- §1. Schwarzschild Curvature
- §2. Rain Curvature
- §3. Einstein Components
- §4. Einstein Divergence
- §5. Metric Divergence in 2d
- §6. Metric Divergence
- §7. RW Curvature
- §8. Birkhoff's Theorem
- §9. Point Charge

### The Stress Tensor for a Point Charge

The electromagnetic field is a 2-form $F$. Maxwell's equations imply that $F$ is closed, that is that \begin{equation} dF = 0 \end{equation} which leads to the assumption that \begin{equation} F = dA \end{equation} where $A$ is the *4-potential*. In special relativity, we have \begin{equation} A = -\Phi\,dt + \AA\cdot d\rr \end{equation} where $\Phi$ is the *electric potential*, also called the *scalar potential*, and the 3-vector $\AA$ is the *magnetic
potential*, also called the *vector potential*. The electric and magnetic fields are then \begin{align} \EE &= -\grad\Phi \\ \BB &= \grad\times\AA \end{align} both of which are 3-vectors, that is, have no component in the $t$ direction. In the language of differential forms, we have \begin{equation} F = E \wedge dt + {*}B \end{equation} where of course $E=\EE\cdot d\rr$, $B=\BB\cdot d\rr$, and where $*$ denotes the *3-dimensional* Hodge dual operator.

The electric field of a point charge should take the form \begin{equation} \EE = \frac{q}{r^2} \>\rhat \end{equation} where in our geometric units we set $4\pi\epsilon_0=1$, so that, like mass, charge has the dimensions of length. We therefore expect the electromagnetic field of a point charge to take the form 1) \begin{equation} F = \frac12 F_{ij} \,\sigma^i\wedge\sigma^j = \frac{q}{r^2} \>\sigma^r\wedge\sigma^t \end{equation}

The components of the stress-energy tensor for an electromagnetic field can be shown to be \begin{equation} 4\pi T^i{}_j = F^{im}F_{jm} - \frac14 \delta^i{}_j F^{mn}F_{mn} \end{equation} which reduces to the diagonal matrix \begin{equation} 4\pi \left(T^i{}_j\right) = \frac12 \begin{pmatrix} -q/r^2 & 0 & 0 & 0 \\ 0 & -q/r^2 & 0 & 0 \\ 0 & 0 & q/r^2 & 0 \\ 0 & 0 & 0 & q/r^2 \end{pmatrix} \end{equation} for the electromagnetic field of a point charge.

*Gauge field theory*, including electromagnetism, can be formulated in the geometric language of

*fibre bundles*; the corresponding notion of curvature in the electromagnetic case is precisely $F$.