- #1
cesiumfrog
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"A manifold (with a metric tensor) is said to be spherically symmetric iff the Lie algebra of its Killing vector fields has a sub-algebra that is the Lie algebra of SO(3)." Why?
The statement is paraphrased from texts such as Schutz or D'Inverno, where it is always expressed like a definition with no explanation.. except to note that this definition is sufficient to calculate the general form of a spherically symmetric metric/line-element (rather than just writing one down intuitively), which then might be used (in combined with the vacuum-EFE) to obtain Schwarzschild's solution. Could anyone point me to an explanation or proof for the statement?
It does make sense that there should be some relationship between the space-time symmetries and the familiar rotation group, but why do we know the relationship to have this specific form?
The statement is paraphrased from texts such as Schutz or D'Inverno, where it is always expressed like a definition with no explanation.. except to note that this definition is sufficient to calculate the general form of a spherically symmetric metric/line-element (rather than just writing one down intuitively), which then might be used (in combined with the vacuum-EFE) to obtain Schwarzschild's solution. Could anyone point me to an explanation or proof for the statement?
It does make sense that there should be some relationship between the space-time symmetries and the familiar rotation group, but why do we know the relationship to have this specific form?