Are there alternative expressions for the d'Alembertian?
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I know that the Christoffel symbols satisfy:
Γaab=1−g−−−√∂b(−g−−−√),Γaba=1−g∂b(−g),
where gg is the usual metric determinant. From this we can get an expression for the divergence of a vector VaVa:
∇aVa=1−g−−−√∂a(−g−−−√Va).∇aVa=1−g∂a(−gVa).
My question is: can get a similar expression for □Vb◻Vb? I know of the usual □≡gab∇a∇b◻≡gab∇a∇b, but I wanted to know if there was an expression like the two above? I can't seem to find anything on this, though I also don't know the best keywords to search for. Any help would be appreciated.
I've seen an expression from another Physics StackExchange answer here, but it was only applicable for an antisymmetric tensor (like the electromagnetic tensor FabFab).
Γaab=1−g−−−√∂b(−g−−−√),Γaba=1−g∂b(−g),
where gg is the usual metric determinant. From this we can get an expression for the divergence of a vector VaVa:
∇aVa=1−g−−−√∂a(−g−−−√Va).∇aVa=1−g∂a(−gVa).
My question is: can get a similar expression for □Vb◻Vb? I know of the usual □≡gab∇a∇b◻≡gab∇a∇b, but I wanted to know if there was an expression like the two above? I can't seem to find anything on this, though I also don't know the best keywords to search for. Any help would be appreciated.
I've seen an expression from another Physics StackExchange answer here, but it was only applicable for an antisymmetric tensor (like the electromagnetic tensor FabFab).
max20:
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Explanation:
Solutions of all completely factorable equations form the linear space Ak ae K of d'Alembertian elements. The order of minimal operator over k which annihilates
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