I recently learned about a great blog by S.C. Kavassalis of the University of Toronto called The Language of Bad Physics. She discusses, among other things, the way language is used in physics. She’s got an interesting piece on the use of the word “theory”. This is always a hot area of discussion, but in physics it has particular resonance because so many non-physicists like to come up with their own “theories” about how nature might work.

I put “theory” in scare quotes not because amateurs can’t make contributions to physics – they can and do – but because there’s a heck of a lot of cranks out there with theories that aren’t actually theories. In physics, if you want to come up with a theory at minimum it has to:

1. Generate numbers.

2. Match those numbers consistently with observation.

This is a guideline rather than a formal definition and there’s a little semantic wiggle room that I don’t pretend to have fully captured. But the idea is that for a theory to be worth anything it has got to accurately describe what nature does. If you’ve decided that, I dunno, “the proton mass and the quark mass originate from a gaseous cloud of micro-particles“, then that’s lovely but how exactly do you propose to tell if the theory is right? Unless there’s something *measurable* it describes (and describes to greater accuracy than current theory, if applicable), you haven’t accomplished anything. Let me give an example of how a person – in this case a fictionalized Isaac Newton – might have an idea and then test it numerically.

Newton had the idea that everything with mass is attracted to everything else with mass. That’s nice – it might even be true – but in and of itself that idea isn’t worth much because it doesn’t predict anything. So Newton of course didn’t just say that masses attracted each other, he expanded on his insight mathematically and guessed that they did so with a force in proportion to the product of their masses divided by the square of the distance between them.

Where M is the mass of one object, m is the mass of the other, G is the gravitational constant, and r is the distance between them. (This is specific to point masses, but Newton showed that spherically symmetric objects obey this equation exactly if you take r to be the distance between their centers of mass . Other objects can be treated as a collection of many tiny point masses and their gravitational influences added up via calculus.)

So what does this predict? Knowing that a force F on an object with mass m produces an acceleration by F = ma,* Newton could say that an object on the surface of the earth would experience an acceleration equal to

Where M is the mass of the earth and r is its radius. Now we know by experiment that the acceleration at the earth’s surface is 9.8 m/s^2. Newton knew this, but he had no way of knowing G or M. Still, he did know r (about 6400 km) and so he could at least solve for the product GM. If you do this, you’ll find GM = 3.99 x 10^14 m^3/s^2, which is some wacky units but that’s ok.

But so far Newton hasn’t really tested his theory. He has calculated what GM must be for his theory to work for objects on earth, but there’s no reason to believe it’s actually a description of physical reality since you can fit pretty much any screwball idea to one data point. He needed to take his theory and see if it worked in other contexts. So that’s what he did. Newton knew that the acceleration required to keep an object moving in uniform circular motion is equal to

We won’t pause to prove it – the curious can Google “uniform circular motion” – but it can be mathematically proved in a straightforward way. Therefore if Newton’s idea about gravity could also work to describe things in a circular orbit, he could equate the two above equations to find the velocity of an object in orbit about the earth with an orbital radius r:

Newton wanted to see if this was right for the Moon. Knowing that velocity is distance divided by time, it’s easy to see that the Moon’s velocity v = 2*pi*r/T, where 2*pi*r is the circular distance it covers in one orbit and where T is the time of one orbit. Solving that for T, we get:

So we can plug in Newton’s previously obtained GM, and the known orbital distance of the Moon (about 384,000 km). That distance, by the way, had been measured with surprisingly good accuracy as far back as the ancient Greeks. Plugging in, we get

T = 27.4 days.

The Moon’s actual orbital period? The well known 27.3 days. In my simplified version of the story I’ve neglected the fact that the Moon’s orbit isn’t actually quite circular and I’ve neglected the effect of the Moon on the earth. Newton neglected neither, of course.

So Newton showed that if his theory described the gravity of the earth on you and me, then the moon’s orbital period would have to be a certain value. If it had been anything else, his theory would have had to be scrapped. If, for instance, his idea had involved a GMm/r^4 force then the two values would have been wildly different.

This is in spirit the way every theory should be developed and checked. The theory has got to generate testable numbers, and those numbers have to pass the test of matching with observation. Indeed this was only the first rather than the last test of Newton’s idea, with people like Cavendish testing it on laboratory-scale masses and so on. It passed every test until eventually Einstein predicted some slight deviations under certain circumstances, and of course Einstein’s theory then had to be tested to see if it matched observations. So far it has every time, though it may be that a yet more precise theory will be developed someday.

So keeping this in mind, theorize away! But do it with math, and check it against measurement.

*In this context this is a definition of force rather than an independent physical law.