All the same, Einstein was bothered by something. We decide that it’s car A whose clock stays running at the same rate, and car B whose clock slows down, because car A is moving inertially. But what does that mean? It means whatever happens in car A satisfies Newton’s first law of motion, namely:
Any object at rest or moving uniformly, remains at rest or moving uniformly, as long as no force acts upon it.
However, that’s not strictly true in car A, because if you’re sitting there in the car and you drop a tennis ball, it doesn’t just hang in mid-air, it falls to the floor. And if you want to pass a drink to someone sitting in the back seat, you can’t simply send it floating in the right direction, because what it will do instead is spill on the floor; instead, you have to carry the darned thing into the person’s waiting hands.
This doesn’t bother most people, because we all know what’s behind this: the force of gravity. But Einstein made it a habit to let things that didn’t bother anyone else bother him, and this time it led him to one of his greatest triumphs—the general theory of relativity. This theory allowed Einstein to do away with gravity as a force, and made it instead a consequence of the way that space and time “want” objects to move, and the various ways in which objects “follow” space and time.
Einstein’s view of gravity may sound pretty strange, but it was motivated by an odd coincidence in Newton’s view of gravity. Since it will be easier to see how Einstein came up with general relativity if we understand this coincidence, let’s look at Newton’s conception first.
In 1665, at the age of 23, Newton had already formulated his three laws of motion. Just so we’re clear on what they are, let’s recap. We’ve already taken a quick glance at Newton’s first law of motion. To put it in mathematical terms, if the net force F on an object is zero, then so is its acceleration a:
(1) F = 0 implies a = 0
Newton’s second law of motion is possibly his most familiar one; it states that if there is a force acting upon an object, then the object accelerates in direct proportion to the size of the force. We can write “in direct proportion to” in mathematics by using a tilde sign, thus:
(2) F ~ a
What this means, in practical terms, is that if you throw a ball, and then throw it twice as hard, it ought to accelerate twice as much and end up with twice as great a speed on the second throw (neglecting friction, air resistance, and other effects). In other words, F and a are related by a ratio, which is inherent in the object—it doesn’t change if you move the object from the Earth to the Moon, or wherever. Newton called this quantity mass. Since this mass represents the tendency of the object to resist acceleration, or remain inert, it is often called inertial mass, but usually it is just called mass, and denoted by the letter m. We can then rewrite Equation 2 in its usual form, as:
(2a) F = ma
Newton’s third and final law of motion is usually stated as the law of action and reaction: “If object A acts upon object B, then B acts equally, and in an opposite direction, upon A.” It should be emphasized that there’s no delay involved. It’s not as if A shoves on B, and then B, in a response of justified anger, shoves back on A. They happen together. We can write this third law as
(3) FA on B = FB on A
As a simple example, consider the act of pushing down on a table with your hand. Your hand is exerting a downward force on the table. At the same time, the table exerts an upward force on your hand. The moment you lift your hand from the table, it no longer exerts a force on the table, and the table no longer exerts any force on your hand. That’s all the third law is: it’s a case of physics bookkeeping.
Back to Newton in 1665. He was at his childhood home, his school at Cambridge having been evacuated to minimize an outbreak of the plague. It’s plausible that he was sitting under a tree one evening when an apple fell, although it probably didn’t hit him on the head. Instead, he likely saw it fall from another tree, at a time when the Moon was visible in the sky. Newton wondered to himself why the apple fell to the Earth, and the Moon didn’t. Was the Moon somehow exempt from the gravity of the Earth? Perhaps the Moon was far enough away such that the gravity of the Earth was insufficiently strong to affect it.
Then it occurred to him that perhaps the Moon was falling, but was simultaneously moving “sideways,” and that that sideways motion was enough to keep the Moon in orbit around the Earth. For example, if you drop a ball, it falls straight to the ground. If you throw it so as to give it some horizontal motion, then it doesn’t drop straight to the ground when you let go of it. Instead, it continues the horizontal motion of your throwing hand, as predicted by Newton’s first law. Gravity pulls on the ball so that it does hit the ground eventually, but by the time that happens, the ball has moved a significant distance. What’s more, the harder you throw, the faster the ball moves forward, and the further it goes before hitting the ground.
Now, if the Earth were flat and infinite in extent, then it wouldn’t matter how hard you threw—the ball would eventually fall to the ground, although you could get it to go as far as you wanted by throwing it harder and harder. But in fact, the Earth isn’t flat and infinite—this was known as far back as Eratosthenes (c. 284-192 B.C.). Instead, it’s roughly a sphere with a radius of 6,400 km.
This makes a big difference! When you throw a ball, it curves back down to the ground, and the faster you throw it, the gentler the curve. What if you threw it so hard that the curve of the ball was just as gentle as the curve of the Earth? Then it would never fall down, and would instead orbit the Earth, just as the Moon does. And as long as nothing slowed the ball down (such as air resistance, but let’s ignore that for the moment), the ball would continue to orbit the Earth, without you having to rethrow it every now and then.
Newton decided, therefore, that the fact that the Moon didn’t come crashing into the Earth didn’t mean that the Earth’s gravity didn’t pull on the Moon, as it did on tennis balls. There didn’t need to be anything up there to hold the Moon up, or keep it moving in orbit against the pull of gravity; all that mattered was that it had enough speed to begin with. And if the Earth was indeed pulling on the Moon, why shouldn’t any object pull on any other object? If Newton could uncover a law of gravity to cover the Earth and the Moon, the same law should also cover the Sun pulling on the Earth, or any of the other planets, or two bowling balls pulling on each other, or whatever. It would be a truly universal law of gravitation.
However, as great an insight as this is, it’s not of much use scientifically until it’s quantified and tested. It’s no good to simply say, “The Sun pulls on the Earth with gravity, and that’s that,” because there’s nothing you can observe to say that it doesn’t. It’s not falsifiable, in other words, and a proposition that isn’t falsifiable isn’t worth the paper it’s written on. One might just as well say, “The Sun pulls the Earth with telekinesis, and that’s that.”
Newton therefore needed to come up with some formula that would predict the strength of gravity’s pull, depending on some parameters of the objects pulling each other. But which parameters should he start with? Newton was aware of the experiments of the Italian physicist and astronomer Galileo Galilei (1564-1642), in which he dropped objects of different weights from the same height, and observed that they all fell in the same amount of time, with the same motions. (Actually, he rolled them down inclined planes, but as long as other factors such as the angular momentum of rotation are properly accounted for, you can translate rolling down an inclined plane into free falling from a height.)
In other words, if you drop an anvil with a mass of 10 kg and a basketball with a mass of 1 kg, they both experience identical accelerations due to gravity, despite the fact that one has a mass 10 times greater than the other. By Newton’s second law, that means that the force of gravity on the anvil also has to be 10 times greater than the force of gravity on the basketball in order to overcome the extra inertia of the anvil. In general, the force of gravity upon any object is therefore proportional to the mass of that object, which we can write as: