The spatial geometry of a uniformly rotating reference frame: the Newtonian and the relativistic point of view
Kostas Papamichalis Dr of Theoretical Physics

The Theoretical Model and Applications

In the two windows of the virtual environment, we see the 'rest plane' Oxy of a coordinate system, uniformly rotating around the z-axis, which is vertical to the plane of the simulation. The angular velocity vR of the rotation is selected by the corresponding tools, at the control panel of the simulation.
The left window depicts a Newtonian world and the right, a relativistic. In both cases, there has been drawn a circle (O,R) whose radius R can be controlled by the user. The user can drag and transfer a vector V parallel to itself along the boundary of the circle. In the relativistic world, the plane Oxy is not Euclidean and the parallel displacing vector when returns at its initial position has a direction which is in general different from the original. By using the theoretical model (see the attached pdf-file: "The Theoretical Model and Applications"), and the measured angles, the user can calculate the angular velocity of the rotating frame.
Furthermore, the relativistic observer measures the length L of the boundary of the circle and its diameter, and he finds out that their ratio is different of pi; it depends on the angular velocity and the radius R.

The user controls the values of the angular velocity vR and the radius R of the circle. The values of vR are selected by using the plus and minus buttons next to the corresponding field, and R takes any positive value less than the maximum value determined by the theoretical model.
The value of the angle AOQ can also be inserted by the user in the corresponding field of the control panel. By pressing the plus and minus buttons, angle AOQ increases or decreases by 2pi/360 rad compared to its current value.
In both windows there are three counters: a) a counter measuring the angle AOQ which determines the position of the transferred vector along the circumference of the circle, b) a counter measuring the angle formed by V and the radial basis vector e_r, c) a counter measuring the the arc-length (AQ) which is determined by the initial to the current position of the transferred vector.
With double-click on any of the windows of the simulation, you can reset it.

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