(a) In a 4-space that is not Euclidean, the D’Alembertian is defined as Here gμv is the co...

(a) In a 4-space that is not Euclidean, the D’Alembertian is defined as

 

Here gμv is the contravariant metric tensor, which in the flat space of special relativity is indeed the same as gμv. For the metric tensor of trace +2 instead of −2 used in Eq. (7.33), find the explicit form of the D’Alembertian so defined.

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(b) A suitable Lagrangian for the charged scalar meson field in this metric is

 

Show that one of the corresponding field equations is

 

Show also that in light of part (a) this equation is actually identical with Eq. (13.99).