This is intended to give those with only high-school math the best insight possible into Einstein’s theory of General relativity, including why it was such a startling idea. Obviously not all aspects of the theory can be made accessible to everyone, but I tried to give a taste..........

In 1905, Einstein had published his Special Relativity theory. The central idea of Special Relativity is that that if you play table tennis on a steadily moving ship, the ball will bounce exactly as it would if the ship was not moving. This had actually been stated centuries before by Galileo, but since then a major problem had arisen. In air of a given temperature and composition, sound travels at a known speed relative to the air. Sounds travel more slowly upwind than down, and this fact can be used to measure the wind velocity (though there are better ways). It was assumed that light waves behaved the same, traveling at a particular speed relative to some universal “air” called the aether which was believed to fill the universe. In the nineteenth century, two American scientists, Michelson and Morley, tried to measure the speed of the aether, but always got the result that the aether was not moving relative to their apparatus, no matter how the earth was moving the apparatus around the sun. Einstein managed to combine this result with Galileo’s principle. To do so, he had to introduce the idea that if one ship was moving East at speed v, and another ship was moving West at speed V, then someone on board one ship measured the speed of the other, the result would not be v+V, as common sense might suggest, but (v+V)/√(1+vV/c²) where c is the speed of light. Because speeds we encounter in everyday life are so much smaller that c, we never noticed the difference until Einstein suggested looking for it. This change messed up many other laws of physics, so Einstein had to make many other adjustments before he had a self-consistent theory. In our discussion, we will be particularly interested in the fact that he had to assume that if an object weighed M when it was at rest, it would weigh M/√(1-V²/c²) when it was moving at speed V. Again, the result is normally too small to measure, but it, and many other startling predictions of the theory, have been dramatically confirmed by experiment, and have proved crucial to our understanding of the world.

Special Relativity took care of the fact that playing table tennis was unaffected by the ship moving in a straight line, but did not address what happened if the ship was spinning around. Obviously, one cannot play table tennis on a carrousel without being aware that it is spinning. This does not seem puzzling until one asks oneself “If I was the only object in the universe, could I tell if I was spinning? If I could, what would I be spinning relative to? If not, how does the existence of the rest of the universe create the sensations of spinning? For example, if I stand on the center of a carrousel with my arms out, I feel the centrifugal force in my hands.” General Relativity tells us that we feel the centrifugal force because we are spinning relative to distant stars and galaxies. But how do they exert this effect?

To set the stage, let us go back to Newton’s understanding of gravity. One of his results was that if the Earth was a uniform hollow shell, someone inside the earth would feel no gravitational field. The only way I can explain this without calculus is to point out that to “escape” from the interior in a straight line from whatever was your initial position, you would have to pass through the same thickness of rock as someone escaping in exactly the opposite direction, and the fact that there’s more area of rock on the more distant side exactly cancels the fact that the distance decreases the effect, so the two pulls balance. The result is that if you and a friend were floating together inside a hollow earth, attached to each other by a rope, there would be no tension on the rope. If you and your friend were spinning around each other, then “centrifugal force” would put tension on the rope. But if you and your friend were stationary, and the hollow earth around you began to spin, that would not put tension on the rope. Or would it? If you two and the earth made up the entire universe, how could there be a difference according to which was doing the spinning?

To simplify the math, let’s replace the hollow earth with six equal weights, equally spaced around you (one in front, one behind, one above, one below, one left, and one right). These are truly massive weights - let’s call them stars - so they each pull on you gravitationally, but because they are in three opposing pairs, the forces balance out, and you stay still. Still doing everything we can to simplify the math, let’s say each star is one unit of distance (light year?) away from you, and pulls on you with one unit of force. Your friend is again floating in front of you at a small distance D, attached by a rope. Will there be tension on the rope? (If there is, our approximation of the universe is clearly not up to my hopes!)

Lets see, the star in front of you is only 1-D from your friend, so the pull on him is now 1/(1-D)² by the inverse square law of gravitation. If you remember your math this is approximately 1+2D for small D. Similarly, the star behind you is 1+D from your friend, pulling him 1/(1+D)² or approximately 1-2D, so there seems to be a net force of 4D on your friend away from you. But wait! There are four other stars. Because the small distance from you to your friend is perpendicular to the distance from you to these stars, his distance from them is still 1, but he is not exactly between these pairs as you are, so the pulls do not exactly cancel. The star above you is pulling him up with a force of 1/(1+D)² and toward you with a force of D. So the net force from these four stars exactly cancels the one from the first two stars (for small D), so there is no tension on the rope. Our approximation of the universe by six stars is successful!

Now let’s start the universe spinning around an axis passing through you and the stars above and below you. We’ll spin it fast enough so that the four stars away from the axis are moving at c√3/2, where c is the speed of light. This would not make any difference according to Newton, so there would still be no tension on the rope. But according to Special Relativity, the moving stars now weigh exactly twice as much, so they are each pulling with twice the gravitational force. You are not affected, since the stars are still symmetrically placed about you, but what happens to your friend?

The stars in front and behind are now twice as heavy, so the net force from them is now 8D instead of 4D away from you. The left and right stars are also twice as heavy, so the net force from them is now 4D instead of 2D toward you. The up and down stars are on the axis and not moving, so the net force from them is still 2D toward you. Therefore your friend is feeling a net force away from you of 8D-4D-2D=2D. Because the universe consists only of two people and six stars, it will seem more reasonable to describe the stars as stationary, and your friend as spinning around you. The 2D force on him will therefore be described as centrifugal. Note that centrifugal forces are indeed proportional to the distance from the center of rotation.

The first amazing thing about this idea is that when you feel a centrifugal force, you are directly detecting the most distant galaxies in the universe. The second amazing thing is that if there was twice as much matter in the universe, it would seem that the centrifugal force would be doubled. But the centrifugal force is also defined purely by geometry – it must cause unconstrained objects to move in straight lines relative to an inertial frame of reference. Therefore gravity is absolutely necessary for the universe to behave self-consistently! Even more amazing, there is only one possible average density for the universe for the measured gravitational constant.

The average density of the universe has been estimated by counting stars, galaxies, etc. and making reasonable estimates of the amount of hydrogen floating about in intergalactic space. It turns out that General Relativity seems to predict a density about 25 times greater than this “observed” density. This may seem like a large error, but it’s very small compared to the minimum (one hydrogen atom in the observable universe) and maximum (neutron star? black hole?) densities the universe could have, so this is actually encouraging. Various theories of "dark matter" have been thought up to try to explain the “missing mass”.