The Geometric Intuition Behind General Relativity

In nineteen oh five, Einstein published special relativity, and at its centre is a speed limit: no influence of any kind travels faster than light. Ten years later he published a theory of gravity. If you had to compress that theory into three words, they would be: not even gravity. This is where we are going. It is the equation Einstein arrived at in nineteen fifteen. The left hand side is a quantity that is zero when spacetime is flat and not zero when spacetime is curved. The right hand side is matter and energy. You do not have to read it yet. Everything between here and there is Newton, one coincidence that Newton himself noticed and shrugged at, and one change of mind about what a straight line is. Here is the Sun, here is the Earth, and this arrow is the pull the Sun exerts on it. Newton's law of gravity says that pull is his constant, times one mass, times the other, divided by the distance between them squared. Now jiggle the Sun. Read that formula literally: the instant the separation changes, the force out at the Earth changes with it. Not eight minutes later. Now. But light itself takes eight minutes to cross that gap. Watch a flash leave the Sun and reach us. Newton's force beat that flash across, which is exactly what the speed limit forbids. One of the two has to give. Einstein had spent years hunting down every way of sending a signal faster than light, and he was not about to allow a gravitational telephone. So the law that has to change is Newton's. There is a precedent for repairing an inverse square law. Electricity has one of exactly the same shape: a constant, times one charge, times the other, over the separation squared. Maxwell had already dressed that law up properly. Once the charges move, magnetic effects appear, and the whole package obeys the speed limit exactly. So copy the trick for gravity and be done. Two like charges push each other apart. Two masses pull each other together. Copy the electric calculation exactly and you get the electric answer, which is that the Sun would shove the Earth away. There is the difference, and it is one character wide. A plus on this side, a minus on that one. That single character is why gravity cannot simply be copied from electricity. The reason lies in the messenger. The electric force is carried by a particle of spin one, the photon. Gravity is carried by a particle of spin two. That difference is exactly the difference in sign. So Einstein could not copy. He had to find another route, and he had one clue: a coincidence sitting in the middle of Newton's own equations.

Look again at the two laws, and at the letter m in each of them. In Newton's second law, the mass measures stubbornness: how hard the body fights being accelerated. That is the inertial mass. In the law of gravity, the mass measures something else entirely: how firmly gravity takes hold of the body. That is a completely different job, and it is called the gravitational mass. In electricity those two jobs are done by two unrelated numbers. A neutron is heavy and carries no charge at all. An electron is light and carries a full unit of it. So there is no relation whatsoever between how much a particle weighs and how strongly it feels an electric field. Now watch what gravity does. In gravity, the number that resists and the number that gets pulled are the same number. Not nearly the same. The same. That statement is the equivalence principle, and here is what it does. Drop a brick and a feather in a vacuum chamber, with no air to slow either one down. Gravity pulls the brick harder, because the brick has more gravitational mass. The brick also resists harder, because it has more inertial mass. The two effects cancel exactly, and the two bodies fall together. Newton noticed this himself. He ran experiments on it, and found the two masses agreeing to roughly one part in a thousand. By Einstein's day the agreement was one part in a billion. Today it is one part in ten to the fifteen, which makes it one of the most precisely tested statements in all of physics. In Newton's physics this is a pure coincidence. Two unrelated quantities, agreeing to fifteen decimal places, for no reason anybody could give. Einstein refused to accept that as an accident.

Here is the other half of Einstein's clue. A bucket of water on a rope, swung in a vertical loop. Down at the bottom nothing is mysterious: gravity pulls the water into the base of it. Swing it fast enough and the water stays put at the top, upside down though it is. Ride along with the water and there is a second account of why. From inside, something presses the water outward, into the base of the bucket. We call that the centrifugal force. It is not a real force. Nothing is pushing. It appears only because the frame you chose to describe things in is spinning, and its size is your speed squared over the radius, times your mass. And look which mass that is. Not some new centrifugal charge. It is the inertial mass, the very same stubbornness from Newton's second law. Here, though, that is no coincidence. It could not have come out any other way. The reason you feel pressed outward is precisely that your body wants to keep going straight. Inertia is the whole cause, so inertia is the whole charge. So write two facts side by side. Any inertial force, by construction, couples to the inertial mass. And gravity, as we measured to fifteen decimal places, couples to the inertial mass as well. Which raises the question Einstein called the happiest thought of his life. What if gravity is an inertial force? Not a force at all, but the price of describing motion in the wrong frame. That guess is even permitted only because the two masses are equal. Try the same move on electricity and it collapses at once, because electric charge is nothing whatsoever like inertial mass. But it sounds mad, and here is exactly why. An inertial force is what you feel when you are not moving along a straight line. Move along a straight line and you feel nothing at all. So if gravity is one of those, then the astronaut floating in orbit, feeling nothing, is the one going straight. And you, held up by your chair, feeling that push in your back, are not. Which means we are about to be very wrong about straight lines.

Here is a graph of height above the ground against time. This flat line is you, sitting in your chair, staying at the same height while time runs on. And this arc is a piece of chalk, thrown up and caught again. The chalk is in free fall the whole way, so it feels nothing, while you feel the chair pressing into your back the entire time. If gravity is an inertial force, then the chalk is the one travelling in a straight line, and you are not. Drawn like this, that is plainly absurd. One of them is straight, the other is bent, and I can see which. Unless the picture is lying to us. Which is a thing that pictures do. You have met this exact lie on an aeroplane. Here is the map on the seat in front of you, with San Francisco on the left and London on the right. The obvious shortest route is the dashed line straight across. But the aircraft flies the red one, up over Greenland, which on this map looks like a wasteful detour. Now put the same two cities on the actual Earth, which is a sphere. The shortest path between two points on a sphere is an arc of a great circle. Here is that arc, and it does run up over the north, exactly where the airline said it would. The detour was never a detour. The flat map bent it, because that map is trying to draw a curved Earth on a flat page, and something always has to give. What gives is your idea of which lines are straight. Now back to our graph, with the same accusation. That picture drew time and height on a flat grid. Spacetime is not flat: matter curves it. On the curved thing, the arc is the straight line, and the level line is the bent one. So the astronaut, feeling nothing, goes straight. You, held up by your chair, are being pushed off the straight line, and the weight you feel is the price. Gravity is not a force in the picture. It is a property of the picture. That leaves exactly one job: say precisely how much matter curves spacetime, and how much. It took Einstein eight further years, and this is the answer he arrived at. On the left, the curvature of spacetime. On the right, all the matter and all the energy there is. Read it as a slogan: matter tells spacetime how to curve, and curved spacetime tells matter how to move. Newton's law of gravity is gone. Newton's first law survives, and reads better than before: with no force on you, you travel in a straight line. We simply had the wrong lines.

Curved spacetime has consequences, and the first is about clocks. Here is a mass, here is a clock sitting close to it, and here is one far away. Neither of them is moving. The one down low ticks slower. Not because of any motion, but because it sits deeper in the gravitational well. The whole effect is this square root, and it is less than one. This has been measured for seventy years. Two atomic clocks at two heights in one building disagree. Satellite navigation has to correct for it, or the position it reports would wander off within minutes. And notice that this is not the symmetric effect of special relativity. There, each of two moving observers sees the other one running slow. Here we both agree who is deeper in the well, so we both agree whose clock is the slow one. Now send a photon up from the low clock to the high one. Everything down there looks slow from up here, and that includes the oscillation of a light wave. So the light arrives with a lower frequency than it left with. Lower frequency means shifted toward the red end of the spectrum, and it means less energy carried. That is the gravitational redshift, and we see it in light climbing out of the Sun. Turn it around, and light falling down arrives blue, carrying more energy than it started with. Whatever the exchange rate is for time between two heights, it is also the exchange rate for energy. The third consequence is that light bends. Here is the Sun, and here is a star far off to the left, sending a ray past it. If gravity did nothing, that ray would run dead flat along the dashed line. It does not. The ray is pulled toward the Sun, and the observer over on the right, looking back along the direction it arrives from, sees the star shifted away from the Sun's edge. Treat light as a fast particle and Newton's gravity predicts a deflection of two G M over c squared b. Einstein's theory predicts four. Exactly twice as much. That factor of two was testable, if only you could see stars right beside the Sun. You cannot, except during a total eclipse, when the Moon covers the disc and the sky beside it goes dark. Eddington's expedition of nineteen nineteen measured it, came back with Einstein's number rather than Newton's, and made him famous overnight. A British expedition confirming a German theory, months after the war. Newton unified the falling apple and the planets. This unified the apple, the planets, and the light.

Now take the whole argument to its limit. To leave a body you need escape velocity: enough kinetic energy to pay off the gravitational binding energy at its surface. Solve that for the speed and you get the square root of two G M over r. For the Earth it comes to eleven kilometres a second. Make the body more compact and the speed climbs. Keep squeezing. Here, for comparison, is the speed of light, and there is the moment the escape velocity matches it exactly. Set the two equal and solve for the radius. The mass of the escaping object cancels, and what is left is two G M over c squared, the Schwarzschild radius. People wrote that expression down in the seventeen hundreds and wondered. Now we know. Newton's law says nothing special happens at that radius. Einstein's does. For an observer trying to hover at rest, the pull is the inverse square law multiplied by one over that same square root. The grey curve is Newton. The red one is the truth, and it sits above the grey one everywhere. Gravity in general relativity is stronger at short range, not weaker, which is the opposite of what would rescue us. Now walk inward. Hover out here and you need a certain thrust. Closer in, more thrust. And at the Schwarzschild radius, the dashed line, the thrust you need runs away to infinity. So there is no hovering at that surface, or anywhere inside it. Fire your rocket as hard as you like and you still go in. That surface is the event horizon. It is not a wall, and nothing about crossing it feels violent. For a big enough hole the tidal stretching there is gentle and you sail through noticing nothing at all. What kills you is the singularity at the centre, much later. And from far away I never see you cross. Your clock, by that same square root, runs slower and slower as you approach. Your light stretches redder and redder. You fade out rather than arrive. Let us put the whole thing back together. Einstein's equation states it in one line, and the argument that got us there came in three steps. First, the speed limit. Nothing carries an influence faster than light, and Newtonian gravity plainly broke that rule. Second, the coincidence Newton shrugged at. The mass that resists being accelerated and the mass that gravity pulls on are one number, agreeing to fifteen decimal places. And third, the price of taking that seriously. Gravity is not a force at all. It is the shape of spacetime, and free fall is what a straight line looks like once you admit the shape. It took Einstein ten years, and it grew out of a coincidence that everyone else had already seen and let pass.