We could. It would just take a tremendous amount of time and effort and money to do it.
Speaking of the physics of it, we do know there are aircraft that can remain relatively stationary in the air (helicopters, some drones, planes like the A-10 and the Harrier Jet) and that some of those craft can appear to move like you describe, while we have yet to observe the ability of anything to both travel at or near light speed and defeat the relativistic effects of doing so. It would be better to keep to the mundane explanations at this point than jump to the extraordinary ones.