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At relativistic speeds there is an optical effect called Terrell rotation causing objects passed by to seemingly rotate.

As bats use sound rather than light when echo-locating, at what degree would the experience the Terrell rotation? We know the speed of sounds in air, and can estimate the speed of bats and a field of view. Given all this, to what extent would the bats experience this effect? Not at all, barely visible, or pronounced?

If this effect is not experienced, what do they experience?

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    $\begingroup$ It's not clear to me why you would expect bats to experience this effect at all - other relativistic effects like time dilation or length contraction obviously don't happen when approaching the speed of sound, why would you think this one is different? $\endgroup$
    – ACuriousMind ♦
    Jun 12, 2017 at 10:41
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    $\begingroup$ They wouldn't experience anything, just as we don't experience light waves, polarization or white balance - our brain processes information and creates an image inside our heads. Sure, I can recognize f.ex diffraction in an oil spill, but at the end of the day, oil spills look like oil spills usually do. The answer to your question is: If they experience it, they are already including it in their heuristic, and therefore are not experiencing it, but unconsciously correcting for it... $\endgroup$
    – Stian
    Jun 12, 2017 at 10:59
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    $\begingroup$ @ACuriousMind no just the fact that you see objects you pass by from different time on differnt parts of the object, similar to what is seen for light. I was under the impression the Terrell effect was optical, e.g. the there is a timelag in the received signal. But you're probably right, but this is why I asked this question. $\endgroup$
    – vidstige
    Jun 12, 2017 at 14:09
  • $\begingroup$ @StianYttervik thanks for the clarification. I very much agree with this. I rather mean whether their sensory organs "see" this (or any other related effects). However, this leads to combersome wording of the question. Do you have a suggestion for alternate formulation to better capture this? $\endgroup$
    – vidstige
    Jun 12, 2017 at 14:10
  • $\begingroup$ Hm. Good question. I would perhaps leave the bats out of it: "Is Terrel Rotation also reasonably detectable (>1 per 1000) with sound waves moving close to the speed of sound?" 1 per 1000 in this case i guess would be related to the distortion, either by radian or by cartesian coordinates. $\endgroup$
    – Stian
    Jun 13, 2017 at 10:50

2 Answers 2

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I don't think it would happen at all - Terrell rotation is specifically due to the Lorentz contraction from special relativity, not just from the Doppler effect (which, in any case, takes a different mathematical form in the nonrelativistic case).

Most common bats usually fly at a rather sedate 16 km/h or so, so they wouldn't experience much of a Doppler shift. But the fastest bat ever recorded has flown at 160 km/h or 13% of the speed of sound, so Doppler effects should indeed be quite noticeable (after all, you can hear the Doppler shift of ambulance and police sirens when they're just moving at ~80 km/h). In fact, bats actually adjust their echolocation pitch based on their flight speed to compensate for the Doppler effect in order to keep the return pulse's frequency within their optimal hearing range.

Extremely rarely, one finds bats capable of even more complicated sonar data processing. But I don't think they would "see" a Terrell rotation.

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    $\begingroup$ Could you pick a set of speed units and stick to it? My brain isn't accustom to switching back and forth between mph and km/hr. =) (Just listing both is a viable alternative, of course, but that's a bunch of extra work. ;) ) $\endgroup$
    – jpmc26
    Jun 12, 2017 at 8:42
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    $\begingroup$ Wouldn't there be some kind of "optical" effects as there is a timelag e.g. the bat would "see" different parts of an object as reflected by the object at different time? $\endgroup$
    – vidstige
    Jun 12, 2017 at 14:41
  • $\begingroup$ @vidstige Yes, but I don't think the distortion would take the form of a rotation. Whatever effect there would be would be much more pronounced for aircraft-detection radar, since planes move at hundreds of km/hr, but I've never heard of radar images appearing rotated. $\endgroup$
    – tparker
    Jun 12, 2017 at 15:51
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    $\begingroup$ I do think the bats would see it (in small degree) as well though. As @vidstige says, it's an optical effect, not a special relativistic one. That is, if we lived in a universe with Galilean invariance but a finite speed of light (that could be exceeded by other objects) then the Terrell effect would still be present. I agree it should hold for sound as well. But the point stands about bats being too slow. $\endgroup$
    – Alex Meiburg
    Jun 12, 2017 at 18:57
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    $\begingroup$ it seems the question is still open then..? Interesting. :-) Well, radar uses radio rather than sounds, so it's no wonder they don't see anything like this. Perhaps high-speed boats with sonars could be interesting to learn more about. $\endgroup$
    – vidstige
    Jun 12, 2017 at 21:04
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I've read paper of R.Penrose linked on wiki page (in fact as it is on subscription basis only first page is available). research note by N.Penrose. It says:

the light from the trailing part reaches the observer from behind the sphere, which it can do since the sphere is continuously moving out of its way

So effect is neither due to Lorentz contraction nor Doppler shift, but from finite speed of light and high (relative to that speed) movement of object relatively to the observer. As other answer already noted speed of bats (relative to sound), the Terrell rotation is possible to experience for bats.

ADDED: after some hard thought I keep final conclusion: Terrell rotation in some form is possible for some settings (distance, angle and relative speed) for echo-location of bats. However I realized two key differences for sound echo-location from original effect: 1) light is relativistic, whereas for sound in air whether object or observer moves matters 2) active echo-location means bat "hears" reflected sound it emitted, whereas for light observer is assumed to see light created by other objects. So time for sound to travel from the bat to the object had to be taken into account to calculate actual "image" of the object.

"Hearing" response from "not seen" from initial position part of the object is possible because sound wave can "go around the corner". Then due the fact that the bat changed position, bat would hear response from that part simultaneously with response from "front" part, therefore experiencing effect similar to Terrell rotation.

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