Geodesics on upper half-plane factory direct

This is a continuation of Einstein the Stubborn.
We have calculated the Christoffel symbols of the Levi-Civita connection of the SL(2,R) invariant metric on the upper half-plane.

We have used the standard formula, the same that physicists and astronomers are using in their calculations of Black Holes, White Holes, Big Bangs and Small Bangs:

(1)   \begin{equation*}\Gamma^{i}_{kl}=\frac{1}{2}g^{im}\left(\frac{\partial g_{mk}}{\partial x^{l}}+\frac{\partial g_{ml}}{\partial x^{k}}-\frac{\partial g_{kl}}{\partial x^{m}}\right).\end{equation*}

With the metric of upper half-plane hyperbolic geometry given by

(2)   \begin{equation*}g=\frac{1}{y^2}\begin{bmatrix}1&0\\0&1\end{bmatrix},\end{equation*}

which is a very simple kind of metric, only four of the six Christoffel symbols are non-zero. They are:

(3)   \begin{eqnarray*} \Gamma^1_{21}&=&\Gamma^1_{12}=-\frac{1}{y},\\ \Gamma^2_{11}&=&\frac{1}{y},\\ \Gamma^2_{22}&=&-\frac{1}{y}. \end{eqnarray*}

In the previous post I have originally written “only three of the six Christoffel symbols are non-zero”, but I have forgotten about the symmetry, as in the first line above.

The Christoffel symbols, that is “the coefficients of the torsion free metric affine connection” serve as the tools for defining “parallel transport” of geometric objects along curves. The transport is, in general, path dependent, when there is a non-vanishing “curvature”.

Curvature is expressed in terms of Christoffel symbols and their derivatives. But “geodesics” are expressed directly in terms of the Christoffel symbols. Here are their equations:

(4)   \begin{equation*}\frac{d^2 x^i}{ds^2}=  -\Gamma^{i}_{jk}\frac{dx^j}{ds}  \frac{dx^k}{ds}.\end{equation*}

One has to remember that the Einstein convention is being used, so that in the above formula summation over the dummy indices j,k is implied.

Let us apply this general formula to our case, with x^1=x,x^2=y. Let us calculate the right hand side for i=1. With i=1 we have \Gamma^1_{12}=\Gamma^1_{21}=-\frac{1}{y}, therefore

(5)   \begin{equation*} \frac{d^2x}{ds^2}=\frac{2\frac{dx}{ds}\frac{dy}{ds}}{y}.\end{equation*}

With i=2 we have \Gamma^2_{11}=1/y, \Gamma^2_{22}=-1/y. Therefore

(6)   \begin{equation*}\frac{d^2y}{ds^2}=-\frac{\left(\frac{dx}{ds}\right)^2-\left(\frac{dy}{ds}\right)^2}{y}.\end{equation*}

These are the geodesic equations as they come directly from the factory, in their original shape.

In Geodesics on the upper half-plane – Part 2 circles we have derived the formula for geodesics from conservation laws, that is “second-hand”. We have obtained the following formulas:

(7)   \begin{equation*} \frac{\dot{x}}{y^2(t)}=\mathrm{const},\end{equation*}

(8)   \begin{equation*} \frac{\dot{x}(t)x(t)+\dot{y}(t)y(t)}{y^2(t)}=\mathrm{const}.\end{equation*}

They are simpler than Eqs. (5,6). But they are consequences of (5,6). Taking derivatives of the left hand sides of Eqs. (7,8) we can easily check that they are automatically zero if Eqs. (5,6) are satisfied! By using conservation laws we have simply taken a short way.

Once we got into the main objects of Riemannian differential geometry, in the next post we will calculate the curvature of our metric.