Geodesics in Minkowski Spacetime

[Oxymoron]

What are Jacobi Fields and how can I better understand geodesics in Minkowski spacetime by knowing what they are?

 

[Oxymoron]

I'm kind of familiar with one-parameter groups of maps (or flows) and how associated with any such flow we can define a coordinate independent vector field (sometimes called a "velocity" field) by differentiation. One may then go in reverse and say that for any vector (velocity) field we can associate a flow by solving a differential equation with initial conditions. Solutions to the D.E. yield integral curves.

Now, the very first thing I do not understand about Jacobi Fields is that they require two-parameter groups of diffeomorphisms instead of one.

Apparently two-parameter diffeomorphisms keep track of "geodesic separation". So how to two-parameter diffeomorphisms work and how are they related to Jacobi Fields?

 

[Tantoblin]

As you pointed out, Jacobi fields measure the separation of geodesics. Suppose you have a bunch of geodesics $$\gamma_\tau$$, where $$\tau$$ is a parameter labelling the individual geodesics. So, we're talking about two-parameter families of curves, and not of diffeomorphisms!

Each of these geodesics is a curve, with its own parametrization (say, arclength), so the family looks like $$\gamma_\tau(s)$$, or with a slight change in notation, $$\gamma(\tau, s)$$, which is probably the two-parameter family you mentioned. If you keep $$\tau$$ fixed and allow $$s$$ to vary, you move along a fixed geodesic. If, on the other hand, you vary $$\tau$$, you hop from one geodesic to the next.

From this point of view, the Jacobi field is just the vector field (along a curve) which you obtain by taking the derivative with respect to $$\tau$$. It's precisely the "velocity field" of the curves $$\gamma(\tau, s)$$ where $$s$$ is kept fixed.

So, I hope you see why the Jacobi field, being related to families of geodesics, involves two parameters: one to label the members of the family, and one to move along each individual member. Bearing that in mind, the Jacobi field is just the velocity field associated with the "family" parameter, and hence measures rate of change in the family.

Now for the interpretation: if you have a bunch of initially parallel geodesics, the Jacobi field gives some measure of their separation at later times. In Minkowski space, geodesics are straight lines and remain parallel, so a priori, the Jacobi field is zero. On the other hand, there is a differential equation for the Jacobi field. Of course, solving this equation will also show us that the Jacobi field is zero, without having to construct a set of geodesics first.

The differential equation for the Jacobi field involves the Riemann curvature tensor, and a very rough interpretation says that on manifolds of negative curvature, initially parallel geodesics will tend to diverge, whereas on manifolds of positive curvature, initially parallel geodesics will oscillate around each other.