# Jacobi Fields and tidal forces

• I
Gold Member
Summary:
Particles in free fall in a gravitational field feel a tidal force that pulls them towards each other, How does once describe this force mathematically? It would seem that some feature of associated Jacobi fields should give the answer.
Given a one parameter family of geodesics, the variation vector field is a Jacobi field. Mathematically this means that the field, ##J##, satisfies the differential equation ## ∇_{V}∇_{V}J =- R(V,J,)V## where ##V## is the tangent vector field and ##R## is the curvature tensor and ##∇## is the covariant derivative operator.

Suppose the variation through geodesics is a one parameter family of particles in free fall in a gravitational field. One would think that the tidal drift could be expressed in terms of the Jacobi field ##J##. If true, how is this done mathematically and what is the physical reasoning?

PeterDonis
Mentor
2020 Award
Particles in free fall in a gravitational field feel a tidal force that pulls them towards each other

They don't feel a force; they are in free fall, so by definition they feel zero force. However, their freely falling worldlines will diverge or converge if there is a gravitational field present due to a massive object, i.e., there will be geodesic deviation.

How does once describe this force mathematically?

The Riemann curvature tensor describes all geodesic deviation.

It would seem that some feature of associated Jacobi fields should give the answer.

IIRC Misner, Thorne, and Wheeler discuss this. They discuss various ways of representing the curvature tensor.

strangerep