Relativistic Cycloids

[LaurieAG]

Relativistic rolling wheels are just as interesting as non relativistic rolling wheels and they both have several things in common.

 

The first link is to "Space geometry in rotating reference frames: A historical appraisal" by Øyvind Grøn

http://areeweb.polito.it/ricerca/relgrav/solciclos/gron_d.pdf

 

The second link is to "The Cycloid: Tangents, Velocity Vector, Area, and Arc Length" http://quadrivium.info/MathInt/Notes/Cycloid.pdf

 

Cycloid arcs have several geometric properties in that (1) the area above the arc equals the area of the circle that created the arc (2) the area below the arc equals 3 times the area of the circle that created it (total area - 4 * circle area) and (3) the length of the arc equals 8 * radius of the circle.

 

I was looking at Gron's Fig 9 and the first Cycloid arc Figure 2.13e and noticed a few things that might be of interest.

 

(1) the sizes of the circles were the same when printed out and overlaid.

(2) the tangent sections shown in Figure 2.13e meant that you could plot the Cycloid arc if you had the tangents.

(3) in A and B in Fig 9. each individual point mapped on A formed a line through its equivalent point on B and (almost) all lines drawn terminated at point T as per Figure 2.13e.

 

To me this meant that a relativistic Cycloid arc could be plotted geometrically.

 

The relativistic Cycloid arc below is plotted from the tangents made between each pair of positions in Fig 9. A and B in the same way as Figure 2.13e. At this stage I just used the next tangent point down as the end point for the tangent so later on I added the distorted circles centered on T (note the red and blue bars on the top and side of the 2 plots) to reflect the varying segment size in the relativistic plot.

 

So it looks like the two Cycloid arcs are similar in shape and length with the only real difference being the varying segment sizes used in the Relativistic plot as opposed to the fixed segment sizes used in the Euclidian plot.

 

As the area under the relativistic cycloid is less than the area of 3 circles (and the area above is greater than the area of one circle) is the difference in arcs related to the extra surface area, contained within a circle with the same radius of a sphere, that is drawn on the spheres surface, which is larger than the area of a circle of the same radius due to the curvature of the sphere?