Geodesic Deviation: Difference between revisions
From Cibernética Americana
Jump to navigationJump to search
| Line 7: | Line 7: | ||
== Overview == | == Overview == | ||
In [[:en:differential | In [[:en:differential geometry|differential geometry]], the '''geodesic deviation equation''' is an equation involving the [[:en:Riemann curvature tensor|Riemann curvature tensor]], which measures the change in separation of neighbouring [[:en:geodesic]]s. In the language of mechanics it measures the rate of relative [[:en:acceleration]] of two particles moving forward on neighbouring geodesics. | ||
From the standpoint of counting indices alone a four-[[:en:Index (mathematics)|index]] [[:en:tensor|tensor]] is needed to measure the change, and that Riemann is the right one. There's a 4-vector velocity along one geodesic, which has to be folded into one index. There's an [[:en:infinitesimal]] separation vector between the two geodesics, which also eats up an index on Riemann. A third (free) index is needed to exit with the rate of change of the displacement. The fourth (summed) index is less obvious. It is assumed that the two geodesics are neighboring ones that start out parallel. So at first there is no relative velocity. But we have to use the velocity vector (4-velocity) along both neighbors, so it comes in twice and the last index is used. Note that it is not just the distance between the geodesics that can change, there can also be [[:en:torsion]] of the bundle and they may twist around each other. | From the standpoint of counting indices alone a four-[[:en:Index (mathematics)|index]] [[:en:tensor|tensor]] is needed to measure the change, and that Riemann is the right one. There's a 4-vector velocity along one geodesic, which has to be folded into one index. There's an [[:en:infinitesimal]] separation vector between the two geodesics, which also eats up an index on Riemann. A third (free) index is needed to exit with the rate of change of the displacement. The fourth (summed) index is less obvious. It is assumed that the two geodesics are neighboring ones that start out parallel. So at first there is no relative velocity. But we have to use the velocity vector (4-velocity) along both neighbors, so it comes in twice and the last index is used. Note that it is not just the distance between the geodesics that can change, there can also be [[:en:torsion]] of the bundle and they may twist around each other. | ||
