Question

From classical mechanics and general relativity, we know that the natural motion of a particle in general curvilinear coordinates, assuming the Levi-Civita connection, is given by the geodesic equation.

d^2 x^μ /dτ^2 = Γ^μ_ν^σ dx^ν/dτ dx^σ/dτ

Where x are the coordinates of the particle and τ is an affine parameter (usually proper time).

I have never seen it discussed, but is it possible to derive the usual non-inertial/fictitious forces in a rotating reference frame, just from say, replacing the derivatives in Newton's second law with a covariant derivative?

Answer 1

In rotating reference frames, the geodesic equation does not directly give the usual non-inertial forces. These forces need to be derived separately, and one way to do so is by replacing the derivatives in Newton's second law with a covariant derivative.

Step-by-step explanation:

From classical mechanics and general relativity, the natural motion of a particle in general curvilinear coordinates, assuming the Levi-Civita connection, is given by the geodesic equation:

d^2 x^μ /dτ^2 = Γ^μ_ν^σ dx^ν/dτ dx^σ/dτ

However, the geodesic equation does not directly give the usual non-inertial/fictitious forces in a rotating reference frame.

These forces, such as the Coriolis force, need to be derived separately. One way to do this is by replacing the derivatives in Newton's second law with a covariant derivative. This approach can help derive the fictitious forces in a rotating reference frame.