Using the politically correct speech, people are discriminating against geometries of mixed signature. Let's try to fix it.
The ordinary two-dimensional sphere may be defined as the set of all points in the flat three-dimensional Euclidean space whose coordinates obey\[
x_1^2+x_2^2+x_3^2 = 1.
\] We have set the radius to one. One of the three coordinates may be expressed in terms of the other two – up to the sign. The remaining surface – e.g. the surface of the Earth – is two-dimensional which means that it can be parameterized by two real coordinates, e.g. the longitude and the latitude.
On the surface, one may measure distances. The right way to measure the length of a path is to cut the path to many infinitesimal (infinitely short) pieces and to add their lengths. The length of the infinitesimal piece is determined by the metric. The metric of the sphere is invariant under the \(SO(3)\) rotations. Locally, this group is isomorphic to \(SU(2)\) which is also the same thing as \(USp(2)\).
But what if we change a sign or two? The equation\[
x_1^2+x_2^2-x_3^2=-1
\] With \(-1\) on the right hand side, we get a two-sheeted hyperboloid. (We would get a one-sheeted one if there were a plus sign.)
Let's take one component of this manifold only. Does it have some symmetries similar to the \(SO(3)\) rotational symmetry of the two-sphere we started with? If we only allow rotations that are also symmetries of the three-dimensional "environment" and if we assume this environment to be a flat Euclidean space that uses the Pythagorean theorem to measure distances, the answer is that the two-sheeted hyperboloid only has an \(SO(2)\) symmetry: we may rotate it around the axis. That's a one-dimensional group isomorphic to \(U(1)\).
But we have only modified one sign. That's not a big change; in some sense, we have only changed a radius to an imaginary value. Doesn't the hyperboloid have a larger, three-dimensional group of symmetries that would be as large as the group \(SO(3)\)? The answer is Yes. But we must allow transformations that don't preserve the distances in the parent three-dimensional Euclidean spacetime. Even more accurately, we must imagine that the parent three-dimensional spacetime is not Euclidean but Lorentzian, like in general relativity, and its distances are given by\[
ds^2 = dx_1^2+dx_2^2 - dx_3^2.
\] The signs defining the hyperboloid respect the relative signs from the metric above so the symmetries of the actual hyperboloid will include the whole \(SO(2,1)\) Lorentz group of the original three-dimensional space – or spacetime, if you want to call it this way.
Such a two-sheeted hyperboloid may be thought of as the space of all allowed energy-momentum vectors of a massive particle in 2+1 dimensions, i.e. all vectors obeying\[
E^2 - p_x^2 - p_y^2 = m^2 \gt 0.
\] The Lorentz transformations, \(SO(2,1)\), act on the vectors' coordinates in the usual way. When we talk about the single component of the two-sheeted hyperboloid as about a "geometry", we call it a "hyperbolic geometry". This concept should be viewed as another example of a non-Euclidean geometry besides the spherical geometry. Non-Euclidean geometries are similar to geometries of the flat Euclidean plane/space but they reject Euclid's axiom about the parallel line: it is no longer true that "there is exactly one straight line going through a given point that doesn't intersect another given straight line". For the spherical geometries, there is usually none (pairs of maximal circles such as two meridians always intersect, e.g. at the poles); for the hyperbolic geometries, there are infinitely many (the lines diverge from each other so there are many ways to adjust their directions so that they still don't intersect).
Is there something we should know about the hyperbolic geometry? How can we visualize it? Much like the sphere, the hyperbolic geometry has an intrinsic curvature so it is not isometric to a piece of the flat plane. Much like in the case of maps of the sphere, i.e. the Earth's surface, we have to choose a method to depict it. Some geometric quantities will be inevitably distorted.
One cute "compact" way to visualize the hyperbolic geometry is the Poincaré disk. Here is an animation of the Poincaré disk equipped with a uniform collection of Escher's batmen.
The hyperboloid had an infinite area – even when you adopt the Lorentzian signature for the metric – because one may "Lorentz boost" vectors indefinitely. Another related fact is that the group \(SO(2,1)\) of the symmetries of the hyperbolic geometry is noncompact; if we define a group-invariant volume form on the group manifold, the volume of the group is infinite. It follows that there have to be infinitely many batmen living on the hyperbolic geometry.
As the name indicates, the Poincaré disk represents the hyperbolic geometry as a disk – it means the interior of a circle. But because there have to be infinitely many batmen, their density has to diverge in some regions. As you see in the animation above, the density of batmen diverges near the boundaries of the disk.
But much like in the case of angle-preserving maps (e.g. the stereographic projection), you see that all the internal angles of the batmen are preserved. The model of the hyperbolic geometry obviously doesn't preserve the areas (batmen near the boundary look smaller). And the Poincaré disk model doesn't make straight lines (geodesics) on the hyperboloid look straight here, either. (Another model, the Beltrami-Klein model or the Klein disk, does, but it doesn't preserve the angles.)
Can we reconstruct the metric on the original hyperboloid from the coordinates \(x_1,x_2\) parameterizing the unit Poincaré disk? Yes, we can.\[
ds^2= 4 \frac{dx_1^2+dx_2^2}{1-x_1^2-x_2^2}
\] It would be straightforward to add additional coordinates if you needed to do so.
Note that up to the factor of \(4\) (which is a convention, an overall scale of the metric, but it is actually helpful to make the curvature radius equal to one) and up to the denominator (a pure scalar), this is nothing else than the metric on the flat plane. Because the metric on the Poincaré disk only differs from the metric on the underlying paper by a scalar, Weyl rescaling, it preserves the angles. The denominator makes it clear that as \(x_1^2+x_2^2\to 1\), the proper distances (and areas) blow up.
The animation shows some transformations that don't change the internal geometry of the hyperbolic geometry. They are elements of \(SO(2,1)\). This group has three generators. The action of one of them is shown by the animation; the action of another one would look the same except that the batmen would be drifting in another, orthogonal direction; the action of the third generation is nothing else than the rotations of the disk which are isometries of the underlying paper, too.
In the case of the sphere, we noticed that \(SU(2)\sim SO(3)\); the groups are locally isomorphic. This fact is related to the existence of spinors which have 2 complex (pseudoreal) components if we deal with the three-dimensional Euclidean space. Are there similar groups isomorphic to \(SO(2,1)\)? Yes, there are. In fact, there are at least two very important additional ways to write \(SO(2,1)\).
Because we are talking about transformations preserving the angles, both of these alternative definitions of \(SO(2,1)\) may be obtained as subgroups of \(SL(2,\CC)\), the group of Möbius transformations. The cute old video below discusses the angle-preserving tranformations of the plane.
All one-to-one angle-preserving transformations of the plane may be written down in terms of a simple function of a complex variable \(z\in\CC\),\[
z\to z' = \frac{az+b}{cz+d}, \quad \{a,b,c,d\}\subseteq\CC.
\] For the transformation to be nonsingular, we require \(ad-bc\neq 0\). In fact, whenever this determinant is nonzero, we may rescale \(a,b,c,d\) by the same complex number to achieve \(ad-bc=1\) without changing the function. So we may assume \(ad-bc=1\) and the group of all transformations of this form is therefore \(SL(2,\CC)\). Just to be sure, if you're annoyed by the nonlinear character of the function \(z\to z'\), don't be annoyed. The variable \(z\) may be represented simply as \(u_1/u_2\), the ratio of two coordinates of a complex vector, and when the Möbius transformations are acting on \((u_1,u_2)\) in the ordinary linear way, they will be acting on \(z=u_1/u_2\) in the nonlinear way depicted by the formula above.
There are four complex parameters underlying the transformation, \(a,b,c,d\), but because we imposed one complex condition \(ad-bc=1\), there are effectively three free complex parameters i.e. six real parameters in the Möbius group. But we're interested in the Poincaré disk. It means that we would like to restrict our focus on the Möbius transformations that map the disk onto itself. If we deal with the boundary i.e. \(zz^*=1\), then we would like to have \(z' z^{\prime *}=1\), too. How does this condition constrain the parameters \(a,b,c,d\)?
One may prove that this restricts the matrix to be inside a smaller group, \(SU(1,1)\). That's a group of matrices \(M\) obeying\[
M\cdot \diag (1,-1)\cdot M^\dagger = \diag (1,-1), \quad {\rm det}\,M = 1.
\] Note that up to the insertion of the diagonal matrix with the \(\pm 1\) entries, this would be a condition for a unitary group. However, the extra diagonal matrix changes the signature so instead of a unitary group, we obtain a pseudounitary group. This group \(SU(1,1)\) is the group of all angle-preserving, one-to-one transformations of the unit disk onto itself, and because we've seen that the unit disk may be viewed as an angle-preserving depiction of the two-sheeted hyperboloid, i.e. the hyperbolic geometry, it follows that this group must be isomorphic to the group of symmetries of the hyperboloid,\[
SU(1,1)\sim SO(2,1).
\] The isomorphism is valid locally. Note that both groups have three real parameters. And I won't spend too much time with it but there's another isomorphism of this kind we may derive from the Poincaré disk model. The disk is conformally equivalent to a half-plane and the group of Möbius transformations that preserve the half-plane (and its boundary, let's say the real axis) is nothing else than the group of Möbius transformations with real parameters \(a,b,c,d\). So we also have\[
SU(1,1)\sim SO(2,1)\sim SL(2,\RR).
\] Both \(SU(1,1)\) and \(SL(2,\RR)\) may be easily visualized as the groups acting on the two-component spinors in 2+1 dimensions.
In the case of the spherical geometry, we may construct Platonic polyhedra and various cute discrete subgroups of \(SO(3)\), i.e. the group of isometries of an icosahedron (which is the same one as the group of isometries of a dodecahedron, the dual object to the icosahedron). Analogously, there are many interesting "polyhedra" and discrete subgroups of \(SO(2,1)\), too. These mathematical facts were essential for Escher to be able to draw his batmen into the Poincaré disk, of course.
There are many things to be said about the Poincaré disk and its higher-dimensional generalizations. And these objects play a very important role in theoretical physics – in some sense, they are as important as the spheres themselves. The importance of the hyperbolic geometry in relativity (on-shell conditions for the momentum vectors) has already been mentioned. But there are many other applications. The world sheet description of string theory depends on conformal transformations which makes the appearance of similar structures omnipresent, too. The geometry of the moduli spaces of Riemann surfaces – starting from the torus – depends on groups such as \(SL(2,\RR)\) which are also analyzed by tools similar to the mathematical games above.
Finally, the anti de Sitter space – the key geometric player of the AdS/CFT correspondence – may be considered as a higher-dimensional generalization of the hyperbolic geometry, too. (But in this case, there is a temporal dimension even "inside" the picture with the batmen.) That's why the Poincaré disk and various "cylinders" that generalize it are a faithful portrait of the AdS spaces. The regions near the boundary where the batmen get very dense become the usual "AdS boundary" which is where the conformal field theory, CFT, is defined.
But I didn't want to beyond the elementary mathematical observations so if you were intrigued by the previous two paragraphs, you will have to solve the mysteries yourself (or find the answers elsewhere in books or on the Internet).
Off-topic: Bad Universe
Tonight, I turned on my Czech Prima Cool TV half an hour too early, before the S05E17 episode of The Big Bang Theory. Whenever I do it, I can see things like the Simpsons, Futurama, Topgear, and others – in Czech dubbing. And they're often nice programs. But what I got tonight was... Phil Plait's Bad Universe ("Divoký vesmír" in Czech, meaning "Wild Universe"). Holy cow, this is an incredibly crappy would-be scientific program!
I have watched it for ten minutes or so but this period of time has saturated my adrenaline reservoir and depleted all my patience. First of all, it's sort of a crazy explosion of exhibitionism if someone looking like Phil Plait – the blogger behind Bad Astronomy – agrees to turn himself into a "TV star". But the content was much worse than that. He was showing some random combinations of scientific concepts – X-rays from outer space, global cooling, solar eruptions, random oxides etc. – as the culprits that have destroyed the trilobites. The program is meant to be catastrophic and in between the lines, the program clearly wants to fill the viewers' heads with many kinds of hypothetical catastrophes that may occur in a foreseeable future, too.
The unlimited combination of random "scientific ingredients" and contrived lab experiments pretending to emulate conditions in the past combined with a nearly complete absence of any explanation or argument or fair judgement or impartial and careful analysis or anything that actually makes any sense is what creates a program that decent people can't possibly like. I like to listen to scientific explanations about chemistry, biology, history, cosmology, geology, and other things – but this weird mixture of everything is just over the edge.
My rating for the program: pure shit.
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