Free Falling Objects in GR: Initial Speed Matters

[majong]

In GR a object follows a geodesic when free falling as I understand.
A object near the sun for instance wil fall to the sun following that line.

If the initial speed is different the path will be different.
How can it be that the geadesic is depending on the initial speed of the object.

Martin

 

[tom.stoer]

A geodesic is a generalization of the notion of a 'straight line'; in Riemannian geometry as used in GR this is equivalent (locally) to the 'shortest line' between two given points(locally, b/c globally a geodesic need not be unique; an example is the S², i.e. the surface of the earth, where you have infinitly many geodesics connecting the north and the south pole).

Now for each point you have infinitly many directions, i.e. infinitly many geodesics.

In navigation you fix two points, e.g. "New York" and "London" and calculate the geodesic between these two cities. But in physics you use a different method, namely initial conditions specifying the initial point "New York" and the initial direction "eastward". Therefore you don't fix "London" but you let the initial conditions (position and idrection) plus the local e.o.m. (the geodesic equation) decide where the journey leads ...

 

[Chestermiller]

In GR a object follows a geodesic when free falling as I understand.
A object near the sun for instance wil fall to the sun following that line.

If the initial speed is different the path will be different.
How can it be that the geadesic is depending on the initial speed of the object.

Martin

In this context, the geodesics are in 4D space-time, and include the time dimension. Changing the initial speed is equivalent to changing the trajectory of the 4 velocity, which switches you to a different geodesic.