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Killing Vectors

From Misner Thorne & Wheeler "Gravitation" part on Killing vectors.

Let $Z_K$ = a Killing vector of a (coordinate?) basis metric.
$Z_K = \frac{\partial}{\partial x^K}$

So $Z^u = \delta^U_K$

Let
$\nabla_v Z_u = g_{ua} \nabla_v Z^a = g_{ua} (e_v(Z^a) + {\Gamma^a}_{vb} Z^b) = g_{ua} e_v(Z^a) + g_{ua} {\Gamma^a}_{vb} Z^b = g_{ua} {\Gamma^a}_{vb} \delta^b_K = \Gamma_{uvK} = \frac{1}{2} (e_K(g_{uv}) + e_v(g_{uK}) - e_u(g_{Kv})) $
TODO explain this part:
$= \frac{1}{2} (e_v(g_{uK}) - e_u(g_{Kv}))$

Killing vectors are eigenvectors to the equation:
$\square Z^u = \lambda Z^u$
$\nabla_v \nabla^v Z^u + {R^u}_v Z^v = \lambda Z^u$
...for $\lambda = 0$

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Useful hypersurface projection equations:
(From my "Differential Geometry" notes, esp the projection and ADM worksheets.)

Let $n_u$ = the hypersurface normal such that $n_u n^u = -1$
Let $\gamma_{ab} = g_{ab} + n_a n_b$ = the hypersurface projection transform and the hypersurface metric tensor
Let $a_u = n^v \nabla_v n_u$
Notice $n^u a_u = 0$
Notice $\gamma_{uv} a^v = a_u$
Notice $\gamma_{uv} n^v = 0$
So $a_u$ is spatial (within the hypersurface), $n_u$ is perpendicular to the hypersurface.
Let $K_{uv} = -{\gamma_u}^a {\gamma_v}^b \nabla_b n_a$
So $K_{uv} = -\nabla_u n_v - n_u a_v$
So $\nabla_v n_u = -K_{vu} - n_v a_u$
So $K_{uv} n^v = 0$ and $n^u K_{uv} = 0$ simply by virtue of the definition being a projection.
And? $K_{uv} = K_{vu}$? This is true if we are using a coordinate basis (which I think I read is a requirement for the ADM decomposition and extrinsic curvature definition.

Normal projection of covariant derivative of spatial vector:
$n^b n_a \nabla_b v^a = n^b \nabla_b (n_a v^a) - n^b v^a \nabla_b n_a$
...using $n_a v^a = 0$ and $n^b \nabla_b n_a = a_a$...
$= -v^a a_a$
Notice that $a_a$ is a spatial vector as well, so the result is a product of spatial vectors.

How about $-\nabla^\perp_b n^a$?
$= -\perp \nabla_b n^a$
$= -\perp \nabla_b (g^{ac} n_c)$
$= -\perp (g^{ac} \nabla_b n_c)$
$= -{\gamma_u}^a {\gamma_b}^v (g^{uc} \nabla_v n_c)$
$= -\gamma^{ca} {\gamma_b}^v \nabla_v n_c$
$= {K^a}_b$

Vector divergence: $\nabla_a v^a = \nabla^\perp_a (v^\perp)^a - a_a (v^\perp)^a - n^a \nabla_a v^\top - K v^\top $

Lie derivative along normal vs time coordinate:
$\mathcal{L}_\vec{n} \Theta = n^u \partial_u \Theta = n^u \nabla_u \Theta$
$\mathcal{L}_\vec{n} = \frac{1}{\alpha} (\partial_t - \mathcal{L}_\vec\beta)$

Lie derivative of a scalar:
$\mathcal{L}_\vec{n} s = n^u \partial_u s = n^u \nabla_u s$

Lie derivative of a vector:
$\mathcal{L}_\vec{n} v^a = n^u e_u ( v^a ) - v^u e_u ( n^a )$
$= n^u \nabla_u v^a - v^u \nabla_u n^a$

Lie derivative of a scalar along the hypersurface normal:
$n^u \nabla_u \Theta$
$= \mathcal{L}_\vec{n} \Theta$
$= \frac{1}{\alpha} (\partial_t \Theta - \beta^i \partial_i \Theta)$

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Z4 Killing Vector decomposition and projections:

Let $Z_u$ be our Killing vector
Let $Z^\perp_u = {\gamma_u}^a Z_a$ be the spatial projection of our Killing vector.
Let $Z^\top = \Theta = -n_u Z^u$

Lie derivative of Killing vector along hypersurface normal:
$\mathcal{L}_\vec{n} Z^a$
$= n^u \nabla_u Z^a - Z^u \nabla_u n^a$
$= n^u \nabla_u ((Z^\perp)^a + n^a \Theta) - ((Z^\perp)^u + n^u \Theta) \nabla_u n^a$
$= n^u \nabla_u (Z^\perp)^a - (Z^\perp)^u \nabla_u n^a + \Theta n^u \nabla_u n^a + n^a n^u \nabla_u \Theta - \Theta n^u \nabla_u n^a $
...notice the n component of Z cancels from the Lie derivative along n.
$ = n^u \nabla_u (Z^\perp)^a - (Z^\perp)^u \nabla_u n^a + n^a n^u \nabla_u \Theta $
$ = \mathcal{L}_\vec{n} (Z^\perp)^a + n^a \mathcal{L}_\vec{n} \Theta$
...compare to expanding Z from the beginning:
$= \mathcal{L}_\vec{n} ((Z^\perp)^a + n^a \Theta)$
...and that looks like some linearity rule of the Lie derivative.
$ = n^u \nabla_u (Z^\perp)^a - (Z^\perp)^u (-{K_u}^a - n_u a^a) + n^a n^u \nabla_u \Theta $
$ = n^u \nabla_u (Z^\perp)^a + (Z^\perp)^u {K_u}^a + n^a n^u \nabla_u \Theta $

same but as one-form:
$\mathcal{L}_\vec{n} (Z^\perp)_a = n^u \nabla_u (Z^\perp)_a + (Z^\perp)_u \nabla_a n^u$
$= n^u \nabla_u (Z^\perp)_a + (Z^\perp)_u (-{K_a}^u - n_a a^u)$
$= n^u \nabla_u (Z^\perp)_a - (Z^\perp)_u {K_a}^u - n_a (Z^\perp)_u a^u$

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Different Literature Z4 Formalisms:

2003 Bona, Ledvinka, Palenzuela, Zacek, eqn. 11:
$R_{uv} + 2 \nabla_{(u} Z_{v)} = 8 \pi (T_{uv} - \frac{1}{2} g_{uv} T_{ab} g^{ab})$ (eqn 11)
$n_u = [\alpha, 0]$ (implied from eqn 13?) = past-pointing normal.
$\Theta = n_u Z^u = \alpha Z^0$ (eqn 13) = normal component for past-pointing normal.

2005 Gundlach et al:
$R_{uv} + 2 \nabla_{(u} Z_{v)} - \kappa_1 (2 n_{(u} Z_{v)} - (1 + \kappa_2) g_{uv} n_a Z^a) = 8 \pi (T_{uv} - \frac{1}{2} g_{uv} T_{ab} g^{ab})$ (eqn 2)
$G_{uv} + 2 \nabla_{(u} Z_{v)} - g_{uv} \nabla_a Z^a - \kappa_1 (2 n_{(u} Z_{v)} + \kappa_2 g_{uv} n_a Z^a) = 8 \pi T_{uv}$ (eqn 3)
$\kappa_1 \ge 0$ (after eqn. 3)
$n_u = [-\alpha, 0]$ (implied from what's after eqn. 4) = future-pointing normal.
$\Theta = -n_a Z^a = \alpha Z^0$ (after eqn. 4) = normal component of future-pointing normal.
"(Note that $n^a$ as defined in [7] is past-pointing, while ours is future-pointing, so that the two definitions of $\Theta$ are the same.)", where [7] is 2003 Bona et al.
... which is "past-pointing" and which is "future-pointing"? $\nabla_u t$ vs. $-\nabla_u t$? Other sources say $n_u = [-\alpha, 0]$ is future-pointing, which makes sense with what I see.

2008 Alcubierre "Introduction to 3+1 Numerical Relativity":
$R_{uv} + 2 \nabla_{(u} Z_{v)} = 8 \pi (T_{uv} - \frac{1}{2} g_{uv} T_{ab} g^{ab})$ (eqn 5.8.7)
$n_u = [-\alpha, 0]$ (eqn. 2.2.8) = future-pointing normal.
$j^i = -\gamma^{iu} n^v T_{uv}$ (eqn 2.4.12) = spatial momentum, sign consistent with future-pointing normal.
Z4 section of the hyperbolic formalism chapter:
$\Theta = n_u Z^u = \alpha Z^0$ (eqn. 5.8.15) = normal component of past-pointing normal.
Ok how do I resolve these two definitions of $n_u$? Did Alcubierre just copy 5.8.15 from earlier Z4 papers by Bona et al, therefore using a different convention of $n_u$ than his own? Is he implying that $Z^u$ contians a negative $Z^0$, so $n_u Z^u = (-\alpha) \cdot (-Z^0) + 0 \cdot 0 ... = \alpha Z^0$? I think the equation was just copied from the Z4 papers without thought of how the convention was different from the rest of the book.

2009 Bona et al "Elements of Numerical Relativity":
$R_{uv} + 2 \nabla_{(u} Z_{v)} = 8 \pi (T_{uv} - \frac{1}{2} g_{uv} T_{ab} g^{ab})$ (eqn 3.69)
$G_{uv} + 2 \nabla_{(u} Z_{v)} - g_{uv} \nabla_a Z^a + \kappa_1 (2 n_{(u} Z_{v)} + \kappa_2 g_{uv} n_a Z^a) = 8 \pi T_{uv}$ (eqn. 3.79) ... this equation is citing 2005 Gundlach, though the $\kappa_1$ sign is flipped, maybe due to the normal sign being flipped as well?
$\kappa_1 > 0; \kappa_2 > -1$
$n_u = [\alpha, 0]$ (eqn 2.26) = past-pointing normal.
$\Theta = n_u Z^u = \alpha Z^0$ (eqn 3.84) = normal component of past-pointing normal.
$S_i = n_u {T^u}_i$ (eqn 2.36) = spatial momentum, sign consistent with past-pointing normal.

2010 Baumgarte & Shapiro "Numerical Relativity: Solving Einstein's Equations on the Computer"
$n_u = [-\alpha, 0]$ (eqn. 2.117) = future-pointing normal.

2011 Cao, Hilditch seems to match 2005 Gundlach et al but with stress-energy rhs added:
$G_{uv} + 2 \nabla_{(u} Z_{v)} - g_{uv} \nabla_a Z^a - \kappa_1 (2 n_{(u} Z_{v)} + \kappa_2 g_{uv} n_a Z^a) = 8 \pi T_{uv}$ (eqn 1)
$\Theta = -n_a Z^a$ (after eqn. 1)
The paper doesn't say, is $n_a = [-\alpha, 0]$ or $n_a = [\alpha, 0]$? By $\Theta$'s definition I'm guessing it is future-pointing, $n_a = [-\alpha, 0]$.
Cites 2005 Gundlach et al as its source.

2012 Alic et al:
$R_{uv} + 2 \nabla_{(u} Z_{v)} + \kappa_1 (2 n_{(u} Z_{v)} - (1 + \kappa_2) g_{uv} n_a Z^a) = 8 \pi (T_{uv} - \frac{1}{2} g_{uv} T_{ab} g^{ab})$ (eqn 2)
$\kappa_1 > 0, \kappa_2 > -1$
$n_u = [\alpha, 0]$ (implied from what's after eqn. 9) = past-pointing normal.
$\Theta = n_a Z^a = \alpha Z^0$ (after eqn. 9) = normal component of past-pointing normal.
$S_i = n_u {T^u}_i$ (after eqn. 9) = spatial momentum, sign consistent with past-pointing normal.
Cites 2005 Gundlach et al as its source.
The trace-reversal of 2011 Cao matches this except that the $\kappa_1$ sign is flipped.

Since 2011 Cao et al uses future-pointing normals, like I do in my worksheets, I'll go with that equation.
$G_{uv} + 2 \nabla_{(u} Z_{v)} - g_{uv} \nabla_a Z^a - \kappa_1 (2 n_{(u} Z_{v)} + \kappa_2 g_{uv} n_a Z^a) = 8 \pi T_{uv}$

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Change to the Einstein Field Equations

Einstein field equations:
$G_{uv} = 8 \pi T_{uv}$

Z4 field equations:
TODO a comment on Killing vectors or on the $k_1, k_2$ constrants would be nice.
$ G_{uv} + 2 \nabla_{(u} Z_{v)} - g_{uv} \nabla_a Z^a - \kappa_1 ( 2 n_{(u} Z_{v)} + \kappa_2 g_{uv} n_a Z^a ) = 8 \pi T_{uv} $

...in normal and spatial components:
$ G_{uv} + 2 \nabla_{(u} Z_{v)} - g_{uv} \nabla_a Z^a - \kappa_1 ( 2 n_{(u} Z_{v)} + \kappa_2 g_{uv} n_a Z^a ) = 8 \pi T_{uv} $
Using $\nabla_u Z_v = \nabla^\perp_u (Z^\perp)_v - n_u n^a \nabla_a (Z^\perp)_v - K_{ua} (Z^\perp)^a n_v - (K_{uv} + n_u a_v) \Theta + n_v \nabla^\perp_u \Theta - n_u n_v n^a \nabla_a \Theta $
Using $\nabla_a Z^a = \nabla^\perp_a (Z^\perp)^a + a_a (Z^\perp)^a + n^a \nabla_a \Theta - K \Theta $
Using $Z_u = (Z^\perp)_u + n_u \Theta$
$ G_{uv} + (\nabla_u Z_v = \nabla^\perp_u (Z^\perp)_v - n_u n^a \nabla_a (Z^\perp)_v - K_{ua} (Z^\perp)^a n_v - (K_{uv} + n_u a_v) \Theta + n_v \nabla^\perp_u \Theta - n_u n_v n^a \nabla_a \Theta ) + (\nabla_v Z_u = \nabla^\perp_v (Z^\perp)_u - n_v n^a \nabla_a (Z^\perp)_u - K_{va} (Z^\perp)^a n_u - (K_{vu} + n_v a_u) \Theta + n_u \nabla^\perp_v \Theta - n_v n_u n^a \nabla_a \Theta ) - (\gamma_{uv} - n_u n_v) ( \nabla_a Z^a = \nabla^\perp_a (Z^\perp)^a + a_a (Z^\perp)^a + n^a \nabla_a \Theta - K \Theta ) - \kappa_1 ( n_u (Z_v = (Z^\perp)_v + n_v \Theta) + n_v (Z_u = (Z^\perp)_u + n_u \Theta) + \kappa_2 (\gamma_{uv} - n_u n_v) n^a ( Z_a = (Z^\perp)_a + n_a \Theta ) ) = 8 \pi T_{uv} $
using $T_{uv} = S_{uv} + n_v j_u + n_u j_v + n_u n_v \rho$
$ G_{uv} + \nabla^\perp_u (Z^\perp)_v + \nabla^\perp_v (Z^\perp)_u + \gamma_{uv} ( - \nabla^\perp_a (Z^\perp)^a - a_a (Z^\perp)^a - n^a \nabla_a \Theta + K \Theta + \kappa_1 \kappa_2 \Theta ) - 2 K_{uv} \Theta + n_u ( - \kappa_1 (Z^\perp)_v - K_{va} (Z^\perp)^a - n^a \nabla_a (Z^\perp)_v - a_v \Theta + \nabla^\perp_v \Theta ) + n_v ( - \kappa_1 (Z^\perp)_u - K_{ua} (Z^\perp)^a - n^a \nabla_a (Z^\perp)_u - a_u \Theta + \nabla^\perp_u \Theta ) + n_u n_v ( \nabla^\perp_a (Z^\perp)^a + a_a (Z^\perp)^a - n^a \nabla_a \Theta - K \Theta - \kappa_1 (\kappa_2 + 2) \Theta ) = 8 \pi ( S_{uv} + n_v j_u + n_u j_v + n_u n_v \rho ) $

Trace of Z4:
$g^{uv} (G_{uv} + 2 \nabla_{(u} Z_{v)} - g_{uv} \nabla_a Z^a - \kappa_1 (2 n_{(u} Z_{v)} + \kappa_2 g_{uv} n_a Z^a)) = 8 \pi g^{uv} T_{uv}$
$ g^{uv} (R_{uv} - \frac{1}{2} g_{uv} R) + 2 g^{uv} \nabla_u Z_v - g^{uv} g_{uv} \nabla_a Z^a - \kappa_1 (2 g^{uv} n_u Z_v + \kappa_2 g^{uv} g_{uv} n_a Z^a) = 8 \pi T$
$ - R - 2 \nabla_a Z^a - 2 \kappa_1 (1 + 2 \kappa_2) n_a Z^a = 8 \pi T$

Trace-reversal of Z4:
$G_{uv} + 2 \nabla_{(u} Z_{v)} - g_{uv} \nabla_a Z^a - \kappa_1 (2 n_{(u} Z_{v)} + \kappa_2 g_{uv} n_a Z^a) - \frac{1}{2} g_{uv} ( - R - 2 \nabla_a Z^a - 2 \kappa_1 (1 + 2 \kappa_2) n_a Z^a ) = 8 \pi T_{uv} - \frac{1}{2} g_{uv} ( 8 \pi T ) $
$R_{uv} + 2 \nabla_{(u} Z_{v)} - \kappa_1 ( 2 n_{(u} Z_{v)} - (1 + \kappa_2) g_{uv} n_a Z^a ) = 8 \pi ( T_{uv} - \frac{1}{2} g_{uv} T_{ab} g^{ab} ) $
(Note: Between 2011 Cao et al and 2012 Alic et al, the only difference in the equations is the sign next to $\kappa_1$, which is equivalently seen as the sign associated with $n_u$, which fits perfectly with the fact that 2011 Cao uses future-pointing normals while 2012 Alic uses past-pointing normals.)

Separate trace-reversal of Z4 field equations into normal and hypersurface components:
$R_{uv} + \nabla^\perp_u (Z^\perp)_v - n_u n^a \nabla_a (Z^\perp)_v - K_{ua} (Z^\perp)^a n_v - n_u a_v \Theta + n_v \nabla^\perp_u \Theta + \nabla^\perp_v (Z^\perp)_u - n_v n^a \nabla_a (Z^\perp)_u - K_{va} (Z^\perp)^a n_u - n_u a_v \Theta + n_u \nabla^\perp_v \Theta - 2 K_{uv} \Theta - 2 n_u n_v n^c \nabla_c \Theta - \kappa_1 ( n_u ((Z^\perp)_v + n_v \Theta) + n_v ((Z^\perp)_u + n_u \Theta) + (1 + \kappa_2) g_{uv} \Theta ) = 8 \pi ( T_{uv} - \frac{1}{2} g_{uv} T_{ab} g^{ab} ) $

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Derivation of Constraints and Evolution Equations:

Contract normals:
$ n^u n^v ( G_{uv} + \nabla^\perp_u (Z^\perp)_v + \nabla^\perp_v (Z^\perp)_u + \gamma_{uv} ( - \nabla^\perp_a (Z^\perp)^a - a_a (Z^\perp)^a - n^a \nabla_a \Theta + K \Theta + \kappa_1 \kappa_2 \Theta ) - 2 K_{uv} \Theta + n_u ( - \kappa_1 (Z^\perp)_v - K_{va} (Z^\perp)^a - n^a \nabla_a (Z^\perp)_v - a_v \Theta + \nabla^\perp_v \Theta ) + n_v ( - \kappa_1 (Z^\perp)_u - K_{ua} (Z^\perp)^a - n^a \nabla_a (Z^\perp)_u - a_u \Theta + \nabla^\perp_u \Theta ) + n_u n_v ( \nabla^\perp_a (Z^\perp)^a + a_a (Z^\perp)^a - n^a \nabla_a \Theta - K \Theta - \kappa_1 (\kappa_2 + 2) \Theta ) ) = n^u n^v 8 \pi ( S_{uv} + n_v j_u + n_u j_v + n_u n_v \rho ) $
$ n^u n^v G_{uv} + \nabla^\perp_a (Z^\perp)^a + a_a (Z^\perp)^a - n^a \nabla_a \Theta - (K + \kappa_1 (\kappa_2 + 2)) \Theta = 8 \pi \rho $
using $n^u n^v G_{uv} = \frac{1}{2} R^\perp + \frac{1}{2} K^2 - \frac{1}{2} K^{ab} K_{ab}$
$ \frac{1}{2} R^\perp + \frac{1}{2} K^2 - \frac{1}{2} K^{ab} K_{ab} + \nabla^\perp_a (Z^\perp)^a + a_a (Z^\perp)^a - n^a \nabla_a \Theta - (K + \kappa_1 (\kappa_2 + 2)) \Theta = 8 \pi \rho $
Using the Einstein field equations' Hamiltonian constraint / normal contraction of the Einstein field equations:
$H = \frac{1}{2} (R^\perp + K^2 - K_{ab} K^{ab}) - 8 \pi \rho$
So our normal-contracted Z4 field equations become
$ n^u \nabla_u \Theta = H + \nabla^\perp_a (Z^\perp)^a + a_a (Z^\perp)^a - (K + \kappa_1 (\kappa_2 + 2)) \Theta $
using $n^u \nabla_u \Theta = \frac{1}{\alpha} (\partial_t \Theta - \beta^i \partial_i \Theta)$
$ \frac{1}{\alpha} (\partial_t \Theta - \beta^i \partial_i \Theta) = H + \nabla^\perp_a (Z^\perp)^a + a_a (Z^\perp)^a - (K + \kappa_1 (\kappa_2 + 2)) \Theta $
$ \partial_t \Theta = \alpha ( H + \nabla^\perp_a (Z^\perp)^a + a_a (Z^\perp)^a - (K + \kappa_1 (\kappa_2 + 2)) \Theta ) + \beta^i \partial_i \Theta $

How about trying the normal-contraction of the trace-reversal of this?
$ n^u n^v ( R_{uv} + \nabla^\perp_u (Z^\perp)_v - n_u n^a \nabla_a (Z^\perp)_v - K_{ua} (Z^\perp)^a n_v - n_u a_v \Theta + n_v \nabla^\perp_u \Theta + \nabla^\perp_v (Z^\perp)_u - n_v n^a \nabla_a (Z^\perp)_u - K_{va} (Z^\perp)^a n_u - n_u a_v \Theta + n_u \nabla^\perp_v \Theta - 2 K_{uv} \Theta - 2 n_u n_v n^c \nabla_c \Theta - \kappa_1 ( n_u ((Z^\perp)_v + n_v \Theta) + n_v ((Z^\perp)_u + n_u \Theta) + (1 + \kappa_2) g_{uv} \Theta ) ) = 8 \pi n^u n^v ( T_{uv} - \frac{1}{2} g_{uv} T_{ab} g^{ab} ) $
$ n^u n^v R_{uv} + 2 n^u n^a \nabla_a (Z^\perp)_u - 2 n^c \nabla_c \Theta + \kappa_1 (\kappa_2 - 1) \Theta = 4 \pi (S + \rho) $
using $n^a n^b \nabla_a v_b = -(v^\perp)^a a_a - n^a \nabla_a v^\top$
$ n^u n^v R_{uv} - 2 (Z^\perp)^a a_a - 4 n^a \nabla_a \Theta + \kappa_1 (\kappa_2 - 1) \Theta = 4 \pi (S + \rho) $
Using $n^a n^b R_{ab} = - K_{ab} K^{ab} + K^2$
$ K^2 - K_{uv} K^{uv} - 2 (Z^\perp)^a a_a - 4 n^a \nabla_a \Theta + \kappa_1 (\kappa_2 - 1) \Theta = 4 \pi (S + \rho) $
... would this Theta ivp be useful too?

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Contract and project:
$ n^u {\gamma_b}^v ( G_{uv} + \nabla^\perp_u (Z^\perp)_v + \nabla^\perp_v (Z^\perp)_u + \gamma_{uv} ( - \nabla^\perp_a (Z^\perp)^a - a_a (Z^\perp)^a - n^a \nabla_a \Theta + K \Theta + \kappa_1 \kappa_2 \Theta ) - 2 K_{uv} \Theta + n_u ( - \kappa_1 (Z^\perp)_v - K_{va} (Z^\perp)^a - n^a \nabla_a (Z^\perp)_v - a_v \Theta + \nabla^\perp_v \Theta ) + n_v ( - \kappa_1 (Z^\perp)_u - K_{ua} (Z^\perp)^a - n^a \nabla_a (Z^\perp)_u - a_u \Theta + \nabla^\perp_u \Theta ) + n_u n_v ( \nabla^\perp_a (Z^\perp)^a + a_a (Z^\perp)^a - n^a \nabla_a \Theta - K \Theta - \kappa_1 (\kappa_2 + 2) \Theta ) ) = 8 \pi n^u {\gamma_b}^v ( S_{uv} + n_v j_u + n_u j_v + n_u n_v \rho ) $
$ n^u {\gamma_b}^v G_{uv} + \kappa_1 (Z^\perp)_b + K_{ab} (Z^\perp)^a + {\gamma_b}^v n^a \nabla_a (Z^\perp)_v + a_b \Theta - \nabla^\perp_b \Theta = -8 \pi j_b $
using $n^a {\gamma_v}^b \nabla_a v_b = n^a \nabla_a (v^\perp)_v - n_v (v^\perp)^a a_a + a_v v^\top $
$ n^u {\gamma_b}^v G_{uv} + \kappa_1 (Z^\perp)_b + K_{ab} (Z^\perp)^a + n^a \nabla_a (Z^\perp)_b - n_b (Z^\perp)^a a_a + a_b \Theta - \nabla^\perp_b \Theta = -8 \pi j_b $
$ n^u {\gamma_b}^v G_{uv} + \kappa_1 (Z^\perp)_b + K_{ab} (Z^\perp)^a + n^a \nabla_a (Z^\perp)_b - n_b (Z^\perp)^a a_a + a_b \Theta - \nabla^\perp_b \Theta = -8 \pi j_b $
using ${\gamma_a}^u n^v G_{uv} = \nabla^\perp_a K - \nabla^\perp_b {K_a}^b$
$ \nabla^\perp_b K - \nabla^\perp_a {K_b}^a + \kappa_1 (Z^\perp)_b + K_{ab} (Z^\perp)^a + n^a \nabla_a (Z^\perp)_b - n_b (Z^\perp)^a a_a + a_b \Theta - \nabla^\perp_b \Theta = -8 \pi j_b $
Using $\mathcal{L}_\vec{n} (Z^\perp)_a= n^u \nabla_u (Z^\perp)_a - (Z^\perp)_u {K_a}^u - n_a (Z^\perp)_u a^u$
$ \mathcal{L}_\vec{n} (Z^\perp)_b + \nabla^\perp_b K - \nabla^\perp_a {K_b}^a + \kappa_1 (Z^\perp)_b + 2 K_{ab} (Z^\perp)^a + a_b \Theta - \nabla^\perp_b \Theta = -8 \pi j_b $
$ \mathcal{L}_\vec{n} (Z^\perp)_b = \nabla^\perp_a {K_b}^a - \nabla^\perp_b K + \nabla^\perp_b \Theta - a_b \Theta - 2 K_{ab} (Z^\perp)^a - \kappa_1 (Z^\perp)_b - 8 \pi j_b $
$ \frac{1}{\alpha} (\partial_t (Z^\perp)_b - \mathcal{L}_\vec\beta (Z^\perp)_b) = \nabla^\perp_a {K_b}^a - \nabla^\perp_b K + \nabla^\perp_b \Theta - a_b \Theta - 2 K_{ab} (Z^\perp)^a - \kappa_1 (Z^\perp)_b - 8 \pi j_b $
$ \partial_t (Z^\perp)_b = \alpha ( \nabla^\perp_a {K_b}^a - \nabla^\perp_b K + \nabla^\perp_b \Theta - a_b \Theta - 2 K_{ab} (Z^\perp)^a - \kappa_1 (Z^\perp)_b - 8 \pi j_b ) + \mathcal{L}_\vec\beta (Z^\perp)_b $

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Project both:
$ {\gamma_a}^u {\gamma_b}^v ( G_{uv} + \nabla^\perp_u (Z^\perp)_v + \nabla^\perp_v (Z^\perp)_u + \gamma_{uv} ( - \nabla^\perp_c (Z^\perp)^c - a_c (Z^\perp)^c - n^c \nabla_c \Theta + K \Theta + \kappa_1 \kappa_2 \Theta ) - 2 K_{uv} \Theta + n_u ( - \kappa_1 (Z^\perp)_v - K_{vc} (Z^\perp)^c - n^c \nabla_c (Z^\perp)_v - a_v \Theta + \nabla^\perp_v \Theta ) + n_v ( - \kappa_1 (Z^\perp)_u - K_{uc} (Z^\perp)^c - n^c \nabla_c (Z^\perp)_u - a_u \Theta + \nabla^\perp_u \Theta ) + n_u n_v ( \nabla^\perp_c (Z^\perp)^c + a_c (Z^\perp)^c - n^c \nabla_c \Theta - K \Theta - \kappa_1 (\kappa_2 + 2) \Theta ) ) = 8 \pi {\gamma_a}^u {\gamma_b}^v ( S_{uv} + n_v j_u + n_u j_v + n_u n_v \rho ) $
$ {\gamma_a}^u {\gamma_b}^v G_{uv} + \nabla^\perp_a (Z^\perp)_b + \nabla^\perp_b (Z^\perp)_a + \gamma_{ab} ( - \nabla^\perp_c (Z^\perp)^c - a_c (Z^\perp)^c - n^c \nabla_c \Theta + K \Theta + \kappa_1 \kappa_2 \Theta ) - 2 K_{ab} \Theta = 8 \pi S_{ab} $
...nah, maybe not so useful.

How about the projection of the trace-reversal?
Start with the trace-reversal of the Z4 equation in hypersurface and normal components
$R_{uv} + \nabla^\perp_u (Z^\perp)_v - n_u n^a \nabla_a (Z^\perp)_v - K_{ua} (Z^\perp)^a n_v - n_u a_v \Theta + n_v \nabla^\perp_u \Theta + \nabla^\perp_v (Z^\perp)_u - n_v n^a \nabla_a (Z^\perp)_u - K_{va} (Z^\perp)^a n_u - n_u a_v \Theta + n_u \nabla^\perp_v \Theta - 2 K_{uv} \Theta - 2 n_u n_v n^c \nabla_c \Theta - \kappa_1 ( n_u ((Z^\perp)_v + n_v \Theta) + n_v ((Z^\perp)_u + n_u \Theta) + (1 + \kappa_2) g_{uv} \Theta ) = 8 \pi ( T_{uv} - \frac{1}{2} g_{uv} T_{ab} g^{ab} ) $
...now project ...
$ {\gamma_a}^u {\gamma_b}^v ( R_{uv} + \nabla^\perp_u (Z^\perp)_v - n_u n^c \nabla_c (Z^\perp)_v - K_{uc} (Z^\perp)^c n_v - n_u a_v \Theta + n_v \nabla^\perp_u \Theta + \nabla^\perp_v (Z^\perp)_u - n_v n^c \nabla_c (Z^\perp)_u - K_{vc} (Z^\perp)^c n_u - n_u a_v \Theta + n_u \nabla^\perp_v \Theta - 2 K_{uv} \Theta - 2 n_u n_v n^c \nabla_c \Theta - \kappa_1 ( n_u ((Z^\perp)_v + n_v \Theta) + n_v ((Z^\perp)_u + n_u \Theta) + (1 + \kappa_2) g_{uv} \Theta ) ) = 8 \pi {\gamma_a}^u {\gamma_b}^v ( T_{uv} - \frac{1}{2} g_{uv} T ) $
$ {\gamma_a}^u {\gamma_b}^v R_{uv} + \nabla^\perp_a (Z^\perp)_b + \nabla^\perp_b (Z^\perp)_a - 2 K_{ab} \Theta - \kappa_1 (1 + \kappa_2) \gamma_{ab} \Theta = 8 \pi ( S_{ab} - \frac{1}{2} \gamma_{ab} (S - \rho) ) $
$ {\gamma_a}^u {\gamma_b}^v R_{uv} = - \nabla^\perp_a (Z^\perp)_b - \nabla^\perp_b (Z^\perp)_a + 2 K_{ab} \Theta + \kappa_1 (1 + \kappa_2) \gamma_{ab} \Theta + 8 \pi ( S_{ab} - \frac{1}{2} \gamma_{ab} (S - \rho) ) $

next start with the last point of Einstein field equations' ADM formalism extrinsic curvature evolution and replace the rhs with the Z4 rhs (when solving for $\perp G_{ab}$):
$\partial_t K_{ab} = -\nabla^\perp_a \nabla^\perp_b \alpha + \alpha ( R^\perp_{ab} + K K_{ab} - 2 K_{ac} {K^c}_b - \perp R_{ab} ) + \mathcal{L}_\vec\beta K_{ab} $
... and substitute the Z4 rhs of $\perp R_{ab}$:
$\partial_t K_{ab} = -\nabla^\perp_a \nabla^\perp_b \alpha + \alpha ( R^\perp_{ab} + K K_{ab} - 2 K_{ac} {K^c}_b + \nabla^\perp_a (Z^\perp)_b + \nabla^\perp_b (Z^\perp)_a - 2 K_{ab} \Theta - \kappa_1 (1 + \kappa_2) \gamma_{ab} \Theta - 8 \pi ( S_{ab} - \frac{1}{2} \gamma_{ab} (S - \rho) ) ) + \mathcal{L}_\vec\beta K_{ab} $
...simplify...
$\partial_t K_{ab} = -\nabla^\perp_a \nabla^\perp_b \alpha + \alpha ( R^\perp_{ab} + K_{ab} (K - 2 \Theta) - 2 K_{ac} {K^c}_b + \nabla^\perp_a (Z^\perp)_b + \nabla^\perp_b (Z^\perp)_a - \kappa_1 (1 + \kappa_2) \gamma_{ab} \Theta - 8 \pi ( S_{ab} - \frac{1}{2} \gamma_{ab} (S - \rho) ) ) + \mathcal{L}_\vec\beta K_{ab} $

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Ok Summary:

- the spatial metric ivp is independent of the field equations / the stress-energy constraint, so it is unchanged
$\partial_t \gamma_{ab} = -2 \alpha K_{ab} + \beta^c \partial_c \gamma_{ab} + \gamma_{cb} \partial_a \beta^c + \gamma_{ac} \partial_b \beta^c $

- Contract field equations with both normals to come up with Z4's form of the hamiltonian constraint. Is it me or does most literature stick to the non-Z4 Hamiltonian constraints. Maybe because the Z4 Hamiltonian constraint is the $\Theta$ ivp:
Yup, so the Z4 field equation double-normal-contracted is the ivp of $\Theta$.
$ \partial_t \Theta = \alpha ( H + \nabla^\perp_k (Z^\perp)^k + a_k (Z^\perp)^k - (K + \kappa_1 (\kappa_2 + 2)) \Theta ) + \beta^k \partial_k \Theta $

- Contract field equations with normal and project to come up with Z4's form of the momenutm constraint, substitute Lie derivative of Z spatial vector to get its evolution equation:
$ \partial_t (Z^\perp)_k = \alpha ( \nabla^\perp_i {K_k}^i - \nabla^\perp_k K + \nabla^\perp_k \Theta - a_k \Theta - 2 K_{ik} (Z^\perp)^i - \kappa_1 (Z^\perp)_k - 8 \pi j_k ) + \mathcal{L}_\vec\beta (Z^\perp)_k $

- project both indexes of (trace-reversed, Ricci-based) field equations to get Z4's form of the extrinsic curvature ivp:
$\partial_t K_{ab} = -\nabla^\perp_a \nabla^\perp_b \alpha + \alpha ( R^\perp_{ab} + K_{ab} (K - 2 \Theta) - 2 K_{ac} {K^c}_b + \nabla^\perp_a (Z^\perp)_b + \nabla^\perp_b (Z^\perp)_a - \kappa_1 (1 + \kappa_2) \gamma_{ab} \Theta - 8 \pi ( S_{ab} - \frac{1}{2} \gamma_{ab} (S - \rho) ) ) + \mathcal{L}_\vec\beta K_{ab} $
Looks like the original Bona et al Z4 comes out for $\kappa_1 = 0$, nice and simple here.
This is based on the geometry's:
$\partial_t K_{ab} = -\nabla^\perp_a \nabla^\perp_b \alpha + \alpha ( R^\perp_{ab} + K K_{ab} - 2 K_{ac} {K^c}_b - \perp R_{ab} ) + \mathcal{L}_\vec\beta K_{ab} $
... mixed with our trace-reversed field equations' rhs. And the trace-reversed 2005 Gundlach equations reduce to the 2003 Bona equations if you just set $\kappa_1 = 0$.

- why are gauge (lapse and shift) ivp's changed? where is the origin of the Bona-Masso lapse function? shift is always the last thing anyone messes with.
TODO.. Show where that $K \rightarrow K - 2 \Theta$ comes from.