Put

h_{ab} = δ g_{ab}

. Varying

g^{a c} g_{c b} = δ_{b}^{a}

gives

δ g^{ab} = - g^{a c} g^{b d} h_{c d} = - h^{ab} .

The determinant identity

δ lo g ∣ det g ∣ = g^{ab} h_{ab}

gives

δ (d vol_{g}) = \frac{1}{2} g^{ab} h_{ab} d vol_{g} .

δ Γ^{a}_{b c} = \frac{1}{2} g^{a d} (\nabla_{b} h_{c d} + \nabla_{c} h_{b d} - \nabla_{d} h_{b c}) .

These are the basic metric variation identities.

ii

Solution

Varying the Ricci tensor and contracting gives

δ R = - R^{ab} h_{ab} + \nabla_{a} (g^{c b} δ Γ^{a}_{c b} - g^{ab} δ Γ^{c}_{c b}) .

Substitution of part (i), followed by metric compatibility, reduces the divergence to

\nabla_{a} \nabla_{b} h^{ab} - \nabla^{a} \nabla_{a} h .

δ R = - R^{ab} δ g_{ab} - g^{ab} \nabla_{c} \nabla^{c} δ g_{ab} + \nabla^{a} \nabla^{b} δ g_{ab},

and consequently

α = - 1, β = 1 .

There are no discussions about this article yet.

There are currently no matching articles.