The gravitational field

Today we will start implementing the 7 point roadmap in the case of the gravitational field. Technically gravity does not form a gauge theory but since it was the starting point of Weyl's insight, I will start with this as well and next time I will show how the program works in case of the electromagnetic field.

1. The gauge group

The "gauge group" in this case is the group of general coordinate transformations in a real four-dimensional Riemannian manifold M. Now the argument against Diff M as a gauge group comes from locality. An active diffeomorphism can move a state localized near the observer to one far away which can be different. However, for the sake of argument I will abuse this today and considered Diff M as a "gauge group" because of the deep similarities (which will explore in subsequent posts) between this and proper gauge theories like electromagnetism and Yang-Mills.

2. The covariant derivative giving rise to the gauge group

For a vector field $f^\alpha$ the covariant derivative is defined as follow:

$D_\rho f^\alpha = \partial_\rho f^\alpha +{\Gamma}^{\alpha}_{\rho\sigma} f^\alpha$

where ${\Gamma}^{\alpha}_{\rho\sigma}$ is called an affine connection. If we demand that the metric tensor is a covariant constant under D we can find that the connection is:

${\Gamma}^{\sigma}_{\mu\nu} = \frac{1}{2}[g_{\rho\mu,\nu} + g_{\rho\nu,\mu} - g_{\mu\nu,\rho}]$

where $f_{\rho,\sigma}  = \partial_\sigma f_\rho$ 

3. The integrability condition

We define this condition as the commutativity of the covariant derivative. If we define the notation: $D_\mu D_\nu f_\sigma = f_{\sigma;\nu\mu}$ we can write this condition as:

$f_{\rho;\mu\nu} - f_{\rho;\nu\mu} = 0$

Computing the expression above yields:

$f_{\rho;\mu\nu} - f_{\rho;\nu\mu} = f_\sigma {R}^{\sigma}_{\rho\mu\nu}$

where

${R}^{\sigma}_{\rho\mu\nu} = {\Gamma}^{\tau}_{\rho\mu}{\Gamma}^{\sigma}_{\tau\nu} - {\Gamma}^{\tau}_{\rho\nu}{\Gamma}^{\sigma}_{\tau\mu} + {\Gamma}^{\sigma}_{\rho\mu,\nu} - {\Gamma}^{\sigma}_{\rho\nu,\mu}$

4. The curvature

From above the integrability condition is ${R}^{\sigma}_{\rho\mu\nu} = 0$ and R is called the Riemann curvature tensor.

5. The algebraic identities

The algebraic identities come from the symmetry properties of the curvature tensor which reduces the 256 components to only 20 independent ones. I am too tired to type the proof of the reduction to 20, but you can easily find the proof online.

6. The homogeneous differential equations

If we take the derivative of the Riemann tensor we obtain a differential identity known as the Bianchi identity:

${R}^{\sigma}_{\rho\mu\nu;\tau} + {R}^{\sigma}_{\rho\tau\mu;\nu} + {R}^{\sigma}_{\rho\nu\tau;\mu} = 0$

7. The inhomogeneous differential equations

This equation is of the form:

geometric concept = physical concept

And in this case we use the stress energy tensor $T_{\mu\nu}$ and we find a geometric object with the same mathematical properties: symmetric and divergenless build out of curvature tensor. The left-hand side is the Einstein tensor:

$G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R$

The constant of proportionality comes from recovering Newton's gravitational equation in the nonrelativistic limit. In the end one obtains Einstein's equation:

$G_{\mu\nu} = 8\pi G T_{\mu\nu}$

Next time I will go through the same process for the electromagnetic field and map the similarities between the two cases. Please stay tuned.