The following is an investigation into an interesting mathematical object I stumbled upon when reading a General Relativity textbook.

Defining the spacetime:

A static universe in which space is hyperbolic can be described by the metric:

, where dτ is the infinitesimal spacetime distance separating two events, is the distance from the origin of the spatial polar coordinate system and the other two coordinates are the standard latitude and longitude of the point specified on the surface of a sphere with radius whose centre is the origin. The factor of is what makes space hyperbolic as opposed to Euclidean, by causing the sphere to expand approximately exponentially in surface area and circumference as its radius increases.

Introducing the model of the spacetime:

If the coordinate system is modified so that and are replaced with lightlike coordinates and , the metric takes the form: .

It may also be expressed, with the use of a hyperbolic trigonometric identity, in terms of hyperbolic functions of the individual coordinates:

A transformation is now applied to this spacetime by mapping u to and to , and retaining the form of the metric apart from the application of the function, producing .

This transformation multiplies by . Clearly this factor approaches at lightlike infinity, suggesting that the transformation squashes the entire spacetime into a finite 'box' . Multiplying the hyperbolic sine factor by this conformal expansion (or contraction) factor : ,, yields the metric above,, which is simply that of Minkowski space time, allowing us to confirm that the transformation is indeed conformal.

Why is it interesting? :

As and range between positive and negative infinity, we see that their counterparts range between 1 and -1, suggesting that the entire hyperbolic static universe is conformally equivalent to a finite causal diamond in the flat spacetime. This volume is to that spacetime as the Poincaré ball is to Hyperbolic space. The light cones surrounding it represent its lightlike boundary at conformal infinity, while their vertices correspond to the points at timelike infinity. The 2-sphere where these light cones intersect can be interpreted as a celestial sphere of directions of light rays emanating from the point at past timelike infinity, or converging on that in the future, at which point (in Minkowski spacetime) an idealized observer might view these light rays. The transformations which take the celestial sphere as perceived by any observer into that experienced by another at the same event but with a different velocity are conformal, which is why it is sometimes interpreted as the Riemann sphere. However, this Riemann sphere also has another interpretation: as the sphere of points at infinity in 3-D hyperbolic space, represented by the surface of the Poincaré ball. The conformality of the transformation ensures that the spacelike slice taken directly through this sphere is indeed the Poincaré ball of dimension 3. If the hyperbolic space of the static universe is translated, the Poincaré ball will undergo its conformal automorphisms generated by inversions in spheres orthogonal to its boundary, which also apply the Möbius transformations to its Riemann sphere boundary. However, as mentioned above, the whole causal diamond is itself a conformal model of a spacetime and therefore analogous to the 4-D Poincaré ball; when the original Hyperbolic static universe experiences translation forwards or backwards in time, what will happen to the Causal diamond? The conformal nature of the map entails that it will also be transformed in a conformal way. This suggests that it will be subject to hyperboloid inversions which transform its ambient Minkowski spacetime in a way analogous to the effect of sphere inversion on Euclidean space. In addition, this transformation must preserve the vertices of the light cones and their sphere of intersection, since these correspond to points at infinity of different types. This suggests that the transformation will permute spacelike surfaces (representing the hyperbolic slices through the static universe) which intersect at the sphere where the light cones meet (the Riemann sphere) along with the Euclidean 3-surface containing the Poincaré ball. As hyperboloid inversions preserve hyperboloids and hyperplanes, with these thought of as hyperboloids of infinite size, this implies that the other surfaces representing moments in the stationary universe's time are themselves spacelike hyperboloids. We can conclude something about the causal structure of the original hyperbolic static spacetime. Clearly, it has only a single point in its infinite past and future, unlike some expanding universes, and has no cosmological event horizon. It is also interesting to note that all events share the same spatial sphere of ideal points at infinity, regardless of time. If observers in this universe could see these points, they would appear frozen, however it is also apparent that no causal signal from them can ever reach the interior, so they could not. However, this doesn't preclude timelike worldlines reaching lightlike infinity, or originating there, which seems to contrast with the intution that most of them will converge on the timelike future event. We can also see that this would require anything travelling along such a trajectory to travel on average at the speed of light.

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1. ^{^}

   They are proportional to the time at which a spherical wavefront of light (with radius r) would need to have been emitted from the origin to arrive at the event they represent (at time t), and the time at which another such light wavefront would converge on the origin respectively. If they're divided by a factor of the square root of two they can also be thought of as the (non -physical) Euclidean distances along the light rays.