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Deriving General Relativity from Field Theory

In this section, we further explore classical field theory. We will see how the Einstein-Hilbert action leads to Einstein's equations.

Previously, we have explored the Klein-Gordon equation, which describes the dynamics of a scalar field. Scalar fields are rank-0 tensors, and they are the simplest type of field. The central object of study in this section is the metric tensor, which is a rank-2 tensor. As such, we are dealing with a rank-2 field theory. This is also related to the fact that the hypothetical graviton is a spin-2 particle.

Table of Contents​

Einstein-Hilbert Action​

To begin, we make the assumption that curvature has a relationship with energy-momentum, and the classical limit yields Poisson's equation . Furthermore, we assume that our expression for action can be split into two parts: one that depends on the metric (the "curvature" part) and one that depends on the matter (the "matter" part);

Recall that to integrate over spacetime, we need to use the invariant volume element , where is the determinant of the metric tensor. The inclusion of the determinant can be though of as cancelling the scaling factor from the coordinate transformation. The curvature part of the action is given by:

where is the Lagrangian density for the curvature part. Because is a scalar, and it has to do with curvature, we conjecture that it is proportional to the Ricci scalar with some constant . We can also add an arbitrary constant to the Lagrangian density (the specific form is not important). Thus, we have:

The Einstein-Hilbert action is then given by:

Applying the Calculus of Variations​

Now we are ready to apply action principles to this action. To begin, we need to vary the action with respect to the metric tensor :

To bring the variation inside the integral, we add a variation of the metric (similar to -substitution in calculus):

Next, we distribute the functional derivative to each term in the integrand:

Because both and depend on the metric, we need to use the product rule to differentiate them:

Next, because is just , we can differentiate it just like a normal function:

Plugging this into the equation, we have:

Because Hamilton's principle demands that this variation is zero:

Calculating the Variations​

Variation of the Matter Action​

First, we will calculate the variation of the matter action . The matter action is given by a spacetime integral of the Lagrangian density :

Then, using the same -substitution trick as before, we can write:

Just like before, we use the product rule and plug in the functional derivative of :

We can now plug this into Equation :

Because we have spacetime integrals on both sides, we can drop them and focus on the integrands. This is because for the integrals to be equal for all variations, the integrands must be equal, and their difference is hence zero.

Multiplying both sides by gives:

And rearranging gives:

Without an explicit form for the Lagrangian density, we cannot proceed further. However, asserting a specific form for the Lagrangian density will make the equation lose generality. Instead, we define the entirety of the right-hand side as a new tensor , which is known as the energy-momentum tensor;

Thus, we have:

Dividing both sides by gives:

Variation of the Metric Determinant​

The next step is to calculate the variation of the metric determinant . Because the metric tensor is essentially a matrix, we can use the following identity:

Taking the functional derivative of both sides with respect to an arbitrary parameter gives:

The left-hand side gives . The right-hand side gives :

The variation fo the determinant is then:

To apply this to the metric, we set , , and :

The trace is just the sum of the diagonal elements, so we can write:

To write this in terms of instead of , we make use of the fact that and take the variation of both sides:

The left-hand side can be expanded using the product rule:

In other words, . We can plug this back into the equation for :

And so the functional derivative of the metric determinant is:

Plugging this into Equation gives:

Variation of the Ricci Scalar​

Finally, we need to calculate the variation of the Ricci scalar . The Ricci scalar is given by the trace of the Ricci tensor :

We shall first derive a lemma that will help us with this calculation, known as the Palatini identity:

Proof: Recall that the Ricci tensor is given by: