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.matumich26 said: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 said: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.
The Poincaré group is a mathematical concept that describes the symmetries of Minkowski spacetime, which is the spacetime used in special relativity. It is a ten-dimensional group that includes translations, rotations, and boosts (Lorentz transformations).
Minkowski spacetime is a mathematical model used to describe the four-dimensional spacetime of special relativity. It incorporates the three dimensions of space and one dimension of time, and is named after the mathematician Hermann Minkowski.
A geodesic in Minkowski spacetime is the shortest path between two points, taking into account the curvature of spacetime. In flat spacetime, geodesics are straight lines, but in curved spacetime, they can be curved paths.
The Poincaré group describes the symmetries of Minkowski spacetime, including the transformations that leave geodesics unchanged. This means that geodesics are invariant under the Poincaré group, and the group can be used to study and understand the properties of geodesics in Minkowski spacetime.
The Poincaré group is important because it is the mathematical framework for understanding the symmetries of spacetime in special relativity. It has applications in various areas of physics, including quantum mechanics, quantum field theory, and particle physics. In mathematics, it is also used to study the properties of geometric objects such as geodesics and spacetime curvature.