Last year, Jim Al-Khalili published a new book called Paradox. I’ve been a fan of his books ever since I read Black Holes, Wormholes and Time Machines as a teenager, and since I’m interested in the topic of this book, I picked it up.
The title of the book is a bit of a misnomer. A paradox is an argument that leads to an inescapable contradiction, but most of the ones presented in the book are not truly paradoxes – they have been resolved. Once the premises are analyzed, the paradox untangles and the apparent contradiction is resolved. The book actually explores questions in physics that began as paradoxes, but more research or better theories have since put them to rest. Al-Khalili eases into these questions in the first chapter by discussing logic problems that demonstrate this false-paradox concept, but don’t require any physics knowledge to make sense of.
Al-Khalili examines nine classic physics paradoxes, but I’ll skip ahead to talk about my favourites.
Chapter 5 – The Ladder Paradox
In chapter 5, he deals with the Ladder paradox (or Pole in the Barn paradox). This paradox is surprisingly obscure: while being a great model for puzzling out the implications of special relativity, it’s rarely encountered outside of textbooks. Here’s how it goes: imagine someone running toward a garage near the speed of light, carrying a ladder that is normally too long to fit in the garage.
Ladder does not fit in the garage
Source: Wikipedia
The first surprise, for someone new to relativity, is that from the perspective of someone inside the garage the ladder fits in the garage. Here’s why: when the ladder’s speed approaches the speed of light, relativistic length contraction comes into play. The ladder, as seen by someone inside the garage, is shorter, and will be able to fit inside the garage.
The ladder, as seen by someone inside the garage, is be able to fit inside the garage
Source: Wikipedia
The second surprise – and our ‘paradox’ – is how all this appears to the runner carrying the ladder. To the runner, it is the garage that appears to be moving at near-light speed, and thus from his perspective the garage’s length is contracted. This means that the garage is even smaller now, and from the runner’s perspective, there is no way that the ladder can fit inside the garage.
From the runner’s perspective the garage size is smaller, so there is no way that the ladder can fit inside the garage.
Source: Wikipedia
How do we reconcile these two perspectives? Does Special Relativity really have such a glaring internal contradiction? The answer is no. In fact, Special Relativity itself gives us the resolution to this apparent paradox. However, to explain that, a few basic concepts in relativity need to be illustrated first, which Al-Khalili does clearly.
When talking about this paradox, Al-Khalili focuses on the effects of relativity on length, leaving time to be the focus of the next chapter. In relativity, time and space are joined together into ‘spacetime’, which points to the fact that time isn’t a constant clock independent of what occurs in space. Rather, time and space are intertwined aspects of a single entity and they do affect each other. Objects moving at near-light speeds appear to be shorter, and time progresses differently for observers moving at different speeds. Al-Khalili’s reasoning for not bringing up time is that time is a hard enough concept to grasp without subjecting it to the abuse of special and general relativity; space is more comfortable, so he starts there.
His plan fell through a little since the idea of simultaneous events is critical to the resolution of the ladder paradox and ends up creeping in anyway. But, whether it was intentional or not, I think that seeing time sneak into what initially seems to be a length-related problem subtly provides a taste of what dealing with relativity is like. As for the resolution, you’ll find it on page 89.
Chapter 6 – The Twins Paradox
The next chapter deals with the well-known ‘paradox’ of The Twins. Consider a pair of twins: one stays on Earth and the other travels for some time in a rocket at near-light speed and then returns to Earth. When the twin returns, he finds that he has aged less than his brother, who stayed on Earth[ref]The exact age difference would depend on the speed, trip length and exact path taken.[/ref]. No problems so far: when traveling at near-light speeds, a dilation of time occurs. In this situation it results in the traveling twin experiencing time moving much slower compared to his twin on Earth. By the time the traveler returns, far less time has passed from his point of view, explaining why he’s aged less. The problem is that this story only seems to make sense from the perspective of the twin on Earth. From the perspective of the twin in the rocket, it’s the twin on Earth who is moving away at near-light speed; so he could conclude that it’s his twin who will be younger when they meet again. Only one of the twins can be right. It turns out that in his explanation the twin in the rocket missed something. What he missed, and why this is yet another non-paradox I leave for you to discover in the book.
As interesting as the paradox of the Twins is, my favourite part of the chapter was Al-Khalili’s presentation of some related relativity concepts. For example, many people, while aware that time is strange in special relativity, have never heard of how general relativity brings in a new way of making it speed up or slow down: massive objects distort the passage of time for nearby objects. As a result, the closer someone is to a very massive object (like a planet or a star), the slower time passes for them. An observer standing at the base of a tall building on Earth will observe her clock running more slowly than a clock at the top of the building. This is because the clock at the top of the building is further away from Earth (a massive planet), thus lessening the time-distorting effect of the planet’s mass.
Chapter 8 – Laplace’s Demon
Chapter 8 is a discussion about determinism and free will, emerging from the concept of Laplace’s demon. This is a demon that has complete understanding of the laws of physics, knows the exact location and state of every particle in the Universe, and has the vast computational ability necessary to work out, at any given time, how the Universe will evolve based on the laws of physics.
The paradox Al-Khalili presents here is that such a demon could decide to do something that changes the future as he predicted it, therefore undermining his own ability to predict the future. There are many ways to side-step this paradox. Although Laplace’s demon is an interesting concept, treating it as a paradox feels a little forced. The real problem Laplace’s demon demonstrates isn’t really paradoxical: it’s about whether our present understanding of physics leaves room for free will. This leads us to appeal to concepts like chaos and quantum uncertainty to look for ways out of the unyielding determinism of classical physics. Al-Khalili does an excellent job of presenting these ideas and tying them together with the concept of an all-knowing entity – making for a great chapter – but I think that the concept had to be stretched to fit the theme of the book[ref]By the way, there is a great short story I recommend by Isaac Asimov called “Alexander the God” that features a Laplace’s demon-like computer.[/ref].
Chapter 9 – Schrödinger’s Cat
Chapter 9 is about the famous thought experiment, Schrödinger’s cat. Quantum mechanics, one of our most successful theories, seems to demand that the cat in the box be in this bizarre state of being both alive and dead. While we might bring ourselves to be comfortable with tiny particles like electrons occupying these strange states, it’s difficult to accept that a macroscopic (and conscious!) entity like a cat could be in a smeared-out state like that. So the ‘paradox’ is that quantum mechanics prescribes this situation for the cat in the box, but our macroscopic, real-world intuition tells us that the cat surely must have a definite, live or dead state at all times (whether we’re looking or not).
The Schrödinger’s cat concept has seen a lot of action Schrödinger mentioned it in passing in a 1935 paper. It’s the stereotypical example used to illustrate ‘quantum weirdness’ to the uninitiated. In the context of Paradox, it’s mainly used to illustrate the Measurement Problem in quantum mechanics.
In essence, the Measurement Problem is that we’re not sure what is physically happening when we make a measurement. Before the cat goes into the box, we say that the cat is alive with 100% probability. Inside the box, what we know about the cat becomes fuzzy: in quantum mechanics terminology, we say that the cat is in a superposition of states. One state is dead; the other is alive. Opening the box forces the cat into one state or the other, and we once again are 100% certain what the cat’s state is (alive or dead).
Several questions arise. At which point did the cat switch from having a definite live state (before closing the box) to being in a fuzzy in-between state? Similarly, when exactly did it return to having a definite live or dead state when the box is opened? What is the mechanism that changed the cat’s state in this way? Human observation? If so, is it consciousness doing the work? Intelligence? Or maybe the cat had a definite state all along, and opening the box just revealed what was already there. In that case, at which point (if any) was the cat in a superposition, and what mechanism made that happen? Quantum mechanics, at least in its basic, functional form used in most textbooks, is evasive on this point.
Much discussion (perhaps even progress!) has occurred since these problems first started being discussed in the 1920s and 30s. By now there are many different interpretations of quantum mechanics which differently answer the Measurement Problem. Each of them agrees experimentally with quantum mechanics: what we actually measure and observe is the same. The difference is in the explanation they offer for what is ‘actually’ happening behind the scenes and how our information and knowledge come into play.
The fundamental point is that the measurement problem (and, as a particular application of the measurement problem, the Schrödinger’s cat paradox) can’t really be solved within basic, vanilla quantum mechanics. Al-Khalili discusses some of the intricacies of dealing with Schrödinger’s cat, and presents some of the recent developments on that front.
I enjoyed Paradox on the whole, though a bit less than “Black Holes, Wormholes and Time Machines”. There’s an inherent difficulty in trying to cover nine challenging topics in physics in a mere 250 pages. I think Al-Khalili did an excellent job of giving just enough information to make each ‘paradox’ comprehensible – but some of the explanations of how they are resolved might be confusing to those new to the topic. I don’t think that’s necessarily a bad thing – there are quite a few books out there talking about each of the subjects Al-Khalili discusses in detail. The confusion may lead to interest in picking up one of those books to learn more. In that sense “Paradox” is a good starting point for someone interested in learning a bit more about physics. It may not have much new information for physics students and enthusiasts, but it provides an interesting perspective.