After completing my review of the equations of general relativity, I was left feeling that something more was needed. The equations themselves are beautiful, but having built up so much mathematical framework, it seemed like I was missing an opportunity by not giving any examples of solutions to the field equations. I intend to rectify that situation today.

As I mentioned at the end of the last article, there are only a handful of solutions to the Einstein field equations, corresponding to specific systems whose symmetry allows the equations to be simplified. There is no general solution known. Of these solutions, there are two that are particularly interesting and informative for beginners.

The first solution was calculated by Schwarzschild in 1916, and represents a system with spherical symmetry. This corresponds to spacetime around a star or planet that is not rotating. (While the derivation is not difficult, it is beyond the level of these review articles.) The metric function is:

ds2 = ( 1 - 2GM/r) dt2 - (1 - 2 GM/r)-1 dr2 - r2 (dq2 + sin2(q) df2)

where r,q,f are spherical coordinates corresponding to radial distance from the centre of the object, and two angular coordinates (similar to longitude and latitude on Earth), while G is the gravitational constant and M is the mass of the spherical object (ie star, planet, or black hole). 

The metric may not look like much, but it has several interesting properties. First, requiring other smaller objects to follow the shortest path between two points results in a trajectory exactly the same as Newtonian gravity. There is no actual force of gravity here, but rather objects follow the shortest path through spacetime and that leads to an apparent curvature in their path that centuries of scientists ascribed to a force of gravity.

The second interesting point is more obvious in the metric function given above. When r = 2GM, which is called the Schwarzschild radius, the time component of the metric goes to zero and the radial component becomes infinite. This means that time appears to slow down, light has its frequency shifted down to zero, and even light is no longer fast enough to escape from this surface. An object smaller than its own Schwarzschild radius is of course now known as a Black Hole.

The other simple solution to Einstein's equation is called the Friedmann-Robertson-Walker metric (or FRW metric, or FLRW metric) and represents a system in which energy and matter are evenly distributed across all of space, and where space looks the same in all directions. When Einstein's equations are solved under these conditions, the resulting metric is:

ds2 = dt2 - a(t)2 (dx2 + dy2 + dz2)

where a(t) is a function only of time. In our Universe, a(t) is currently increasing and from Einstein's equations we also know that it had to be zero at some time in the past. In physical terms, this means that space itself is expanding over time and that at some time in the past the distance between any two points in space was zero. This moment is what we now call the Big Bang. It is worth noting here that, contrary to popular belief, the Big Bang does not mean that matter is expanding out through space, but rather than space itself is expanding.  And while we do not know what a(t) will look like in the future, it is now generally believed that it will continue to increase forever. 

Those are the two simplest solutions of general relativity, and two of the most interesting because of the wide range of systems that they describe. The first describes all stars and planets, while the second describes the entire Universe.

And so ends this review of the theory of general relativity. I truly hope that it has been interesting and understandable to my readers. I have had great fun writing it, and I found myself challenged to keep my explanations non-mathematical while still presenting the overwhelming beauty of the theory. And I hope that it will inspire some of you to delve deeper into the theory and learn more about it.

General Relativity is one of the most beautiful theories ever created by the human mind, and it is one well worth studying. I hope you have enjoyed reading it as much as I have enjoyed writing it!