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Thanks.

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- Thread starter Tangent87
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- #1

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Thanks.

- #2

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The metric is a constant wrt to the covariant derivative. So the 'wave'/D'Alembert operator is defined by

[tex] \Box =: g^{ab} \nabla_a \nabla_b = \nabla^{b} \nabla_b = \nabla_a \nabla^{a} [/tex]

which is to be applied on any tensor object.

The contravariant derivative is

[tex] \nabla^{b} = g^{ba} \nabla_a [/tex]

so

[tex] \nabla^{b} T_{ac} = g^{bd} \nabla_d T_{ac} [/tex]

The last part you can compute using the normal rules for covariant differentiation.

[tex] \Box =: g^{ab} \nabla_a \nabla_b = \nabla^{b} \nabla_b = \nabla_a \nabla^{a} [/tex]

which is to be applied on any tensor object.

The contravariant derivative is

[tex] \nabla^{b} = g^{ba} \nabla_a [/tex]

so

[tex] \nabla^{b} T_{ac} = g^{bd} \nabla_d T_{ac} [/tex]

The last part you can compute using the normal rules for covariant differentiation.

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