A Sameness Both Bitter and Cruel

Thursday is Science Day here at Frothing-at-the-Mouth.

Today I'm going to talk about a simplifying coincidence in physics that makes a lot of basic physics a lot easier but which has absolutely no reason to be the way it is. That's the statement that gravitational mass is the same as inertial mass. I'll elaborate.

Newton's laws specify, inter alia, and in modern form, that F=ma, that is, force equals mass times acceleration. This equation forms the basis for the branch of physics called mechanics, which is concerned with how objects move or are balanced in equilibrium. Throw a baseball to your buddy and you're doing something described by mechanics. Newton's m, inertial mass, shows up all over mechanics. Kinetic energy, for example, is 1/2 mv^2, where v is the velocity of hte object.

Digression: One thing physicists, or any specialists, do is to create jargon. Jargon has some social aspects--it's used to identify members of the group-- but more importantly, it's the process of assigning very specific meanings to pariticular terms. Ordinary English (or any language) tends to have squishy words that are more general than specific. For example, "energy", in English, means a variety of different things--how much get-up-and-go you've got today, the tone of an advertisement, the nature of your personal relationships. The differences between "energy" and "force" and "power" are contextual. In physics, however, "energy" has an extremely specific meaning. It is a quantity whose units are mass times distance squared divided by time squared; it is the distance integral of force and the time integral of power; it is the sum of potential and kinetic energy; it is the sum in quadrature of mc-squared and c- squared-p-squared (in special relativity). Similarly, "velocity" and "speed" are basically the same thing in English, but they have distinct meanings in physics. So when I talk about "energy" or "force" or other physics jargon, you don't necessarily need to know the specific meaning, but you should know that I am talking about something very specific. Back to the topic.

One of the lines along which mechanics is developed is based on looking at where forces come from. For example, the force a spring exerts is kx, that is, the "spring constant" times the distance the spring has been compressed or expanded. From there, you can get that the potential energy of a spring is 1/2 kx^2, and that if you just let the spring go, it's going to vibrate at a frequency of square-root k/m.

Here's where that simplifying coincidence first pokes its head up. The force due to gravity is mg, where g is the constant gravitational acceleration at the surface of the Earth. The m in mg is the gravitational mass, but that's the same m as in F=ma, so you can see immediately that the acceleration an object feels doesn't depend on its own mass. The object's potential energy is mgh, where h is the height about whatever level you defined as zero.

Take a pendulum. It's moving under the influence of gravity. Because the m in mgh is the same m as in 1/2 mv^2, you get that a pendulum's frequency is square root g/l, where l is the length of the pendulum. It doesn't depend on the mass of the pendulum at all, unlike the spring's frequency.

Basically, because the m in F=ma is the same as the m in F=mg, you get the result that the way an object moves under the effects of gravity doesn't matter at all on what the object is.

So what? It's a nice coincidence, and it makes some math easier, but so what? It can't be a big deal. Except that it is, and here's why: It doesn't have to be that way.

I already gave you a hint of that when I talked about a spring--a spring uses this k number, not m, so the way a spring moves ends up depending on both k and m. But let's go a little more fundamental and jump from the first semester of mechanics into the second semester. Electric charges produce a force between them F=q1*q2/r^2 for q1 the charge of the first object, q2 the charge of the second object, and r the distance between the objects (in a particular set of units in which 4*pi*epsilon-nought is one, but I digress). "Charge" is the particular characteristic of an object that tells you how much it interacts with electricity. Since F=ma, you can see that the way a charge moves relative to another charge depends on the charge of them both and the charge's mass.

Once you get away from the surface of a planet, gravity has a force equation which looks an awful lot like the electric charge equation: the force between two objects due to gravity is F=G*m1*m2/r^2, for G a universal constant, m1 and m2 the masses of the objects, and r the distance between them. But if you're talking about the force on the first object, F=m1*a, and the m1's cancel. The way m1 moves depends only on the mass of m2. This makes orbital mechanics a lot simpler than working with the mechanics of charged particles.

By now, hopefully, you've got the obvious question forefront in your head: Each object has a particular characteristic (charge) which tells us how it interacts with electricity, and a particular characteric (gravitational mass) which tells us how it interacts with gravity. Why in the heck is the quantity that tells us how a particle interacts with gravity the same as the quantity that tells us how a particle moves under a force (force of gravity or any other force)? Why is gravitational mass the same as inertial mass?

It doesn't have to be that way. Electric charge doesn't have anything to do with inertial mass. Neither does the spring constant. Neither does the "color" (which determines how an object interacts with the strong nuclear force). Neither does the "spin" (which is a quantity that varies from particle to particle and affects how it behaves under quantum physics).

There isn't an answer. That's just the way the world works. Physicists hate having to say that--there should always be a deeper answer. We weren't satisfied when we knew how electricity and magnetism worked, we had to show that they were really the same thing; we weren't satisfied then, we had to show that electromagnetism was the same thing as the weak nuclear force. And now we're looking for ways to show that electroweak is the same as the strong nuclear force, because if things are different or the same, we ought to be able to figure out why they're different or the same.

Me, I wonder if the answer doesn't lie in general relativity. In general relativity, gravitational mass is the quantity which tells us how much an object distorts time and space around it. Maybe it's a natural consequence that the way an object distorts space (its gravitational mass) is also the way it moves around in distorted space (inertial mass). I don't know. General relativity is way beyond me, and the people who do know it haven't figured out a connection.

Ultimately, right now, we're stuck with "It's a coincidence. That's just the way the world works." It happens to be the case that this particular m quantity characteristic of an object determines both how it moves under force and how it interacts with gravity, regardless of how weird it seems to have a number do double duty like that.

But someday, we'll figure out why.

Posted by Greg at December 5, 2002 2:06 PM