# Lorentz contraction of object in circular motion

Apparently a bunch of people totally misunderstood my previous question and choose to ignore the clarifying comments. Let me change the conditions to remove all the confusion.

A group of particles forming a circle of uniform density is moving in a circular path (following that circle shape), at a moderate speed $v_1$ (as observed from a frame of reference bound to the center of the circular path, O1). Let's say this is a cloud of ions filling a cyclotron that had accelerated them to 0.2c and then just maintains their circular trajectory with magnetic field without accelerating them any further. (depicted is momentary speed vector of one of these particles)n accelerator.

An observer (O2) moving at $v_2$ near speed of light (say, 0.9c relative to the previous observer (O1) / the cyclotron) observes the positions and movement of these particles. How will they appear from that observer's point of view - what is their apparent their trajectory and how does their density vary from that observer's point of view?

• Are you asking what observer 2 will see i.e. including the travel time of the light reaching observer 2's eyes, or are you asking what the shape of the circle will be in observer 2's frame. Also what is the significance of $v_1$? Can we assume $v_1$ is negligable i.e. the particles are effective stationary in observer 1's frame? Jan 25, 2016 at 9:03
• @JohnRennie: Preferably both, regarding your first case. As for the second, v1 is not entirely negligible. I set it to a relatively small value to stop the suspicions of "infinitely rigid rod with ends moving faster than speed of light" (that had murdered my previous question), but as I wrote, let's assume it's 0.2c relative to O1's frame: The relativistic behaviors relative to O1 are negligible, but they are definitely pronounced relative to O2. They are definitely not stationary in any of these frames though. (also, it's a momentary linear velocity in rotary motion.)
– SF.
Jan 25, 2016 at 9:39
• @SF: how does it matter that the particles in the ring move in a loop? Unless you consider general relativistic effects (which would require specifying if the particles are moving in a gravitational or other field) the problem is equivalent to just looking at how a moving observer sees a moving element of the ring. Or am I missing something? Jan 25, 2016 at 10:59
• @Numrok: You're not missing anything. Observing one single particle moving in a circular path would be sufficient. Introducing a ring instead of a single particle is just to avoid the mess with simultaneousness as in any moment of movement of the observer the single particle. And I used a cloud of particles instead of a ring because I definitely wanted to avoid having any solid object here, as it would inevitably invoke the curse of 'infinitely rigid rod'. Assume charged particles in a magnetic field, no significant gravitational effects.
– SF.
Jan 25, 2016 at 11:16
• Any object seen from a relativistic observer appears rotated, i.e. a round object will appear... round but it will be seen under a different angle than by a resting observer. The circular motion also causes a slight redshift on one side and a slight blueshift on the other, but that's independent of the movement of the observer. Jan 25, 2016 at 11:17

I tried to put it into a figure the results from reasoning in the framework of Special Theory of Relativity, this is what the observer would see:

Where you should notice several effects:

• length contraction along observer's motion causes the ring to look like an elipse
• equal velocities on equator where particles are crossing from top to bottom or vice-versa their velocities are perpendicular to observer's hence not affected (0.2c on either side)
• fast transit through "bottlenecks" the apparent concentration of particles on top and bottom, which I mean by bottlenecks, are just an effect of length contraction, but particles are not really stuck there. Actually they are faster than in the rest of the ring (see velocity colorbar). Also the peak velocity on top is higher (.93c) than on bottom (.85c) because on the former they travel towards observer, while on the latter they travel away from him. But the observer sees all of them travelling towards him in both cases with the mentioned peak velocities.
• spacing asymmetry top vs bottom the different speeds top vs bottom will affect the spacing between particles, the top looking more crowded than the bottom. However the time contraction will counter this effect and and at all times there should be the same amount of them on either side.
• slanting also due to speed differences top vs bottom there will be a kind of slanting/forward-fall effect on the shape of the circle.

Note The figure is a representation of the effects and might exaggerate some of them, since is not made by calculating the actual values. The speeds-colorbar only shows the value of horizontal component and is also exaggerated, since minimum values should be .9c instead of 0.

• Thank you very much! May I ask what software were you using?
– SF.
Jan 25, 2016 at 19:22
• I used a python and matplotib which I wrote, but as I explained it does not calculate from the equations, but rather creates a plot that shows the effects I mention in a visible way. The functions I used probably differ from the actual solutions that would be obtained from the relativistic equations. Jan 25, 2016 at 19:26
• Thanks. I really wish there was some relativistic Algodoo. It would make visualising what goes on much easier.
– SF.
Jan 25, 2016 at 20:22
• Ok I will add some equations soon. Jan 25, 2016 at 20:28
• You don't have to. I can look up any equations just fine, it's just that going from an equation to a picture with a setting more complex than a couple points takes unreasonable amount of time and work.
– SF.
Jan 25, 2016 at 21:50