The universe exists in space and time. Decades before Einstein, writers such as Gustav Fechner (under the pen-name of Dr. Mises) and CH Hinton proposed that Time was a dimension and that the underlying fabric of reality was a four-dimensional "spacetime". Based on a idea by Henri Poincaré, Hermann Minkowski proposed a mathematical description of spacetime, now known as Minkowski space, in which time was treated as an imaginary variable. (In this context it is crucial to realise that "imaginary" is just a technical mathematical term, in much the some way that mathematicians talk of "irrational" numbers which are in fact perfectly reasonable. Stay grounded!)
Einstein seized on Minkowski's spacetime to express his theory of relativity. However his General Relativity introduced a complication, in that spacetime was forced into curves by the presence of matter and energy, it was no longer the kind of "flat" infinite space such as the Euclidean space taught in schools. His equations showed broadly how space would curve under certain circumstances, but did not offer a unique solution for the actual universe. For example if there was enough matter present, then spacetime would close up on itself in the way that the surface of a ball does, with no boundary. On the other hand, if there was insufficient matter then spacetime would extend on outwards indefinitely and never end.
The study of such curved spaces and their overall shapes is called topology, and the various-shaped spaces are called manifolds. Often it is convenient to lay down some kind of coordinate system over the manifold, so that we can use the techniques of analytic geometry to describe it. For example the surface of the Earth is a sphere, a particular kind of two-dimensional manifold. We customarily lay down lines of latitude and longitude so that we can draw maps or charts and navigate from one place to another, with a complete set of charts collected into an atlas. Such a coordinate system must have a scale or metric with an origin and orientation. But there is no absolute coordinate system, we have to choose one. For centuries nations such as France, Britain, Russia and China all used different systems. In the end, you pick the coordinate system that is most useful to you at the time, and you may find that when in different places, it is most convenient to pick different systems. But then, it becomes important to know how these systems are related where they overlap, so that you can stitch together a complete atlas of your manifold.
In the case of General Relativity, it is customary to use Minkowski coordinates in which time is imaginary.
Now we can sensibly ask, what shape is spacetime? Is it infinite or finite, open or closed? Might it be more like a hosepipe, with infinite time but finite space, or like a steel plate with infinite space but finite time? All of these and more may be found as solutions to Einstein's equations.
So far so good, but now I want to take a couple of diversions and come at the problem from a different angle.
The whole point of making time imaginary is that its mathematics behaves differently from ordinary "real" numbers. Then, one you have lots of dimensions, you will want to keep track of this behaviour. Recall that a coordinate system has a scale or metric as well as an origin and orientation. The imaginary treatment of one coordinate is an aspect of its metric, and we keep track of these imaginary dimensions by defining a "metric signature" for the geometry. By convention we use a + for an ordinary "real" dimension and a − for an imaginary one. (No, you can't add and subtract dimensions! The symbols may be stolen but their meanings are quite different) Thus, the metric signature of real three-dimensional space is simply +++, while that of Minkowski spacetime is +++−.
This at least is how physicists like it. But mathematically, there can be advantages in having a metric signature of −−−+, sometimes flipped to +−−−, which is that of special numbers called quaternions. So mathematicians such as Roger Penrose will often work with this signature.
Euclidean space is famous for having parallel lines. Given a plane on which are drawn a point and a line not on that point, there is just one other line which passes through the point but does not meet the plane - a parallel line. But parallelism is not absolute as Euclid thought, we can have spaces with no parallels or with multiple lines through that point. The definition just given, of a line through a point, is known as Euclid's parallel postulate. By varying the postulate we obtain the various non-Euclidean spaces; more than one parallel through a point leads to hyperbolic spaces, no parallels leads to elliptic spaces. There is a fourth option, to simply abandon any kind of parallel postulate at all. This leads to projective geometries. Because they lack the complications of the extra restriction of parallelism they are the simplest and most elegant geometries. They can also be seen as containing the others, since the restrictions of parallelism ban certain constructions which are allowed in projective spaces. From the point of view of constructing the simplest and most elegant theory of reality possible, projective spaces (i.e. manifolds) are thus the obvious starting point.
A complex number is a number which is part real and part imaginary. A complex space has a metric with complex coordinates. The real and imaginary parts of each number are treated as separate coordinates or dimensions, just as say the x and y dimensions in the plane are. This leads to the awkwardness that a complex line, i.e. of one complex dimension, actually has two coordinate dimensions and we represent it as a plane known as an Argand diagram. The phrase "complex plane" thus becomes ambiguous; does it mean the plane of the Argand diagram, a plane representing the complex line with metric signature +−, or does it mean a plane of two complex dimensions with metric signature ++−−? Mathematicians regularly tangle themselves up with this one, so I thought you'd better know. Thus a complex plane of two complex dimensions has four dimensions in all, it is a four-dimensional manifold. Similarly, complex three-space has the metric signature +++−−− and is a six-dimensional manifold.
Penrose is best remembered by physicists for his development of what is superficially a quite different kind of space, called twistor space.
Twistor space can be obtained by applying a mathematical operation known as a transform (specifically the Penrose-Ward transform) to conventional spacetime.
To get some idea of how the mathematics of a transform works, a simple example is the use of the Fourier transform for sound. If you watch a loudspeaker in slow-motion, you can see its cone moving in and out, creating a sound wave in the air. If you draw a graph of movement (amplitude) against time, it does indeed look like a wiggly wave. On the other hand, a typical sound comprises a spectrum of frequencies from deep bass through the mid-range to high treble. So we can also draw a graph of amplitude against frequency to obtain the sound spectrum. Early music synthesisers such as the Moog can create any given sound by generating all these individual frequencies and superimposing them, before feeding them to the loudspeaker as an electrical wave. The little hairs in our inner ear then pick out the individual frequencies of the spectrum and from that our brain creates an overall tonal quality or timbre for the sound. Neither the wave nor the spectrum is any more fundamental as a mathematical description, they both work and we simply use the description that most suits our purpose at the time. Mathematically, we can switch from the one description to the other using a Fourier transform. So it is with conventional spacetime and twistor space, via the Penrose-Ward transform.
Twistor space is a complex three-space, meaning it has six dimensions with the metric signature +++−−−. Remarkably (at least for me), twistor space is also a projective space.
Even more remarkably for physicists, calculations of certain quantum particle interactions and the probable outcomes (scattering amplitudes) are far more elegant and easier to solve in twistor space than in conventional spacetime. There is even a close relationship, via the Penrose-Ward transform, between twistor and string theories, where strings typically involve tightly-closed Calabi-Yau manifolds, also of metric signature +++−−−.
All this appears immensely beguiling, the mathematical elegance is stunning and any variance from it almost unacceptable by definition. The next question must surely be, if twistor space is just a six-manifold with a funny metric, what shape is it?
Here, we reach an impasse. Twistor space simply does not fit with Relativity. Putting massive objects into it doesn't just bend the mathematics, it breaks it. All those particle-interaction shortcuts only work when energies are so high that the particle mass is irrelevant and we can ignore it. We have absolutely no idea how to tackle the problem.
Another loose end is the prediction by twistor-string theory of a supersymmetry between fundamental particles - where are all those supersymmetric partners?
The answer will probably involve the fabled unification of gravity and quantum physics, which in turn will depend on adding in the "dark enegry" behind the expansion of the universe and the "dark matter" hoilding galaxies together so tightly. Perhaps then we will we be able to reconcile twistor theory with massive particles and then pass it through the Penrose-Ward transform. And until that is done, we may never know what shape the universe is.
Updated 21 Jun 2022