## 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.