Some thoughts on spacetime transformation theory


There is a modern preoccupation with transformation optics [1], which means connecting frames of reference in such a way that a distinct advantage can be obtained. It is now globally understood and recognised in all walks of life that hiding objects, and areas, is not only very entertaining but it can be of great value both to ourselves and to the environment. Indeed, the idea of hiding things is taking hold in many disciplines now, with a major effort being in acoustics [2]. The general colloquial term for this type of hiding is called cloaking, and thermal cloaking is now [3] beginning to take off. With respect to the latter, one of the dreams we could have is to create a spring garden in the Antarctic, for example. Going back to optics, the physical cloaking outcome can be expressed very simply. If you do not wish to see a person or an object in front of you, it is necessary to persuade the light coming towards you to bend around the object and then enter your eye, as if the person or object just does not exist. This could be achievable if the space you are sitting in can be filled with suitably designed, spatially dependent permittivity and permeability.

All of this seems a very modern way of thinking and yet we can go all the way back to 1911 and look at what Paul Langevin said, even then, about the work of Lorentz: “He showed that the fundamental equations of electromagnetism allow a group of transformations that enables them to resume the same form when a transition is made from one reference system to another”. This is precisely the point being made today concerning transformation optics, and it does seem remarkable that it has taken a century to get to this stage.

The preliminary discussion above, in terms of cloaking, involves spatial coordinates, but the addition of the time coordinate turns the whole discussion into what can be called spacetime transformations and the tremendous possibility of creating spacetime invisibility. By adding the time coordinate to the spatial coordinates, we are now seeking to manipulate, or cloak, an event. Again, this was a preoccupation over 100 years ago, when relativity and general moving frames of reference were a major topic for investigation. Nevertheless, this is a key issue today and the paper by Kinsler and McCall is absolutely fascinating [4]. It involves concepts that go beyond the original paper “A spacetime cloak, or a history editor” [5] and it introduces the new and exciting idea called the “bubbleverse” – an idea that will stimulate researchers and laymen alike.

As stated above, the concept of the spacetime cloak was first introduced by McCall et al. in 2011 [5], and it immediately opened up an exciting cloaking window on events, rather than just objects. This leap forward also shows how such cloaks can, in principle, be realised in the laboratory, using ideas based on refractive index control in optical fibres. Such was the impact of this paper that it really inspired experimentalists to implement the concepts rapidly, first by hiding a single period of 12ps [6], and then by achieving a rapid cloak repetition at 40kHz [7].

In their current paper Kinsler and McCall are now elaborating on some of the implications of their discoveries. They show, for example, how to write down a general wave model that can be, alternately, applied to electromagnetics and acoustics. An acoustic spacetime cloak could be very important to our environment and may be deployed by an object at large, or employed in a working space to avoid sonar detection.

The intrinsic directionality of the spacetime cloak means that, when viewed backwards, an observer sees events speeded up and slowed down, rather than redacted from view. Exploiting this concept further, the concept of a ‘causality editor’ is discussed in which suitable spacetime cloaking can appear to alter the very causal ordering of events. This may seem like party tricks but it could find application in data processing where not just the bits and bites but the order in which they are streamed to a processor can influence the outcome. This is just one generalisation of the so-called ‘interrupt-without-interrupt’ functionality heralded by the spacetime cloak.

Beyond cloaks, in the final section the authors sketch out some fascinating thoughts on realising laboratory versions of various evolving spacetimes – so-called ‘bubbleverses’ that may enable some of the most exotic cosmological models to be realised through metamaterials technology. The question of blowing bubbles in spacetime is addressed through a spacetime version of Maxwell's fisheye lens. It is very impressive that this lens was invented by James Clerk Maxwell in 1860, using only geometrical optics [8]. This work preceded, by just one year, the publication of his famous electromagnetic theory of light. Maxwell's fisheye is based upon a novel refractive index for a sphere. He showed that rays of light emerging into a particular direction from a given point on the sphere follow a circular path all the way round until they meet, perfectly, on the opposite side. Converted into a planar lens, we find that a source at any point emits rays that re-converge to an opposing point on the other side of the lens' centre. The necessary index distribution can be designed by projecting the geodesics of a sphere on to a circle in the plane, since any point source on the sphere images to its antipodal point. Indeed, the projected circle can represent a form of closed universe, or bubbleverse, the size of which is governed by the level at which the plane intersects the sphere. Moving a sphere up and down allows us to vary, dynamically, the size of the planar fisheye region from zero to infinity, effectively ‘blowing a bubble’ in an otherwise flat two-dimensional plane. With several bubbles being blown simultaneously, this paper by Kinsler and McCall, as shown in Fig. 1, investigates the effect of a ring of bubbles surrounding a source radiating throughout the plane. It is clear that weaker (i.e. flatter) bubbles hardly affect the wavefronts at all, whilst stronger (i.e. highly curved) bubbles cause internal focussing. All of this illustrates the potential for cosmological modelling using transformational ideas.

Figure 1.

A brave new world from a Maxwell fisheye: imagine a separate bubbleverse drifting away, as if on the wind (picture courtesy of Paul Kinsler, Imperial College London).