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 Incoming links: Bona--Masso slicing condition

Stationarity makes

\partial_{t} α = 0

, while

\partial_{r} α = \frac{M}{( r + M ) ^{2}} .

The left side of the Bona--Masso slicing condition is therefore

- β^{r} \partial_{r} α = - \frac{M ^{2} r}{( r + M ) ^{4}} .

Its right side is

- α^{2} f (α) K = - \frac{M r ^{2}}{( r + M ) ^{4}} f (α) .

Equality requires

f = \frac{M}{r} .

Since

α = r / (r + M)

f (α) = \frac{1 - α}{α} .

Consequently $lim_{α \to 1} f (α) = 0$ , as expected in the asymptotically flat region $r \to \infty$ .

For

λ = 0

- g = α γ

g^{00} = - α^{- 2}

, and

g^{0 i} = β^{i} α^{- 2}

\partial_{t} (- \frac{γ}{α}) + \partial_{i} (\frac{γ β ^{i}}{α}) = 0.

Using

\frac{\partial _{t} γ}{γ} = β^{i} \partial_{i} lo g γ + \partial_{i} β^{i} - α K

from the determinant evolution, all shift-divergence and volume terms cancel, leaving

(\partial_{t} - β^{i} \partial_{i}) α = - α^{2} K .

f (α) = 1 .