General Relativity - Double Covariant Derivative

  • Thread starter Tangent87
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I know that for a scalar [tex]\nabla^2\phi=\nabla_a\nabla^a\phi=\nabla^a\nabla_a\phi[/tex]. However what is [tex]\nabla^2[/tex] for a tensor? For example, is [tex]\nabla^2T_a=\nabla_b\nabla^bT_a[/tex] or is it [tex]\nabla^2T_a=\nabla^b\nabla_bT_a[/tex]? Because I don't think they're the same thing.

Thanks.
 

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  • #2
dextercioby
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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.
 
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