[Physics] Rotations in space-time


In Landau's Classical Theory of Fields, one finds the statement:

Every rotation in the four-dimensional space can be resolved into six
rotations, in the planes $xy,zy,xz,tx,ty,tz$ (just as every rotation
in ordinary space can be resolved into three rotations in the planes

How can I prove this statement? Thanks.

Best Answer

Take a system of four mutually orthogonal vectors lined up with the four axis x,y,z,t in 4D space. Apply an arbitrary rotation to move the four axis to some other positions. They remain mutually orthogonal of course. We now need to show that we can rotate the vectors back to their starting position using rotations in the six planes. The inverse of these would then be the six rotations we require to match the original rotation.

Start with the vector that was originally aligned with the x-axis. We need to rotate it back there. First use a rotation in the xy plane to rotate it so that it moves into the xzt hyperplace. Then use a rotation in the xz plane so that it moves into the xt plane. Then use a rotation in the xt plane so that it lines up with the x axis as required.

The other three orthogonal vectors will have been rotated as well, but since they remain orthogonal to the first vector they are now in the yzt hyperplane orthogonal to the x axis. You can now apply use the three rotations in the remaining planes yz, zt and yt to line them up with their original axis using the solution for 3d space. So it's done.

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