# Relativistic wheel and orbit connection?

1. Dec 9, 2007

### yuiop

I have noticed an apparent correlation between two seemingly unconnected phenomena, but I do not have the mathematical ability to prove the connection.

I was wondering what a rolling wheel would look like at relativistic speeds. To investigate this I created a java simulation that solves the problem numerically. It was more difficult than I first imagined, as it was not a simple case of applying the Lorentz transformations as the situation is complicated by simultaneity. When one spoke is opposite the other at 3 o’clock and 9 o’clock to an observer commoving with the wheel axle, they are not opposite each other to the observer that is stationary with respect to the road. (The two spokes are only opposite each other in both reference frames when they are at the 12 o’clock and 6 o’clock positions. The simulation also shows that in the road reference frame, the angular velocity of the rim at side nearest the road is much greater than the angular velocity furthest from the road. The simulation ignores the visual effect due to light travel times that causes an apparent rotation out of the rolling frame.

I ran the relativistic wheel simulation alongside a gravitational orbital simulation. I adjusted the orbit to a similar elliptical shape and radius as the wheel in the road frame. I noticed that the motion of the orbiting body appears to match the motion of a point on the rim of the rolling wheel, (when the forward motion of the wheel is deducted)

I am sure there is some interesting physics and maths to be discovered here for someone who has the ability.

One possible connection is that angular momentum is conserved in both scenarios. However there is the complication that a gravitational orbiting body has its centre of angular momentum at one focus of the ellipse, while the rolling wheel has its centre in the normal centre.

There is another geometrical connection too. Imagine a cone with a circle and an ellipse formed from planes intersecting with the cone. A point on the circumference of the circle is made to follow the circumference at constant velocity. A ray drawn from the apex of the cone to the moving point on the circle intersects the edge of the ellipse to form a second point following the circumference of the ellipse. The motion of the intersecting point on the ellipse appears to follow the same complex motion of the orbiting gravitational body and the point on the rim of the relativistic wheel. (Assuming all 3 have similar elliptical shapes with similar eccentricity)

Is there anyone out there with the physics/maths/geometry ability and knowledge to make the connections?

2. Dec 9, 2007

### A.T.

You schould checkout this 4D ray tracing movies. They not only take the Lorentz transformations into account, but also the light travel times, to predict what an observer would see.

3. Dec 9, 2007

### yuiop

Thanks A.T. for the great link.

The green wheel in the third video down is closest to the type of motion I am talking about. I am still hoping that someone will come up with a mathematical formula for the motion of a point on the rim of the relativistic wheel. It may be that there is no simple solution, just as there is no simple formula for the length of the perimeter of an ellipse or for the motion of an body in an eliptical orbit. The usual solutions involve series and aproximations. I assume the 4D simulations in the link were created using numerical solutions similar to the method I used or possibly tables. My method is suitable to create a simulation, but a mathematical formula would give an insight to the connection between relativistic rotation and the elliptical gravitational orbital motion of a body.

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