  When a gamma or cosmic ray strikes the top of Earth's atmosphere, a shower of
secondary particles moves toward the surface. Some of these secondary particles
are charged muons that subsequently enter Water Cherenkov Detectors (WCDs) on
the ground. Many of these muons, traveling near the speed of light in vacuum,
are moving faster than the speed of light in water and so trigger isotropic
Cherenkov radiation in the WCDs. Inside many WCDs are photomultiplier tubes
(PMTs) that detect this Cherenkov radiation. When the radial component of the
speed of a muon toward a PMT drops from superluminal to subluminal, the PMT
will record Cherenkov light from a little-known optical phenomenon called
Relativistic Image Doubling (RID). Were the RID-detecting PMTs replaced by high
resolution video recorders, they would see two Cherenkov images of the muon
suddenly appear inside the tank, with one image moving with a velocity
component toward the recorders, the other away. Even without a video, the RID
phenomenon will cause different PMTs to record markedly different light curves
for the same muon. In this paper, we present a study hoping to inspire the
explicit detection and reporting of RID effects in WCDs. We consider three
example cases of muon RIDs in High-Altitude Water Cherenkov (HAWC)-like
systems: vertical, horizontal and oblique. Monte Carlo simulations show that
RID effects in HAWC-like systems are not rare -- they occur for over 85 percent
of all muon tracks.
