PMU199S notes

Scale and powers of ten

The first half of the first lecture is taken up with course organization matters. The second half is where we start introducing physics ideas.

Why size matters

Physicists have a great job: we find an interesting system (shiny object) to study, then we poke it and see what happens. What most fascinates us is how properties of the system change over time and across space. For instance, the patterns of winds on planet Earth.

There are three principal aspects of the modern physics enterprise: measurement a.ka. experiment, mathematical modelling a.k.a. theory, and computing, on personal computers or supercomputers. Physics as a structure is a bit like a three-legged stool, in that it requires all three legs to be structurally sound and to work together harmoniously. Different types of physicists specialize in different areas.

In my own work, I do theoretical physics research using pen and paper and on personal computers, performing a lot of symbolic algebra manipulations and some numerical computations as needed. My specific research interest is in string theory, but most of the course will not be about that specifically. My aim here is to give you a flavour of modern physics as a whole without making you suffer, mathematically or otherwise.

Physicists start analyzing a system by first determining its size. We do this for a very simple reason: size is really important to what tools a physicist should use to poke it and measure its response. For instance, if you wanted to do brain surgery, you would pick precision brain surgery instruments with delicate tips, rather than an axe, a chainsaw, and an industrial drill press. Picking the right-sized tool for the job in physics is just as crucial: it can mean the difference between making an original discovery and not making one. Precision is important to being able to discern patterns.

A few images of different measuring apparatus commensurate with different distance scales.

Powers of 10

Because we are interested in physical principles operating over a wide variety of length scales, physicists need to be able to discuss ultra-teeny microscopically small numbers with lots of decimals -- and huge macroscopically large numbers with loads of zeroes -- and somehow keep our paper consumption to a minimum! It turns out that there is a really smart way to do this: use a concept called powers of 10. The basic idea of powers of 10 is that every time we zoom in/out by a factor of ten, we lose/gain one in the power of 10. We also define $1=10^0$. Let us do a couple of examples to illustrate this. Suppose that we zoom in from one metre by a power of ten. That would take $10^0$m to $10^{-1}$m, i.e., one tenth of a metre. Or suppose that we zoomed out by three powers of ten, in other words by a factor of a thousand ($10\times 10\times 10$). That would take $1$m$=10^0$m to $10^3$m, commonly known as $1$km. In the SI metric system, zooming in or out by factors of a thousand is so handy that those powers of ten get their own SI prefixes.

The powers of 10 notation really comes into its own when we are talking about extremely small or extremely large numbers. The smallest distance ever measured by humans is about $10^{-18}$m. The very edge of the visible known universe is about $10^{+25}$m away. If we did not use powers of 10 to write them, those two numbers would be $0.000 000 000 000 000 001$m and $10 000 000 000 000 000 000 000 000$m respectively. As you can see, both are pretty cumbersome to write longhand. The powers of 10 notation is compact and powerful.

Powers of 10 are illustrated very nicely in a classic 9-minute Powers of Ten movie by Charles and Ray Eames. It is really helpful to watch, to help you get your head around how big the universe is and how small subatomic particles are. Here are a few samples of what the world looks like at bigger or smaller length scales than ourselves:-

Some examples of what the world looks like at different length scales

NASA has a beautiful collection of pictures of Earth from space which you may want to peruse.

What do physicists seek?

There is an old science joke that goes: sociology is just applied psychology, psychology is just applied biology, biology is just applied chemistry, and chemistry is just applied physics. This makes physicists feel warm, fuzzy, and important. (Mathematicians and philosophers think they are even more pure than physicists. The cheek!)

[Pictorial representation of the above joke.]

But the real moral of the story is that different tools are appropriate to different length scales -- e.g. societies, mitochondria, or Higgs bosons. Humans are optimized for mm to km distance scales. If we want to go smaller or larger, we need microscopes or telescopes, like the Large Hadron Collider near Geneva or the Hubble space telescope in orbit.

Physicists like me want to explain the structure and origin of particles, forces, and spacetime, all the way from subatomic to cosmological scales. In other words, we dare to seek the operating system of the entire universe at once -- not just a killer app! What dynamic range is involved? So big it pushes the limits of human imagination: about sixty powers of ten ($10^{-33}$cm to $10^{+28}$cm). In musical terms, this would correspond to ranging over about two hundred octaves. For comparison: the Guinness world record for human vocal range is ten octaves.

Building a theory in physics is a bit like wiring up a mixer. Dials and sliders on your dashboard indicate how strongly the various components interact; physicists call them couplings.

[picture of sound mixer with lots of sliders, inputs, and outputs.]

Subatomic structure

Have you ever wondered why the atomic nucleus does not explode? Inside the nucleus there are positively charged protons and electrically neutral neutrons. The problem is, if you have an atom of (say) Carbon, which has 6 protons, then the protons in the nucleus are all repelling each other electrically, because like charges repel. This is just static electricity like happens to your hair after you take your hat off in winter. Now, since the nuclei in our bodies are not constantly exploding, there must be another force so strong that it overwhelms the electric repulsion of the protons in the nucleus and binds them together tightly inside the tiny nucleus in a stable way. This other powerful force is called the strong nuclear force. Its signature properties are that it is strong and short range -- confined to nuclear scales.

[picture of sound mixer with lots of sliders, inputs, and outputs.]

What other forces operate in the universe? You already know about gravity, which keeps your feet on the ground and Earth in orbit around the Sun. You also know about electromagnetism, which makes your radio go and which moves a compass needle during a lightning storm. We just talked about the strong nuclear force that holds atomic nuclei together. That makes three so far. Are there any more? As far as we know, just one. The fourth force humans study, apart from strong nuclear, electromagnetic, and gravitational, is known as the weak nuclear force. This ultra-short range force drives the fusion reaction that powers our Sun, and is also responsible for aspects of radioactivity. We will say more about all four forces when we discuss spacetime, particle physics, and big bang cosmology later in the course. This is just an aperitif to whet your appetite.