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This page is a continuation of the "normal projection" stuff, but now for the ADM metric, lapse, shift, and time vector.
From that, it gives the time derivative of spatial metric (based on the normal derivative that was provided back in the "normal projection" worksheet).
TODO for this worksheet...
- Move as much non-component-description stuff to the top.
- Move as much of the constraints on our manifold to the top (must be commutation-free, torsion-free)
- Move as much ADM-component-description stuff to the bottom.
- Move as much non-EFE-related stuff back to "normal projections" in 12.8
- Don't use dots over variables.
NOTICE: most LQG papers use the original ADM variable standard: $N$ = lapse, $N^a$ = shift, $g_{ab}$ = spatial metric
MTW switches up the 4D metric from $g_{uv}$ to ${^{(4)}g}_{uv}$ to alleviate ambiguity.
Other sources use $q_{uv}$ or $h_{uv}$ to represent the spatial metric.
My guess on why $q$ is so that, combined with the representation of the conjugate momentum tensor (originally $\pi^{ab}$ but now represented as $p^{uv}$),
the Poisson bracket is now expressed in terms of its traditional variables $p$ and $q$.
I've stuck with the numerical relativity crowd's representations of $\alpha$ = lapse, $\beta^u$ = shift, $\gamma_{uv}$ = spatial metric.
Unlike the NR crowd, I did change the spacetime indexes from Latin back to Greek.
let $\alpha$ be our lapse, be the normalizing factor of our unit hypersurface normal.
let $ \beta^a $ be our shift, the spatial components of the normal
let $ \gamma_{ab} $ serve as both the spatial metric and the hypersurface projection
ADM metric:
Previously we showed:
$g^{tt} = -\frac{1}{\alpha^2}$ for normal coordinate t.
$g_{ij} = \gamma_{ij}$ for spatial coordinates ij.
Next let $\beta_i = g_{ti}$
Define our metric and inverse in terms of lapse $\alpha$, shift $\beta^i$, and spatial hypersurface subset metric $\gamma_{ij}$ as:
For this worksheet I'm putting the hypersurface normal dimension first instead of last, because I'm using the txyz ordering of coordinates.
$[g_{uv}] = \overset{u(a)\downarrow v(b)\rightarrow}{\left[ \matrix{
-\alpha^2 + \beta_c \beta^c & \beta_b \\
\beta_a & \gamma_{ab}
} \right]}$
$[g^{uv}] = \overset{u(a)\downarrow v(b)\rightarrow}{\left[ \matrix{
-1/\alpha^2 & \beta^b / \alpha^2 \\
\beta^a / \alpha^2 & \gamma^{ab} - \beta^a \beta^b / \alpha^2
} \right]}$
where $ \beta_b = \gamma_{ab} \beta^a = \gamma_{u b} \beta^u = g_{u b} \beta^b $ courtesy of the fact that $ g_{ab} = \gamma_{ab} $ and $ \beta^t = 0 $.
ADM spatial metric determinant: $ \gamma = det[\gamma_{ab}] $
ADM metric determinant: $ g = -\alpha^2 \gamma $
Volume element: $e = \sqrt{-g} = \alpha \sqrt\gamma$
hypersurface normal:
$n_u = -\alpha \nabla_u t$
If we assume time is a coordinte:
$n_u = -\alpha \delta_u^t$
normal in the ADM metric:
$n_u = \overset{u(a)\rightarrow}{\left[ \matrix{ -\alpha & 0 } \right]} $
$n^u = \overset{u(a)\downarrow}{\left[ \matrix{ 1/\alpha \\ -\beta^a/\alpha } \right]} $
normal is unit:
$n_u n^u
= \overset{u(a)\rightarrow}{\left[ \matrix{-\alpha & 0} \right]}
\overset{u(a)\downarrow}{\left[ \matrix{1/\alpha \\ -\beta^a/\alpha} \right]}
= -1$
4D shift vector:
$ \beta^u = \overset{u(a)\downarrow}{\left[ \matrix{0 \\ \beta^a} \right]} $
$ \beta_u = g_{uv} \beta^v = \overset{u(b)\rightarrow}{\left[ \matrix{\beta^c\beta_c & \beta_b} \right]} $
timelike vector:
$t^u = \delta^u_t$
$t^u = \alpha n^u + \beta^u$ (2008 Alcubierre, eqn. 2.2.14)
Notice that:
$t^u \nabla_u t = (\alpha n^u + \beta^u) (-\frac{1}{\alpha} n_u)$
$t^u \nabla_u t = -n^u n_u - \frac{1}{\alpha}\beta^u n_u$
using $\beta^u$ is spatial so $n_u \beta^u = 0$
$t^u \nabla_u t = 1$ (2008 Alcubierre, eqn. 2.2.15)
Or simpler:
$t^u \nabla_u t = \delta^u_t \delta^t_u = 1$
Also:
$\gamma_{uv} t^v = \beta_u$ (2008 Alcubierre, eqn. 2.2.16)
... in ADM components:
$t^u
= \alpha \overset{u(a)\downarrow}{\left[ \matrix{ 1/\alpha \\ -\beta^a/\alpha } \right]} + \overset{u(a)\downarrow}{\left[ \matrix{ 0 \\ \beta^a } \right]}
= \overset{u(a)\downarrow}{\left[ \matrix{1 \\ 0} \right]} $
$ t_u = g_{uv} t^v = \overset{u(b)\rightarrow}{\left[ \matrix{-\alpha^2 + \beta_c \beta^c & \beta_b} \right]} $
spatial metric components using in ADM metric:
${\gamma_u}^v = \delta^v_u + n_u n^v$
$ = \overset{u(a)\downarrow v(b)\rightarrow}{\left[ \matrix{ 1 & 0 \\ 0 & \delta_a^b } \right]}
+ \overset{u(a)\downarrow}{\left[ \matrix{ -\alpha \\ 0 } \right]}
\overset{v(b)\rightarrow}{\left[ \matrix{ 1/\alpha & -\beta^b/\alpha } \right]} $
$ = \overset{u(a)\downarrow v(b)\rightarrow}{\left[ \matrix{ 1 & 0 \\ 0 & \delta_a^b } \right]}
+ \overset{u(a)\downarrow v(b)\rightarrow}{\left[ \matrix{ -1 & \beta^b \\ 0 & 0 } \right]} $
$ = \overset{u(a)\downarrow v(b)\rightarrow}{\left[ \matrix{ 0 & \beta^b \\ 0 & \delta_a^b } \right]} $
$ \gamma_{uv} = g_{uv} + n_u n_v$
$ = \overset{u(a)\downarrow v(b)\rightarrow}{\left[ \matrix{-\alpha^2 + \beta_c \beta^c & \beta_b \\ \beta_a & \gamma_{ab}} \right]}
+ \overset{u(a)\downarrow}{\left[ \matrix{ -\alpha \\ 0 } \right]}
\overset{v(b)\rightarrow}{\left[ \matrix{ -\alpha & 0 } \right]} $
$ = \overset{u(a)\downarrow v(b)\rightarrow}{\left[ \matrix{-\alpha^2 + \beta_c \beta^c & \beta_b \\ \beta_a & \gamma_{ab}} \right]}
+ \overset{u(a)\downarrow v(b)\rightarrow}{\left[ \matrix{ \alpha^2 & 0 \\ 0 & 0 } \right]} $
$ = \overset{u(a)\downarrow v(b)\rightarrow}{\left[ \matrix{\beta_c \beta^c & \beta_b \\ \beta_a & \gamma_{ab}} \right]} $
$ \gamma^{uv} = g^{uv}+n^u n^v $
$ = \overset{u(a)\downarrow v(b)\rightarrow}{\left[ \matrix{-1/\alpha^2 & \beta^b / \alpha^2 \\ \beta^a / \alpha^2 & \gamma^{ab} - \beta^a \beta^b / \alpha^2} \right]}
+ \overset{u(a)\downarrow}{\left[ \matrix{ 1/\alpha \\ -\beta^a/\alpha } \right]}
\overset{v(b)\rightarrow}{\left[ \matrix{ 1 / \alpha & -\beta^b / \alpha } \right]} $
$ = \overset{u(a)\downarrow v(b)\rightarrow}{\left[ \matrix{-1/\alpha^2 & \beta^b / \alpha^2 \\ \beta^a / \alpha^2 & \gamma^{ab} - \beta^a \beta^b / \alpha^2} \right]}
+ \overset{u(a)\downarrow v(b)\rightarrow}{ \left[ \matrix{ 1/\alpha^2 & -\beta^b/\alpha^2 \\ -\beta^a / \alpha^2 & \beta^a \beta^b / \alpha^2 } \right]} $
$ = \overset{u(a)\downarrow v(b)\rightarrow}{\left[\matrix{0&0\\0&\gamma^{ab}}\right]} $
All that is maintained is a collection of properties of the variables used to construct the metric:
$ \gamma_{uv} = g_{uv} + n_u n_v $
$ t^u = \alpha n^u + \beta^u $
$ n_u n^u = -1 $
spatial metric Levi-Civita connection:
${(\hat\Gamma^\perp)^i}_{jk} = \gamma^{im} (\hat\Gamma^\perp)_{mjk}$
$= \gamma^{im} \cdot \frac{1}{2} ( \partial_k \gamma_{mj} + \partial_j \gamma_{mk} - \partial_m \gamma_{jk})$
...using $\gamma_{ij} = g_{ij}$ within our hypersurface...
$= \gamma^{im} \cdot \frac{1}{2} ( \partial_k g_{mj} + \partial_j g_{mk} - \partial_m g_{jk})$
$= \gamma^{im} \hat\Gamma_{mjk}$
TODO can I redo this without invoking specific indexes?
${(\hat\Gamma^\perp)^a}_{bc} = \gamma^{ad} (\hat\Gamma^\perp)_{dbc}$
$= \gamma^{ad} \cdot \frac{1}{2} ( \partial_c \gamma_{db} + \partial_b \gamma_{dc} - \partial_d \gamma_{cb})$
$= \frac{1}{2} (g^{ad} + n^a n^d) (
\partial_c (g_{db} + n_d n_b)
+ \partial_b (g_{dc} + n_d n_c)
- \partial_d (g_{cb} + n_c n_b)
)$
$=
\frac{1}{2} g^{ad} (
\partial_c g_{db} + \partial_c (n_d n_b)
+ \partial_b g_{dc} + \partial_b (n_d n_c)
- \partial_d g_{cb} - \partial_d (n_c n_b)
)
+ \frac{1}{2} n^a n^d (
\partial_c g_{db} + \partial_c (n_d n_b)
+ \partial_b g_{dc} + \partial_b (n_d n_c)
- \partial_d g_{cb} - \partial_d (n_c n_b)
)
$
$=
\frac{1}{2} g^{ad} (
\partial_c g_{db}
+ \partial_b g_{dc}
- \partial_d g_{cb}
)
+ \frac{1}{2} g^{ad} (
\partial_c (n_d n_b)
+ \partial_b (n_d n_c)
- \partial_d (n_c n_b)
)
+ \frac{1}{2} n^a n^d (
\partial_c g_{db}
+ \partial_b g_{dc}
- \partial_d g_{cb}
)
+ \frac{1}{2} n^a n^d (
\partial_c (n_d n_b)
+ \partial_b (n_d n_c)
- \partial_d (n_c n_b)
)
$
$=
{\hat\Gamma^a}_{bc}
+ \frac{1}{2} (
g^{ad} \partial_c (n_d n_b)
+ g^{ad} \partial_b (n_d n_c)
- g^{ad} \partial_d (n_c n_b)
)
+ \frac{1}{2} n^a n^d (
\partial_c g_{db}
+ \partial_b g_{dc}
- \partial_d g_{cb}
+ \partial_c (n_d n_b)
+ \partial_b (n_d n_c)
- \partial_d (n_c n_b)
)
$
spatial projection of Levi-Civita connection:
$\perp {\hat\Gamma^i}_{jk} = {\gamma_a}^i {\gamma_j}^b {\gamma_k}^c {\hat\Gamma^a}_{bc}$
$ = (\delta_a^i + n_a n^i) (\delta_j^b + n_j n^b) (\delta_k^c + n_k n^c) {\hat\Gamma^a}_{bc}$
...using $n_i = 0$...
$ = (\delta_a^i + n_a n^i) (\delta_j^b) (\delta_k^c) {\hat\Gamma^a}_{bc}$
$ = (\delta_a^i + n_a n^i) {\hat\Gamma^a}_{jk}$
$ = {\hat\Gamma^i}_{jk} + n_a n^i {\hat\Gamma^a}_{jk}$
...using $n_i = 0$...
$ = {\hat\Gamma^i}_{jk} + n_t n^i {\hat\Gamma^t}_{jk}$
$ = {\hat\Gamma^i}_{jk} + (-\alpha) (-\frac{1}{\alpha} \beta^i) {\hat\Gamma^t}_{jk}$
$ = {\hat\Gamma^i}_{jk} + \beta^i {\hat\Gamma^t}_{jk}$
$ = {\hat\Gamma^i}_{jk} + n^i (-\alpha) (-\frac{1}{\alpha} K_{jk})$
$ = {\hat\Gamma^i}_{jk} + n^i K_{jk}$
Levi-Civita connection in terms of spatial:
${\hat\Gamma^i}_{jk} = g^{it} \hat\Gamma_{tjk} + g^{im} \hat\Gamma_{mjk}$
$= \frac{1}{\alpha^2} \beta^i \hat\Gamma_{tjk} + (\gamma^{im} - \frac{1}{\alpha^2} \beta^i \beta^m) \hat\Gamma_{mjk}$
$= \gamma^{im} \hat\Gamma_{mjk} + \frac{1}{\alpha^2} \beta^i (\hat\Gamma_{tjk} - \beta^m \hat\Gamma_{mjk})$
$= \gamma^{im} \hat\Gamma_{mjk} - \beta^i (g^{tt} \hat\Gamma_{tjk} + g^{tm} \hat\Gamma_{mjk})$
$= \gamma^{im} \hat\Gamma_{mjk} - \beta^i {\hat\Gamma^t}_{jk}$
$= \gamma^{im} \hat\Gamma_{mjk} - n^i K_{jk}$
So ${\hat\Gamma^i}_{jk} - {(\hat\Gamma^\perp)^i}_{jk} = \gamma^{im} \hat\Gamma_{mjk} - n^i K_{jk} - \gamma^{im} \hat\Gamma_{mjk}$
$= -n^i K_{jk}$
And $\perp {\hat\Gamma^i}_{jk} - {\hat\Gamma^i}_{jk} = {\hat\Gamma^i}_{jk} + n^i K_{jk} - {\hat\Gamma^i}_{jk}$
$= n^i K_{jk}$
So $\perp {\hat\Gamma^i}_{jk} - {(\hat\Gamma^\perp)^i}_{jk} = -n^i K_{jk} + n^i K_{jk} = 0$
Therefore $\perp {\hat\Gamma^i}_{jk} = {(\hat\Gamma^\perp)^i}_{jk}$
Spatial Riemann tensor:
Is the projection covariant derivative connection the Levi-Civita connection of the spatial metric?
We know $\gamma_{ij} = g_{ij}$.
For coordinate basii (such that the commutation of the Levi-Civita connection ${\hat{c}_{ij}}^k = 0$) this means $(\hat\Gamma^\perp)_{ijk} = \hat\Gamma_{ijk}$
In fact, so long as $e_i = (e^\perp)_i$, $(\Gamma^\perp)_{ijk} = \Gamma_{ijk}$ ...
Since, even if ${\hat{c}_{ij}}^t \ne 0$, which means $(e_i(e_j(f)) - e_j(e_i(f))) \cdot e_t(f) \ne 0$, this only produces a ${\hat{c}_{ij}}^t$ value, which would not be used in the calculations of $\Gamma_{ijk}$
But does $\perp {\Gamma^i}_{jk} = {(\Gamma^\perp)^i}_{jk}$?
Because, if no, then this means that projection introduces torsion.
I saw somewhere that the extrinsic curvature tensor formula is defined for torsion-free manifolds. So, there you go.
Want to redefine it for a nonzero torsion curvature tensor?
Riemann curvature of projected covariant derivative connection:
${(R^\perp)^c}_{dab} =
2 e_{[a} ({(\Gamma^\perp)^c}_{b]d})
+ 2 {(\Gamma^\perp)^c}_{[a|u} {(\Gamma^\perp)^u}_{b]d}
- {(\Gamma^\perp)^c}_{ud} {(c^\perp)_{ab}}^u
$
${(R^\perp)^c}_{dba} v_c = 2 \nabla^\perp_{[a} \nabla^\perp_{b]} v_d + {(T^\perp)^u}_{ab} \nabla^\perp_u v^c$
From here on, $\nabla$ will only refer to the Levi-Civita connection.
extrinsic curvature components, in ADM metric:
$ K_{ab} = -{\gamma_a}^c \nabla_c n_d {\gamma_b}^d $
$ [K_{ab}] =
-\overset{a(i)\downarrow c(k)\rightarrow}{\left[ \matrix{ 0 & \beta^k \\ 0 & \delta_i^k } \right]}
\left(
\overset{ c(k) \downarrow d(l) \rightarrow}{\left[ \matrix{ e_t (n_t) & 0 \\ e_k (n_t) & 0 } \right]}
- \overset{ c(k) \downarrow d(l) \rightarrow}{\left[ \matrix{ {\Gamma^t}_{tt} & {\Gamma^t}_{tl} \\ {\Gamma^t}_{kt} & {\Gamma^t}_{kl} } \right]} n_t
\right)
\overset{d(l) \downarrow b(j) \rightarrow}{\left[ \matrix{ 0 & 0 \\ \beta^l & \delta_j^l } \right]}$
$= \overset{a(i) \downarrow c(k) \rightarrow}{\left[ \matrix{ 0 & \beta^k \\ 0 & \delta_i^k } \right]}
\overset{c(k) \downarrow d(l) \rightarrow}{\left[ \matrix{ {\Gamma^t}_{tt} n_t - e_t (n_t) & {\Gamma^t}_{tl} n_t \\ {\Gamma^t}_{kt} n_t - e_k (n_t) & {\Gamma^t}_{kl} n_t } \right]}
\overset{d(l) \downarrow b(j) \rightarrow}{\left[ \matrix{ 0 & 0 \\ \beta^l & \delta_j^l} \right]} $
$= -\alpha \overset{a(i) \downarrow b(j) \rightarrow}{ \left[ \matrix{ \beta^k \beta^l {\Gamma^t}_{kl} & \beta^k {\Gamma^t}_{kj} \\ \beta^l {\Gamma^t}_{il} & {\Gamma^t}_{ij} } \right]} $
$K_{ij} = -\alpha {\Gamma^t}_{ij}$ for spatial $ij$
Antisymmetry of extrinsic curvature
$K_{[ij]} = -\alpha {\Gamma^t}_{[ij]} = - \frac{1}{2} \alpha {{c}_{ij}}^t$
For (Levi-Civita) commutation ${{c}_{ab}}^c$
So $K_{[ij]} = 0$
And $K_{ij} = K_{(ij)}$
Metric in terms of ADM components (again):
$g_{tt} = -\alpha^2 + \beta^i \beta^j \gamma_{ij}$
$g_{ti} = \beta^k \gamma_{ik}$
$g_{ij} = \gamma_{ij}$
Metric inverse in terms of ADM components (again):
$g^{tt} = -\frac{1}{\alpha^2}$
$g^{ti} = \frac{1}{\alpha^2} \beta^i$
$g^{ij} = \gamma^{ij} - \frac{1}{\alpha^2} \beta^i \beta^j$
Metric partial in terms of ADM components:
$g_{tt,u} = -2 \alpha \alpha_{,u} + 2 \beta_l {\beta^l}_{,u} + \beta^l \beta^m \gamma_{lm,u}$
$g_{ti,u} = {\beta^l}_{,u} \gamma_{li} + \beta^l \gamma_{li,u}$
$g_{ij,u} = \gamma_{ij,u}$
Lower connection in terms of ADM metric components:
${\Gamma}_{ttt}
= \frac{1}{2} g_{tt,t}
= -\alpha \alpha_{,t} + \beta_l {\beta^l}_{,t} + \frac{1}{2} \beta^l \beta^m \gamma_{lm,t}
$
${\Gamma}_{tti}
= {\Gamma}_{tit} = \frac{1}{2} g_{tt,i}
= -\alpha \alpha_{,i}
+ \beta_l {\beta^l}_{,i}
+ \frac{1}{2} \beta^l \beta^m \gamma_{lm,i}
$
${\Gamma}_{tij}
= \frac{1}{2} (g_{ti,j} + g_{tj,i} - g_{ij,t})
= \frac{1}{2} (
{\beta^l}_{,j} \gamma_{li} + \beta^l \gamma_{li,j}
+ {\beta^l}_{,i} \gamma_{lj} + \beta^l \gamma_{lj,i}
- \gamma_{ij,t}
)
$
${\Gamma}_{ktt}
= g_{tk,t} - \frac{1}{2} g_{tt,k}
= {\beta^l}_{,t} \gamma_{lk}
+ \beta^l \gamma_{lk,t}
+ \alpha \alpha_{,k}
- \beta_l {\beta^l}_{,k}
- \frac{1}{2} \beta^l \beta^m \gamma_{lm,k}
$
${\Gamma}_{kti} = {\Gamma}_{kit}
= \frac{1}{2} (g_{tk,i} + g_{ki,t} - g_{ti,k})
= \frac{1}{2} (
{\beta^l}_{,i} \gamma_{lk} + \beta^l \gamma_{lk,i}
+ \gamma_{ki,t}
- {\beta^l}_{,k} \gamma_{li} - \beta^l \gamma_{li,k}
)
$
${\Gamma}_{kij}
= \frac{1}{2} (g_{ki,j} + g_{kj,i} - g_{ij,k})
= \frac{1}{2} (\gamma_{ki,j} + \gamma_{kj,i} - \gamma_{ij,k})
$
Upper connection in terms of ADM meric components:
${\Gamma^t}_{tt}
= g^{tt} \Gamma_{ttt} + g^{tn} \Gamma_{ntt}
= -\frac{1}{\alpha^2} \cdot (
- \alpha \alpha_{,t}
+ \beta_l {\beta^l}_{,t}
+ \frac{1}{2} \beta^l \beta^m \gamma_{lm,t}
)
+ \frac{1}{\alpha^2} \beta^n \cdot (
{\beta^l}_{,t} \gamma_{ln}
+ \beta^l \gamma_{ln,t}
+ \alpha \alpha_{,n}
- \beta_l {\beta^l}_{,n}
- \frac{1}{2} \beta^l \beta^m \gamma_{lm,n}
)
$
$ = \frac{1}{\alpha^2} (
\alpha \alpha_{,t}
+ \frac{1}{2} \beta^l \beta^m \gamma_{lm,t}
+ \alpha \beta^l \alpha_{,l}
- \beta_l \beta^m {\beta^l}_{,m}
- \frac{1}{2} \beta^l \beta^m \beta^n \gamma_{lm,n}
)
$
${\Gamma^t}_{ti} = {\Gamma^t}_{it}
= g^{tt} \Gamma_{tti} + g^{tn} \Gamma_{nti}
= -\frac{1}{\alpha^2} \cdot (
- \alpha \alpha_{,i}
+ \beta_l {\beta^l}_{,i}
+ \frac{1}{2} \beta^l \beta^m \gamma_{lm,i}
)
+ \frac{1}{\alpha^2} \beta^n \cdot \frac{1}{2} (
{\beta^l}_{,i} \gamma_{ln}
+ \beta^l \gamma_{ln,i}
+ \gamma_{ni,t}
- {\beta^l}_{,n} \gamma_{li}
- \beta^l \gamma_{li,n}
)
$
$ = \frac{1}{\alpha^2} (
\frac{1}{2} \beta^n \gamma_{ni,t}
+ \alpha \alpha_{,i}
- \frac{1}{2} \beta_l {\beta^l}_{,i}
- \frac{1}{2} \beta^m {\beta^l}_{,m} \gamma_{li}
- \frac{1}{2} \beta^l \beta^m \gamma_{li,m}
)
$
${\Gamma^t}_{ij}
= g^{tt} \Gamma_{tij} + g^{tn} \Gamma_{nij}
= -\frac{1}{\alpha^2} \cdot \frac{1}{2} (
{\beta^l}_{,j} \gamma_{li}
+ \beta^l \gamma_{li,j}
+ {\beta^l}_{,i} \gamma_{lj}
+ \beta^l \gamma_{lj,i}
- \gamma_{ij,t}
)
+ \frac{1}{\alpha^2} \beta^n \cdot \frac{1}{2} (
\gamma_{ni,j}
+ \gamma_{nj,i}
- \gamma_{ij,n}
)
$
$ = \frac{1}{\alpha^2} (
\frac{1}{2} \gamma_{ij,t}
- \frac{1}{2} {\beta^l}_{,j} \gamma_{li}
- \frac{1}{2} {\beta^l}_{,i} \gamma_{lj}
- \frac{1}{2} \beta^l \gamma_{ij,l}
)
$
${\Gamma^k}_{tt}
= g^{kt} \Gamma_{ttt} + g^{kn} \Gamma_{ntt}
= \frac{1}{\alpha^2} \beta^k \cdot (
-\alpha \alpha_{,t}
+ \beta_l {\beta^l}_{,t}
+ \frac{1}{2} \beta^l \beta^m \gamma_{lm,t}
)
+ (
\gamma^{kn}
- \frac{1}{\alpha^2} \beta^k \beta^n
) \cdot (
{\beta^l}_{,t} \gamma_{ln}
+ \beta^l \gamma_{ln,t}
+ \alpha \alpha_{,n}
- \beta_l {\beta^l}_{,n}
- \frac{1}{2} \beta^l \beta^m \gamma_{lm,n}
)
$
$ = \frac{1}{\alpha^2} (
- \beta^k \alpha \alpha_{,t}
- \frac{1}{2} \beta^k \beta^l \beta^m \gamma_{lm,t}
- \alpha \beta^k \beta^l \alpha_{,l}
+ \beta^k \beta^m \beta_l {\beta^l}_{,m}
+ \frac{1}{2} \beta^k \beta^l \beta^m \beta^n \gamma_{lm,n}
)
+ {\beta^k}_{,t}
+ \beta^l \gamma^{km} \gamma_{lm,t}
+ \alpha \gamma^{km} \alpha_{,m}
- \beta_l \gamma^{km} {\beta^l}_{,m}
- \frac{1}{2} \beta^l \beta^n \gamma^{km} \gamma_{ln,m}
$
${\Gamma^k}_{ti} = {\Gamma^k}_{it}
= g^{kt} \Gamma_{tti} + g^{kn} \Gamma_{nti}
= \frac{1}{\alpha^2} \beta^k \cdot (
-\alpha \alpha_{,i}
+ \beta_l {\beta^l}_{,i}
+ \frac{1}{2} \beta^l \beta^m \gamma_{lm,i}
)
+ (
\gamma^{kn}
- \frac{1}{\alpha^2} \beta^k \beta^n
) \cdot \frac{1}{2} (
{\beta^l}_{,i} \gamma_{ln}
+ \beta^l \gamma_{ln,i}
+ \gamma_{ni,t}
- {\beta^l}_{,n} \gamma_{li}
- \beta^l \gamma_{li,n}
)
$
$ = \frac{1}{\alpha^2} (
- \frac{1}{2} \beta^k \beta^l \gamma_{li,t}
- \beta^k \alpha \alpha_{,i}
+ \beta^k \beta_l {\beta^l}_{,i}
- \frac{1}{2} \beta^k \beta^m {\beta^l}_{,i} \gamma_{lm}
+ \frac{1}{2} \beta^k \beta^m {\beta^l}_{,m} \gamma_{li}
+ \frac{1}{2} \beta^k \beta^m \beta^l \gamma_{li,m}
)
+ \frac{1}{2} \gamma^{kl} \gamma_{li,t}
+ \frac{1}{2} {\beta^k}_{,i}
- \frac{1}{2} \gamma^{km} {\beta^l}_{,m} \gamma_{li}
+ \frac{1}{2} \gamma^{km} \beta^l \gamma_{lm,i}
- \frac{1}{2} \gamma^{km} \beta^l \gamma_{li,m}
$
${\Gamma^k}_{ij}
= g^{kt} \Gamma_{tij} + g^{kn} \Gamma_{nij}
= \frac{1}{\alpha^2} \beta^k \cdot \frac{1}{2} (
{\beta^l}_{,j} \gamma_{li}
+ \beta^l \gamma_{li,j}
+ {\beta^l}_{,i} \gamma_{lj}
+ \beta^l \gamma_{lj,i}
- \gamma_{ij,t}
)
+ (
\gamma^{kn}
- \frac{1}{\alpha^2} \beta^k \beta^n
) \cdot \frac{1}{2} (
\gamma_{ni,j}
+ \gamma_{nj,i}
- \gamma_{ij,n}
)
$
$ = \frac{1}{2} \frac{1}{\alpha^2} \beta^k (
- \gamma_{ij,t}
+ \gamma_{li} {\beta^l}_{,j}
+ \gamma_{lj} {\beta^l}_{,i}
+ \beta^l \gamma_{ij,l}
)
+ \frac{1}{2} \gamma^{kl} (
\gamma_{li,j}
+ \gamma_{lj,i}
- \gamma_{ij,l}
)
$
Start with $K_{ij} = -\alpha {\Gamma^t}_{ij}$, substitute the definition of ${\Gamma^t}_{ij}$ in terms of $\alpha, \beta^i, \gamma_{ij}$:
$K_{ij} = -\alpha \cdot \frac{1}{\alpha^2} (
\frac{1}{2} \gamma_{ij,t}
- \frac{1}{2} {\beta^l}_{,j} \gamma_{li}
- \frac{1}{2} {\beta^l}_{,i} \gamma_{lj}
- \frac{1}{2} \beta^l \gamma_{ij,l}
)$
$K_{ij} = \frac{1}{2 \alpha} (
-\gamma_{ij,t}
+ {\beta^l}_{,j} \gamma_{li}
+ {\beta^l}_{,i} \gamma_{lj}
+ \beta^l \gamma_{ij,l}
)$
solve for $\gamma_{ij,t}$:
$ \gamma_{ij,t}
=
-2 \alpha K_{ij}
+ {\beta^l}_{,j} \gamma_{li}
+ {\beta^l}_{,i} \gamma_{lj}
+ \beta^l \gamma_{ij,l}
$
Voila. Coordinate-aligned initial-value formulation of $\gamma_{ij}$.
Now with this definition of $K_{ij}$ in terms of $\gamma_{ij,t}$, substitute it back into our ${\Gamma^a}_{bc}$ definition:
${\Gamma^t}_{tt} = \frac{1}{\alpha^2} (
\alpha \alpha_{,t}
+ \alpha \beta^l \alpha_{,l}
- \beta_l \beta^m {\beta^l}_{,m}
- \frac{1}{2} \beta^l \beta^m \beta^n \gamma_{lm,n}
+ \frac{1}{2} \beta^l \beta^m (
-2 \alpha K_{lm}
+ {\beta^n}_{,m} \gamma_{nl}
+ {\beta^n}_{,l} \gamma_{nm}
+ \beta^n \gamma_{lm,n}
)
)
$
$= \frac{1}{\alpha} (
\alpha_{,t}
+ \beta^l \alpha_{,l}
- \beta^l \beta^m K_{lm}
)
$ (2008 Alcubierre, eqn. B.7)
${\Gamma^t}_{ti} = \frac{1}{\alpha^2} (
\frac{1}{2} \beta^n (
-2 \alpha K_{in}
+ {\beta^l}_{,n} \gamma_{li}
+ {\beta^l}_{,i} \gamma_{ln}
+ \beta^l \gamma_{in,l}
)
+ \alpha \alpha_{,i}
- \frac{1}{2} \beta_l {\beta^l}_{,i}
- \frac{1}{2} \beta^m {\beta^l}_{,m} \gamma_{li}
- \frac{1}{2} \beta^l \beta^m \gamma_{li,m}
)
$
$= \frac{1}{\alpha} (
\alpha_{,i}
- \beta^l K_{il}
)
$ (2008 Alcubierre, eqn. B.8)
${\Gamma^t}_{ij} = \frac{1}{\alpha^2} (
\frac{1}{2} (
-2 \alpha K_{ij}
+ {\beta^l}_{,j} \gamma_{li}
+ {\beta^l}_{,i} \gamma_{lj}
+ \beta^l \gamma_{ij,l}
)
- \frac{1}{2} {\beta^l}_{,j} \gamma_{li}
- \frac{1}{2} {\beta^l}_{,i} \gamma_{lj}
- \frac{1}{2} \beta^l \gamma_{ij,l}
)
$
$ = -\frac{1}{\alpha} K_{ij}$ (2008 Alcubierre, eqn. B.9)
${\Gamma^k}_{tt} = \frac{1}{\alpha^2} (
- \beta^k \alpha \alpha_{,t}
- \frac{1}{2} \beta^k \beta^l \beta^m (
-2 \alpha K_{lm}
+ {\beta^n}_{,m} \gamma_{nl}
+ {\beta^n}_{,l} \gamma_{nm}
+ \beta^n \gamma_{lm,n}
)
- \alpha \beta^k \beta^l \alpha_{,l}
+ \beta^k \beta^m \beta_l {\beta^l}_{,m}
+ \frac{1}{2} \beta^k \beta^l \beta^m \beta^n \gamma_{lm,n}
)
+ {\beta^k}_{,t}
+ \beta^l \gamma^{km} (
-2 \alpha K_{lm}
+ {\beta^n}_{,m} \gamma_{nl}
+ {\beta^n}_{,l} \gamma_{nm}
+ \beta^n \gamma_{lm,n}
)
+ \alpha \gamma^{km} \alpha_{,m}
- \beta_l \gamma^{km} {\beta^l}_{,m}
- \frac{1}{2} \beta^l \beta^n \gamma^{km} \gamma_{ln,m}
$
$ = \frac{1}{\alpha} \beta^k (
- \alpha_{,t}
- \beta^l \alpha_{,l}
+ \beta^l \beta^m K_{lm}
)
+ {\beta^k}_{,t}
- 2 \alpha \beta^l {K_l}^k
+ \alpha \gamma^{km} \alpha_{,m}
+ \beta^l {\beta^k}_{,l}
+ \beta^l \beta^n \gamma^{km} \gamma_{lm,n}
- \frac{1}{2} \beta^l \beta^n \gamma^{km} \gamma_{ln,m}
$
$ = \frac{1}{\alpha} \beta^k (
- \alpha_{,t}
- \beta^l \alpha_{,l}
+ \beta^l \beta^m K_{lm}
)
+ {\beta^k}_{,t}
- 2 \alpha \beta^l {K_l}^k
+ \alpha \gamma^{km} \alpha_{,m}
+ \beta^l ({\beta^k}_{,l} + \beta^m {(\Gamma^\perp)^k}_{lm})
$
$ = \frac{1}{\alpha} \beta^k (
- \alpha_{,t}
- \beta^l \alpha_{,l}
+ \beta^l \beta^m K_{lm}
)
+ {\beta^k}_{,t}
- 2 \alpha \beta^l {K_l}^k
+ \alpha \gamma^{km} \alpha_{,m}
+ \beta^l \nabla^\perp_l \beta^k
$ (2008 Alcubierre, eqn. B.10)
${\Gamma^k}_{ti} = \frac{1}{\alpha^2} (
- \frac{1}{2} \beta^k \beta^l (
-2 \alpha K_{il}
+ {\beta^m}_{,l} \gamma_{mi}
+ {\beta^m}_{,i} \gamma_{ml}
+ \beta^m \gamma_{il,m}
)
- \beta^k \alpha \alpha_{,i}
+ \beta^k \beta_l {\beta^l}_{,i}
- \frac{1}{2} \beta^k \beta^m {\beta^l}_{,i} \gamma_{lm}
+ \frac{1}{2} \beta^k \beta^m {\beta^l}_{,m} \gamma_{li}
+ \frac{1}{2} \beta^k \beta^m \beta^l \gamma_{li,m}
)
+ \frac{1}{2} \gamma^{kl} (
-2 \alpha K_{il}
+ {\beta^m}_{,l} \gamma_{mi}
+ {\beta^m}_{,i} \gamma_{ml}
+ \beta^m \gamma_{il,m}
)
+ \frac{1}{2} {\beta^k}_{,i}
- \frac{1}{2} \gamma^{km} {\beta^l}_{,m} \gamma_{li}
+ \frac{1}{2} \gamma^{km} \beta^l \gamma_{lm,i}
- \frac{1}{2} \gamma^{km} \beta^l \gamma_{li,m}
$
$ = \frac{1}{\alpha} \beta^k (
- \alpha_{,i}
+ \beta^l K_{il}
)
- \alpha {K_i}^k
+ {\beta^k}_{,i}
+ \frac{1}{2} \gamma^{km} \beta^l (
\gamma_{lm,i}
+ \gamma_{im,l}
- \gamma_{il,m}
)
$
$ = \frac{1}{\alpha} \beta^k (
- \alpha_{,i}
+ \beta^l K_{il}
)
- \alpha {K_i}^k
+ {\beta^k}_{,i}
+ {(\Gamma^\perp)^k}_{li} \beta^l
$
$ = \frac{1}{\alpha} \beta^k (
- \alpha_{,i}
+ \beta^l K_{il}
)
- \alpha {K_i}^k
+ \nabla^\perp_i \beta^k
$ (2008 Alcubierre, eqn. B.11)
${\Gamma^k}_{ij} = \frac{1}{2} \frac{1}{\alpha^2} \beta^k (
- (
-2 \alpha K_{ij}
+ {\beta^l}_{,j} \gamma_{li}
+ {\beta^l}_{,i} \gamma_{lj}
+ \beta^l \gamma_{ij,l}
)
+ \gamma_{li} {\beta^l}_{,j}
+ \gamma_{lj} {\beta^l}_{,i}
+ \beta^l \gamma_{ij,l}
)
+ \frac{1}{2} \gamma^{kl} (
\gamma_{li,j}
+ \gamma_{lj,i}
- \gamma_{ij,l}
)
$
$ = \frac{1}{\alpha} \beta^k K_{ij}
+ \frac{1}{2} \gamma^{kl} (
\gamma_{li,j}
+ \gamma_{lj,i}
- \gamma_{ij,l}
)
$
$ = \frac{1}{\alpha} \beta^k K_{ij}
+ {(\Gamma^\perp)^k}_{ij}
$ (2008 Alcubierre, eqn. B.12)
Acceleration vector:
$a_u = n^v \nabla_v n_u$
Acceleration vector in ADM covariant components:
$a_u = n^v \nabla_v n_u$
$a_u = n^v (\partial_v n_u - {\Gamma^a}_{vu} n_a)$
separate into timelike t and spatial ijk indexes:
$a_t = n^t (
\partial_t n_t
- {\Gamma^t}_{tt} n_t
- {\Gamma^k}_{tt} n_k
)
+ n^j (
\partial_j n_t
- {\Gamma^t}_{jt} n_t
- {\Gamma^k}_{jt} n_k
);
a_i = n^t (
\partial_t n_i
- {\Gamma^t}_{ti} n_t
- {\Gamma^k}_{ti} n_k
)
+ n^j (
\partial_j n_i
- {\Gamma^t}_{ji} n_t
- {\Gamma^k}_{ji} n_k
)
$
using $n_k = 0$
$a_t = -\frac{1}{\alpha} \partial_t \alpha
+ \frac{\beta^j}{\alpha} \partial_j \alpha
+ {\Gamma^t}_{tt}
- \beta^j {\Gamma^t}_{jt}
;
a_i = {\Gamma^t}_{ti}
- \beta^j {\Gamma^t}_{ji}
$
Using our component calculations of the ADM connections ...
: ${\Gamma^t}_{tt} = \frac{1}{\alpha} (\partial_t \alpha + \beta^m \partial_m \alpha - \beta^m \beta^n K_{mn})$
: ${\Gamma^t}_{ti} = \frac{1}{\alpha} (\partial_i \alpha - \beta^m K_{im})$
: ${\Gamma^t}_{ij} = -\frac{1}{\alpha} K_{ij}$
... we get:
$a_t = \frac{1}{\alpha} (
- \partial_t \alpha
+ \partial_t \alpha
+ \beta^m \partial_m \alpha
- \beta^m \beta^n K_{mn}
)
+ \frac{\beta^j}{\alpha} (
\partial_j \alpha
- \partial_j \alpha
+ \beta^m K_{jm}
)
;
a_i = \frac{1}{\alpha} (\partial_i \alpha - \beta^m K_{im})
- \beta^j (
-\frac{1}{\alpha} K_{ij}
)
$
$ a_t = \frac{1}{\alpha} \beta^j \partial_j \alpha;
a_i = \frac{1}{\alpha} \partial_i \alpha
$
$a_t = \frac{1}{\alpha} (\partial_t \alpha - \partial_t \alpha + \beta^j \partial_j \alpha);
a_i = \frac{1}{\alpha} \partial_i \alpha$
$a_t = \frac{1}{\alpha} (\partial_t \alpha - \alpha \frac{1}{\alpha} \partial_t \alpha - \alpha (-\frac{\beta^j}{\alpha}) \partial_j \alpha);
a_i = \frac{1}{\alpha} \partial_i \alpha$
$a_t = \frac{1}{\alpha} (\partial_t \alpha + n_t n^v \partial_v \alpha);
a_i = \frac{1}{\alpha} (\partial_i \alpha + n_i n^v \partial_v \alpha)$
$a_u = \frac{1}{\alpha} (\partial_u \alpha + n_u n^v \partial_v \alpha)$
$ = \frac{1}{\alpha} (\delta_u^v + n_u n^v) \partial_v \alpha$
$ = \frac{1}{\alpha} {\gamma_u}^v \nabla_v \alpha$
$ = \frac{1}{\alpha} \nabla^\perp_u \alpha$
$a_u = \nabla^\perp_u log(\alpha)$ (2010 Baumgarte & Shapiro, exercise 2.13)
Notice that this is also conveniently used in the NR literature as a hyperbolic syste 1st derivative state variable.
Since time is a distinct coordinate, and since $t^u = \delta^u_t$, we can assert:
$\mathcal{L}_\vec{t} = \partial_t$
Lie derivative of timelike vector:
$\mathcal{L}_\vec{t} = \mathcal{L}_{\alpha \vec{n} + \vec\beta}$
$\mathcal{L}_\vec{t} = \alpha \mathcal{L}_\vec{n} + \mathcal{L}_\vec\beta$
$\mathcal{L}_\vec{n} = \frac{1}{\alpha} (\mathcal{L}_\vec{t} - \mathcal{L}_\vec\beta)$
From here, lots of numerical relativity papers will use the definition $\frac{d}{dt} = \mathcal{L}_\vec{t} - \mathcal{L}_\vec\beta = \alpha \mathcal{L}_\vec{n}$
Time derivative of spatial metric:
$ \mathcal{L}_{\vec{n}} \gamma_{ab}
= -2 K_{ab}
= \frac{1}{\alpha} (\mathcal{L}_\vec{t} - \mathcal{L}_\vec\beta) \gamma_{ab}
$
$\mathcal{L}_\vec{t} \gamma_{ab} - \mathcal{L}_\vec\beta \gamma_{ab} = -2 \alpha K_{ab}$
$\partial_t \gamma_{ab} - \mathcal{L}_\vec\beta \gamma_{ab} = -2 \alpha K_{ab}$
$\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
$
using $\mathcal{L}_\vec\beta \gamma_{ab}
= \beta^c \partial_c \gamma_{ab} + \gamma_{cb} \partial_a \beta^c + \gamma_{ac} \partial_b \beta^c
= \beta^c \nabla^\perp_c \gamma_{ab} + \gamma_{cb} \nabla^\perp_a \beta^c + \gamma_{ac} \nabla^\perp_b \beta^c
$ and $\nabla^\perp_c \gamma_{ab} = 0$
$\partial_t \gamma_{ab} = -2 \alpha K_{ab}
+ 2 \nabla^\perp_{(a} \beta_{b)}
$
The initial-value formulation of the spatial metric is based on our hypersurface properties / decomposition, not on the Einstein Field Equations.
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