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?
Best Answer
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.