3.1 Sweeping beam with constant angular speed

The first canonical scenario involving superluminal spot pair events considered here will be that of a beam sweeping across a spherical body of radius R at distance D from an observer, with R ≪ D, and with scattered light observed from very nearly the same direction as the outgoing beam. Assume that the angular size of the sphere is much larger than the angular size of the beamed spot on the sphere. Further, assume that the beam sweeps linearly across the sphere once, through the central point of its projected disk, at a constant angular speed ω on the observer’s sky. This projected hypothetical speed across a flattened disk at the distance of the sphere will be labeled w _⊥ = ωD. Let ϕ be the angle between the incoming beam and a given point on sphere in the beam illumination path, with the angle vertex being at the center of the sphere. Here, x labels the coordinate distance on the plane of the sky at distance D from the observer along the sweeping beam, while y labels a radial coordinate distance into the sky at distance D from the observer. The origin of these (x, y) coordinates is the center of the sphere, while x = Rsin ϕ and y = Rcos ϕ. As the beam sweeps across the sphere, ϕ goes from − π/2 to π/2, x goes from − R to R, while y goes from zero to R and back to zero. The geometry is diagrammed in Figure 3.

Figure 3. The geometry of a sweeping beam that creates a spot or spots on a sphere.

Considering the sphere a flat disk, the time it takes for the light beam to sweep across half the disk is t_sweep = R/w _⊥. Assuming that sweeping begins pointing at t_delay = 0 toward ϕ = −π/2 radians, then the time it takes for the light beam to point toward the location ϕ on the sphere can be quantified to be t_delay = (R + x)/w _⊥ = R(1 + sin ϕ)/w _⊥.

Flying direct, the time it takes for light to cross half the sphere is t = R/c. The time it takes for light to go from the beam source to a location with coordinate y on the sphere is t_path = D/c + (R − y)/c, where the first term is the time it takes for light to reach the closest point on the sphere, and the second term is the time it takes for light to cross distance (R − y) of the sphere. Written in terms of ϕ, then t_path = D/c + R(1 − cos ϕ)/c and so the total time it takes before position ϕ is illuminated will be

(1) \begin{equation} t_{real} = t_{delay} + t_{path} = R (1 + \sin \phi ) / w_\perp + D / c + R (1 - \cos \phi ) / c . \end{equation}

Which part of the sphere is illuminated first? In general, this is not ϕ = −π/2 but rather is found from setting dt_real/dϕ = 0. The first illuminated point is therefore at $\phi _{first}^{\rm real} = \arctan (-c / w_\perp )$. When w _⊥ ≫ c then ϕ^real_first goes to zero, the closest point on the sphere to beam source. When w _⊥ ≪ c then ϕ_first goes to − π/2, the location on the sphere where the beam points first. When w _⊥ = c, then ϕ^real_first goes to − π/4.

Since every ϕ will be illuminated eventually, ϕ values on either side of ϕ^real_first are illuminated later, with pairs of ϕ locations being illuminated simultaneously. Therefore ϕ^real_first is also ϕ^real_pair, and the first illuminated place on the sphere is actually a diverging pair of spots!

The real illumination pattern of a beam sweeping across a sphere can now be described. A real pair of beam spots is first illuminated at ϕ^real_pair with each spot moving in opposite directions. One spot moves toward the closest limb and disappears there, while the other spot crosses the rest of the sphere.

How fast do these spots move across the surface of the sphere? The speeds are computed from

(2) \begin{equation} v = R {\ d\phi \over d t_{real} } = { w_\perp c \over c \ \cos \phi + w_\perp \sin \phi }. \end{equation}

It is easily shown from the above Equation (2) that the surface speed v diverges at ϕ^real_pair, with one spot moving out with initially infinite surface speed toward the ϕ = −π/2 limb, with the other spot moving with initially infinite surface speed in the other direction—toward ϕ = 0.

To find the perceived illumination pattern by an observer, it will be useful to decompose v into radial and perpendicular components. Then, v _⊥ = vcos ϕ perpendicular to the observer and

(3) \begin{equation} v_r = - v \sin \phi = {- w_\perp \ c \ \sin \phi \over c \ \cos \phi + w_\perp \ \sin \phi } , \end{equation}

radially toward the observer. Just as v is unlimited in magnitude, the magnitude of v _⊥ and v_r can exceed c.

Starting from the time the light beam begins its journey at the source toward the sphere, to when the beam is measured back at the source by the observer, the time that angular position ϕ is observed to be illuminated is when

(4) \begin{equation} \begin{array}{l} t_{obs} = t_{delay} + 2 t_{path}\\ \qquad = R (1 + \sin \phi ) / w_\perp + 2 D / c + 2 R (1 - \cos \phi ) / c . \end{array} \end{equation}

Also t_obs = t_real + t_path. Which ϕ on the sphere is observed to be illuminated first? This will be when dt_obs/dϕ = 0, which occurs when $\phi _{\rm pair}^{\rm virtual} = \arctan (- c / 2 w_\perp )$. The superscript ‘virtual’ highlights that no real spot pairs are created on the scatterer at this ϕ location. In general, the observed perpendicular speed of a spot will be

(5) \begin{equation} u_\perp = R {d \phi \over d t_{obs} } = { w_\perp \ c \ \cos \phi \over c \ \cos \phi + 2 \ w_\perp \ \sin \phi } . \end{equation}

Note that when ϕ = ϕ^virtual_pair, then u _⊥ formally diverges. Consequently, when ϕ is slightly less than ϕ^virtual_pair then u _⊥ has a very large negative value, meaning that the observed spot is initially moving very rapidly in the opposite direction than w _⊥. Alternatively, when ϕ is slightly greater than ϕ^virtual_pair, then u _⊥ has a very large positive value, meaning that the observed spot is initially moving very rapidly in the same direction as w _⊥.

No matter how small the angular sweep speed across the observer’s sky ω, so long as it is finite, there is a location near the edge of the sphere where v_r drops from greater than c to less than c. Therefore, in all cases, an observer can perceive, in theory, a virtual spot pair creation event. This will also be the first light of any kind that an observer will see from the sweeping beam.

The angle ϕ^virtual_pair where a virtual pair of spots is first perceived is straightforward to compute. In Equation (3), v_r is set equal to c. One then finds that $\phi _{\rm pair}^{\rm virtual} = \arctan (- c / 2 w_\perp )$, as indicated above. When the angular sweep speed ω is large, the arctangent goes to zero and therefore, so does ϕ, meaning that the spot pair creation event appears near the projected center of the sphere, the nearest point as seen by the observer. Conversely, when the sweep speed ω is low, then the arctangent goes to − π/2 meaning that the virtual spot pair creation event is perceived to occur near the limb of sphere first pointed toward by the source.

The illumination pattern perceived by the observer of a beam sweeping across a sphere can now be adequately described. The very first thing the observer sees is a spot pair creation event with two spots simultaneously created at ϕ^virtual_pair. One spot of this pair moves toward the nearby first-pointed-toward edge, counter-intuitively in the opposite direction from the actual motion of the sweeping beam. Simultaneously, a second spot moves toward the last-pointed-toward edge, in the same direction as the actual sweeping beam. The spot moving toward the first edge disappears at that limb before the spot moving toward the last edge. There is no spot pair annihilation event in this scenario.

A perhaps surprising feature is that one virtual spot appearing at ϕ^virtual_pair will subsequently be perceived to pass over ϕ^real_pair, the ϕ location where real spot pair creation occurred, without anything unusual appearing to happen. Even though two real spots were created at ϕ^real_pair, one of the virtual spots appears to move smoothly across. Therefore, the only spot pair creation event witnessed by the observer is the one at ϕ^virtual_pair. The observer sees nothing unusual happen at ϕ^real_pair.

Information about the angle of virtual spot pair creation, ϕ^virtual_pair, is recoverable, theoretically, in at least three ways. The first detection method is by using both angular and temporal information—by angularly resolving the spot pair creation event with sufficiently high speed imaging. The second method is purely temporal, by measuring the resulting light curve with sufficient detail. The third detection method utilizes polarization measurements of sufficiently high temporal sampling, discerning the changing polarization content of the scattered light.

3.2 Sweeping beams across spheres in astronomical settings

As alluded to above, a popular example of a beam sweeping a spot across a sphere is a laser sweeping across Earth’s Moon. The angular radius of the Moon is about 0.25°, while the Moon’s physical radius is about 1740 km. An easily noticeable spot pair creation event should occur when the linear sweep speed at the average distance of the Moon is w _⊥ = c ~ 300 000 km s⁻¹. Were the Moon a flat disk, the time it would take for a laser to sweep across the Moon at this speed would be t_sweep = 2R_Moon/c ~ 0.0116 s, which is just the light travel time across the Moon. At this rotation rate, a laser could sweep from one Earth horizon to the other, 180° , in about 4.2 s. Creating spot pair events on the Moon with the average laser point is therefore simple and does not require expensive apparatuses.

The relative brightness of spots created by a single beam sweep across the Moon is now estimated given three assumptions. The first assumption is that the Moon is a Lambertian reflector such that it returns the same brightness at all viewing angles, the second is that the beam size is large compared to surface scattering features such as craters and mountains but small compared to the Moon itself, and the third is that the beam sweeps with a constant angular speed on the observer’s sky. Given these assumptions, the observed instantaneous brightness of a sweeping spot is proportional to u _⊥. The theory behind this simple relation starts by noting that each location along the swept path is both illuminated uniformly and scatters uniformly. Were the Moon a flat disk, it would just return a spot of unchanging brightness. The Moon’s depth does not change the integrated brightness of each ϕ value. However, the depth of the truly three-dimensional Moon changes the timing and duration of when different ϕ values are illuminated and subsequently seen to be illuminated by the observer. Therefore, relatively, some illuminated swaths appear instantaneously bright for a short time, while others appear instantaneously dim for a long time.

Perceived instantaneous spot brightness on a sphere is depicted in Figure 4, which plots this brightness as a function of time for a beam sweeping with the speeds w _⊥ = 0.1c, 0.2c, and 1.0c , respectively. Figure 4 was created under the assumption that no angular information is recovered and so gives the gross light curve measured instantaneously over the entire Moon. The faster sweeps show a higher early brightness—formally infinite at t = 0—just as a superluminal virtual pair creation of spots is perceived. The slowest sweep speed w _⊥ = 0.1c takes longer but still shows the initial virtual spot pair peak. The light curve then quickly flattens out to the brightness intuitively expected for a flat two-dimensional Moon, where u _⊥ = w _⊥, which is also the normalized value.

Figure 4. A light curve of the instantaneous brightness of the spots created by a beam swept with constant angular speed across the Moon, as measured back on Earth. The curve labels refer to the spot sweep speed across the closest lunar point, where ϕ = 0. The high initial brightnesses derive from perceived spot pair creation events being the first light that reaches the observer.

The formally infinite brightness appears because u _⊥ in Equation (5) diverges for ϕ angles that make the denominator zero. In reality, the divergence would be muted by several factors including the infinitesimal amount of time that u _⊥ diverges, the finite size of the spot, and the limited amount of energy emitted and scattered by the beam.

Unfortunately, as indicated in Figure 4, the time scale for the virtual spot pair creation episode at the start of the light curves is a bit too brief to be discernable with the human eye. Still, effects of superluminal spot pair events should be discernable rapid imaging and a powerful laser sweeping past the lunar reflectors left by the NASA Apollo missions (Bender et al. Reference Bender1973). Although, these reflectors are too small to show a significant length of any sweeping beam, precisely kept times when specific discrete reflections are seen would test an underlying tenant of this analysis.

A large spot boundary commonly observed to sweep across the Moon is the shadow of the Earth during a lunar eclipse. Given the above analysis, it should be clear that a lunar eclipse actually starts out as a virtual pair of dark shadow edges that suddenly appear very near a limb of the full Moon. One of these dark edges is perceived to move ‘backwards’ toward the closest limb and quickly disappears there, while the other appears to move progressively across the Moon as usually depicted. Towards the end of a lunar eclipse there occurs a creation event of a virtual pair of bright edges, again very near the Moon’s limb. Again, one of the bright edges is perceived to move ‘backwards’ to the closest limb and very quickly disappears there, while the other edge appears to move progressively across the Moon as usually depicted. Unfortunately, all of this occurs within a small fraction of a second (~10⁻⁸ s) after a given eclipse edge begins to cross the Moon and numerous effects including the fuzziness of Earth’s shadow, due to the Earth’s atmosphere, likely make the effect practically unobservable. In principle, this scenario works for other eclipse situations, for example eclipses of Jupiter by its moons. At Jupiter, the effect would last longer but still only be visible for the order of microseconds.

A controlled sweeping beam could be used, in theory, to determine geometric surface characteristics of passing objects, including, for example, asteroids and comet nuclei, to determine how non-spherical they are. Most discerning would be a rapid series of sweeps, possibly in the radio or microwave bands, cycling through a range of orientations and stepping through an array of useful sweeping speeds. Each sweep in the series might itself be repeated numerous times to increase signal strength and to allow detection with a matched-frequency chopped or strobed detector. It may also be possible to change the shape and width of the sweeping beam to optimize sensitivity to surface characteristics slightly larger than the beam size.

Regarding more distant astronomical settings, the beam of a pulsar may sweep a spot across the spherical surface of a companion star and hence create superluminal virtual spot pair creation events. As suggested by Milgrom & Avni (Reference Milgrom and Avni1976) and further analyzed by Chester (Reference Chester1979), some of the X-rays from the binary pulsar 3U 0900-40 may be scattering off the surface of the primary companion and creating a signal possibly misinterpreted as orbital eccentricity. Recent theoretical work modeling this effect subluminally has been done by Dementyev (Reference Dementyev2014).

4.1 Sweeping beam with constant linear speed

A simple but informative scenario is that of a single spot moving with a constant linear speed v across a scattering wall. The distance between the observer and a given point on the line of illumination will be labeled D with minimum distance D_min. It will be assumed that v > c so that superluminal spot pair effects can be demonstrated. The direction with a component toward the observer will be considered the positive v direction. The angle between the closest position on the sweeping beam line to the observer and the position of the beam on the line will be designated ϕ, and the beam will be defined as moving from ϕ = −π/2 to π/2. The geometry is diagrammed on the right of Figure 5.

Figure 5. The geometry of a sweeping beam that creates a spot or spots on a planar wall.

The illuminated spot will start its motion at the ϕ = −π/2 infinitely distant end of the swept line. The spot will then move along a line with a radial component toward the observer, pass the ϕ = 0 point closest to the observer, and then move toward the ϕ = +π/2 infinitely distant end of the line.

Initially, almost the entire spot velocity is directly radially toward the observer, so v_r ~ v. This is depicted in Figure 6. Given that v > c, then v_r > c at the start, but v_r will decrease monotonically with increasing ϕ such that v_r = −vsin ϕ. Clearly, v_r = 0 when ϕ = 0. Therefore, at some point, v_r must cross from being greater than c to being less than c, passing a location where v_r = c. This location is defined by ϕ^virtual_pair = arcsin( − c/v). When v ≫ c, then ϕ^virtual_pair goes toward zero, the closest point on the sweeping beam line to the observer.

Figure 6. The geometry of a spot moving with constant linear speed when the spot is far from the observer. Note that in this situation, most of the spot’s speed v is radially toward the observer, so that v ~ v_r.

Speed v_r becomes equal to c only once on this line. When the spot on the filament is at its closest to the observer, at ϕ = 0, then by definition all of its speed is tangential, so that v = v _⊥ and v_r = 0. After the spot has passed ϕ = 0, its projected speed is away from the observer and so v_r < 0. For the rest of this spot’s trip, ϕ will be greater than zero, and v_r < 0 since the spot is headed away from the observer. In fact, for all positive and higher ϕ values, the spot’s radial speed will always be negative. At some point v_r will drop to below − c, but nothing unusual will be seen by the observer at the v_r = −c location. As the spot finally approaches the end of the infinite filament, its speed is directly entirely away from the observer, so that v_r ~ −v.

To better quantify what an observer would see, it is useful to find the light travel times to the observer from different locations on the spot’s path. The time it takes for a photon to go from a point on the spot’s path to the observer will be designated t_path and is equal to the path length of the light divided by c. At angle ϕ, the distance between the observer and the spot is D = D_min/cos ϕ. Therefore, t_path = D/c = D_min/(c cos ϕ).

Next, define the delay time t_delay as the time between when the spot starts down the filament and the time when the filament element at angle ϕ is illuminated. Distance from the closest point to the observer to the point being considered along the filamentary line can be parameterized as $D_{filament} = \scriptsize\sqrt{ D^2 - D_{min}^2 } = D_{min} \ \tan \phi$. It is further assumed that the filament has total length L which may be infinite. Then, t_delay = L/2v − D_filament/v = L/2v + D_min tan ϕ/v, where a negative ϕ indicates the spot is seen during approach.

From the time the spot started down the filament to reach ϕ, to the time light reaches the observer from the spot at ϕ is t_obs = t_delay + t_path such that

(6) \begin{equation} t_{obs} = {L \over 2 v} + { D_{min} \over c \ \cos \phi } + {D_{min} \ \sin \phi \over v \ \cos \phi } . \end{equation}

The observer will first see the spot at the ϕ location where the total time it takes for light to reach the observer, t_obs, is at a minimum. Mathematically, this occurs when dt_obs/dϕ = 0. Since v > c, this does not occur infinitely far up the filament, and since it is a temporal minimum, spots will be perceived by the observer on both sides of this location at future times. Therefore, the ϕ value at this location will be referred to as ϕ^virtual_pair. One spot of the pair appears to the observer to move along the filament with a component toward the observer, while the other appears to move in the opposite direction. These two spots will appear to diverge to opposite ends of the filament.

The transverse speed of the spot will be observed to be

(7) \begin{equation} u_\perp = D {\rm d \phi \over dt_{obs}} = { c \ v \cos \phi \over v \ \sin \phi + c} . \end{equation}

This transverse speed diverges when ϕ = ϕ^virtual_pair. Specifically, when ϕ is slightly less than ϕ^virtual_pair then u _⊥ is negative and very large, meaning that one image of the spot is seen to start its motion away from the observer quite quickly. Also, when ϕ is slightly greater than ϕ^virtual_pair, u _⊥ is very large and positive, meaning that a second image of the same spot also appears to start its motion quite quickly, but in this case toward the observer.

In sum, even though only a single superluminal spot ever existed on the wall, the first thing the observer sees is a spot pair creation event at ϕ^virtual_pair. The two perceived spots move away from each other, each, at first, with infinite angular speeds, but each quickly slowing. These two virtual spots will always remain visible to the observer, each always moving toward opposite ends of the filament. Note that when v < c then ϕ^virtual_pair is not defined, meaning that subluminal real spots are never seen to create virtual pairs. In the above v > c case, there is never any real spot pair creation event—the existence of virtual spot pairs in this case is purely perceptual.

4.2 Sweeping beam with constant angular speed

Another useful example occurs when a beam sweeps across a planar scattering wall at a constant angular speed, here parameterized as ω. As before, the beam swept line on the wall is conceptually similar to an illuminated filament. The source of this spinning beam is considered at rest with respect to the scattering wall and the observer. With respect to the beam source, distances along the filament are given by the parameter H, with H_min being the closest point on the filament to the source. With a vertex at the beam source, angles on the filament are labeled with the parameter θ, with the furthest point on the filament in the initial direction of the beam to be θ = −π/2, the closest point to the source as θ = 0 and the furthest point opposite the initial direction of the beam to have θ = π/2. Note that $\omega = {\dot{\theta }}$. The geometry is shown diagrammatically on the left part of Figure 5.

This scenario starts with the light beam pointing parallel to the wall. As the beam tilts toward the wall, the first illuminated part of the wall will not be infinitely far from the source, at θ = −π/2, because it will take an infinite time for light to reach that far from the beam source. Starting from the time when the beam points toward θ = −π/2, the delay time it will take for the rotating beam to point toward position θ on the filament will be t_delay = (π/2 + θ)/ω. The time it takes for light to travel from the source to position θ on the filament will be the path length divided by the speed of light, so that t_path = H_min/(c cos θ). From the start, the time that a filament position at θ will be illuminated (and so host a ‘real’ spot) will be

(8) \begin{equation} t_{real} = t_{delay} + t_{path} = { \pi / 2 + \theta \over \omega } + {H_{min} \over c \ \cos \theta } . \end{equation}

Now, the first illuminated point of the filament occurs when dt_real/dθ = 0, which occurs when 1/ω + (H_min/c)(sin θ/cos ²θ) = 0. Therefore, the θ of first illumination occurs when

(9) \begin{equation} { \sin \theta \over \cos ^2 \theta } = {- c \over H_{min} \ \omega } , \end{equation}

which has solution

(10) \begin{equation} \theta _{\rm pair}^{\rm real} = \arcsin \left( {H_{min} \omega \over 2 c} - \sqrt{ {H_{min}^2 \omega ^2 \over 4 c^2} + 1 } \right) . \end{equation}

Since θ locations on either side of θ^real_pair will be illuminated after θ^real_pair, a spot on one side of θ^real_pair will become illuminated at the same time as a spot on the other side. For this reason, θ^real_pair is considered the location of the creation of a real pair of spots. Interrogation of the t_real Equation (8) above indicates that the further that θ is from θ^real_pair, the later in time it becomes illuminated. Therefore, the spots created at θ^real_pair will move on the filament away from θ^real_pair and each other. The analogous quantity to θ^real_pair in the previous section on beam-illuminated spheres is ϕ^real_pair.

The speed of the beamed real spot on the filament perpendicular to the direction to the source is Hdθ/dt_real. Therefore, the speed of the beamed real spot on the scattering wall is v = (H/cos θ)dθ/dt _real which gives

(11) \begin{equation} v = { c \ H_{min} \ \omega \over c \ \cos ^2 \theta + \omega \ H_{min} \ \sin \theta } . \end{equation}

When θ = θ^real_pair, v diverges. Also, v changes sign on either side of θ^real_pair, meaning that each real spot in the created pair moves in opposite directions on the filament, as indicated above. Even if v at θ = 0 is subluminal, v may exceed c at other values of θ.

A plot of the absolute value of real spot speed on the wall as a function of θ is shown in Figure 7 for three values of the speed across the closest section. Formally, v_closest = ω D_min. In general, the lower the sweep speed, the closer the spot pair creation event will be to θ = −90° = −π/2 rad. Conversely, the higher the sweep speed, the closer the real spot pair creation event will be to θ = 0. The two lines on each plot depict the speed of each spot during a single sweep of constant angular speed. The real spot with the most negative θ will move in the opposite direction to that of the sweeping beam. This real spot is created at θ^real_pair with formally infinite speed and will always drop toward ∣v∣ = c as θ drops to − π/2 radians, formally reaching ∣v∣ = c at θ = −π/2.

Figure 7. The absolute value of the speed of real spots moving across a planar wall or linear filament is plotted against the beaming angle, for the case when the spots are created by a single fixed beam rotating with a constant angular speed in a plane perpendicular to the wall. The beam first points toward θ = −90°, moves to point toward the closest point on the scatterer at θ = 0°, and ends at θ = 90°. A divergent spike results from a real spot pair creation event and occurs for a beam with any finite angular speed. The superluminal spots mark the first section of the wall actually illuminated by the beam. The plot labels refer to the real spot speed at θ = 0.

The real spot with the larger θ will move in the same direction as the sweeping beam and will also be created at θ^real_pair at formally infinite speed and at the same time as the other spot. Although, this real spot will at first have its speed drop below ∣v∣ = c as θ further increases, its speed will rise toward ∣v∣ = c as θ rises toward π/2 radians, formally reaching ∣v∣ = c at θ = π/2.

What does an observer see? For the didactic purpose of enhancing the prevalence of superluminal spot pair effects, the observer is considered to be closest to the later part of the beam sweep, at positive θ, as depicted in Figure 5. As seen by the observer, angular placement of the illuminated sections of the filament will be labeled ϕ with ϕ starting at − π/2 and extending to + π/2. The distance between the observer and position ϕ on the filament is labeled D. The minimum distance between the observer and the illuminated filament on reflecting wall is labeled D_min which occurs at ϕ = 0. The spot’s velocity along the filament can be broken up into components perpendicular and radial to the observer such that

(12) \begin{equation} v_\perp = { c \ H_{min} \ \omega \ \cos \phi \over c \ \cos ^2 \theta + \omega \ H_{min} \ \sin \theta } , \end{equation}

and

(13) \begin{equation} v_r = { c \ H_{min} \ \omega \ \sin \phi \over c \ \cos ^2 \theta + \omega \ H_{min} \ \sin \theta } . \end{equation}

Note that θ and ϕ are not independent. Given positions of the source and observer, one can compute ϕ given θ and vice versa. Also, when ϕ is near − π/2 or + π/2 then θ will approach the same value, and vice versa.

To decipher what an observer would see, it is important to first find how v_r changes as the beam sweeps through one cycle. First considering when ϕ is near − π/2, v_r will be negative, meaning that a spot is perceived moving away from the observer. As ϕ increases, there will be discontinuous jump in v_r where v_r suddenly goes from negative infinity to positive infinity. For yet larger ϕ values, v_r is positive, as a spot is moving toward the observer, but decreasing in magnitude. Eventually, as a spot’s v_r decreases, it will drop from being superluminal to subluminal. This v_r = c location will be referred to as the spot ‘virtual pair creation’ location: ϕ^virtual_pair. As ϕ increases further, v_r will continue to decrease to zero and then past zero into negativity. Since as ϕ continues to increase, v_r will only become more negative, then never again will v_r drop from (positive) superluminal to subluminal. Therefore, no more virtual spot pair events will be observed. As ϕ approaches π/2, v_r will continue to drop asymptotically toward − c as the illuminating beam becomes increasingly parallel to the planar scattering wall.

To better quantify what the observer will see, the timing of arriving photons will be calculated. Starting from the time when the source first started releasing photons as it pointed toward θ = −π/2, the time it takes for a photon to reach the observer is

(14) \begin{eqnarray} t_{obs} &=& t_{delay} + t_{path}^{illum} + t_{path}^{obs} = t_{\rm real} + t_{path}^{obs} \nonumber\\ &=& { \pi / 2 + \theta \over \omega } + {H_{min} \over c \ \cos \theta } + { D_{min} \over c \ \cos \phi } , \end{eqnarray}

where t_delay is the time it takes, since the start, for the beam to point at position θ, t^illum_path is the time it takes for light to go from the beam source to illuminate the scattering wall at position θ, and t^obs_path is the time it takes for light to go from position ϕ to the observer. The location of the observed (and hence virtual) spot pair event will be the ϕ^virtual_pair angle that satisfies dt_obs/dϕ = 0. Detailed inspection of how t_obs changes with ϕ reveals what the observer will see and when.

Assuming that the wall is a Lambertian scatterer, it is straightforward to compute the relative brightness changes of the virtual spots visible to the observer. For increased simplicity, it will be further assumed here that the observer is at the source so that θ = ϕ. Specifically, using reasoning similar to the above lunar scenario, the instantaneous perceived brightness of a sweeping virtual spot as a function of θ is proportional to b∝u _⊥(D_min/D)². This instantaneous perceived brightness is depicted in Figure 8. In this Figure, the unusual peak in brightness is caused by the perceived virtual spot pair creation event at ϕ^virtual_pair. The high instantaneous perceived brightness is essentially caused by the relatively short time scale during which a relatively large part of the (nearly) uniformly bright scattering surface is scattering back light.

Figure 8. The instantaneous perceived brightness of virtual spots perceived moving across a planar wall or linear filament, as observed from the beam source, is plotted against the beaming angle. Here, the spots are created by a single fixed beam rotating with a constant angular speed in a plane perpendicular to the wall. The beam first points toward θ = −90°, moves to point toward the closest point on the scatterer at θ = 0°, and continues on to 90°. The divergent spike results from a virtual spot pair creation event and occurs for a beam with any finite angular speed. The virtual spot pair creation event is the first light seen by the observer. The plot labels refer to the spot speed at θ = 0.

Given all of this detail, what an observer sees in this rotating beam scenario is surprisingly simple. The first phenomenon observed is a spot pair creation event at ϕ^virtual_pair. The two virtual spots appear to move away from each other, each, at first, with infinite angular speeds, but each quickly slowing. These two virtual spots will always remain visible to the observer, each always moving toward the opposite distant ends of the wall or filament, and fading. This case is descriptively similar, as seen by the observer, to the previous case where the real spot speed was constant.

4.3 Sweeping beams across walls or filaments in astronomical settings

Similar but more complex scenarios than those considered above include eclipse light boundaries moving across the surface of reflection nebulae. Such mechanisms are thought to be the root cause of variable nebulae. Possibly the most notable variable nebula is Hubble’s Variable Nebula (HVN: NGC 2261). The HVN, first noted by Hubble (Reference Hubble1916), lies at a distance of about 2 500 light years, estimated by an assumed association to the nearby open cluster NGC 2264 (Jones and Herbig Reference Jones and Herbig1982). Changes in the nebula’s brightness have been attributed to shadows of opaque clouds moving between the bright star R Monocerotis and a reflection nebula (Johnson Reference Johnson1966, Lightfoot Reference Lightfoot1989). Seeming shifts in angular structure on the order of 0.5 arcmin have been recorded over the time scale of tens of days. If attributable to single occulting events, these shifts indicate spot motions on the order of one light year per year, equivalent to c. This speed is an estimate of v _⊥ and not v or v_r and so does not directly reveal the key v_r parameter that determines the perceived spot pair creation and annihilation events. Nevertheless, it seems reasonable to assume that given a long enough reflection train, virtual spot pair creation or annihilation events do occur as described above in the planar reflecting model. Additionally, sloping or bumpy terrain on the reflection nebula could well give rise to one or more v_r = c crossings, and so yield virtual spot pair creation or annihilation events similar to that described in the spherical reflecting model.

Although, observers may be unlikely to see a spot pair creation event on the HVN without prior warning, pairs of spots moving away (toward) from each other might be observable from which a virtual spot pair creation (annihilation) event might be inferred. So long as the geometry of the scattering surface and the direction to the beam source remains relatively unchanged, the location of one spot pair event may also be the location of future spot pair events. For example, eclipses may show both ingress and egress events, and a single cloud moving near the source star might have multiple areas of high and low opacity.

Other variable nebulae with potentially similar geometries that might show superluminal spot pair events include Hines Variable Nebula (see, for example, Moore Reference Moore2005), Gyulbudaghian’s Variable Nebula (the variable nebula associated with the variable star PV Cephei; Boyd Reference Boyd2012), infrared variable nebula IN L483 (Connelley, Hodapp, & Fuller Reference Connelley, Hodapp and Fuller2009), and NGC 6729 (the rapidly variable nebula associated with the star R Coronae Australis; see, for example Graham & Phillips Reference Graham and Phillips1987). Additionally, the system UW Cen is a candidate for observerd virtual spot creation and annihilation events as it features an R Coronae Borealis star hypothesized to act as a lighthouse shining through gaps in thick dust clouds near its surface, illuminating changing portions of a surrounding reflection nebula (Clayton Reference Clayton2005).

High frequency monitoring of variable nebulae might be able to find locations where virtual spot pair creation and annihilation events are occurring, and use these to build up information about the geometry of the surrounding reflecting surfaces. It is beyond the scope of this work to model these nebulae in detail but rather to raise the possibility that such effects might be occurring, discoverable in practice, and could yield information about the nebulae. A more specific investigation will be given in Zhong and Nemiroff (Reference Zhong and Nemiroff2015).

Besides variable nebulae, another astronomical system that might show superluminal spot pair events are planetary nebulae. In particular, knots of optically thick dust in planetary nebulae are thought to cast shadows from the central star creating regions where specific ionizations do not occur. These shadows may move quickly as ionization fronts cross background gas, and could, in theory, move superluminally. Prominent possibilities include the NGC 7293 the Helix Nebula (O’Dell, Henney & Ferland Reference O’Dell, Henney and Ferland2007), NGC 6543 the Cat’s Eye Nebula (Balick Reference Balick2004), and bi-polar planetary nebula M2-9 (van den Bergh Reference van den Bergh1974; Trammell, Goodrich, & Dinerstein Reference Trammell, Goodrich and Dinerstein1995; Corradi, Balick, & Santander-García Reference Corradi, Balick and Santander-García2011).

Another system type that might show superluminal spot pair events are circumstellar disks. Specifically, a bright star could create a silhouette of an opaque disk onto more distant reflecting material, enabling a shadow magnified in angular size by as much as a factor of 100 (Pontoppidan & Dullemond Reference Pontoppidan and Dullemond2005). Potentially, rapidly moving but subluminal inhomogeneities in the interior circumstellar disk could cast shadows moving superluminally onto a background. One example system is the Serpens Reflection Nebula and Ced 110 IRS 4 in the Chamaeleon I molecular cloud (Pontoppidan & Dullemond Reference Pontoppidan and Dullemond2005). Another is the case of HH 30 where an inner circumstellar disk is casting a large and variable shadow on an outer disk (Watson & Stapelfeldt Reference Watson and Stapelfeldt2007).

Pulsars surrounded by ionized shells provide yet another type of candidate system to cast superluminal shadows. In the radio, unusual ‘moving’ pulses from the Crab pulsar have been seen during several epochs and interpreted to be reflections of the beam off of ionized shell(s) in the outer part of the nebula (Lyne, Pritchard, & Graham-Smith Reference Lyne, Pritchard and Graham-Smith2001).