Geodesics on the upper half-plane – Part 2 circles

In the last note Geodesics on the upper half-plane – Part 1 Killing vectors we verified that SL(2,R) transformations are isometries of the upper half-plane endowed with the Riemannian metric defined by the line element

(1)   \begin{equation*}ds^2=\frac{dx^2+dy^2}{y^2}.\end{equation*}

The corresponding scalar product between tangent vectors being

(2)   \begin{equation*}(\xi,\eta)_{z}=\frac{\xi_x\eta_x+\xi_y\eta_y}{y^2}.\end{equation*}

The idea is to use a “classical result” from differential geometry (a version of Noether’s theorem) that we have quoted from the online book by Sean Carroll:

Before applying this theorem to our case let us first see how it works for the standard plane Euclidean geometry. Translations are certainly isometries there. Streamlines of the vector fields generating horizontal and vertical translations form a square grid.

That straight lines intersect these grid lines under a constant angle is evident. In fact, that could serve as an alternative definition of a straight line.

So far so good.

But rotations are also isometries of the Euclidean geometry.

Looking at the image that shows the intersections of a straight lines with streamlines of the vector field generating rotation – nothing is evident. The angle of intersection is certainly not constant. We have to look at the theorem more closely.

The theorem states that the scalar product between Killing vector field (generating isometry) and the tangent vector to the geodesic is constant.

A general straight line \gamma(t)=(x(t),y(t)) on the plane has equation:

(3)   \begin{eqnarray*} x(t)&=&at+x_0,\\ y(t)&=&bt+y_0. \end{eqnarray*}

The tangent vector (\dot{x},\dot{y}) has components (a,b)

Rotation is defined by

(4)   \begin{eqnarray*} x(\phi)&=&\cos(\phi)x-\sin(\phi)y,\\ y(\phi)&=&\sin(\phi)x+\cos(\phi)y. \end{eqnarray*}

The corresponding vector field X(x,y) is obtained by taking derivatives with respect to \phi at \phi=0:

(5)   \begin{equation*} X(x,y)=(-y,x).\end{equation*}

It is streamlines of this vector field that we have drawn in the figure above.

Let us now calculate the scalar product of X and the vector tangent to our straight line at a point (x(t).y(t)) on our line:

(6)   \begin{equation*} (\dot{\gamma}(t),X(\gamma(t))=a(-y(t))+b(x(t))=a(-bt-y_0)+b(at+x_0)=-ay_0+bx_0=\mathrm{const}.\end{equation*}

Somewhat miraculously the non-constant (t-dependent) terms have cancelled out, and the scalar product is indeed constant. Not so evident, but nevertheless true. Perhaps there is some simple argument predicting this result, but I do not know it. Either use Noether’s theorem, or use direct calculation as above.

Let us now move from the Euclidean geometry of the plane to the hyperbolic geometry of upper half-plane. In SL(2,R) generators and vector fields on the half-plane we have focused our attention on two Killing vector fields (now we know that they are “Killing”), we will call them K_1,K_2 here:

(7)   \begin{eqnarray*} K_1(x,y)&=&(1,0),\\ K_2(x,y)&=&(x,y). \end{eqnarray*}

The first one corresponds to horizontal translation, the second one to uniform dilation. Their streamlines are

Suppose now \gamma(t)=(x(t),y(t)) is a geodesic, with tangent vector \dot{\gamma}(t)=(\dot{x}(t),\dot{y}(t)). Its scalar products with K_1 and K_2 should be constant. But now we remember that in calculating the scalar product we must use Eq. (2), so there will be denominator:

(8)   \begin{equation*} (\dot{\gamma}(t),K_1(\gamma(t)))=\frac{\dot{x}}{y^2(t)}=\mathrm{const},\end{equation*}

(9)   \begin{equation*} (\dot{\gamma}(t),K_2(\gamma(t)))=\frac{\dot{x}(t)x(t)+\dot{y}(t)y(t)}{y^2(t)}=\mathrm{const}.\end{equation*}

Their ratio should be therefore constant:

(10)   \begin{equation*}\frac{\dot{x}(t)x(t)+\dot{y}(t)y(t)}{\dot{x}(t)}=\mathrm{const},\end{equation*}


(11)   \begin{equation*}\dot{x}(t)x(t)+\dot{y}(t)y(t)=\mathrm{const }\, \dot{x}(t).\end{equation*}

This looks rather simple. In fact it should remind us about the equation of a circle. A circle with center on some point x_0 on x-axis and radius r has equation:

(12)   \begin{equation*}(x-x_0)^2+y^2=r^2.\end{equation*}

Assuming x=x(t),y=y(t) and differentiating with respect to t we get

(13)   \begin{equation*} 2\dot{x}(t)(x(t)-x_0)+2\dot{y}(t)y(t)=0,\end{equation*}


(14)   \begin{equation*} \dot{x}(t)x(t)+\dot{y}(t)y(t)=x_0\,\dot{x}(t).\end{equation*}

This is exactly Eq. (11)

Thus: geodesics are circles. Or better: straight lines are circles! In fact: half-circles, because their centers are on the x-axis, and our arena is only upper half-plane.

Except that we have missed some solutions. In Conformally Euclidean geometry of the upper half-plane there was the following sentence:

In the next note we will start calculating “straight lines”, or “geodesics” of our geometry. Some of them are almost evident candidates: vertical lines, perpendicular to the real line. But what about the other ones?

Well, we have these other ones, but how the vertical lines fit our reasoning above?

This is a homework. It is somewhat tricky though ….

Geodesics on the upper half-plane – Part 1 Killing vectors

According to Wikipedia

In differential geometry, a geodesic is a generalization of the notion of a “straight line” to “curved spaces”.

The first line in the online Encyclopedia of mathematics is similar

The notion of a geodesic line (also: geodesic) is a geometric concept which is a generalization of the concept of a straight line (or a segment of a straight line) in Euclidean geometry to spaces of a more general type.

Wolfram’s MathWorld is somewhat more original:

A geodesic is a locally length-minimizing curve. Equivalently, it is a path that a particle which is not accelerating would follow. In the plane, the geodesics are straight lines. On the sphere, the geodesics are great circles (like the equator). The geodesics in a space depend on the Riemannian metric, which affects the notions of distance and acceleration.

It is time for us to study geodesics on the upper half-plane, and do it in a semi-rigorous way. That would require a rigorous definition of geodesics, which would take us into differential geometry and variational calculus. That would certainly not be a length-minimizing and straight way of achieving our goal. For us, interested mainly in Lie groups and their actions, there is a shorter way. It is like in classical mechanics, where in many important cases we do not have to solve complicated Newton’s differential equations, it is enough to use the law of conservation of energy, or momentum.

That is what we will do. We will use conservation laws. Usually these come as theorems in courses of differential geometry (Noether’s theorem). For instance Sean Carroll in his online book Lecture Notes on General Relativity, in Chpater 5, More geometry has this piece:

And that is what we will use. And how to use it, in detail, we will see below.

Let us start with this sentence:

If a one-parameter family of isometries is generated by a vector field V^{\mu}(x), then V^{\mu} is known as a Killing vector field.

We have our candidates for Killing vector fields. We were plotting some of their streamlines in SL(2,R) generators and vector fields on the half-plane .

But are we sure that they generate “isometries”? Till now we have only a roundabout argument: metric on the upper half-plane comes from the metric on the disk, metric on the disk comes from geometry on the hyperboloid, metric on the hyperboloid comes from flat space-time metric of signature (2,1) and the SL(2,R) group comes from the SO(2,1) group of linear transformations preserving the flat space-time metric. That could be enough for a while, but can’t we check directly if indeed we have isometries?

Yes, we can check, and that is, in fact, quite easy. Generators of the SL(2,R) group form the Lie algebra sl(2,R) of real 2\times 2 matrices of trace zero. After exponentiation they generate one-parameter groups of SL(2,R) matrices. SL(2,R) acts on the upper-half plane \mathbb{H} by linear fractional transformations. If A is in SL(2,R)

(1)   \begin{equation*}A=\begin{bmatrix}\alpha&\beta\\ \gamma&\delta,\end{bmatrix}\end{equation*}

with \det A=\alpha\delta-\beta\gamma=1, then A acts on \mathbb{H} through

(2)   \begin{equation*}z\mapsto \tilde{z}=A\cdot z=\frac{\delta z+\gamma}{\beta z+\alpha}\end{equation*}

Is the transformation defined in Eq. (2) an isometry? The formula looks relatively simple when written in terms of complex variables. But if we write z=x+iy, \tilde{z}=\tilde{x}+i\tilde{y}, then the coordinates (\tilde{x},\tilde{y}) of the transformed point become not that simple functions of the coordinates (x,y) of the original point:

(3)   \begin{equation*}\tilde{x}=\frac{\alpha  \gamma +\alpha  \delta  x+\beta  \gamma  x+\beta  \delta  x^2+\beta  \delta  y^2}{\alpha^2+2 \alpha  \beta  x+\beta^2 x^2+\beta^2 y^2},\end{equation*}

(4)   \begin{equation*}\tilde{y}=\frac{y (\alpha  \delta -\beta  \gamma )}{\alpha^2+2 \alpha  \beta  x+\beta^2 x^2+\beta^2 y^2}.\end{equation*}

Is it an isometry? And what is isometry?

In Conformally Euclidean geometry of the upper half-plane we have derived the formula for calculating the length of a given curve:

(5)   \begin{equation*}s(t_0,t_1)=\int_{t_0}^{t_1}\frac{\sqrt{\dot{x}(t)^2+\dot{y}(t)^2}}{y(t)}\,dt.\end{equation*}

Now, suppose, we transform our curve using the matrix A. The length of the transformed curve is then given by the formula

(6)   \begin{equation*}\tilde{s}(t_0,t_1)=\int_{t_0}^{t_1}\frac{\sqrt{\dot{\tilde{x}}(t)^2+\dot{\tilde{y}}(t)^2}}{\tilde{y}(t)}\,dt,\end{equation*}

where the relation between (\tilde{x},\tilde{y}) and (x,y) is given by Eqs. (3,4).

The transformation is an isometry if \tilde{s}((t_0,t_1)=s(t_0,t_1) for any segment of any curve. Is true in our case? In order to verify it some little calculations are needed. If we listen to Leibniz:

“It is unworthy of excellent men to lose hours like slaves in the labor of calculation which could be relegated to anyone else if machines were used.”
— Gottfried Leibniz

we use our computer. I used Mathematica. Here is the result:

To summarize: after somewhat lengthy calculation we end up with

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

even without using the \det A=1 condition. Therefore SL(2,R) transformations are indeed isometries (for our metric). Therefore our vector fields are “Killing vector fields”. Therefore we can use their properties in our derivation of geodesic equations. Which we will continue in the following post.

Conformally Euclidean geometry of the upper half-plane

In Deriving invariant hyperbolic Riemannian metric on the half-plane we have calculated the line element for the SL(2,R) invariant hyperbolic geometry on the upper half-plane:

(1)   \begin{equation*}ds^2=\frac{dx^2+dy^2}{y^2}.\end{equation*}

Using this formula we can easily calculate the length of any segment of any curve. If (x(t),y(t)) is a parametric equation of the curve, then dx=\frac{dx}{dt}dt=\dot{x}(t)dt,\, dy=\frac{dy}{dt}\,dt=\dot{y}(t)\,dt, and so

(2)   \begin{equation*}ds=\sqrt{\frac{dx^2+dy^2}{y^2}}=\sqrt{\frac{\dot{x}^2+\dot{y}^2}{y^2}}dt,\end{equation*}

(3)   \begin{equation*}s(t_0,t_1)=\int_{t_0}^{t_1}\frac{\sqrt{\dot{x}(t)^2+\dot{y}(t)^2}}{y(t)}\,dt.\end{equation*}

In this last formula we have used the fact that in the upper half-plane we have y(t)>0.

From the studies of geometry of surfaces and its generalization developed mainly by Riemann we know that the formula for the line element allows us not only to find the length, but also the angles.

From Wikipedia:
Bernhard Riemann – see Modern Science Map

Riemannian geometry originated with the vision of Bernhard Riemann expressed in his inaugural lecture “Ueber die Hypothesen, welche der Geometrie zu Grunde liegen” (“On the Hypotheses on which Geometry is Based”). It is a very broad and abstract generalization of the differential geometry of surfaces in \mathbf{R}^3. Development of Riemannian geometry resulted in synthesis of diverse results concerning the geometry of surfaces and the behavior of geodesics on them, with techniques that can be applied to the study of differentiable manifolds of higher dimensions.

It allows us to calculate scalar products of tangent vectors. Let us rewrite Eq. (1) in the form:

    \[ds^2=\begin{bmatrix}dx&dy\end{bmatrix}\begin{bmatrix}g_{xx}&g_{xy}\\g_{yx}&g_{yy}\end{bmatrix}\begin{bmatrix}dx\\dy\end{bmatrix}= g_{xx}dx^2+2g_{xy}dx\,dy+g_{yy}dy^2,\]

using the symmetric (i.e. g_{xy}=g_{yx}) “Riemannian metric” matrix g


Thus, from Eq. (1), we have g_{xx}=1/y^2, g_{xy}=g_{yx}=0, and g_{yy}=1/y^2. Or

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

In general, if \xi=(\xi_x,\xi_y) and \eta=(\eta_x,\eta_y) are two tangent vectors at z=x+iy, then their scalar product is given by a general formula

(5)   \begin{equation*}(\xi,\eta)_{z}=\begin{bmatrix}\xi_x& \xi_y\end{bmatrix}g\begin{bmatrix}\eta_x\\ \eta_y\end{bmatrix}=g_{xx}\xi_x\eta_x+g_{xy}(\xi_x\eta_y+\xi_y\eta_x)+g_{yy}\xi_y\eta_y,\end{equation*}

which in our case specializes to

(6)   \begin{equation*}(\xi,\eta)_{z}=\frac{\xi_x\eta_x+\xi_y\eta_y}{y^2}.\end{equation*}

The numerator is exactly the same as for the Euclidean scalar product. It has the consequence that the angles in the hyperbolic upper half-plane geometry are the same as in the Euclidean geometry. That it is so follow from the definitions. Angles in any Riemannian geometry are defined the same way as in the Euclidean geometry. If \xi, \eta are two tangent vectors at z, then the angle \phi between them is defined through the formula:

(7)   \begin{equation*}\cos\phi=\frac{(\xi,\eta)}{||\xi|| ||\eta||},\end{equation*}

where ||\xi||=\sqrt{(\xi,\xi)} and ||\eta||=\sqrt{(\eta,\eta)} are their norms. If we use the scalar product given by Eq. (6), then the denominator y^2 in (\xi,\eta) cancels out with the product of denominators is the norms. The end result is

(8)   \begin{equation*}\cos(\phi)=\frac{\xi_x\eta_x+\xi_y\eta_y}{\sqrt{\xi_x^2+\xi_y^2}\sqrt{\eta_x^2+\eta_y^2}}.\end{equation*}

Therefore, when we look at the picture like the one obtained in the last post:

Esher’s Limit Circle IV developed onto upper half-plane. Click on the image to open it in full resolution.

all hyperbolic geometry angles between various lines at any particular joining point are the same as the perceived ones by our eyes ones – contrary to the sizes, where SL(2,R) invariant geometry and Euclidean geometry we are used to are different.

Whenever we have two metrics, g and g' that differ a scalar factor g'=c\,g, we say that they are conformally related, or conformal to each other. In our case the metric g in Eq. (4) is proportional to the identity matrix, our geometry is conformally Euclidean.

In the next note we will start calculating “straight lines”, or “geodesics” of our geometry. Some of them are almost evident candidates: vertical lines, perpendicular to the real line. But what about the other ones?

Our geometry is a toy geometry, as simple as possible. The group S(1,1) (or SL(2,R) ) is too simple. But one step further there is SU(2,2) – which became the basis for the “chronometric cosmology” developed by the late mathematician Irving Ezra Segal.

From American Astronomical Society, Irving Ezra Segal (1918 – 1998)

… The number of astronomers convinced was small, with experimentalists providing a slightly more favorable reception than theoreticians. Occasionally, the controversy was quite theatrical. A younger colleague of Segal was assaulted on Massachusetts Avenue by an astrophysicist still fuming from a Segalian conversation, and the publication of all the correspondence to and from Cambridge would be delightful. Particularly vituperative referee reports were always posted on his office door, some years overflowing onto the walls. Tee-shirts emblazoned with “Save Energy—Stop the Expansion of the Universe” can still be seen in Harvard Square. Whatever the final judgment is on the chronometric theory, Segal’s contributions to mathematics and mathematical physics have already had profound influence. Segal will be remembered for his conviction that the proper role of mathematics is to illuminate physics and his constant, irascible charm. We will also remember the perfect host, pouring wine for many a dinner, discussing physics, mathematics, philosophy, the proper preparation of coffee, and life.