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bravelittlemu
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In general, what can one say about the relationship between geodesic motion of (massive and massless) particles and the killing vectors associated with the metric?
The relationship between killing vectors and geodesics is that killing vectors are vector fields that generate isometries on a manifold, while geodesics are the paths that represent the shortest distance between two points on a manifold. Killing vectors can be used to determine and preserve geodesic curves.
Killing vectors affect geodesics by preserving their shape and length. This means that when a killing vector is applied to a geodesic, the resulting curve will still represent the shortest path between two points on the manifold.
Yes, killing vectors can be used to find geodesics. In fact, the existence of a killing vector is a necessary condition for a geodesic to exist. By solving the equations of motion for the killing vector, one can determine the geodesic curves on the manifold.
The relationship between killing vectors and geodesics is important because it allows us to study and understand the geometry of a manifold. By using killing vectors, we can determine the isometries and symmetries of a manifold, and by studying the geodesic curves, we can gain insight into the curvature and topology of the manifold.
In physics, killing vectors and geodesics are used to study the behavior of particles and fields in curved spacetime. By understanding the relationship between killing vectors and geodesics, we can determine the trajectories of objects in a gravitational field, or the behavior of light rays in a black hole. This has important applications in general relativity and cosmology.