# Foucault pendulum with General Relativity

It is well-known that the plane of the swing of a Foucault pendulum exhibits parallel transport (wrt to the Levi-Civita connection) of the round sphere.

Seeing as Einstein's relativity theories geometrize inertial motion as autoparallel transport in pseudo-Riemannian geometry, it seems a bit suspicious that the above should just be a coincidence.

In the following, I propose an explanation via GR. Any thoughts are welcome to try to make the following argument formal.

Namely, assuming that the frequency of the Foucault pendulum is very small (compared to the time of one revolution of the Earth), treat the pendulum as being attached to the reference frame $$(x \circ \gamma) (\tau) = (ct(\tau) , R_{Earth}(\tau), \theta(\tau) = \theta_0 , \omega \tau)$$ where $$\omega$$ is $$360^{\circ} / 24$$ hours, $$\theta_0$$ is the colatitude of the pendulum in the Schwarzschild geometry (let us assume the Kerr corrections are negligible); i.e. the pendulums base point rotates around with the Earth as usual. Note we are using the usual Schwarzschild chart.

One can check as expected the proper acceleration of this curve is always totally in the $$\frac{\partial}{\partial r} \big|_{\gamma(\tau)}$$ direction: i.e. as expected by the equivalence principle, standing on a slowly rotating, massive sphere is at each proper time locally the same as acceleration in flat spacetime - the acceleration being at each time normal to the surface of the Earth.

So that means during one (short) swing, one should from the accelerated reference frame see a pendulum that behaves exactly as the basic examples in a undergrad mechanics text. Now importantly, the net acceleration of the pendulum bob during one swing is entirely normal to the surface of the spacelike hypersurface representing the Earth's regular surface.

Thus one may get a "often defined" vector along $$\gamma$$ which is the spacetime tangent (4-velocity) to the bob each time it visits the bottom of its swing (so a vector in $$T_{\gamma(\tau_i)} S^2 \subset T_{\gamma(\tau_i)} M$$ for various evenly spaced $$t_i$$). Furthermore, in each time chunk between visits to the bottom of the swing, the covariant derivative (force acting on the bob) is totally normal to the $$S^2$$, so one gets that the tangent to the bob is the unique parallel vector field along $$\gamma$$ with the right initial conditions.

Have such problems with a discretized notion of parallel transport emerging from a continuous setting been studied?

• You should use the stellar day (the Earth's mean rotation period relative to the ICRF inertial frame), ~86164.1 seconds. See en.wikipedia.org/wiki/… And since you're investigating GR effects, at that level of precision, Earth's rotation rate isn't regular. See en.wikipedia.org/wiki/%CE%94T_(timekeeping) & en.wikipedia.org/wiki/DUT1 Commented Jun 7 at 22:32
• Thank you. The GR corrections would be certainly interesting to consider: at this point, though, I'm just trying to reproduce/justify the naive Newtonian result as a limit of GR. (I think even considering an even semi realistic gravitational model of the Earth-moon system would make corrections to a Newtonian Foucault pendulum a massive headache - I doubt approximations could be to produce anything observable.) That's to say, what I am interested in is the classical idealized setting where a) the Earth is a perfect sphere and it rotates without bulging b) the pendulum has a small frequency Commented Jun 7 at 23:19
• But instead of the usual Newtonian analysis like you see in a book like Landau and Lifschitz, how can we explain the pendulum's behaviour via GR (i.e. using 4-force and 4-velocity) Commented Jun 7 at 23:22