Local Isometries
Our spacetime is modeled by a Lorentzian manifold which has some remarkable symmetries. Namely, our spacetime admits a six dimensional Lie group of self-isometries. This group is generated by a six dimensional Lie algebra of Killing vector fields. A convenient basis consists of one null vector field,
three spacelike vector fields,
and two additional vector fields,
Here, generate the Euclidean group, acting within each planar wavefront, which justifies the name plane wave for this solution. Also show that all nontranverse directions are equivalent. This corresponds to the well-known fact that in flat spacetime, two colliding plane waves always collide head-on when represented in the appropriate Lorentz frame.
For future reference we note that this six dimensional group of self-isometries acts transitively, so that our spacetime is homogeneous. However, it is not isotropic, since the transverse directions are distinguished from the non-transverse ones.
Read more about this topic: Monochromatic Electromagnetic Plane Wave
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