In light of the recent (*very incredible!*) discovery of gravitational waves, I thought it would be fun and interesting to describe a derivation of the (basically) fundamental result of general relativity: the Einstein Field Equations (EFEs). Note that this was written in the *plural* form; this is because it’s really a collection of 10 partial differential equations, written in the form of a single equation using tensors. (Why *ten* equations? Well, general relativity makes heavy (this is a pun) use of the metric tensor. This is a four-by-four matrix, and is also symmetric, giving us *ten* independent coefficients.) The post assumes knowledge of differential geometry.

Before deriving the field equations, we will talk about some physics. The *principle** **of stationary action* says that varying the action of a system gives us the equations of motion for that system. Let’s be a little more precise. Suppose is the Lagrangian of a physical system. The *principle of least action* says that if is the action of the mechanical system, then . You can check that in a classical conservative system (when is the kinetic energy minus the potential energy) that this yields Newton’s second law.

Let be the spacetime (a smooth, connected Lorentzian manifold with a metric, which, by the Lorentzian condition, means that it has signature, for eg., +++-) For general relativity, the gravitational part of the action is, in natural units:

Here , and . This is called the *Einstein-Hilbert action*. The whole action is, therefore, simply given by

Let’s note here that the energy-momentum tensor is .

The principle of least action tells us that . What is ? First, let’s consider . This is simply given by . We will expand each term separately. Since , varying this gives ; but the latter term vanishes, so this is just .

What about ? We will use Jacobi’s determinant formula, . Now, (basic calculus), and hence, . Since , this is equivalent to .

Now, let’s go back to the general action. The principle of stationary action tells us that:

This is obtained by setting the terms inside the expression for equal to , and then multiplying the terms by . Using the above equations for and , we end up with the famed Einstein equations:

(For example, in a vacuum, if we expand out in terms of the Christoffel symbols, which are derivatives of the metric tensor, we end up with a second-order partial differential equation.) Let’s be honest, though – this was more of a *mathematical* derivation than a physical derivation. In the next post, we’ll describe a physical derivation of the Einstein field equations, and show that the equation indeed reduces to the Poisson equation for the Newtonian potential.

Have fun,

SKD