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I've been playing around a bit with the Kerr orbit program I wrote, and have been thinking about ways to set the initial conditions. One thing I'd like to be able to do is specify a launch from some event in terms that would be convenient for an observer at that event with some given four-velocity - but the orbit program works with global coordinates (Boyer-Lindquist in this case, although I don't think that's important). I think I've worked out (in painful detail below) how to transform between local coordinates and global, but would appreciate a sanity check (apologies if this better belongs in the homework section).
The metric at the launch point is ##g_{ab}## specified in some global coordinate system, and the launching observer has four-velocity ##U^a##, specified in that same system. What I'm looking for is the transform from this system to local Minkowski coordinates with time axis parallel to ##U^a##. I'm going to work with the metric with upper indices so I can apply the results to four-velocities without further mucking around.
Diagonalising the inverse metric with a matrix of eigenvectors, ##\mathbf E## is textbook. The result, ##D^{a'b'}=E^{a'}{}_{a}E^{b'}{}_bg^{ab}##, has the eigenvalues of ##g^{ab}## on the diagonal (##D^{ii}=\lambda^{(i)}## where no summation is assumed). Presumably one will have the opposite sign to the other three. I construct another matrix, ##S^{a''}{}_{a'}##, which is diagonal with ##S^i{}_i=|\lambda^{(i)}|^{-1/2}## (where, again, no summation is assumed), and then ##S^{a''}{}_{a'}E^{a'}{}_{a}S^{b''}{}_{b'}E^{b'}{}_bg^{ab}## should be a Minkowski metric.
Now I can apply this transform to ##U^a## to get the launching observer's four velocity in this Minkowski frame. I can then find a Lorentz transform, ##\Lambda^{a'''}{}_{a''}##, so that ##\Lambda^{a'''}{}_{a''}S^{a''}{}_{a'}E^{a'}{}_{a}U^a## has no spatial components. I may also wish to apply this composite transform to vectors parallel to the global spatial coordinate axes and append a rotation, ##R^{a''''}{}_{a'''}##, to enforce a simple relationship between the global and local spatial coordinates - that their spatial projections are parallel, for example.
Then I have a Jacobean ##J^{a''''}{}_{a}=R^{a''''}{}_{a'''}\Lambda^{a'''}{}_{a''}S^{a''}{}_{a'}E^{a'}{}_{a}## that transforms global coordinates to local ones, and whose inverse takes local coordinates to global ones.
Right? Or am I over-complicating things?
The metric at the launch point is ##g_{ab}## specified in some global coordinate system, and the launching observer has four-velocity ##U^a##, specified in that same system. What I'm looking for is the transform from this system to local Minkowski coordinates with time axis parallel to ##U^a##. I'm going to work with the metric with upper indices so I can apply the results to four-velocities without further mucking around.
Diagonalising the inverse metric with a matrix of eigenvectors, ##\mathbf E## is textbook. The result, ##D^{a'b'}=E^{a'}{}_{a}E^{b'}{}_bg^{ab}##, has the eigenvalues of ##g^{ab}## on the diagonal (##D^{ii}=\lambda^{(i)}## where no summation is assumed). Presumably one will have the opposite sign to the other three. I construct another matrix, ##S^{a''}{}_{a'}##, which is diagonal with ##S^i{}_i=|\lambda^{(i)}|^{-1/2}## (where, again, no summation is assumed), and then ##S^{a''}{}_{a'}E^{a'}{}_{a}S^{b''}{}_{b'}E^{b'}{}_bg^{ab}## should be a Minkowski metric.
Now I can apply this transform to ##U^a## to get the launching observer's four velocity in this Minkowski frame. I can then find a Lorentz transform, ##\Lambda^{a'''}{}_{a''}##, so that ##\Lambda^{a'''}{}_{a''}S^{a''}{}_{a'}E^{a'}{}_{a}U^a## has no spatial components. I may also wish to apply this composite transform to vectors parallel to the global spatial coordinate axes and append a rotation, ##R^{a''''}{}_{a'''}##, to enforce a simple relationship between the global and local spatial coordinates - that their spatial projections are parallel, for example.
Then I have a Jacobean ##J^{a''''}{}_{a}=R^{a''''}{}_{a'''}\Lambda^{a'''}{}_{a''}S^{a''}{}_{a'}E^{a'}{}_{a}## that transforms global coordinates to local ones, and whose inverse takes local coordinates to global ones.
Right? Or am I over-complicating things?