# Equivalence

I want to talk a little about the central idea that led Einstein to the concept of General Relativity. To get there, I want you to perform a little experiment.

Take a moment to think carefully about the forces you feel on yourself right now. If you’re sitting, you feel the chair pushing on your back, which pushes on the rest of your body. You feel the floor pushing your feet up. You might feel the muscles and tendons in your shoulder holding your arm up, or strain your neck holding your head up.

But do you feel the force of gravity? No. You feel the floor and chair keeping your feet and backside from falling through the floor, and you therefore feel those parts of yourself pulling on the other parts of your body. But you don’t actually feel the force of gravity. If you were in free fall, with no floor holding you up (and no air resistance), you’d never notice you were accelerating, even though the Earth would be pulling you downwards (and when you hit the ground, you will notice a rather significant force).

How can you not notice a force? What makes gravity special?

Let’s compare to another force: if I grabbed your arm and gave you a sharp tug, you’d feel the electromagnetic force of atoms in my hand pulling on the atoms in the skin of your arm, trying to cause them to move in the direction I am pulling you. Those atoms exert electromagnetic forces on the rest of the atoms in your arm, accelerating them, which in turn transmits forces which accelerate your shoulder, torso, and so on. The point is you feel a force when different parts of your body are accelerated at different rates.

If I could instead pull individually on each atom in your body with exactly the same acceleration, you’d never feel a thing. Each part of your body would be moved in some direction at the same rate, and so no individual part of you would be trying to “run off” away from the rest.

Newton taught us the relation between a force and an acceleration:
$F = m a$

Force is the mass times acceleration. Or, acceleration is force divided by mass. When I yank on your arm, I’m exerting forces on your body; those forces are not precisely calibrated so that more massive parts of your body get the bigger forces. Thus, different parts of your body are pulled at different rates, and you feel forces attempting to tear you apart.

However, Newton also taught us how one object exerts a gravitational force on another:
$F_g = \frac{G M m}{r^2},$
where one object is mass $M$, the other is mass $m$, separated by distance $r$. $G$ is a constant of Nature, called (appropriately) Newton’s Constant. So, combining these two, we see that the object with mass $m$ is accelerated at
$a_g = \frac{GM}{r^2}.$
independent of its mass. Thus, when I sit in the gravitational field of the Earth, each part of me is pulled downward with a force precisely calibrated so that the more massive parts of my body get a larger force, the less massive get smaller forces, and each part of my body gets accelerated downward at exactly the same rate.

If I tied you to a rocket sled, and fired you off at $20 g$ of acceleration (that is, about 200 m/s$^2$ of acceleration), I would likely kill you (or at very least, you’d be seriously injured). However, if you were instead falling towards a massive planet with a gravitational pull of $20 g$, you’d be fine. Until you hit the ground, of course.

Thus I cannot feel the force of gravity. Only when I am prevented from moving towards the object exerting the large gravitational force (the Earth) do I feel forces, caused not by gravity, but by the floor impeding my motion.

For additional evidence of this truth, look no further than the astronauts on the International Space Station: floating “in zero gravity.” But the ISS is only a few hundred kilometers above the surface of Earth. The force of gravity is only a few percent smaller up there than down here: those astronauts are no more in “zero gravity” than we landlubbers are. Instead, astronauts are falling towards the Earth. They just happen to be moving sideways fast enough that, when they have fallen 1 meter towards the center of the Earth, the Earth has curved 1 meter away from them. That is, they are in orbit. Being in orbit is just a fancy way of saying you are throwing yourself at the ground and missing.

In principle, you could do the same trick down here at low altitudes, but two things would get in the way: first mountains would get in the way. Second, the way you avoid the ground is by moving very fast sideways, and down here there is a lot of air. The friction from the air would slow you down (and probably catch you on fire) if you tried to go into orbit at sea-level. Only by moving a few hundred kilometers up can you get low enough air densities to avoid that problem. (The key difficulty in getting into orbit is less the “up” bit, but rather the “moving sideways very fast” part. The energy requirements to move fast enough to stay in orbit are much larger than the requirements to launch something straight up to orbital altitudes. SpaceX can do the former, Blue Origin can only do the latter)

So, gravity pulls on matter with a force exactly calibrated to mass, what does this matter?

Well, the fact that the mass of an object controls both how it reacts to forces and how it reacts to gravity is odd. No other force is like that (objects react to electric forces via their charge, not their mass, for example). For Newton, this was just a strange fact he had to accept. We have since done precise experiments demonstrating that the deviation from the inertia mass (entering into $F = m a$) and the gravitational mass is less than one part in a trillion. But Newton has no explanation for why this is true.

Enter Einstein. Einstein took this observation, and figured out a why that it would follow naturally from a theory of gravity.

Einstein realized that, if you are freely falling and can’t feel the force of gravity, then there is no experiment you can do that demonstrates you are in a gravitational field. That is, if you are in a closed elevator, and I cut the rope holding you up, then there is no experiment you can do in that box (short of cutting a hole and looking outside) that would demonstrate you are in an elevator shaft on Earth, rather than floating freely in deep space, far away from any star or planet. Of course, when you hit the ground, that’s another issue. We’ll get to the ground thing soon, I promise.

There is another famous example of such an equivalence. Long ago, Galileo had observed that, if you were in the hold of a ship and that ship was moving with constant velocity, there is no experiment you can do in that hold that demonstrates you are moving. This is Galilean Relativity: if you move with constant velocity, you don’t notice anything special locally around you (unless you hit something moving at a different speed, but that’s not a local effect). Einstein himself had improved this idea to accommodate the fact that everyone measures the same speed of light, giving us Special Relativity.

So Einstein’s little thought experiment demonstrated to him that someone being accelerated by gravity was equivalent to someone moving with a constant velocity. After all, no experiment they could do would tell the difference.

So, Einstein reasoned, if a freely falling observer can’t tell the difference between gravity and moving with a constant velocity, then that observer must be moving in a straight line at constant velocity.

But that’s crazy right? Astronauts freely falling in orbit move in orbit. They circle, they don’t move in straight lines. A falling elevator accelerates: it moves faster and faster.

But Einstein saw the resolution to this paradox. A freely falling observer near the Earth is moving in a straight line, it is just that the Earth is changing what a “straight line” is. That is, what we call gravity is a bending of “straight lines.” Furthermore, this bending would have to occur in both space and time: that is the only way I can see someone moving faster and faster as they move in their “straight line” towards the center of the Earth. Thus, gravity is a bending of space-time.

What Einstein realized is that we are all trying to move in “straight lines” through time and space. These paths are called geodesics. We feel a force when we are diverted from the geodesic we are currently on. If you were far from any mass, spacetime would be “flat,” and your geodesic would be “travel in a straight line in space, with time ticking by at a constant rate, as measured by a distant observer.”

However, near a mass (or any form of energy, Einstein realized), under the influence of gravity, things get more complicated. The Earth (or any mass), bends spacetime. Thus, even though at each point you think you are traveling in a straight line, a distant observer will see your path curve. Just as an airplane can be traveling “straight” at each point on its Great Circle route, yet end up curving with the arc of the Earth. This curving occurs in time as well as space, so a distant observer sees clocks on Earth’s surface tick slower than a clock far away from the Earth does.

Thus, when you sit in your chair, and feel the chair and floor push up on you, you are not feeling gravity. You are feeling electromagnetic forces diverting from your geodesic, which would otherwise carry you in a straight line through spacetime towards the center of the planet. Similarly, if you fall in Einstein’s elevator, you follow your geodesic, up until you reach the ground. At which point the ground will express its pointed disagreement with your path through spacetime.

This explains why the inertia mass is identical to the gravitational mass. Inertial mass is just a form of energy (via $E = mc^2$). Energy also causes spacetime to bend. But that bending is what we call “the force of gravity,” invented because we wanted to describe what it looks like when objects follow their geodesics, rather than get prevented from doing do by the floor.