We all know the (probably apocryphal) story. Galileo Galilei, all around physics bad-ass, went up to the top of Leaning Tower of Pisa and dropped stuff off the top. He found that objects of vastly different weights, like bowling balls and feathers for example, would fall at exactly the same rate and hit the ground at exactly the same time. Air resistance gets in the way, of course. But if you perform the experiment in vacuum, as these guys did, then you do find the bowling ball and the feather land at exactly the same time:

This leads to a fundamental truth we’ve all memorized in school: The acceleration due to gravity is constant. But there’s a more fundamental truth underneath that one, a truth that sat unrecognised until the time of Einstein: Gravity is not a force. To get the full story, you’ll need to wait until next time, when I start to describe general relativity. But for now, let’s explore how Galileo’s experiment shows that gravity is incredibly special.

Electric Bowling Ball, Electric Feather

To understand why gravity is weird, we have to understand how the other forces work. So let’s set up an experiment analogous to Galileo’s, but with electricity, and see what happens. So here’s the experiment (shown in figure 2).

We take two metal plates out into space, far enough away that there’s no gravity. Then we connect the plates to a battery so that one plate gets a positive charge (red) and one gets a negative charge (blue). This creates a constant electric field, much like the constant gravitational field near the Earth. Finally, we place two particles of equal mass at rest at the same position between the plates. We give one particle a very large positive charge (right), and one particle a smaller positive charge (left). Opposite charges attract and like charges repel, so both particles will move towards the blue plate.

The particle on the right will absolutely reach the plate before the particle on the left.

Okay, that’s strange. In this experiment, electric charge played the role of “mass” in the sense that it controls how strong the electric force that acts on the particles is. In electromagnetism, unlike in gravity, the stronger force resulted in a bigger acceleration. But is that the whole story? To find out, let’s try the same experiment, but with a twist.

 Electric Dumb-bell

Let’s take the same battery-powered metal plates into space. But this time, we put two particles of equal electric charge between the plates, as shown in figure 3. But we give one particle (on the left) much more mass by attaching some extra weights to it. Now the electric forces on the two particles are the same. Should they fall at the same rate?

Not so fast. Now the less massive particle (on the right) falls faster.

There Are Two Types of Mass

What we’ve discovered with these two experiments is that there are two types of mass. There’s the gravitational mass, which controls how strong the force of gravity is for a particular object. This is directly analogous to electric charge. The bigger an electric charge, the bigger the electric force. The bigger a gravitational mass, the stronger gravity is. This is what we saw in our first experiment.

But there’s also the inertial mass. This mass controls how difficult it is to change an object’s motion. This is the mass in Newton’s second law of motion, . The bigger the mass is, the more force is required to move it. We all know this intuitively: the heavier an object, the harder it is to push. This is what we observed in our second experiment. More mass means less motion.

But when Galileo performed his experiment at Pisa, he discovered something incredible: for gravity and only gravity, these two types of mass are the same. This is unlike every other force. The electromagnetic force, the strong force, and the weak force, all have a charge, which controls how strong the force is, that is separate from their inertial mass. But the gravitational “charge” is the same as the inertial mass.

And if you allow inertial and gravitational mass to be the same, something changes. Instead of “the gravitational field due to the Earth is the same everywhere,” we get “the acceleration due to the gravity of the Earth is always the same everywhere.”

Gravity becomes the same as acceleration.

That’s very weird. And very special. And it is this fact that lead Einstein to develop general relativity. But through his experiment, Galileo almost found it first.

To learn how this leads to general relativity. Tune in next week.

Related reading

The thought experiments I described here are modifications of Einstein’s famous elevator thought experiments. You can find many descriptions of those thought experiments online. Here’s a few:

• My description of Einstein’s elevator
• A cool video demonstration of the elevator at work, and how gravity and acceleration are one and the same.
• A wonderful article by Paul Halpern on the history of the thought experiment.