As you may know, Einstein’s theory of relativity is actually two related theories. I wrote about the Theory of Special Relativity here, and that’s where most of the counter-intuitive weirdness is (time dilation, length contraction, and the unification of space and time into spacetime). In contrast, General Relativity has only one big weirdness, the assertion that the “force” of gravity is an illusion, and that what we observe is just a manifestation of the fact that our spacetime is curved. Hopefully you’ll be able to get your head around this by the end of the essay. The math of general relativity is pretty advanced and hairy, but luckily it’s not necessary to understand this to get the gist of the theory, so I’m going to gloss over it. It’s also not necessary to grok the concept of “spacetime” for our purposes here, so I’m only going to refer to “space”.
General Relativity gets its name from the fact that it describes all kinds of motion, while Special Relativity only describes the special case of inertial motion, which is motion that doesn’t include acceleration. Einstein’s great insight about acceleration and gravity is that they’re the same thing. Therefore, General Relativity (GR) is a theory about accelerated motion and gravity.
Before we get into all that, let’s review the theory that GR replaces, Newton’s Law of Universal Gravitation, which is usually given by this equation:
However, this equation is useless on its own. To understand how gravity acts on massive bodies, we need to add Newton’s Second Law, which can be written this way:
You may be more familiar with the simple F=ma form, but since acceleration is just the change in velocity over time, the middle term should be understandable to anyone with a passing familiarity with calculus. (For bonus points, understand that acceleration is the second derivative of position). The reason to emphasize this is to show that the law of motion involving forces is a differential equation, which is an equation that relates quantities and their rates of change. Einstein’s equations will also take the form of differential equations.
The two equations above work together to tell us what to expect from gravity. The first one says how to combine the masses of two objects with the distance between them and the mysterious constant G to calculate the force exerted on the body of interest. We then plug the force into the second equation to calculate how the body of interest will move under the influence of this force.
Simple and effective, Newton’s laws reigned for hundreds of years, and were utilized to calculate everything from the paths of celestial bodies and cannonballs to the interactions of billiard balls. However, there were anomalies that Newton’s equations couldn’t explain. In particular, the orbit of Mercury was slightly different from what was predicted by Newton’s laws. The orbit precessed, which means the axis of rotation moved in a regular way, and nobody could explain this.
It took Einstein ten agonizing years to work his way from the Theory of Special Relativity to the Theory of General Relativity. Reportedly, when he used his completed equations to calculate Mercury’s orbit and the precession was perfectly predicted, he was so excited he couldn’t sleep for days.
Einstein’s equations have the same two-part structure as Newton’s, but instead of a “force” being the currency traded between the two parts, Einstein’s equations use the curvature of space. While curved space is a mental hurdle we’ll attack shortly, this is actually more intuitive, since it’s very hard to say exactly what a gravitational “force” is, beyond vacuous definitions like “the thing that makes masses move”. Without further ado, here are the equations:
The first equation plays the role of Newton’s first equation, but instead of an “other mass” and a distance, the equation describes the curvature of space in terms of the matter density (expressed by the T term). As I said, I’m not going to attempt to explain the math, but the quantities with the Greek indices are special matrices called tensors that contain factors for every dimension of space. The second equation describes how objects move in a curved space. This is clearly a differential equation, equivalent to F = ma, which is of course a special case solution when the space is flat and the second term vanishes. In Newton’s universe, masses create forces and these forces make other masses move. In Einstein’s universe, masses make space curve, and curved space makes masses move on paths called geodesics, which follow this curvature unless a force compels them otherwise.
Visualizing curved spaces is actually pretty easy if you limit yourself to two dimensions. The surface of a sphere (such as the earth) is a great example you can intuitively explore. If you compare a flat plane to the surface of a sphere, some pretty stark differences arise immediately. On the plane, if you walk in a straight line forever, you never come back. On a sphere, if you walk in a straight line you arrive back where you started in a finite time. On a plane, parallel lines never meet. On a sphere, parallel lines do meet (think of the meridian/longitude lines on earth meeting at both poles).
Three-dimensional curved space is harder to visualize, and we’re not going to try. It’s enough to accept that a creature trapped on a two dimensional surface who believes it is flat will be surprised at some of the phenomena of curved 2-D space described above. If you can tentatively accept that you are a creature trapped in a curved 3-D space that you think is flat, you can anticipate that some surprising things will be observed if that space is actually curved.
The main counter-intuitive result of the fact that we live in a curved 3-D space is that we have mistaken concepts of what “at rest” and “accelerating” mean. We think we’re at rest when we’re standing, sitting, or lying down on something connected to the surface of the earth. According to Einstein, this is not a rest state, it’s an accelerated state. According to GR, the only time you’ve not been accelerating in your life is when you’ve been freely “falling” in earth’s gravity. Probably the longest freefall you’ve experienced is the time between stepping off the high dive and hitting the water in the pool.
How can anyone say lying in bed is acceleration? In classic Einstein style, we use thought experiments. Imagine you’re in a closed box that appears to be an elevator car. If this box was placed out in space an infinite distance away from any mass, you’d float freely, with no particular tendency to go to one side or another. If you held a ball and then released it, the ball wouldn’t move. Now imagine some force grabs one side of the elevator car and starts pulling at it, accelerating at 9.8 m/s^2. Immediately, you would find one of the sides of the car rushing towards you. As you hit the side, it would immediately seem to become the “floor”, and the side the force was pulling on would seem to become the “ceiling”. If you regained your footing, it would feel just like standing up on earth. If you held a ball in your hand, your hand would impart an acceleration to the ball which comes from the acceleration of the floor traveling through your body. If you let go of the ball, it would stop accelerating and inertia would cause it to travel in the same direction at a constant speed. Since the elevator car is accelerating, it would soon catch up to the ball and the “floor” would hit it. Unless you knew what was happening, you would interpret this as the ball “falling” “down”. This is how acceleration and gravity are the same thing, or at least indistinguishable.
Back to you lying in bed, the matter comprising the earth and your house and bed are constantly accelerating and pushing on you. This explains the force you feel the bed pressing against you, just like the ball experienced the force of your hand pressing against it in the elevator car. If an empty shaft to the center of the earth suddenly appeared under your bed, you’d no longer feel the force of anything pressing against you. Very relaxing and comfortable! You might think that you’d start “falling” but this is only because you have a habit of comparing your motion to things on the surface of earth. Relative to the curved space you inhabit, you’d finally be at rest (or in inertial motion, actually, so at rest in some suitable inertial reference frame), while objects on or in the earth accelerate away from you.
That’s really all there is. It’s taken me quite a long time to become even remotely comfortable with the explanation above. But if you want to understand how curved space explains gravity, changing your habitual notions of “at rest” and “accelerating” is the only way.
As remarkable as the Theory of General Relativity is, many mysteries about the nature of gravity remain. Matter following geodesics through curved space makes some sense, and allows you to keep your intuitive idea of space as a set of locations that may or may not contain matter. But the other half of the theory, where matter causes space to curve, doesn’t really have an intuitive explanation behind it. What is matter and what is space? How exactly does matter cause space to curve? It may well be that space is just another mathematical fiction like Newton’s force. It may be that there are particles called gravitons that transmit the force of gravity, and their discovery may make curved spacetime unnecessary. We also haven’t united General Relativity and Quantum Mechanics into a theory of Quantum Gravity, so there will doubtless be many surprises when that happens.
As remarkable as the Theory of General Relativity is, many mysteries about the nature of gravity remain. Matter following geodesics through curved space makes some sense, and allows you to keep your intuitive idea of space as a set of locations that may or may not contain matter. But the other half of the theory, where matter causes space to curve, doesn’t really have an intuitive explanation behind it. What is matter and what is space? How exactly does matter cause space to curve? It may well be that space is just another mathematical fiction like Newton’s force. It may be that there are particles called gravitons that transmit the force of gravity, and their discovery may make curved spacetime unnecessary. We also haven’t united General Relativity and Quantum Mechanics into a theory of Quantum Gravity, so there will doubtless be many surprises when that happens.
