### Abstract

- Top of page
- Abstract
- 1 Introduction
- 2 Speed cloaks
- 3 Spacetime cloaks
- 4 Space-time transformations
- 5 Blowing bubbles in spacetime
- 6 Conclusion
- Acknowledgements
- References

Spacetime or ‘event’ cloaking was recently introduced as a concept, and the theoretical design for such a cloak was presented for illumination by electromagnetic waves [McCall et al., J. Opt. 2011]. Here it is described how event cloaks can be designed for simple wave systems, using either an approximate ‘speed cloak’ method, or an exact full-wave one. Further, details of many of the implications of spacetime transformation devices are discussed, including their (usually) directional nature, spacetime distortions (as opposed to cloaks), and how leaky cloaks manifest themselves. More exotic concepts are also addressed, in particular concepts that follow naturally on from considerations of simple spacetime transformation devices, such as spacetime modeling and causality editors. A proposal for implementing an interrupt-without-interrupt concept is described. Finally, the design for a time-dependent ‘bubbleverse’ is presented, based on temporally modulated Maxwell's Fisheye transformation device (T-device) in a flat background spacetime.

### 1 Introduction

- Top of page
- Abstract
- 1 Introduction
- 2 Speed cloaks
- 3 Spacetime cloaks
- 4 Space-time transformations
- 5 Blowing bubbles in spacetime
- 6 Conclusion
- Acknowledgements
- References

At first sight, spacetime cloaking might just seem like an esoteric variant of standard cloaking theories. Indeed, if taking a mathematician's view of the design procedure for a spacetime transformation device, this seems true – to construct a perfect spacetime event cloak [1] we simply used the fully covariant form of Maxwell's equations, and derived event cloak material parameters that required controlled magneto-electric effects.

However, event cloaks are in many respects conceptually simpler than the widely studied types of ordinary spatial cloak [2-6], and other spatial transformation devices (T-devices) such as illusion generators [7-9], beam control [10, 11] geodesic lenses [12, 13], and hyperbolic materials [14, 15]. This conceptual simplicity is most obvious if we step back from an insistence on *perfect* spacetime T-devices. This is because ordinary spatial cloaks required an undetectable diversion around a cloaked region, and the subsequent perfect reassembly of ‘undisturbed’ light signals, just as other types of spatial T-device also rely on careful directional control or sensitivity.

In contrast, event cloaks work simply by speeding up and slowing light down – no diversion or realignment is necessary. This was evident from the optical fibre implementation suggested originally in [1], and also from the quickly achieved experimental demonstration using an improved time-lens technique [16]. Other event cloak inspired ideas have been also proposed [17, 18], but the purist might debate whether or not they count as true event cloaks, since they rely on spatial re-routing of the light signals.

In this paper we will largely dispense with a complete treatment of spacetime transformation theory in a full 1+3D spacetime, and instead focus on the concepts both implemented by and revealed by it; nevertheless, the theory presented here generalizes quite naturally. Further, we consider full-wave transformations, and not conformal ones [3, 19], which are restricted in what transformations they can implement. After showing how simple wave speed control can be used to generate spacetime T-device components in Section 'Speed cloaks', we will implement a full spacetime cloak in 1+1D using a straightforward transformation as an instructive example in Section 'Spacetime cloaks'. We will discuss some of the new physical possibilities opened up by the advent of spacetime cloaking in Section 'Space-time transformations'. Before concluding, we show in Section 'Blowing bubbles in spacetime' how time-dependent spatial transformations might be used as simple cosmological models.

### 2 Speed cloaks

- Top of page
- Abstract
- 1 Introduction
- 2 Speed cloaks
- 3 Spacetime cloaks
- 4 Space-time transformations
- 5 Blowing bubbles in spacetime
- 6 Conclusion
- Acknowledgements
- References

Transformations in spacetime quite naturally alter the (apparent) speeds of objects between the original or reference view and the new, transformed view. Although we might consider the transformation of discrete particles or ray trajectories, here we will use a wave picture, albeit one restricted to one dimension. This then enables us to convert our results directly over to a 1D plane polarized electromagnetism (EM), or a simple 1D acoustic wave, with equal ease. Generalizations to 2 or 3 spatial dimensions then follow in a relatively straight forward manner.

Therefore let us start by considering a simple one dimensional (1D) wave equation based on two coupled fields. The differential parts of the wave equations are

- (1)

- (2)

which will combine to form a wave theory with the addition of a constitutive (or state) equation

- (3)

We can do this with a relative velocity profile

- (7)

- (8)

For a more realistic proposal, we can mimic the refractive index profile proposed by McCall et al. [1] and shown in their Fig. 3. We have simulated this using a simple 1+1D FDTD [23] code, with a smoothed refractive index profile that varies between n = 1 and n = 2. The results are shown on Fig. 1. Note that when smoothing, it is the velocity profile which needs to be smoothed before conversion into an index profile. In these simulations, the cloak opening process proceeds smoothly, but the closing process tended to generate numerical difficulties; this was mollified by applying a weak loss to the fast/slow index transition region. This loss causes a brief dip in the intensity of the field propagating from the point where the cloak closes.

### 3 Spacetime cloaks

- Top of page
- Abstract
- 1 Introduction
- 2 Speed cloaks
- 3 Spacetime cloaks
- 4 Space-time transformations
- 5 Blowing bubbles in spacetime
- 6 Conclusion
- Acknowledgements
- References

Having established the basic principles in the previous section, we can now attempt to design a better spacetime cloak. This has already been done, using a curtain map contained within three compounded transformations [1], which has the advantage of working at arbitrary speeds. Here we take a more direct but less elegant approach, and define a free-space spacetime cloak, i.e. one not reliant on a reflecting surface, as ‘carpet’ or ground-plane cloaks are [20]. In a medium of background wave speed *c*, we can design a cloak using a Galilean-style coordinate transformation, i.e.

- (10)

- (11)

This cosine-like transformation applied over a single cycle is chosen because it is smooth and localized, and enables easy matching of first derivatives across the boundary between cloak halo and the exterior as shown on fig. 2.

In the cloak halo region where all instances of *C* (and its derivative *D*) retain their trigonometric form, we can combine them to replace with

- (20)

Further, we could adapt *C* and retain the full periodic variation to give us a chain of spacetime cloaks along the line surrounded by oscillatory wave propagation (or oscillatory ray trajectories). Other variations could model a chain of cloaks in time only, as in the recent experiment of Lukens et al. [24].

Before applying this cloaking transformation to our simple waves, we first write the differential equations as

- (21)

- (22)

but allow the widest possible range of linear interactions between field components, so that the constitutive relations are

- (23)

Ordinary materials would be expected to have and , but cross-couplings are possible, and are often induced by spacetime transformations. In EM, these cross-couplings represent either a medium in motion or one having intrinsic magnetoelectric properties, whereas in acoustics they are unconventional couplings between the scalar pressure and the velocity-field density, and between population density and momentum density.

Since this is the result of a Galilean-style transformation, there is an implicit restriction that any wave speed modulations induced by the cloak will be much smaller than the wave speed. Thus even if we had designed our cloak using a Lorentz transform, clocks at different points within that cloak would differ by only a negligible amount. This is particularly relevant for descriptions of EM cloaks, since non relativistic limits need to be applied carefully in EM (see e.g. [25]).

Also, when thinking of how to modulate wave speeds, as we need to for these spacetime cloaks, we might consider using a group velocity modulation (see e.g. [26, 27]), rather than the phase velocity modulation used here. In fact, it is possible to do cloaking in this way, but since a group velocity is in essence a pulse velocity, such a ‘group velocity’ cloak would work only for illumination consisting of a train of pulses, and not a constant incident wave field. As described below, however, this can still be a useful process.

### 4 Space-time transformations

- Top of page
- Abstract
- 1 Introduction
- 2 Speed cloaks
- 3 Spacetime cloaks
- 4 Space-time transformations
- 5 Blowing bubbles in spacetime
- 6 Conclusion
- Acknowledgements
- References

In this section we will discuss a variety of spacetime transformations, and T-device concepts derived from them. Although electromagnetic applications are perhaps the most obvious ones to consider, since that is the most active T-device field at present, there is no reason to restrict ourselves to only light. Of course, electromagnetism has considerable advantages, both in technology (e.g. that of microwaves and nonlinear optics) and as convenient analogies to other systems – for example, hyperbolic spacetimes [14] or supersymmetry [28]. However, even water waves can be used as the substrate systems for transformation aquatics T-device concepts [13] as well as convenient analogies [29-31].

In addition to allowing for different types of wave, we can also apply the spacetime T-device concept to types of ‘illumination’ other than a continuous background intensity. We can apply it to streams of illuminating pulses, which might be slowed or speeded as desired, or where telegraph-like 0-to-1 and 1-to-0 transitions in a clock signal have their timings adjusted. Further, we might also imagine carpet/ground-plane versions of T-device concepts in other waves and illumination-types (see [20]).

One point to note is that although a composition of spatial and spacetime cloaks might suggest enhanced cloaking, it does not really add extra utility. However, it might be used in cases where it was desirable to temporarily alter a spatially cloaked region – e.g. by expanding or contracting it. And perhaps, even if the spatial part of the cloak was detected, its additional spacetime capacity might be missed, deceiving even a careful observer.

#### 4.1 Through a cloak, backwards

Space-time cloaks are intrinsically directional, not only temporally but also spatially; this is required because the deformation that eases open a spacetime shadow region for wave or rays traveling forward, has a different effect on those traveling backwards. We can make a forward spacetime cloak by aligning the cloaked region along the orientation of the forward rays, but this then is hopelessly mismatched to the backward rays, requiring them (in places) to go backwards in time – and for spacetime cloaks, which are intrinsically dynamic, we cannot disguise this failing by retreating to the steady-state behaviour as we could for ordinary spatial cloaks. This remains true for approximate implementations which use speed modulation by means of a refractive index contrast or experimental ‘time lens’ implementations; as shown in Fig. 3, in such cases it is impossible to disguise the presence of a (forward) cloak from a backwards observer.

What this means is that although a forward observer may be both unaware of the speed modulated event cloak *and* the events it hides, a backward observer will be able to see both. However, if it is possible to decouple the speed of the backward (uncloaked) waves from that of the forward waves, then the presence of a forward cloak *can* be hidden from the backwards observer. An agent provocateur could then use a spacetime cloak to hide a contentious event from one trusted, honest observer, while nevertheless revealing it to a different but equally respectable observer.

If it were also possible to design independent, overlapping forward and backward cloaks whose hidden core regions intersected – and it would be in EM – even then, parts of the forward cloak's core would remain visible to a backward observer, and vice versa.

#### 4.2 Not cloaking, but distorting

We will see next, when considering the visibility of radiating events inside the cloak leaking out, that such leakage would be seen as a burst of speeded up history. This emphasizes that spacetime cloaking is only a very specific application of a much more general process – that of speeding or slowing signals, and therefore speeding or slowing the pace at which events will finally be perceived. We can see this in Fig. 4, where, if taken in isolation, parts of the cloak can be seen to perform these more elementary spacetime transformations.

We could, for example, re-design the cloak so that a dark shadow was not created, but only a regime of slowed illumination; temporarily changing how the illuminating waves interacted with some object or event. We could then reverse the slowing back to normal, leaving only the illusion of a speeded up event. This would be the spacetime analog of a spatial T-device that does not cloak but instead shrinks the apparent size of an object. Likewise, we might first compress the illumination in time before the event, then restore it. Either transformation could be directly implemented with the following transformation

- (30)

where δ specifies the degree of expansion or contraction, σ the spatial extent, and τ the temporal extent. Using this, or a similar transformation, to slow down or speed up events would be the spacetime analog of a spatial T-device that magnifies or shrinks the apparent size of an object. We might even dispense with the post-event restoring phase and just have converging or diverging time lens T-devices.

For spatial T-devices that generate apparent shrinking or magnification, this applies not just to the object in the interior of the T-device, but also to the space represented by the T-device itself. The spatially magnifying T-device can appear bigger on the inside than on the outside – a sort of ‘tardis’ illusion. Then, a spacetime tardis would be a T-device that allows more time to pass than would be expected – unfortunately not a time machine, only the illusion of one.

#### 4.3 Leaky cloaks

One distinctive difference between ordinary spatial cloaks and spacetime cloaks is that in a spacetime cloak, the wave or ray trajectories cannot be trapped, they must always move towards larger times. This means that any emission from events inside the cloaked region, if not deliberately absorbed, must also escape. In contrast, in a spatial cloak which exists for all time, emission can be trapped (e.g. by reflection, or by being guided around in circles) forever with no limitation.

Thus non-absorbed emission from any events inside the cloak must exit it, and will do so in the vanishingly small gap between the early and late halves of the cloak. If, for example, the cloak neither absorbed or closed perfectly, then these emissive events would be visible as a burst of speeded up history from inside the cloak. This ‘champagne cork’ effect on signals from internal events is shown on Fig. 5. In a leaky but otherwise ideal EM cloak, this speeding up would blue shift the escaping light, in an acoustic cloak it would (likewise) raise the pitch of the escaping sound.

#### 4.4 The causality editor?

An observer will attempt to deduce cause and effect from light or sound signals providing information about the environment. Since the proposal of a spacetime cloak shows that we can interfere with those signals in an (in principle) undetectable way, we might also consider manipulating an observer's view with the aim of confusing cause and effect – by reversing them, for example. Figure 6 shows schematically how this might be achieved for light signals, using polarization switching to separate and distinguish between the cause and effect segments of the light stream forming the observer's view. Other methods of distinguishing between segments are possible, such as frequency conversion or even a physical separation by interposing routing into different waveguides [18].

Figure 6 shows five stages of the manipulation of the light stream seen by the observer. Its original state is seen in (i), containing the view (B)efore, the (C)ause, the (E)ffect, and the (F)inal view. In (ii) the effect segment is switched into the perpendicular polarization, so that in (iii) the two can pass by each other without interfering. In (iv), the effect (E) segment is now before the cause (C), so that in (v) the original polarization can be restored. Thus the observer will now see a view of history, containing all of the expected data, but in a misleading sequence. As long as we chose carefully, and picked edit times when the signals matched up, this could be even made seamless. With separated events, linked by some undisturbed background view, multiple segments might be reordered in this way.

If we further allowed a continuous re-modulation, such as the time-to-frequency mapping used in the spacetime cloak experiment of Fridman et al. [16], we could straighforwardly reverse the order of events seen by an observer; as shown schematically in Fig. 7. Again, as long as the ‘before’ and ‘after’ cuts in the signal stream were at places where the observer's view was identical, this could be made seamless and undetectable. Such a situation might lead us to imagine the possibility of free re-editing of a given spacetime event sequence, although of course the technical challenges are formidable. Nevertheless, the speed at which the spacetime cloak experiment of Fridman et al. was achieved after the publication of the theoretical scheme and proposed optical fibre implementation of McCall et al. suggests that simple causal editing – e.g a simple reversal – could be rapidly implemented.

#### 4.5 Applications

Perhaps inevitably, the end-user applications of spacetime cloaking will seem rather mundane in comparison to the history editor concept promised by the original paper [1]. However, rapid progress is being made towards making those applications more achievable. For example, consider the recent paper by Lukens et al. [24], where the time-domain Talbot effect is used as a means to open a periodic array of spacetime cloaks in the background illumination, enabling the smuggling of data (as bits or symbols) through the system in the periodic gaps created in the illumination. Although not in itself a practical application, the demonstration of spacetime cloaking at telecommunications data rates is suggestive of future success.

Another avenue for applications would be to apply the velocity-modulation approach to spacetime cloaking of pulse trains. This would require a time-dependent (i.e. dynamic) group velocity control [26, 27], where a much larger than normal gap between previously regularly spaced pulses is used to construct the cloaking region, before the process is (as usual) reversed to return the illuminating pulse train back to its original state. If such a pulse train was being used as a clock signal to control the behaviour of some signal processing unit (SPU), then this would be the starting point for a interrupt-without-interrupt functionality as proposed by McCall et al. [1]; and as outlined in Fig. 8. The advantage of a spacetime cloak method over a simple temporary hijacking of the SPU is not only the potential for stealth and lack of any disruption to the ordinary processing, but also the ability to do this whilst only over-clocking the processor both slightly and gradually (by e.g. smoothly tweaking the timing of ten ordinary clock cycles to insert one extra priority computation). Further, given the straightforward nature of this concept, one could as easily adapt the idea to telegraph-like electrical or electronic clock signals, as to other wave models such as acoustics; or, indeed, to particle-like ‘illumination’ such as cars on a road [32] or even pedestrians.

### 5 Blowing bubbles in spacetime

- Top of page
- Abstract
- 1 Introduction
- 2 Speed cloaks
- 3 Spacetime cloaks
- 4 Space-time transformations
- 5 Blowing bubbles in spacetime
- 6 Conclusion
- Acknowledgements
- References

Although current implementations of spacetime cloaks do not rely on metamaterials, such technology would seem to be the obvious solution to building more accurate versions. But if spacetime metamaterials technology were developed to any degree, then we can imagine investigating spacetime engineering in an experimental setting. This would be along the lines of the works of Smolyaninov and others (see e.g. [14]), where various causal and cosmological features have had metamaterial-based T-devices proposed. A specific one useful to consider here is that of a 2D spatial metamaterial with hyperbolic dispersion along one axis [14]. This spatial direction then acts as a time axis, allowing the forward-only light cone structure of the fields from an emitting source to be generated, as seen in Smolyaninov's Fig. 3. Here we can imagine imposing a spatial modulation on the metamaterial that matches the spacetime structure of Smolyaninov's metamaterial, but adds in spacetime cloaking properties. We could then see, in either simulation or experiment, a version of McCall et al.'s [1] (spacetime) Fig. 3 laid out as a spatial pattern.

This idea has been taken further using a ferrofluid in which thermal fluctuations can lead to the anisotropic dispersion that creates such a Smolyaninov ‘spatial-spacetime’ [14]. Each such fluctuation is then a spacetime-like patch that we might like to think of as mini-universe, or ‘miniverse’. Since many such patches can grow and shink over time in the usual (and less exotic state) of the ferrofluid, this then provides an ad hoc model of a time-dependent multiverse [33].

Here, however, we take a different tack and consider a time-dependant, spacetime version of the Maxwell's Fisheye transformation. Usually, this just enables us to project the surface of a sphere (or hypersphere) onto a plane (hyperplane), whilst preserving the properties of the original manifold by means of a spatially varying refractive index. For a spherical surface of radius *r*_{0} and index *n*_{0}, (see e.g. [13]), the index profile on the plane is simply

- (31)

where *r* is the in-plane radius. Such a device can also be made outside optics using transformation aquatics on shallow water waves, as in e.g. the Maxwell's Fishpond [13]. The projection, constructed entirely on the (hyper) plane, could then represent a self contained expanding spherical universe using time-dependent radial scale factor properties (i.e. ), rather like the curvature is used in simple cosmological models that are spatially homogeneous and isotropic at any given time [34].

To make it more interesting, we now take a disc of radius *r*_{1}, containing a medium with a fisheye index profile, as shown in Fig. 9. If, as before, the index on the sphere is *n*_{0}, so that the physical line element on the sphere is , then the required index in the plane, determined by relating to (the radial increment in the plane), is found to be

- (32)

where , and is defined in Fig. 9. By embedding this inhomogeneous disc as an inclusion in some otherwise uniform planar material with constant index , we will have a model for waves that propagate in a new spacetime – one that is flat, except where distorted (at the inclusion) into a bubble-like spherical cap. We might describe this bubble region as a ‘fisheye miniverse’ or ‘bubbleverse’. Matching indexes at the boundary requires . Furthermore, by introducing a suitable mollifier function to soften the transition between the bubble and the flat exterior region, we obtain an index profile for the inclusion which is

- (33)

where . A suitable mollifier is

- (34)

where β sets the gradient of the transition region. Of course, this static situation might be constructed using a purely spatial transformation optics.

Although the 2D ‘spacetime bubbleverse’ phrasing gives a very compelling visualization, this model is not limited to the 1+2D case of a surface changing in time. The concept is equally valid in 1+1D, where the cross-sectional pictures in Figs. 10 and 11 would become an accurate representation of a now 1D linear detour rather than a bubble. We might again combine this with Smolyaninov's spatial-spacetime [14] to lay out the 1+1D time dependent situation here on the 2D plane. Further, the general ‘blowing bubbles’ scenario equally well extends to higher dimensions, such as the 1+3D case which has a time dependent spherical inclusion.