The **theoretical description** of the movement of a particle in some curved 3D-space and the movement of it on a 2D-surface are equivalent. Both entities are described by a **metric** and the equation of motion is the **geodesic equation**. Further giving the particle a charge lets it further interact with an **electric field**. In this problem you can learn to **derive** the corresponding geodesic equation.

## Problem Statement

Consider the movement of a particle with charge \(q\) and mass \(m\) attached to a **two-dimensional surface** \(S\) in an external **electrostatic field**. Let us assume that the surface is described by the **metric** \(g\left(\mathbf{r}\right)=g_{ij}\left(\mathbf{r}\right)dx^{i}dx^{j}\). Using the definition of the **Christoffel symbols**, \[\Gamma_{kl}^{i}\left(\mathbf{r}\right) \equiv \frac{1}{2}g^{im}\left(\mathbf{r}\right)\left(g_{mk,l}\left(\mathbf{r}\right)+g_{ml,k}\left(\mathbf{r}\right)-g_{kl,m}\left(\mathbf{r}\right)\right)\ ,\]find an expression for the **equation of motion** of the particle. Compare your result to the equations of motion for a free particle in an electric field. Explain the meaning of the arising force terms.

##

Relativistic Notation

Note that in this problem it is helpful to use the condensed **Einstein notation** for which we leave out the summation symbol if an index appears both up- and downwards of a quantity:\[\sum_{i}u^{i}v_{i} = u^{i}v_{i}=g_{ij}\left(\mathbf{r}\right)u^{i}v^{i}\ .\]As we can see, indices can be “moved” with the metric - lower und upper indices correspond to a representation in cotangent and tangent space, respectively. The **length of a vector** can be calculated using \(g_{ij}\): \(\mathbf{u}^{2}=g_{ij}u^{i}u^{j}\equiv g\left(\mathbf{u},\mathbf{u}\right)\). For a surface in a cartesian (flat) x-y-plane we would simply have \(g_{ij}=\delta_{ij}\) and the squared length of a vector would be just \(\mathbf{u}^{2}=u_{x}^{2}+u_{y}^{2}\).

Furthermore, **partial derivatives** can also be denoted by a comma, \(\partial f\left(x\right)/\partial x=\partial_{x}f\left(x\right)=f_{,x}\left(x\right)\). See also our section on relativistic electrodynamics for more in-depth informattion.

## Background: Geodesic Motion in General Relativity

In the case of a vanishing electric field we will derive a very general and important result. It explains the motion of a particle not only on surfaces but holds for motions on “generalized surfaces” called **manifolds** with a given metric \(g_{\mu\nu}\left(x^{\alpha}\right)\). The equation can also be derived from the requirement of the minimum of the length\[L = \int_a^b \sqrt{g_{\mu\nu}\left(x^{\alpha}\left(\tau\right)\right)\frac{dx^{\mu}}{d\tau}\frac{dx^{\nu}}{d\tau}}d\tau\]of a curve \(x^{\alpha}\left(\tau\right)\) connecting two points on that manifold. Such a shortest path is called a **geodesic** and the equation we will derive consequentially termed **geodesic equation**.

In general relativity, the **metric** itself represents the **gravitational field**. However, it is not the metric of just space but also includes time components. Roughly speaking, all kinds of matter and energy affect the metric as described by the **Einstein field equations**. In our problem we assumed that the mass or velocity of our charge does not influence the surface, i.e. the metric. In general relativity this is an approximation - the geodesic motion is assumed to not influence the metric. The geodesic equation thus describes the movement of a test particle through spacetime.

Electromagnetic fields itself can be the source of gravitation since they contain energy. This can also be incorporated into general relativity. The corresponding theory is called **Einstein-Maxwell** theory. In our problem, the metric is given ad-hoc and not influenced by the electrostatic field.