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Proof.
This follows from similar, and simpler arguments as above. First,
(3.249) d r 2 = 2 r d r = 2 y α η α + O ~ ( r 4 ) , dr^{2}=2rdr=2y_{\alpha}\eta_{\alpha}+\widetilde{O}(r^{4}),
so it follows that
(3.250) ∗ d r 2 = 2 y α η α ^ ∧ dvol T + O ~ ( r 3 ) *dr^{2}=2y_{\alpha}\eta_{\widehat{\alpha}}\wedge\dvol_{T}+\widetilde{O}(r^{3})
and
(3.251) d ∗ d r 2 = 6 dvol N ∧ dvol T + O ~ ( r 2 ) = 6 dvol g + O ~ ( r 2 ) . d*dr^{2}=6\dvol_{N}\wedge\dvol_{T}+\widetilde{O}(r^{2})=6\dvol_{g}+\widetilde{O}(r^{2}).
Hence
(3.252) Δ ( r 2 ) = d ∗ d r 2 = − ∗ d ∗ d r 2 = − 6 + O ~ ( r 2 ) . \Delta(r^{2})=d^{*}dr^{2}=-*d*dr^{2}=-6+\widetilde{O}(r^{2}).
∎