.

We have just proved that one equals zero. Now on to the Imaginary.

As I said earlier, Hendrik Lorentz figured out an equation that could be used to calculate the ratio of the mass of our one pound ball on a rocket to the mass of the same ball if it had remained on earth. This equation is:

MASSonrocket / MASSonearth = 1 / SQROOTof ( 1 - ( V times V ) / ( C times C ) ).

Or, as Lorentz said, LTR = MASSonrocket / MASSonearth.

When V (velocity) exceeds the speed of light, SQROOTof becomes, as we have said before, an imaginary number. And it is at this point that I want to rebel against relearning the rules that typically govern calculations involving imaginary numbers.

One of these rules is that if we multiply the imaginary number "i", which is our SQROOTof negative one, by itself, the results is minus one ( i squared = -1 ). I ask where have they (math or science people) proven this "fact" relates to reality.

i times i = i squared = -1.

(i times i times i) or (i squared times i) or (i cubed) = - i.

(i times i times i times i) or (i raised to the fourth power) = i.

Other rules assert that you can break up imaginary numbers into real number parts plus imaginary parts. Mathematicians can then use regular math on the real numbers.

I know this is hard to follow so let me give a couple of examples where these rules might be applied, one where V is slightly greater than light speed and the other where V is almost twice light speed.

For slightly greater:

MASSonrocket / MASSonearth = 1 / SQROOTof ( 1 - ( V times V ) / ( C times C ) ).

LTR = 1 / SQROOTof ( 1 - (a number slightly greater than 1) ).

LTR = 1 / SQROOTof ( a very small negative number ).

When V almost reaches twice light speed, ( V times V ) / ( C times C ) becomes almost ( 2C times 2C ) / (C times C) = ( 4C squared) / (C squared) = 4 and:

MASSonrocket / MASSonearth = 1 / SQROOTof ( 1 - 4 ) and

LTR = 1 / SQROOTof ( almost negative 3 ).

In both these cases (small negative number, almost negative 3), the math rules say LTR has a real number component and an imaginary number component. I do not believe reality follows these rules. All we can say is that both LTRs are imaginary. Are they equal? Maybe yes, maybe no. It may be analagous to asking if you add one to infinity, is the new infinity equal to the old infinity.

Remember the alternate universe of Schrodinger's cute kitten and old cat? In this story, alternate universes could be easily created. Maybe imaginary numbers in our equations are implying there are also many alternate realities. To see how, let us embark on more thought experiments.

If we think about our rocket configuration which includes a one pound ball, we can calculate the mass of the ball when the velocity is less than the speed of light. If the velocity reaches 99.99 percent of the speed of light, the mass of the ball will be a little over seventy pounds. LTR is still a real number and we could conclude that the ball still exists in what we call our reality.

Light in a vacuum travels at (for this discussion, we will say) exactly 186,282 miles per second. If the rocket / ball was traveling one mile per second slower, or 186,281 miles per second, the observer on earth would say light is outrunning the ball at a rate of one mile (or 5,280 feet) per second. The mass of the ball has increased significantly since it left earth, but can still be calculated and the ball is still in our reality.

We can say the same thing about the ball if its speed increases again to (186,281 miles plus 5,279 feet) per second or even to (186,281 miles plus 5,279 feet plus 11 inches) per second. In the latter case, the observer on earth would say light is moving one inch per second faster that the ball and LTR is a very large, but real, number, which is still equal to the mass of the ball. The ball is still in our reality.

We can continue this process of imagining the ball is getting closer and closer to the speed of light. We, in this case, might say light is moving one inch per second faster than the ball, then light is moving one-tenth of an inch per second faster than the ball, then light is moving one one hundredth of an inch per second faster than the ball, and so on. We cannot, however, continue this process forever.

To understand why, consider the following: One Angstrom is defined as 10 billion meters which means that one inch is equal to 254 million angstroms. When we say, as we did above, that light is moving one inch per second faster than the ball and light is traveling at, well, the speed of light, we would think we could ignore the difference. When we say the same thing, but express it as light is moving 254 million angstroms per second faster than the ball, we may want to take a closer look. If we do, we will find that a lot happens when we try to increase the speed of the ball a mere one inch per second.

Until now, we haven't cared what the ball on the rocket is made of. We were just concerned that when it was on earth, it had a mass equal to one pound. Now we realize that it is a steel ball made up primarily of an alloy of iron and carbon atoms. Our scientist friends could probably tell us about how many atoms are in the ball, but I just want to say to there are less than an infinite number of atoms in the ball - but, to our way of thinking, not much less.

We can, of course, continue the process of increasing the speed of the rocket ball so that it is only one inch or 254 million angstroms per second slower than light, then 127 million angstroms slower, then 64 million angstroms slower, and so on. At some point, maybe when we are down to a few thousand or a few hundred angstroms, interesting things begin to happen.

At some point in the latter stages of this process, when the ball is almost traveling at light speed, we need to stop thinking of the ball as a ball and start thinking of it as a vast collection of atoms, or, since we are looking at String Theory, vibrating strings.

Let us take atoms first, looking at an individual atom speeding away from the same atom in the ball if it had stayed on earth. When we look at an atom on earth, it will display quantum properties. Heisenberg's Uncertainty Principle asserts, more or less, that you can never know exactly where an atom or its components are. You can only assign a probability to the position of the atom's nucleus or, for example, one of its electrons.

We have discussed an unbound electron (not part of an atom) being fired at a steel pole. Heisenberg said we could never be sure the electron would hit the pole - it might miss by various amounts. In fact, there was an extremely remote possibility that you might miss by a mile, or by a hundred million miles. I am not sure if this is true for a bound electron, but it would be very difficult to say the electron was closer to or further away to the observer on earth than the associated neutron.

Now what happens if we put the ball on the rocket and it eventually reaches a speed a few angstroms per second short of light speed?

By the way, I am pretty sure we cannot measure the speed of light in angstroms. Calculations show it is equal to 2.99792458 quintillion angstroms per second, which can also be written as 2,997,924,580,000,000,000 angstroms per second. The "true" speed of light in a vacuum could be, for example, 2,997,924,580,123,456,789.123456789 angstroms per second.

According to quantum mechanics, the ball and by extension a particular atom in the ball doesn't exist until an observer looks at it. If the atom is moving close enough to the speed of light, the observer on earth may decide that the atom is a few angstroms closer or a few angstroms further away. In the first case, the observer will note that in the last second the atom has traveled a few angstroms per second less than light speed and the atom will still be in our reality. On the other hand, the distance the atom has traveled in the last second will be greater than light speed. To the observer on earth, the atom would have evaporated. It would have entered a new reality. This would be analogous to, or even an explanation of, quantum tunneling (refers to the quantum mechanical phenomenon where a particle tunnels through a barrier (in this case, the speed of light) that it classically could not surmount).

As the ball came closer and closer to light speed, the probability that atoms would land "on the other side" would increase until the ball faded from our reality.

We might get another effect if we looked at the situation from the standpoint of an electron. As we get closer and closer to light speed, the probability increases that an electron that is furthest from its neutron will cross over into the land of Oz. The atom left in our reality would display a positive electrical charge. Again, as above, as speed increased, more and more atoms would lose an electron, and the ball would become more and more highly charged.

Wikipedia states that the conservation of energy, a bedrock of our physics, means that the total energy of an isolated system remains constant—it is said to be conserved over time. Energy can neither be created nor destroyed; rather, it transforms from one form to another. For instance, chemical energy can be converted to kinetic energy in the explosion of a stick of dynamite.

It seems likely to me that electrons dropping out of existence might violate the law of the conservation of energy.

Now is a good time to remind the reader we are still talking about Bosonic String Theory under which our ball that is made up of fermions could not exist. I don't know why this is true, either because I don't have the time to research it, or more likely, the mathematical model that supports this "fact" is beyond my comprehension - so I am turning to faith. Nevertheless, my arguments about disappearing atoms or electrons could apply under other theories.

Our next question should be "What is happening on the String level when our subatomic particles are dropping out of existence?". After all, the vibration of strings define these particles. One obvious problem, however, crops up when we try to do this. When we look at light in a classical sense, we think it takes a few quintillionths of a second to cross atomic distances. The scale we are dealing with when discussing strings is a million billion times smaller and thus light must cross string related distances a million billion times faster.

I want to use the phrase "hard to imagine" in the next paragraph, but first I want to say what I mean. If I asked you to describe something imaginary, you might describe a ghost, looking like a sheet floating through the air. This is not what I mean. The ghost is imaginary, but you have described it using common terms like floating, air, and sheet. By "hard to describe", I mean you can't imagine what it is you are talking about or how to describe it.

To the human mind, which can't really even visualize a billion, considering a quintillion seems hopeless. Now we are talking about a million billion quintillion. In fact, I don't even know or care if a million billion quintillion is equal to a quintillion million billion. Such big number make even visualizing what form a thought experiment should take hard to imagine.

But let's try anyway.

In our reality, when we are discussing subatomic particles, whether they are stationary or moving away from us at nearly the speed of light, we calculate their mass using the familiar Lorentz equation:

MASSonrocket/MASSonearth = LTR = 1/SQROOTof (1-(V times V)/(C times C)).

In this equation, we can ask what is the velocity V, and for that matter, what is the velocity C. The simple answers are V is the distance the particle travels in one second while C is the distance light travels in one second. For the significant part of the above equation, namely:

(V times V) / ( C times C ) we have

((the distance the particle travels in one second) times (the distance the particle travels in one second)) divided by ((the distance light travels in one second) times (the distance light travels in one second)).

As long as the distance that the particle travels is less than the distance light travels, LTR may be arbitrarily large, but it remains a real number and we stay in the Reality that we, more or less, recognize. If, on the other hand, the particle travels the same distance or a greater distance than light, LTR becomes infinite, or even worse, imaginary. And we have said goodbye to the Reality we love so much.

We are now faced with a question that is rarely asked: What is distance?

We build great mathematical models to describe our universe. The problem is these mathematical models are made up of building blocks that are all approximations.

A good example - one that involves distance - is the definition of PI, a ubiquitous symbol in many mathematical equations and is also a symbol intimately related to the circle.

When we see a circle, what we usually think is "that is a circle", but what we are really seeing is a line, called the circumference of the circle, which runs around the center of the circle.

When the mathematician imagines a perfect circle, he sees a point in the center of the circle, called a focal point, and an infinite number of points that make up the circumference. To be a perfect circle, every point on the circumference must be the same distance from the focal point. This distance is called the radius of the circle.

The distance across a circle is called the circle's diameter and is equal to twice the radius. By "across", we mean we start drawing a straight line anywhere on the circumference and then pass that line through the focal point before stopping when we again intercept the circumference. By the way, this line will divide the circle into two equal halves.

PI is defined as the ratio of the circumference of a circle to the circle's diameter. PI has a value slightly greater than three and this value is not affected by the size of the circle.

To clarify this, imagine the value of PI is exactly three. You could draw two circles side by side on the ground, one with a diameter of two hundred feet, the other with a diameter of four hundred feet. You could walk across the smaller circle by traveling 200 feet or you could get to the same point by following the circumference for 300 feet (if you continued following the circumference for another 300 feet, you would be back where you started) . By the same token, You could walk across the larger circle by traveling 400 feet or you could get to the same point by following the circumference for 600 feet.

In our reality, PI is not equal to 3, but is more accurately listed as 3.14159. PI, however, is said to be an irrational number, that is, a number that cannot be expressed as a ratio of two whole numbers. If PI were exactly 3.14159, this could be expressed as the ratio of two whole numbers 314,159 and 100,000. It is said that PI is irrational and the digits to the right of the decimal point never repeat themselves no matter how accurate you try to calculate the value.

PI has been said to have been calculated to a least a million digit, but I will just show a few. PI =

3.141592653589793238462643383279502884197169399375105820974944592307.

I would now like to assert that in the reality we live in, which includes the gibberish of quantum mechanics, any more exact definition of PI than the above is meaningless.

To start to show this, let me first quote a Google snippet defining the wave particle duality of matter.

Wave-particle duality is the concept that every elementary particle or quantic entity may be partly described in terms not only of particles, but also of waves. It expresses the inability of the classical concepts "particle" or "wave" to fully describe the behavior of quantum-scale objects.

To a non-physicist like me, this definition means, and I hope I am accurate, that even a particle, like an atom of iron, has certain wave-like properties, and by extension, so does our one pound steel ball.

I also wanted to bring up one more definition to help prove my assertion about PI, but my research indicates that I should really talk about three related definitions. At the same time, since I like to think of myself as someone willing to take things to extremes, I found a new hero. This hero is the German theoretical physicist who won the Nobel Prize in 1918 for his work on quantum theory. My new hero is Max Karl Ernst Ludwig Planck and let us look at three definitions: Planck's Constant, Planck's Length, and Planck's Time.

We can look to Google for a definition of Planck's Constant:

Planck's constant, symbolized h, relates the energy in one quantum (photon) of electromagnetic radiation to the frequency of that radiation. In the International System of units (SI), the constant is equal to approximately 6.626176 x 10-34 joule-seconds.

Just for the fun of it, let's write this as

0.000000000000000000000000000000000626176 joule-seconds and let me sum up by saying this is a very, very, very, very small amount of energy.

Before going on to Planck's Length and Planck's Time, I would like to point out that Werner Heisenberg used Planck's constant in his equations that correctly predict the probability that a particular electron would be found at a particular place - that is, there was a possibility an electron shot at a pole would hit the pole, a possibility it would miss by a foot, and another possibility it would miss by ten miles. Heisenberg was saying you couldn't know in advance where an electron would be at a particular time. It wasn't a matter of measuring more precisely, it was the result of the wave-like nature of the electron. And now we are saying that iron atoms and even entire one pound steel balls also have a wave-like property. Although the probability is exceedingly small, there is the possibility that the ball we think is here is really over there.

Planck's length is the smallest unit of distance that can have any meaning in the "real" world. Any length less than Planck's length cannot exist because quantum effects dominate. The diameter of a proton is roughly 100,000,000,000,000,000,000 greater than Planck's length. Planck's length is 1.61622837 E-35 meters, which we can write as:

0.0000000000000000000000000000000000161622837 meters.

The word quantum means the smallest possible discrete unit of any physical property. Once you have stated that there is a smallest possible unit of length, Planck's length, there must also be the smallest possible unit of time, Planck's time. This is the time it takes for light to travel across one Planck's length. It might be better to visualize light as making a "quantum leap" across Planck's length - if light did not leap, it would travel a distance less than Planck's length which is impossible because no distance can be less than Planck's length.

Planck's time is equal to 5.39116 E-44 seconds.

Time makes 539,116,000,000,000,000,000,000,000,000,000,000,000,000,000 quantum leaps every second.

Now, finally, we are ready to make an argument about the meaninglessness of calculating PI to a million digits, or even a thousand digits, to the right of the decimal point.

The diameter of any circle is determined by starting at a point on the circumference and drawing a straight line through the center of the circle and continuing until we again intercept the circumference. If, however, we are looking at this process at a super microscopic level, namely, measuring in Planck lengths, we only have probabilities that the line will start exactly on the circumference, pass exactly over the center of the circle, and then stop exactly when the circumference is again reached.

If we imagine that we have created the circumference by laying down carbon (graphite) atoms with a very sharp pencil, we would be trying to connect one atom to the corresponding atom that is on the other side of the circle. Quantum effects plus the wave-like property of the carbon atoms would make determining each atoms precise location impossible. Every time we measured the distance, we could get a slightly different answer. When we calculated the circumference, we would also get different answers and it would not help to increase the precision (number of decimals to the right of the decimal point).

There is another way to get an exact value for PI.

We start by visualizing the smallest possible circle - one with a diameter of one Planck's length. What is the circumference of this circle? The answer is three Planck's lengths. On this scale, the value of PI is exactly 3.00 (If you multiple the diameter by a value of PI equal to 3.14159, you get 3.14159 Planck's lengths for the circumference, but this is clearly impossible - all lengths have to be whole number multiples of Planck's length).

The value of PI seems to remains 3.00 for Planck's length diameters of 2 and 3. On the other hand, if we have a diameter of 4, the impossible results (that is, using PI = 3.14159) is 12.56636. The question then becomes is the "true" circumference equal to 12 or is it 13. And is the "true" value of PI 3.00 or 3.14159?

It seems to me that, when we are dealing with subatomic scales, we must always remember that everything is probabilities. For a diameter of 4, the circumference is slightly more likely to be 13, but there is still a significant chance it is 12. All that extending the precision of PI is doing is letting us look at the probabilities more closely - instead of saying there is a 30% chance of a particular outcome, we can say there is a 32% chance, or even a 32.648% chance.

What happens if we try to calculate, in Planck's lengths, the circumference of a slightly larger circle. Believe it or not, we know the approximate diameter of the observable universe. It is many, many light years, or 2.7199622 E+61 Planck's lengths. This number is not usually written as:

27,199,622,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000.

When I see this number written out, it makes me, reluctantly, think that professionals, with their "scientific notation" might have a point. I can't really visualize E+61 or E-35, but who wants to spend all day writing zeroes.

The thing that makes 2.7199622 E+61 so long when written out is what I want to discuss, namely the 54 zeroes.

I said earlier that I don't know or care if a million billion quintillion is equal to a quintillion million billion. In my student days, when I thought in scientific notation, I knew they were equal. A million is E+6, a billion is E+9, and a quintillion is E+18. A million billion quintillion is equal to 1.0 E+(6+9+18) or 1.0 E+33.

Our 54 zeroes can be expressed as 1.0 E+54 or 1.0 E+(6+6+12+12+18) or, as we like to say, a million million trillion trillion quintillion. We can therefore say that the diameter of the observable universe, in Planck's lengths, is 27,199,622 million million trillion trillion quintillion.

No reasonable person, if they think about it, would ever say this is the exact number of Planck's lengths in the diameter of the universe. The 54 zeroes actually show how uncertain we are of the true value. A better way to state the diameter of the universe is to say it is:

27,199,622 million million trillion trillion quintillion plus or minus one million million trillion trillion quintillion.

If we want to calculate the circumference of the universe, we multiply by PI = 3.1415926 and get:

85,450,131.198 million million trillion trillion quintillion plus or minus 3.1415926 million million trillion trillion quintillion.

Note that we could make this calculation other ways, by varying the precision of the value of PI, multiplying first by 3.14, then, as we just did, by 3.1415926, and then by 3.141592653 (and so on). In each case, we are getting a more exact value, but our uncertainty is also rising from (leaving out million million trillion trillion quintillion) 3.14 to 3.1415926 to 3.141592653.

We can easily calculate PI to the necessary precision, but until we can also determine the diameter of the universe to the same precision, we cannot get a "true" value of the circumference.

The value we need for the diameter of the universe is a number that has 61 significant digits and begins 27,199,622, and continues for 53 more digits with zeroes appearing only when there should be zeroes. We, of course, have no idea how to get such a number.

If we then used a PI with a precision of 62, this should be adequate for any calculations involving distances less than ten times the width of the observable universe. We might treat this PI like we do the speed of light - in some calculations, PI might be less precise than this Super PI, but it could never be greater.

(Super) PI =

3.1415926535897932384626433832795028841971693993751058209749445

Return to INTRODUCTION TO THINKING LONGER AND HARDER. Or, if you liked this, send an email to Mike Stewart. - mike@esearchfor.com