7.3: Hyperbolic Geometry with Curvature k < 0
We may do the same gentle scaling of the Poincaré model of hyperbolic geometry as we did in the previous section to the disk model of elliptic geometry. In particular, for each negative number \(k \lt 0\) we construct a model for hyperbolic geometry with curvature \(k\text{.}\)
We define the space \(\mathbb{D}_k\) to be the open disk of radius \(\dfrac{1}{\sqrt{|k|}}\) centered at the origin in \(\mathbb{C}\text{.}\) That is, \(\mathbb{D}_k\) consists of all \(z\) in \(\mathbb{C}\) such that \(|z| \lt \dfrac{1}{\sqrt{|k|}}\text{.}\) In this setting, the circle at infinity is the boundary circle \(|z| = \dfrac{1}{\sqrt{|k|}}\text{.}\)
The group \({\cal H}_k\) consists of all Möbius transformations that send \(\mathbb{D}_k\) to itself. The geometry \((\mathbb{D}_k,{\cal H}_k)\) with \(k \lt 0\) is called hyperbolic geometry with curvature \(k\text{.}\) Pushing analogy with the elliptic case, we may define the group of transformations to consist of all Möbius transformations with this property: if \(z\) and \(z^*\) are symmetric with respect to the circle at infinity then \(T(z)\) and \(T(z^*)\) are also symmetric with respect to the circle at infinity. Noting that the point symmetric to \(z\) with respect to this circle is \(\displaystyle z^* = \dfrac{1}{|k|\overline{z}} = -\dfrac{1}{k\overline{z}}\text{,}\) we draw the satisfying conclusion that \(T \in {\cal H}_k\) if and only if the following holds:
\[ \text{if}~z^* = -\dfrac{1}{k\overline{z}}~~ \text{then}~~ T(z^*) =-\dfrac{1}{k\overline{T(z)}}\text{.} \]
Thus, the group \({\cal H}_k\) in the hyperbolic case has been defined precisely as the group \({\cal S}_k\) in the elliptic case. Furthermore, one can show that transformations in \({\cal H}_k\) have the form
\[ T(z) = e^{i\theta}\dfrac{z-z_0}{1+k\overline{z_0}z}\text{,} \]
where \(z_0\) is a point in \(\mathbb{D}_k\text{.}\)
Straight lines in this geometry are the clines in \(\mathbb{C}^+\) orthogonal to the circle at infinity. By Theorem \(3.2.3\), a straight line in \((\mathbb{D}_k,{\cal H}_k)\) is precisely a cline with the property that if it goes through \(z\) then it goes through its symmetric point \(\dfrac{-1}{k\overline{z}}\text{.}\)
The arc-length and area formulas also get tweaked by the scale factor, and now look identical to the formulas for elliptic geometry with curvature \(k\text{.}\)
The arc-length of a smooth curve \(\boldsymbol{r}\) in \(\mathbb{D}_k\) is
\[ {\cal L}(\boldsymbol{r}) = \int_a^b \dfrac{2|\boldsymbol{r}^\prime(t)|}{1 + k|\boldsymbol{r}(t)|^2}~dt\text{.} \]
The area of a region \(R\) given in polar coordinates is computed by the formula
\[ A(R) = \iint_R \dfrac{4r}{(1+kr^2)^2}dr d\theta\text{.} \]
As in Chapter 5 when \(k\) was fixed at \(-1\), the area formula is a bear to use, and one may convert to an upper half-plane model to determine the area of a \(\dfrac{2}{3}\)-ideal triangle in \(\mathbb{D}_k\text{.}\) The ambitious reader might follow the methods of Section 5.5 to show that the area of a \(\dfrac{2}{3}\)-ideal triangle in \((\mathbb{D}_k,{\cal H}_k)\) (\(k \lt 0\)) with interior angle \(\alpha\) is \(-\dfrac{1}{k}(\pi - \alpha)\text{.}\)
With this formula in hand, we can derive the area of any triangle in \(\mathbb{D}_k\) in terms of its angles, exactly as we did in Chapter 5.
In hyperbolic geometry with curvature \(k\text{,}\) the area of a triangle with angles \(\alpha, \beta\text{,}\) and \(\gamma\) is
\[ A =\dfrac{1}{k}(\alpha+\beta+\gamma -\pi)\text{.} \]
Observing Negative Curvature
Suppose we are located at a point \(z\) in a hyperbolic universe with curvature \(k\text{.}\) We see in the distance a hyperbolic line \(L\) that seems to extend indefinitely. We might intuitively see the point \(w\) on the line that is closest to us, as suggested in Figure \(7.3.1\). Now suppose we look down the road a bit to a point \(v\text{.}\) If \(v\) is close to \(w\) the angle \(\angle wzv\) will be close to \(0\). As \(v\) gets further and further away from \(w\text{,}\) the angle will grow, getting closer and closer to the angle \(\theta = \angle wzu\text{,}\) where \(u\) is an ideal point of the line \(L\text{.}\)
A curious fact about hyperbolic geometry is that this angle \(\theta\text{,}\) which is called the angle of parallelism of \(\boldsymbol{z}\) to \(\boldsymbol{L}\), is a function of \(z\)'s distance \(d\) to \(L\text{.}\) In Section 5.4 we saw that \(\cosh(d) = \dfrac{1}{\sin(\theta)}\) in \((\mathbb{D},{\cal H})\text{.}\) In particular, one may deduce the distance \(d\) to the line \(L\) by computing \(\theta\text{.}\) No such analogy exists in Euclidean geometry. In a Euclidean world, if one looks farther and farther down the line \(L\text{,}\) the angle will approach \(90^{\circ}\text{,}\) no matter one's distance \(d\) from the line. The following theorem provides another formula relating the angle of parallelism to a point's distance to a line.
In hyperbolic geometry with curvature \(k\text{,}\) the hyperbolic distance \(d\) of a point \(z\) to a hyperbolic line \(L\) is related to the angle of parallelism \(\theta\) by the formula
\[ \tan(\dfrac{\theta}{2}) = e^{-\sqrt{|k|}d}\text{.} \]
- Proof
-
For this proof, let \(s = \dfrac{1}{\sqrt{|k|}}\text{.}\) Note that \(s\) is the Euclidean radius of the circle at infinity in the disk model for hyperbolic geometry with curvature \(k\text{.}\) Since angles and lines and distances are preserved, assume \(z\) is the origin and \(L\) is orthogonal to the positive real axis, intersecting it at the point \(x\) (with \(0 \lt x \lt s\)).
Recall the half-angle formula
\[ \tan(\dfrac{\theta}{2}) =\dfrac{\tan(\theta)}{\sec(\theta)+1}\text{.} \]
According to Figure \(7.3.2\), \(\tan(\theta) = \dfrac{r}{s} \text{,}\) where \(r\) is the Euclidean radius of the circle containing the hyperbolic line \(L\text{.}\) Furthermore, \(\sec(\theta) = \dfrac{(x+r)}{s}\text{,}\) so
\[ \tan(\dfrac{\theta}{2}) = \dfrac{r}{x+r+s}\text{.} \label{7.3.8} \]
We may express \(x\) and \(r\) in terms of the hyperbolic distance \(d\) from \(0\) to \(x\text{.}\) In Exercise \(7.3.2\) we prove the hyperbolic distance from \(0\) to \(x\) in \((\mathbb{D}_k,{\cal H}_k)\) is
\[ d = s \ln\left(\dfrac{s + x}{s-x}\right) \]
so that
\[ x = s \cdot \dfrac{e^{d/s} - 1}{e^{d/s}+1}\text{.} \]
Express \(r\) in terms of \(d\) by first expressing it in terms of \(x\text{.}\) Note that segment \(x x^*\) is a diameter of the circle containing \(L\text{,}\) where \(x^* = \dfrac{-1}{kx}\) is the point symmetric to \(x\) with respect to the circle at infinity. Thus, \(r\) is half the distance from \(x\) to \(x^*\text{:}\)
\[ r = -\dfrac{1 + kx^2}{2kx}\text{.} \]
Replacing \(k\) with \(-\dfrac{1}{s^2}\text{,}\) we have
\[ r=\dfrac{s^2-x^2}{2x}\text{.} \]
One checks that after writing \(x\) in terms of \(d\text{,}\) \(r\) is given by
\[ r = \dfrac{2se^{d/s}}{e^{2d/s}-1}\text{.} \]
Substitute this expression for \(r\) into the equation labeled (\(\ref{7.3.8}\)) in this proof, and after a dose of satisfactory simplifying one obtains the desired result:
\[ \tan(\dfrac{\theta}{2}) = e^{-d/s}\text{.} \]
Since \(s = \dfrac{1}{\sqrt{|k|}}\) this completes the proof.
Parallax
If a star is relatively close to the Earth, then as the Earth moves in its annual orbit around the Sun, the star will appear to move relative to the backdrop of the more distant stars. In the idealized picture that follows, \(e_1\) and \(e_2\) denote the Earth's position at opposite points of its orbit, and the star \(s\) is orthogonal to the plane of the Earth's orbit. The angle \(p\) is called the parallax , and in a Euclidean universe, \(p\) determines the star's distance from the Sun, \(D\text{,}\) by the equation \(D = \dfrac{d}{\tan(p)}\text{,}\) where \(d\) is the Earth's distance from the Sun.
We may determine \(p\) by observation, and \(d\) is the radius of Earth's orbit around the Sun (\(d\) is about \(8.3\) light-minutes.) In practice, \(p\) is quite small, so a working formula is \(D = \dfrac{d}{p}\text{.}\) The first accurate measurement of parallax was recorded in \(1837\) by Friedrich Bessel (\(1784\)-\(1846\)). He found the stellar parallax of \(0.3\) arc seconds (\(1\) arc second \(= \dfrac{1}{3600}^{\circ}\)) for star \(61\) Cygni, which put the star at about \(10.5\) light-years away.
If we live in a hyperbolic universe with curvature \(k\text{,}\) a detected parallax puts a bound on how curved the universe can be. Consider Figure \(7.3.3\). As before, \(e_1\) and \(e_2\) represent the position of the Earth at opposite points of its orbit, so that the distance between them is \(2d\text{,}\) or about \(16.6\) light-minutes. Assume star \(s\) is on the positive real axis and we have detected a parallax \(p\text{,}\) so that angle \(\angle e_2se_1 = 2p\text{.}\)
The angle \(\alpha = \angle e_1e_2s\) in Figure \(7.3.3\) is less than the angle of parallelism \(\theta = \angle e_1e_2u\text{.}\) Noting that \(\tan(x)\) is an increasing function and applying Lobatchevsky's formula it follows that
\[ \tan(\dfrac{\alpha}{2}) \lt \tan(\dfrac{\theta}{2}) = e^{-\sqrt{|k|}2d}\text{.} \]
We may solve this inequality for \(|k|\text{:}\)
\begin{align*} \tan(\dfrac{\alpha}{2}) & \lt e^{-\sqrt{|k|}2d} & \\ \ln(\tan(\dfrac{\alpha}{2})) & \lt -\sqrt{|k|}2d & \ln(x) ~\text{is increasing}\\ \bigg[ \dfrac{\ln(\tan(\dfrac{\alpha}{2}))}{2d}\bigg]^2 & > |k|.& x^2~ \text{is decreasing for}~x\lt 0 \end{align*}
To get a bound for \(|k|\) in terms of \(p\text{,}\) note that \(\alpha \approx \dfrac{\pi}{2} - 2p\) (the triangles used in stellar parallax have no detectable angular deviation from \(180^{\circ}\)), so
\[ |k| \lt \left[\dfrac{\ln(\tan(\dfrac{\pi}{4}-p))}{2d}\right]^2\text{.} \]
We remark that for values of \(p\) near \(0\), the expression \(\ln(\tan(\dfrac{\pi}{4}-p))\) has linear approximation equal to \(-2p\text{,}\) so a working bound for \(k\text{,}\) which appeared in Schwarzschild's \(1900\) paper [27] , is \(|k| \lt (\dfrac{p}{d})^2\text{.}\)
Exercises
Prove that for \(k \lt 0\text{,}\) any transformation in \({\cal H}_k\) has the form
\[ T(z) = e^{i\theta}\dfrac{z-z_0}{1+k\overline{z_0}z}\text{,} \]
where \(\theta\) is any real number and \(z_0\) is a point in \(\mathbb{D}_k\text{.}\)
- Hint
-
Follow the derivation of the transformations in \({\cal H}\) found in Chapter 5.
Assume \(k \lt 0\) and let \(s = \dfrac{1}{\sqrt{|k|}}\text{.}\) Derive the following measurement formulas in \((\mathbb{D}_k,{\cal H}_k)\text{.}\)
-
The length of a line segment from \(0\) to \(x\text{,}\) where \(0 \lt x \lt s\) is
\[ d_{k}(0,x) = s\ln\left(\dfrac{s+x}{s-x}\right)\text{.} \]
Hint: Evaluate the integral by partial fractions. - The circumference of the circle centered at the origin with hyperbolic radius \(r\) is \(c=2\pi s\sinh(\dfrac{r}{s})\text{.}\)
- The area of the circle centered at the origin with hyperbolic radius \(r\) is \(A = 4\pi s^2 \sinh^2(\dfrac{r}{2s})\text{.}\)
Let's investigate the idea that the hyperbolic formulas in Exercise \(7.3.2\) for distance, circumference, and area approach Euclidean formulas when \(k \to 0^-\text{.}\)
- Show that the hyperbolic distance \(d_{k}(0,x)\) from \(0\) to \(x\text{,}\) where \(0 \lt x \lt s,\) approaches \(2x\) as \(k \to 0^-\text{.}\)
- Show that the hyperbolic circumference of a circle with hyperbolic radius \(r\) approaches \(2\pi r\) as \(k \to 0^-\text{.}\)
- Show that the hyperbolic area of this circle approaches \(\pi r^2\) as \(k \to 0^-\text{.}\)
Triangle trigonometry in \((\mathbb{D}_k,{\cal H}_k)\text{.}\)
Suppose we have a triangle in \((\mathbb{D}_k,{\cal H}_k)\) with side lengths \(a,b,c\) and angles \(\alpha, \beta,\) and \(\gamma\) as pictured in Figure \(5.4.4\). Suppose further that \(s = \dfrac{1}{\sqrt{|k|}}\text{.}\)
-
Prove the hyperbolic law of cosines in \((\mathbb{D}_k,{\cal H}_k)\text{:}\)
\[ \cosh(\dfrac{c}{s})=\cosh(\dfrac{a}{s})\cosh(\dfrac{b}{s})-\sinh(\dfrac{a}{s})\sinh(\dfrac{b}{s})\cos(\gamma)\text{.} \]
-
Prove the hyperbolic law of sines in \((\mathbb{D}_k,{\cal H}_k)\text{:}\)
\[ \dfrac{\sinh(\dfrac{a}{s})}{\sin(\alpha)}=\dfrac{\sinh(\dfrac{b}{s})}{\sin(\beta)}=\dfrac{\sinh(\dfrac{c}{s})}{\sin(\gamma)}\text{.} \]
As \(k \lt 0\) approaches \(0\), the formulas of hyperbolic geometry \((\mathbb{D}_k,{\cal H}_k)\) approach those of Euclidean geometry. What happens to Lobatchevsky's formula as \(k\) approaches \(0\)? What must be the angle of parallelism \(\theta\) in the limiting case? Is this value of \(\theta\) independent of \(d\text{?}\)
Bessel determined a parallax of \(p = .3\) arcseconds for the star \(61\) Cygni. Convert this angle to radians and use it to estimate a bound for the curvature constant \(k\) if the universe is hyperbolic. The units for this bound should be light years\(^{-2}\) (convert the units for the Earth-Sun distance to light-years).
The smallest detectable parallax is determined by the resolving power of our best telescopes. Search the web to find the smallest detected parallax to date, and use it to estimate a bound on \(k\) if the universe is hyperbolic.