THERE ARE SOME things in common between gravity and the other forces. The ideas of symmetry play an important role in both. Any time we change something (say, rotate a shape) and it makes no difference (if the shape we rotated was a sphere), there is a symmetry. In quantum mechanics, there is a symmetry in that changing the phase of all quantum waves at the same time makes no difference to the physics. In General Relativity, moving around between different accelerating or moving frames, in or out of gravitational fields, makes no difference to the physics. A lot of theoretical effort has gone into trying to exploit such similarities, with some very powerful results, but so far no one has managed to make a fully Standard Model-like quantum theory of gravity which would work at very small distances and high energies.
Stepping back from the quantum world, however, there is a very simple similarity between gravity and electromagnetism.
The Sun pulls the Earth towards it with gravitational force. If the Earth were twice as far away from the Sun, the force would be four times weaker. If the Earth were three times nearer to the Sun, the force would be nine times stronger. This is the famous ‘inverse square’ law. Multiply the distance by two, and the force gets weaker by the square of two – that is, by a factor of four. Shrink the distance to a third, and the force gets stronger by the square of three, i.e. nine.
It tickles me that even though the theories behind the forces are so very different, the electric force does exactly the same thing. There is an attraction between a negatively charged electron and the proton in a hydrogen atom. Double the distance between them and the force drops by four; it follows an inverse square law, just like gravity.
This is no coincidence. We can think of a mass, or an electric charge, as the source of a force. Physicists often draw them as lines of force.fn1 The density of the lines is proportional to the strength of the force. The force is spread out over a bigger and bigger sphere as you move further and further from the source. If the total number of force lines stays the same, then the force at any given point will drop as the area increases. The area of a sphere is four times π times the square of the radius of the sphere, so the amount of force at any given point on the sphere is divided by this. The most important thing in that expression is the radius. The force is divided by the radius squared. This is the ‘inverse square law’, which works just as well for any long-range force, whether the theory behind it is based on quantum mechanics or warped space–time.
This is a good example of the fact that some of the general features that we encounter on our travels across the map are in a sense more fundamental than the details. The principles that emerge are in some ways more fundamental than the underlying theory. Conservation laws and symmetries are other examples of the same effect. You don’t need to know all the internal details of water molecules to have a good idea of what will happen if you boil a kettle. You don’t need to understand QED or General Relativity to know that the inverse square law is probably a good bet to describe how the force falls off with distance.