Undergrad How Do Jacobi Fields Explain Tidal Forces in General Relativity?

[lavinia]

TL;DR

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]

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]

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?

Adding just a little more to Peter's answer,... what you know as a "Jacobi field" is known in GR as the "relative displacement vector field", or just "deviation vector field". The steps to derive the equation you mentioned (involving the Riemann tensor) are exactly what you'd expect: by assuming the tangent vector field along the geodesics and the deviation field to be independent -- meaning that the Lie bracket between them is zero -- one derives the "geodesic deviation equation", aka the"Jacobi equation".