Hassaan Saleem

This post summarizes the reasoning that goes in deriving Einstein’s field equation. There are more than one ways to derive this field equation but I will focus on the most conventional path to derive it. Basic understanding of tensor analysis and special realtivity is assumed to understand this post.

What we need is an equation that relates “how spacetime curves” to “how matter and energy exist and move” in spacetime. So, one side of the equation should be the “geometric” side and one side should be the “matter” side.

Let’s start with the matter side. The relevant quantity here is called the stress-energy tensor. It is a second rank tensor whose components carry certain meanings. These meanings are shown in the figure below.

The expression of stress-energy tensor depends on the type of material but the form of stress tensor is not important for our discussion. The important things are the following;

• Stress-energy is symmetric
• Stress-tensor satisfies the continuity equation

So the natural guess is to have stress-energy tensor in the “matter” side of the equation. Now, what about the “geometry” side? For this question, we first ask the question, “How do we determine if the spacetime is flat?”. The answer lies in moving vectors around loops.

Choose a point on a space and pick a vector at that point, Now, move that vector around a loop and bring it back to the starting point. The catch is that we need to keep this vector parallel to itself while doing it. This procedure is called parallel transport. For readers interested in the mathematical meaning of parallel transport, it means that the covariant derivative of the vector, along the loop vanishes (its components don’t change i.e. it is parallel to itself).

Now, the criterion to determine if a space is flat or not is to determine if all vectors, parallel transported around all possible loops come back to themselves or not. If the vectors come back to themselves, the space is flat and vice versa.

We can calculate the difference between the final vectors if we parallel transport a given initial vector from point A to point B through two different paths. If this difference is zero, then the space is flat and vice cersa. The difference turns out to be the proportional to a commutator of covariant derivatives.

Now, we can write down this commutator of covariant derivatives in terms of a very important quantity called the Riemann tensor.

Therefore, we can say that if Riemann tensor is zero, then the space is flat and vice versa. All the reasoning that we did above was for space but it is also true for spacetime.

Now, we have a candidate (i.e. Riemann tensor) that may go in the “geometric” side of Einstein’s equation. However, it just can’t be Riemann tensor because it is fourth rank tensor but we have a second rank tensor (stress-energy tensor) on the other side of equation.

We can make second order tensors using Riemann tensor and the metric tensor. Using some identities that Riemann tensor satisfies, it turns out that there is only one interesting tensor is a tensor called the Ricci tensor. We can further contract the Ricci tensor with the metric tensor to get a quantity called the Ricci scalar.

So, a natural guess is to put Ricci tensor in the “geometry” side of Einstein’s equation. If this guess is to work, then Ricci tensor should be symmetric just stress-tensor. It can be checked that it is symmetric. The second requirement is that Ricci tensor should satisfy the contiuity equation just like stress-energy tensor does.

So see if this requirement is fulfilled, we use a property of Riemann tensor called the Bianchi identity and use the fact that covariant derivative of the metric tensor is zero.

So we see that Ricci tensor itself doesn’t satisfy the contuity equation but the combination mentioned above does satisfy it. Moreover, this combination is symmetric too. So, the expression that should go to the geometric side is this combination and it can be at most proportional to the stress-energy tensor. So, the preliminary form of Einstein’s equation is derived.

Now we need to determine the value of the constant K. To determine it, we need to make sure that for low speeds and small gravitational fields, Einstein’s equation reduces to Newton’s law of gravity. I will just quote the equation for low speeds and weak gravity that we get from Einstein’s equation and match it with Newton’s equation for gravity.

Now, we have determined the constant K as well and our work here is almost complete. A last point that I want to address is that since covariant derivative of the metric tensor is zero, we can add a term proportional to the metric tensor on the “geometric” side of the equation and the “geometric” side will still satisfy the continuity equation. This extra term is called the cosmological constant term.

Now, we have the form of Einstein’s equation that is usually presented. Mostly, speed of light is set to 1. We will set c=1 as well. However, this isn’t the form of Einstein’s equation that physicists normally use. The form that is actually used is called the “trace-reversed” form. In this form, we contract Einstein’s equation with thre metric tensor to eliminate the Ricci scalar in favour of the trace of stress-tensor. Einstein’s equation in this form is much easier to solve as compared to the original form.

This trace reversed form is the form used to derive solutions of Einstein’s equation e.g. the Schwarzchild solution.