Category: Hyperbolic geometry

Curvature of the upper half-plane

Posted on by arkajad

In Geodesics on upper half-plane factory direct we used the Christoffel symbols and identified geodesics on the upper half plane endowed with the hyperbolic geometry metric. The formulas for Christoffel symbols contain derivatives of the metric tensor components:

(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*}

From these connection coefficients one constructs the Riemann curvature tensor. Let us take the expression from Wikipedia:

(2)   \begin{equation*} R^{i}_{\phantom{i}jkl}=\frac{\partial \Gamma^{i}_{lj}}{\partial x^k}-\frac{\partial \Gamma^{i}_{kj}}{\partial x^l}+\Gamma^{i}_{km}\Gamma^m_{lj}-\Gamma^{i}_{lm}\Gamma^m_{kj}. \end{equation*}

From its construction several symmetries follow, so that only n^2(n^2-1)/12 really independent components remain. For n=4 one has to calculate 20 components. But for n=2 it is only one. But even this one, why should we calculate it? We better listen to Leibniz.

“It is unworthy of excellent men to lose hours like slaves in the labour of calculation which could safely be relegated to anyone else if machines were used.“

Nowadays software does all that. Right now I ma using Mathematica, so I type

riemann = Simplify[Table[
D[christ[[i, j, l]], coord[[k]] ] –
D[christ[[i, j, k]], coord[[l]] ] +
Sum[christ[[s, j, l]] affine[[i, k, s]] –
christ[[s, j, k]] affine[[i, l, s]],
{s, 1, n}],
{i, 1, n}, {j, 1, n}, {k, 1, n}, {l, 1, n}] ]

With the result that

(3)   \begin{equation*}R^1_{\phantom{1}221}=-R^2_{\phantom{2}121}=\frac{1}{y^2}.$ From the curvature we usually calculate the Ricci tensor $R_{jl}$ \begin{equation*}R_{jl}=R^{i}_{\phantom{i}jil}.\end{equation*}

The Ricci tensor is obtained by contraction (summation) over two indices of the Riemann tensor. The Ricci tensor is always automatically symmetric. In our case we get that Ricci is simply the negative of the metric

(4)   \begin{equation*}R_{ij}=-g_{ij}.\end{equation*}

Spaces in which this happens are known under a special name: Einstein spaces (or “Einstein manifolds“).

From Ricci tensor by contraction with the inverse metric one calculates the scalar curvature. In our case the result is -2:

(5)   \begin{equation*}R=g^{ij}R_{ij}=-2\end{equation*}

While Riemann and Ricci are “tensors”, they may have different components in different coordinate systems, the scalar curvature is an invariant. Its value is independent of the coordinates. In our case the result -2 would be the same if we have calculated it with the coordinates and metric that we have used on the Poincare disk.

From two dimensions of the Poincare disk and the upper half-plane we will now move to three-dimensions of the group SL(2,R) itself.

Below is the view of the Mathematica notebook doing the calculations described in this post.

Riemann curvature calculations using Mathematica

Einstein the Stubborn

Posted on by arkajad

Before developing his 1915 General Theory of Relativity, Einstein held the “Entwurf” theory. Tullio Levi-Civita from Padua, one of the founders of tensor calculus, objected to a major problematic element in this theory, which reflected its global problem: its field equations were restricted to an adapted coordinate system. Einstein proved that his gravitational tensor was a covariant tensor for adapted coordinate systems. In an exchange of letters and postcards that began in March 1915 and ended in May 1915, Levi-Civita presented his objections to Einstein’s above proof. Einstein tried to find ways to save his proof, and found it hard to give it up. Finally Levi-Civita convinced Einstein about a fault in his arguments. However, only in spring 1916, long after Einstein had abandoned the 1914 theory, did he finally understand the main problem with his 1914 gravitational tensor. In autumn 1915 the Göttingen brilliant mathematician David Hilbert found the central flaw in Einstein’s 1914 derivation. On March 30, 1916, Einstein sent to Hilbert a letter admitting, “The error you found in my paper of 1914 has now become completely clear to me”.

That is what Weinstein writes about Einstein in Einstein the Stubborn: Correspondence between Einstein and Levi-Civita

Finally Einstein learned what he needed to learn and worked out his “General Relativity Theory” – the theory of gravitation based on the mathematics of (pseudo) Riemannian metric tensor. From metric tensor one calculates the “Levi-Civita connection”, encoded in what are called “Christoffel symbols“. From them one calculates the curvature: ‘Matter tells space how to curve, space tells matter how to move.’ That is the essence of Einstein’s theory of gravitation. Einstein was, for a while, happy with this picture. But only for a while. Many physicists are happy with it even today.

But that is not the subject of my post today. My plan is simply to calculate the Christoffel symbols for the SL(2,R) invariant metric that we have discussed in Conformally Euclidean geometry of the upper half-plane

Let me recall the metric:

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

So, it is a 2\times 2 symmetric matrix that depends on the coordinates of a point z=x+iy in the upper half-plane of complex numbers z with positive imaginary part y. We usually write g as a matrix with entries g_{ij} where i,j=1,2. We have two coordinates, we may call x=x^1,y=x^2. It is customary in differential geometry to use the coordinate index as the upper index. It is a convention, but a useful convention. Usually it is accompanied with another convention, so called “Einstein summation convention”. This Einstein convention is that whenever we see a term that contains one symbol with index down and another symbol with index up – it means that this is an unwritten sum from i=1 to n, where n is the number of dimensions. We are dealing with n=2, so whenever we see something like x^i y_i or x_ky^k it means: x^1y_1+x^2y_2 or x_1y^1+x_2y^2. The name of the repeated index does not matter, it can be any letter, just different from other letters present in the given term. We call it a “dummy index”.

The metric is written with indices i,j as lower indices. We call them “covariant”. Upper indices are usually called contravariant. The metric should always be invertible, otherwise terrible things can happen, the universe may cease to have its ordinary meaning. The inverse metric is usually written as a contravariant tensor g^{ij}. In our case the inverse metric exists because we assume that y>0:

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

For y=0 it would become zero, and the different parts of the Universe would not know what to do. There would be a confusion, the door to paranormal could get opened. And I am not joking. I wrote about it in an unpublished paper. The content of this paper was too scary for the referees of Physics Letters. The paper is available here: Vanishing Vierbein in Gauge Theories of Gravitation. As I said – it was never published – but it is being cited by others, for instance here: Phys.Rev. D62 (2000) 044004 DOI: 10.1103/PhysRevD.62.044004 \url{}, or here: Found.Phys.38:7-37,2008 DOI: 10.1007/s10701-007-9190-0 \url{}

Let us now calculate the Christoffel symbols for our metric. We use the standard formulas from differential geometry, they can be found in Wikipedia at Christoffel symbols of the second kind

(3)   \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*}

The expression for \Gamma^{i}_{kl} is symmetric in kl – the theory has no torsion. There are 2\times 3 =6 symbols that must be computed. If four dimensions one needs to calculate 4\times 10=40 of these symbols – lot of calculation, as each of these 40 symbols is a sum of four terms (sum over m). In old times people were not as lazy as they are today. They were calculating. Today I am using Mathematica (or Maple, or whatever). Using Mathematica, for instance, it goes as follows:

Christoffel sysmbols for the upper half plane with Mathematica

Only four of the six symbols are different from zero, and they are very simple. Christoffel symbols enter geodesic equations, when they are different from zero – the geodesic lines curve. They may curve because coordinates are curved, or because the space is curved. We will discuss it next in our toy model.

Dedekind tessellation on the Poincaré disk

Posted on by arkajad

This is a continuation from Dedekind tessellation or circles all the way down. I am studying the very interesting paper by J. Kocik, aka Jurek, “A note on the Dedekind tessellation”. It is a pity that the paper is unpublished. I am not sure how much of its content I am allowed to reveal. I will take a risk and reveal just one goodie from this paper.

Here is the goodie:

Dedekind tessellation on the Poincaré disk and its neighborhood. Based on Ä note on the Dedekind tessellation”by J. Kocik.
Click on the image to open in full resolution.

And I will explain now how I produced this picture.

As I have explained it in Dedekind tessellation or circles all the way down, we have a nice tessellation of the upper half-plane. The upper half-plane is tessellated by circles which, in hyperbolic geometry are “straight lines”. Using Cayley transform we can move these circles to the unit disk. Kocik, in his paper, gives the algorithm for constructing the centers and the radii of these circles. I used Mathematica to implement this algorithm as follows:

Mathematica implementation based on Kocik’s algorithm

I did not follow exactly Kocik’s algorithm. Moreover I have rotated the data from his paper by 90 degrees, so that it agrees with the Cayley transform I was using before. I stored the list of positions and radii of the circles that I was using in a file: c200.m.

The red circle is the unit circle. Strictly speaking the Dedekind tessellation should be restricted to the inside of the unit circle. But It is amazing to plot the circles also in the outside region. After all the unit circle is just a boundary between two universes. Why should we restrict ourselves to only one side? The universe we are living in is, perhaps, also a boundary between different multidimensional universes of hyperdimensional physics.