# Tautochrone curve and Theory of Relativity

I just came across the concept of a tautochrone curve (or isochrone curve). My question stems from the intuition that a body starting from a higher point would need to be traveling at a greater velocity in comparison to a body which is closer to the lowest point. Now does this mean that if we were to extrapolate, we could actually have a curve on which the body would be moving at the speed of light at some point?

## 1 Answer

The tautochrone problem is formulated for classical mechanics, in classical mechanics there is no speed limit, so yes, if we follow the classical modelling the sliding mass points will surpass $c$ if started form some height.

But the classical modelling does not describe reality correctly. The classical kinetic energy does only hold for velocities small compared to the speed of light, so the solution to the classical equations will not actually be a tautochrone in our physical world due to relativistic effects. (The approximation is, however, exceedingly good at everyday velocities).

Note, however, that other idealizations in the modelling become relevant long before relativistic effects. For one the equations ignore friction (which might be alleviated by evacuated pipes and good bearings). Worse, they assume a homogeneous gravitational field along the entire curve, but earth's gravitational field is not homogeneous, this approximation only holds for small height differences. As you might know, the escape velocity of earth is around $11\,\mathrm{km/s}$, this is also the maximal velocity one can gain when falling from infinity into earth's gravity well (ignoring the gravitation exerted by the sun, our galaxy, and so on), if you had a gravitating body where the escape velocity reached the speed of light (and therefore it were possible according to classical mechanics to reach it by falling down on it), general relativity kicks in: Such an object would form a black hole.

In consequence we can state, that a variety of effects will prevent an object to reach the speed of light on an tautochrone. The mathematical expressions derived from classical mechanics imply this, but they only model nature well for a certain parameter range.