Who was this Adventure in Space-Time Written For?
The challenge of writing any popular science book is that the audience has different levels of knowledge. The author needs to choose the appropriate level of knowledge to aim the writing at. It follows that the reader’s appreciation of the book depends on what they know. To understand my perspective, you should know my background:
A long time ago I completed first year university science before switching into computers. I have since read a number of popular science books on relativity and quantum mechanics. I am presently taking on-line university physics courses to get a deeper understanding. I have thus seen many of the ideas in this book before, and understand the mathematics.
The book begins with the claim, “We do not assume any prior scientific knowledge.” While the language is often simplistic, and they even explain the meaning of a divide sign, I think this is a compromise that does not really work. Frequently they give a hokey apology (I got very tired of chalk dust) and then present complex mathematics in prose. Important steps are often skipped. I can tell you that I had to read much of this book several times, and do the math on paper to try to understand it.
Electricity and Magnetism Dance at the Speed of Light
Einstein’s insight into the special role of the speed of light derives from Maxwell’s equations of electricity and magnetism. Following Einstein, the authors devote a chapter to this subject. This is fine with me, because I am taking my electricity course for precisely this reason. Briefly, experiments in the 19th century showed that a moving magnetic field generates electric current, and an electric current generates a magnetic field. These mutually reinforcing forces create an energy wave, which can be described by a wave equation. This equation gives us the speed at which the wave must travel, which turns out to be the same as the speed of light. Thus light is just another type of electromagnetic wave.
Einstein assumed all movement is relative, and that the laws of physics are always the same no matter what is perceived to be doing the moving. Maxwell’s equations give us the speed of light without mentioning the motion of the source or the observer. His genius was to take this result literally, and conclude that the speed of light must always be constant no matter how it is measured.
The book claims (on page 27) the speed of light is determined by the ratio of the strengths of the electric and magnetic fields. But peeking into the actual wave equation, I see that c = 1 / √(ε*μ), where ε and μ are the electrical “permittivity” and magnetic “permeability” constants respectively. That looks like a product to me, which makes intuitive sense if they are reinforcing each other. But this stuff is over my head, so did they oversimplify, or do I have it wrong?
Riding the Relativity Railroad with Pythagoras
Now we get to the classic thought experiment of measuring the relative passage of time on a train from the perspective of an observer on the platform. The passenger measures time by shining a light from one position to another one meter above it. The observer on the platform sees the light taking a longer path because the light pulse is moving along with the train. As the speed of light is constant, the passenger’s clock must therefore appear to be running slower. We can use the Pythagorean theorem to calculate the rate at which time appears to slow down, known as time dilation. While it is cool that such basic math can be used to derive relativity, for some reason the author chooses not to call this by its usual name of Lorentz factor.
This stuff seems to make sense when I read it, then I wake up in the middle of the night and it does not make sense any more. For example, I wondered if the above result is only true at the exact moment when the train passes the observer. No, it turns out that time dilation does not depend on the direction of motion. This is still not obvious to me, and I could have used an explanation in the book.
I find it easy to get confused about which clock is running slower. I have to remind myself of (what I call) the Spoiled Princess Principle: You are the center of the universe. You do not move, everything else moves relative to you. Your clock always runs at the same rate. It is everybody else’s clock that is different.
Stretching time does not seem as strange as the fact that space contracts in front of you when you are moving. The example he gives is that if we go fast enough, we can get to the Andromeda galaxy, which is three million light years away, in fifty years of our time. Are we going faster than light? No, my spoiled princess, remember that is only how it appears to you. How it looks to the people who sent you is coming up next.
According to the equation, light, which goes the speed of light, takes no time at all to reach its destination. I don’t know why they don’t talk more about this. It suggests that a photon traveling between you and Andromeda is everywhere on its path simultaneously. Does this not mess with causality? Is there any connection with wave nature of particles in quantum mechanics?
The Imaginary Single Speed in Minkowski Space-time
I was aware that travelling in space means that you also travel in time, and nothing can go faster than light. But the introduction of Minkowski space-time was a revelation. It begins with the observation we just made that distance in space changes depending on the relative motion of the observer. We can add a fourth dimension in time such that some quantity (distance “s”) is invariant no matter what speed the observer is moving. This new dimension is multiplied by the speed of light (c*t), so it becomes expressed in the same distance units as the other dimensions. But if we assume this is a Euclidean space, so that the Pythagorean triangle relationship
s^2 = x^2 + (c*t)^2
holds, we can get events that finish before they start. This violates causality, which means this model does not work. He then tells us “there is no other option” than to assume that
s^2 = x^2 - (c*t)^2
Not true, it is only one of many possibilities. But, of course, he knows the answer in advance. He then constructs a hyperbolic curve to demonstrate this relationship. It solves the causality problem, but visually the arithmetic is clearly false.
I had to hunt around to discover that Minkowski space-time uses an (unfortunately so-called) “imaginary” time dimension, meaning a multiple of the square root of -1. Squaring it gives a negative number, hence the minus sign. Now it makes mathematical sense, but why did he not explain this? I still do not understand why, given that it is the time dimension that is “imaginary”, he subtracting the space dimension instead.
Anyway, the amazing result is that the speed of light is not just a limit, it is the only possible speed! When we think we are standing still, we are zooming through the time dimension at the speed of light. When we move in space we have to slow down our movement through time to compensate.
The Mystery of the Time Travelling Twins
I think the twin paradox is the key mystery in special relativity. One twin takes a round trip to Andromeda on a fast spaceship while his sister remains behind on Earth. When the travelling twin returns, he thinks it took one hundred years, but the Earth, including his sister, is now six million years older. The apparent speed-up in space is compensated by a vast increase in elapsed time, at least from someone’s point of view.
After all the work we have done, I thought I was finally going to understand it. But at the end we are told the whole calculation fails because it does not take into account the acceleration required to turn the spaceship around. How can he do this to me?
Lets jump ahead to the final chapter on General Relativity. We learn that acceleration and gravitational attraction are equivalent. We also learn that clocks run faster in weaker gravitational fields. This must mean that clocks run slower under higher acceleration. Is this why the travelling twin’s clock slowed down while he was turning around? Then why did he not make the connection?
Well, lets go ask Mr. Internet. I’m back! Yikes, you would think this would have been figured out by now, but apparently not. The most common view is that, contrary to this book, the acceleration is irrelevant. Apparently the special relativity formula is correct, and there are ways to explain it with simple math. It would have been nice if he did this for us. So then, why do the effects of General Relativity make no additional difference? Ah, forget it, nobody answers my dumb questions.
So wow, we can go anywhere in space as quickly as we want, and travel far into the future! The problem is that we are not photons – we have mass. Why does he not mention that it takes an ever increasing amount of energy to accelerate mass? Near the speed of light the energy required becomes infinite. I think of it as all the energy is going into moving in time rather than space, but I can’t connect this thought to what I learned about Minkowski space-time. Anyway, don’t expect to book your sci-fi trip anytime soon. Lose some weight first (like all of it).
The Energy of Momentum in Space-Time
To introduce the concept of momentum, we get a crash course on vectors, mass, Newton’s second law, and energy. Then we realize that a momentum vector in space does not work in relativity, so we must find an equivalent in space-time.
The now familiar distance vector in space-time is transformed into a momentum vector, which is mass times velocity. To get there, we multiply by mass and by the velocity of light, then divide out the distance. Figuring this out from the text took a lot of work on my part. By trying to simplify, he actually made it a lot more difficult than it needed to be. In the end, we get of the standard momentum component (mass * velocity) in the space direction, and mass times the speed of light (m*c) in the time direction, both multiplied by the Lorentz factor (γ).
Because the total momentum must be conserved, the time direction (γ*m*c) must also be conserved. Therefore γ*m*c^2 is also conserved. Now he makes an approximation for the Lorentz factor, which is reasonably accurate at low speeds. The result is:
m*c^2 + (1/2)*m*v^2
This is interpreted as the sum of the energy contained in the mass (m*c^2) plus its kinetic energy. At zero velocity, we still have energy in the mass. We have just achieved the stated goal of the book. But what happens if we don’t cheat on the math? I can’t help thinking there is an important consequence at high speeds, such as what happens when you accelerate mass?
A Good Learning Experience with Room for Improvement
After proving that E = m*c^2 he cannot resist going on to presenting the Entire Standard model, with its master equation that shows how every particle reacts with every other particle. So far I have been feeling sorry for the scientifically challenged trying to read this book, but now I experienced being one of them. The verbal description made sense, but without knowing what the symbols or operators mean, constantly referring back to the equation just got in the way.
I think this book could have been better if they accepted they are writing for two different levels of knowledge. The equations could then be presented properly in a text box, accompanied by a text description of what they mean, followed by a philosophical summary for the non-mathematical reader. Better still, they could provide supplementary material on the web to answer my questions.
I suppose this review is really a long confession of ignorance. I learned a lot from this book, and more from the research to make sense of it. While I still don’t completely get it, this journey into the mystery of space and time was well worth it.