The Poincaré Group and Geodesics in Minkowski Spacetime

[Demon117]

The poincare' group is the group of isometries of Minkowski spacetime, in a nutshell. In terms of an actual physical definition it is the group of all distance preserving maps between metric-spaces in Minkowski spacetime. What is the difference between this and geodesics?

 

[tom.stoer]

The poincare' group is the group of isometries of Minkowski spacetime, in a nutshell. In terms of an actual physical definition it is the group of all distance preserving maps between metric-spaces in Minkowski spacetime. What is the difference between this and geodesics?

A geodesic is a curve in an arbitrary curved Riemann manifold generalizing the "straight line" in flat space. A geodesic in a Riemann manifold is both the straightest curve and the shortest curve connecting two points A and B. Poincare symmetry is not a symmetry of arbitrary Riemann manifolds but a symmetry of flat Minkowski spacetime space only.

 

[dextercioby]

Poincare symmetry connects observers on different worldlines in a flat space-time, irrespective whether the worldlines are geodesics or not. One observer describes physics through one system of space-time coordinates x, another has x' for that. x and x' are linked through Poincare transformations. Observer's motion needn't be along a geodesic.

 

[Matterwave]

Geodesics...don't form a group, they are just curves in the space-time. I don't believe there is a natural group operation that would make geodesics into a group...

 

[robphy]

Maybe the OP is asking about characterizing the Poincare transformations as [determinant 1] symmetries that preserve the set of geodesics of Minkowski space.

 

[Demon117]

A geodesic is a curve in an arbitrary curved Riemann manifold generalizing the "straight line" in flat space. A geodesic in a Riemann manifold is both the straightest curve and the shortest curve connecting two points A and B. Poincare symmetry is not a symmetry of arbitrary Riemann manifolds but a symmetry of flat Minkowski spacetime space only.

Very nice. This does it.