Model Problem(Kerr-Schild metric)

g_{μν} = η_{μν} + 2Hl_{μ}l_{ν}

where H is a scalar, and l^{μ}l_{μ} = 0 wrt η_{μν} and g_{μν}.

• Schwarzschild spacetime(ingoing Eddington-Finkelstein coordinates)

• Kerr

• Boosted Schwarzschild/Kerr

• 3+1 Decomposition of Kerr-Schild metric

α = 1/\sqrt{1 + 2Hl_{t}^{2}}
β_{i} = 2Hl_{t}l_{i}
g_{ij} = δ_{ij} + 2Hl_{i}l_{j}
K_{ij} = −\frac{1}{2α} (∂_{t}g_{ij} − D_{i}β_{j} − D_{j}β_{i}) .

• Algebraic condition on lapse derived from l^{μ}l_{μ} = 0 as α = \sqrt{1/ (1 + \sqrt{β_{x}^{2} + β_{y}^{2} + β_{z}^{2}})}

• For each case we specify H, l_{μ}, ∂_{μ}H and ∂_{μ}l_{ν} and use the resulting 3-metric and extrinsic curvature as initial data.

• β^{i} specified analytically for now

• α is specified algebraically in terms of the shift and 3-metric components.

Model Problems

• Schwarzschild spacetime

H = M/r
l_{t} = 1
l_{i} = x_{i}/r
r = \sqrt{x^{2} + y^{2} + z^{2}}

• Kerr

H = Mr^{3}/ (r^{4} + a^{2}z^{2})
l_{t} = 1
l_{x} = (rx + ay) / (r^{2} + a^{2})
l_{y} = (ry − ax) / (r^{2} + a^{2})
l_{z} = z/r

where r is now determined from

\frac{x^{2} + y^{2}}{r^{2} + a^{2}} + \frac{z^{2}}{r^{2}} = 1

• Boosted Schwarzschild

1. The Kerr-Schild spacetime metric is form-invariant under a boost.

l̄_{μ} → l_{μ} and H̄ → H. The spacetime metric remains of the Kerr-Schild form.

H = M/r
l_{t} = γ(l̄_{t} − vl̄_{z})
l_{x} = l̄_{x}
l_{y} = l̄_{y}
l_{z} = γ(l̄_{z} − vl̄_{t})

where r = \sqrt{x^{2} + y^{2} + γ^{2}(z − vt)^{2}} and

l̄_{t} = 1
l̄_{x} = x/r
l̄_{y} = y/r
l̄_{z} = γ(z − vt)/r