After that long journey through the mathematics of curved spacetime, we have finally arrived at Einstein's Equations of General Relativity. And where before it was all mathematical foundations, today we arrive at the real physics. And in keeping with my intentions of keeping this entire review intuitive rather than mathematically rigourous, I will introduce the theory using the less common Einstein-Hilbert action, which was introduced by Hilbert rather than Einstein and in my opinion is easier to understand than the more common derivations.

Suppose we define a new quantity S, as the sum of the curvature at every point in spacetime. For curvature, we will use the Ricci scalar defined in the previous article. Now S gives us the total curvature of the entire Universe averaged over all time. And suppose we insist that for any particular region of spacetime we require this curvature to be minimized. The mathematics required to make the total curvature minimal is beyond what I can include in this article, but the end result is the condition:

Ri(A,V) = 0

for all vectors A and V. This is Einstein's equation for a vacuum. When there is no matter present at a particular point in spacetime, all of general relativity reduces to the the condition that the Ricci tensor is zero for all vectors.

So what happens if there is matter and energy present?

In this case we must minimize not only the total curvature, but also a second quantity called the Lagrangian. I have described the Lagrangian before in other articles, so I will not go into too much detail here. In the simplest terms one can think of the Lagrangian as the difference between the kinetic energy and the potential energy of all matter, energy, and other fields. And in classical non-relativistic physics, the sum of the Lagrangian over space and time must be minimized. This one condition, known as the principle of least action, leads to all of the equations of classical physics.

However in general relativity we do not require the curvature to be minimized, or the Lagrangian to be minimized, but rather the sum of the two quantities. By summing the Ricci scalar, which represents curvature of spacetime, and the Lagrangian, which represents the properties of all matter and fields, and then summing over all points in spacetime, we define the Einstein-Hilbert action.

And once more, the mathematics to calculate the minimum is beyond the level of this article, but the end result is the equation:

Ri(A,B) - g(A,B) Rc/2 = k T(A,B)

where A and B are arbitrary vectors, Ri and Rc are the Ricci tensor and scalar respectively, g is the metric function, k is a constant which must be measured experimentally, and T(A,B) is called the energy-momentum tensor, and simply represents the energy and momentum of all matter and fields, as measured by someone traveling along the vector A. (Technically it is much more complicated than this simple description, however this definition works for this rough overview).

That is Einstein's field equation, and represents all of general relativity. This equation contains all of the physics necessary to understand general relativity. And unfortunately in practice it is such a complicated equation that there are only a handful of solutions known, and those are only for very special, highly symmetric systems.

And so after all of this build up, and all of these mathematical tools, we have the general theory of relativity. And one century after it was first published, it is still one of the most interesting and beautiful equations in all of the fields of science.