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Special Relativity:
Why Can't You Go Faster Than Light?

An Essay By W. Daniel Hillis

Charles Simonyi, Lee Smolin, W. Daniel Hillis, and Phil Anderson on Why Can't You Go Faster Than Light? by W. Daniel Hillis

From: Charles Simonyi
Date: 3-9-99

I recognize the importance of popularizing science and Danny Hillis' patient explanation how special relativity works is well written and well reasoned for its audience.

However it reminded me of my favorite complaint about the use of notations when addressing lay people.

I am not referring to the idiosyncratic *2* notation although **2 would have worked just as well and would have been much more standard.

I am talking about the use of decimal notation (apparently one of the greatest inventions) when discussing very large or very small numbers. It is only marginaly better than what I call the abominable "unary" system of writing "9 hundred million trillion billion million" and similar stuff.

What's wrong with writing 90,000,000,000,000,000 m or 0.0000000001 grams ? One is that the relationship is hidden. Probably because the relationship is hidden, it seems to be also wrong. I count 16 zeros in the c**2 number, that is fine, since 3*10**5 /km sec or 3*10**8 m /sec, if squared gets us 9*10**16. To invert this I get 1/9 * 10**(-16). OK, so 1/9 is like .1, so we have 10**(-17). Multiply this by the 10**6 Joules in the example, we get 10**(-11) kg, or 10**(-8) g. We seem to be off by a factor of 100. If I am wrong, the article does not help me see where my error is.

I am also a little skeptical of putting a million (OK to use the word here, I think) Joules into a battery. A Joule is a watt for a second. A million Joules will keep a thousand watt oven hot for 1000 seconds, more than a quarter of an hour. I would guesstimate my (car)battery would last maybe 100 seconds at most when connected to an 1KW oven (of the right design). Or, 1 Joule could operate a lift of 1 Newton force for 1 m. That is about 1/10 kg. So 1 MJoule could lift a 1000kg car vertically 100 meters. That is a lot for a DieHard, although electric cars need quite a bit more than that.

Mind you, I am not criticising mistakes. Frankly, I am more likely to be wrong here with my guesses and random recollections of physics I've heard 35 years ago than Danny Hillis with his carefully written note. But I could not tell whether I am right or wrong from the article.

I am critical of the widespread pedagogical approach of effectively considering the lay audience as idiots who could not possibly understand exponential notation. We think they are interested in relativity, in energy, they want to know about Joules, about kilograms. But to say to them ten to the sixteen? The editor will slap your hand right away (a slap is about 1 Joule.) By the way, contrary to popular belief, calculators do not work with such large or small numbers, and even if they did, inputing numbers by repeated pushing of 0's would be not only error prone but also a return to the primitive times when people kept "score" by marking each unit individually. I've heard educators saying "People use computers and calculators these days". This is a red herring at best.

I think authors writing to lay audiences about big numbers owe their audience a line or two on exponential notation. ("one followed by sixteen zeroes" is commonly said). Also when units are introduced, the examples should include something that is the right size for the units. But most of all, we must have respect for the lay audience who have been abused (dumbed down) for too long by educators and editors.

From: Lee Smolin
Date: 3-9-99

I very much admired Danny Hillis's book on computers, so it is only with some hesitation that I express my opinion about his pedagogical article about relativity. But, I have to say that, as a physicist who works with relativity and a teacher who has often taught it to non-scientists, I liked it less. What he says is very clear, but the problem is that he gives the illusion of explaining something when what he really does is replace one counterintuitive fact-that there is a speed nothing can exceed- for another, that mass increases with velocity. What Hillis does is show that the first is not quite so surprising if you know the second, but he really does not explain either. This is because he avoids mentioning the essential facts and ideas which make the whole theory simple and easy to understand.
This is an example of a problem, not just with a few, but with most attempts at physics pedagogy. This is to try to teach physics without requiring the student or audience to think. What makes physics really interesting is that several of its central discoveries contradict our ordinary intuitions. These are replaced by new ideas, which turn out to be much closer to the truth. Once one knows them the new ideas are just as intuitive as the old ones. This is why physics is much easier than it looks, once one knows relativity it is easy. But the process by which one replaces the old intuitive idea by the new one cannot be avoided if one is to reach the stage where one genuinely understands physics. This is a step which in my experience as a teacher is accessible to anybody who wants to take it. At the same time it is not easy, in the sense that it requires concentrated thought.
Think of learning to ride a bicycle. It is definitely counter-intuitive that a two wheeled object will not fall down. One tries it and learns it works, and in the process learns to trust that riding a bike is safe. Now think of learning why it is safe. One cannot avoid learning a new concept, angular momentum. Fine, but to really understand this one has to understand why there is a quantity like angular momentum that is conserved. Memorizing the formulas for it cannot substitute for the actual understanding of why there is a directional quantity that never changes. It turns out there is a reason, but I cannot imagine most students would ever guess at the right reason, for it represents one of the great, and usually unmentioned, discoveries of physics. I will say it, but I cannot explain why in this short space. The reason angular momentum is conserved is that the laws of physics do not pick out any preferred direction of space.
The connection between symmetries and conservation laws is one of the great discoveries of twentieth century physics . But I think very few non-experts will have heard either of it or its maker — Emily Noether, a great German mathematician. But it is as essential to twentieth century physics as famous ideas like the impossibility of exceeding the speed of light.
It is not difficult to teach Noether's theorem, as it is called; there is a beautiful and intuitive idea behind it. I've explained it every time I've taught introductory physics. But no textbook at this level mentions it. And without it one does not really understand why the world is such that riding a bicycle is safe.
Unfortunately, physics pedagogy is full of cases like this, which is why, I believe, physics is so disliked by students. Most introductory textbooks, as well as popularizations, do not sufficiently stress the key insight of classical physics, which is the relativity of inertial frames. This is the idea that velocity is a purely relational quantity, dependent on the observer. There is no notion of velocity except relative to what an observer sees. Nor is there a notion of being at rest. Without this basic insight nothing in physics is comprehensible. This includes the notion of mass.
What is mass? We have some kind of intuition that it is a measure of "stuff" or "resistance to motion". But none of these captures it. An essential insight of Newton and contemporaries is that there is a conserved quantity of motion, which is momentum. Mass is nothing but the ratio between this quantity and velocity.
Thus to understand mass you have to understand momentum, and why it is conserved. This also requires Emily Noether's insight, that conserved quantities have to do with symmetries of natural law. In this case the symmetry is that one location in space is as good as another.
To understand relativity one needs to think about this and then ask what happens to the measure of mass when different observers look at an object in motion. Is there any reason for them to agree on the ratio between the momentum and the velocity?
The answer turns out to be no. Moreover, it turns out they cannot in a world where light travels at a speed which is independent of the observer. This is Einstein's central discovery.
I will not go through the steps of how one gets from here to the fact that there is an ultimate speed, I have already gone on too long and there are many good books for non-scientists that walk you through it.
I have only gone on this long to make a point: relativity can be understood by anyone. But nothing important in physics can be understood without going through a mental process which begins with an insight that one's thinking is based on an idea which is intuitive but still might be wrong. One then goes through a process in which one reasons one's way to a new intuitive idea that contradicted the old one, but is much better supported by the evidence. One cannot learn physics, at any level, without going through this process.
It is a bit like learning to draw. Anyone can learn to draw, I am told, but it takes some hundreds of hours of practice. Anyone who wants to can learn the basic insights that physics is based on. The problem is not math, the math can be skipped. But the process of thinking cannot.
My impression as a teacher is that attempts to teach physics as a set of cool facts, without taking the time to walk the student through the journey that is required for real understanding, in the end mystifies it. Most physics textbooks make this mistake; because they seem more interested in producing students who can do well on tests than students who genuinely understand the great insights that our science has brought to light.
This means that those of us who popularize physics have a non-trivial task, for we really want to explain, but we do not want to turn off our audience. I am not saying that Hillis does worse than most people; I think he does far better than many, including almost all textbook writers. He writes wonderfully clearly and I hope that his essay will inspire many people to be interested in relativity. But if they want to really understand why there is an ultimate speed they will have to dig a bit deeper.

From: Danny Hillis
Date: 3- 25-99

In commenting on my essay "Special Relativity: Why Can't You Go Faster Than Light?", Lee Smolin points out that I made no attempt at explaining many of the key aspects Einstein's theory of special relativity. I plead guilty. As Smolin says, really understanding a theory of physics requires some real work. The problem is that most people are unwilling to go do that work until they at least get a taste of what there is to be understood. I remember once as child seeing a mummy in a museum. I was so interested in it that I wanted to become an archeologist. I began to learn about Egyptian history, but not by looking at the mummy. I learned by reading and asking questions. A short science essay, like mine on relativity, is like a museum exhibit. It can display a curiosity, and maybe even teach you something about it, but it is not a replacement for the hard, fun work of real learning.

- Danny Hillis

From: Philip W. Anderson
Date: 4-1-99

Hooray for Lee Smolin's comment! I have thought this about physics pedagogy for decades without putting it into words so well. Physics is an intuitive, not a mathematical, science — one of the reasons why so many physicists do it so badly, because they too have been taught in this conventional way.

- Phil Anderson

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