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TODO shoudn't the operator that projects parallel to the hypersurface be the symbol $\top$, or even better, the parallel symbol $\parallel$, and not the perpendicular symbol $\perp$ ?
Shouldn't the component that is perpendicular to the hypersurface by the perpendicular symbol $\perp$ ?
This means all the mainstream numerical relativity literature symbols of $\perp$ and $\top$ are mixed up.
It also means that if I were to adopt the corrected convention that the perpendicular $\perp$ symbol is the component perpendicular to the hypersurface that I would have to swap them in my own worksheets.
For this worksheet, $\nabla$ and ${\Gamma^a}_{bc}$ will refer to the Levi-Civita connection.
Let manifold $\mathcal{M}$ have dimension n and be spanned by basis elements $e_u \in \{ e_1 ... e_n \}$
Let hypersurface $\Sigma$ be a subset of our manifold, spanned by coordiante basis operators $e_i \in \{ e_1 ... e_{n-1} \}$.
Let $n_a$ be the hypersurface normal proportional to $e_n(x)$ (where the subset n denotes the n'th dimension).
Let $\alpha$ be the normalization proportion factor such that:
$n_a = -\alpha e_a (x^n)$ (for $x^n$ the n'th coordinate)
...for a coordinate basis where $e_a (x^b) = \delta_a^b$...
$n_a = -\alpha \delta_a^n$
Let $n_a n^a = -1$
Therefore:
$-1 = \alpha^2 g^{ab} e_a (x^n) e_b (x^n)$
$-\frac{1}{\alpha^2} = g^{ab} e_a (x^n) e_b (x^n)$
$-\frac{1}{\alpha^2} = g^{ab} \delta^n_a \delta^n_b$
$-\frac{1}{\alpha^2} = g^{nn}$
And here's the start of our ADM metric definition.
Thinking about it, is the rest of this worksheet dependent on needing $n_a$ to lie along a coordinate line? Maybe not.
However I remember reading somewhere that it is necessary to use a coordinate basis if you want to invoke extrinsic curvature ... TODO more picking that apart that later I guess.
spatial metric / projection tensor:
$\gamma_{uv} = g_{uv} + n_u n_v $
Notice that this operates as a projection onto the hypersurface, and the hypersurface components/sub-tensor operate as the metric of the hypersurface.
Also notice, for covariant indexes, $n_i = 0$ for $i \ne n$, and therefore $\gamma_{ij} = g_{ij}$ for $i,j \ne n$.
That is the next piece to our ADM metric definition.
spatial metric identity:
${\gamma_u}^a {\gamma_a}^v$
$= (\delta_u^a + n_u n^a)(\delta_a^v + n_a n^v)$
$= \delta_u^v + n_u n^v + n_u n^v + n_u n^a n_a n^v$
$= \delta_u^v + 2 n_u n^v - n_u n^v$
$= \delta_u^v + n_u n^v$
$= {\gamma_u}^v$
projection is orthogonal to normal:
$\gamma_{uv} n^v$
$= g_{uv} n^v + n_u n_v n^v$
$= n_u - n_u$
$\gamma_{uv} n^v = 0$
projection operation:
$\perp {T^{a_1 ... a_p}}_{b_1 ... b_q} = {\gamma^{a_1}}_{c_a} \cdot ... \cdot {\gamma^{a_p}}_{c_p} \cdot {\gamma^{d_1}}_{b_1} \cdot ... \cdot {\gamma^{d_q}}_{b_q} \cdot {T^{c_1 ... c_p}}_{d_1 ... d_q}$
spatial covariant derivative:
scalar: $\nabla^\perp_u f = \perp \nabla_u f = {\gamma_u}^v \nabla_v f$
tensor: $\nabla^\perp_u {T^v}_a = \perp \nabla_u {T^v}_a = {\gamma_u}^b {\gamma_c}^v {\gamma_a}^d \nabla_b {T^c}_d$
proof $\nabla^\perp_a$ is a metric-cancelling covariant derivative of $\gamma_{ab}$:
$\nabla^\perp_a \gamma_{bc}$
$={\gamma_a}^e {\gamma_b}^g {\gamma_c}^f \nabla_e \gamma_{gf}$
$={\gamma_a}^e {\gamma_b}^g {\gamma_c}^f \nabla_e (n_g n_f)$
$={\gamma_a}^e {\gamma_b}^g {\gamma_c}^f (n_f \nabla_e n_g + n_g \nabla_e n_f)$
$={\gamma_a}^e {\gamma_b}^g ({\gamma_c}^f n_f) \nabla_e n_g + {\gamma_a}^e {\gamma_c}^f ({\gamma_b}^g n_g) \nabla_e n_f$
$={\gamma_a}^e {\gamma_b}^g \cdot 0 \cdot \nabla_e n_g + {\gamma_a}^e {\gamma_c}^f \cdot 0 \cdot \nabla_e n_f$
$\nabla^\perp_a \gamma_{bc}=0$
unit normal is orthogonal to rotation:
$0 = \nabla_v (-1) = \nabla_v (n^u n_u) = n_u \nabla_v n^u + n^u \nabla_v n_u = 2 n^u \nabla_v n_u$
$n^u \nabla_v n_u = 0$
normal acceleration vector:
$a_u = n^v \nabla_v n_u$
acceleration is orthogonal to normal:
$n^u a_u = n^u n^v \nabla_v n_u = n^v (n^u \nabla_v n_u) = n^v \cdot 0 = 0$
${\gamma_u}^v a_v = \delta_u^v a_v + n_u n^v a_v = a^v$
Extrinsic curvature:
(some sources intentionally symmetrize $K_{ab} = -\nabla^\perp_{(a} n_{b)}$)
$K_{ab} = -\nabla^\perp_a n_b$
$= -{\gamma_a}^c {\gamma_b}^d \nabla_c n_d$
$= -(\delta_a^c + n_a n^c)(\delta_b^d + n_b n^d) \nabla_c n_d$
$= -\nabla_a n_b - n_a n^c \nabla_c n_b - n_b n^d \nabla_a n_d - n_a n_b n^c n^d \nabla_c n_d$
$= -\nabla_a n_b - n_a a_b$
By projection definition of the extrinsic curvature, we can assert:
$n^a K_{ab} = -n^a {\gamma_a}^c {\gamma_b}^d \nabla_c n_d = 0$ since $n^a {\gamma_a}^c = 0$
$n^b K_{ab} = -n^b {\gamma_a}^c {\gamma_b}^d \nabla_c n_d = 0$ since $n^b {\gamma_b}^d = 0$
What is the spatial covariant derivative of the spacetime metric?
$\nabla^\perp_a g_{uv}$
$= \nabla^\perp_a (\gamma_{uv} - n_u n_v)$
$= \nabla^\perp_a \gamma_{uv} - \nabla^\perp_a n_u n_v - n_u \nabla^\perp_a n_v$
using $\nabla^\perp_a \gamma_{uv} = 0$
$= -n_v \nabla^\perp_a n_u - n_u \nabla^\perp_a n_v$
$= n_v K_{au} + n_u K_{av}$
Therefore:
$K = K_{ab} \gamma^{ab} = K_{ab} g^{ab} = (-\nabla_a n_b - n_a a_b) g^{ab} = -\nabla_a n^a$
Decomposing a vector into its projection and normal components:
Let vector $v_a$ have spatial components $(v^\perp)_a = {\gamma_a}^u v_u$ and normal component $v^\top = n^a v_a / n^a n_a = -n^a v_a$
So $v_u = \delta^a_u v_a = ({\gamma_u}^a - n_u n^a) v_a = (v^\perp)_u + n_u v^\top$
Covariant derivative of vector in projection and normal components:
$\nabla_u v_v$
$= \nabla_u ((v^\perp)_v + n_v v^\top)$
$= \nabla_u (v^\perp)_v + v^\top \nabla_u n_v + n_v \nabla_u v^\top$
$= \nabla_u (v^\perp)_v - v^\top (K_{uv} + n_u a_v) + n_v \nabla_u v^\top$
$= \nabla_u (v^\perp)_v - v^\top K_{uv} - v^\top n_u a_v + n_v \nabla_u v^\top$
Vector covariant derivative, projection of covariant derivative index:
${\gamma_u}^a \nabla_a v_v$
$= {\gamma_u}^a (\nabla_a (v^\perp)_v - v^\top K_{av} - v^\top n_a a_v + n_v \nabla_a v^\top)$
$=
{\gamma_u}^a \nabla_a (v^\perp)_v
- v^\top K_{uv}
+ n_v {\gamma_u}^a \nabla_a v^\top
$
$=
{\gamma_u}^a \nabla_a (v^\perp)_v
- v^\top K_{uv}
+ n_v \nabla^\perp_u v^\top
$
$=
{\gamma_u}^a \delta_v^b \nabla_a (v^\perp)_b
- v^\top K_{uv}
+ n_v \nabla^\perp_u v^\top
$
$=
{\gamma_u}^a ({\gamma_v}^b - n_v n^b) \nabla_a (v^\perp)_b
- v^\top K_{uv}
+ n_v \nabla^\perp_u v^\top
$
$=
{\gamma_u}^a {\gamma_v}^b \nabla_a (v^\perp)_b
- {\gamma_u}^a n_v n^b \nabla_a (v^\perp)_b
- v^\top K_{uv}
+ n_v \nabla^\perp_u v^\top
$
...using $\nabla_a (n^b (v^\perp)_b) = 0$ therefore $n^b \nabla_a (v^\perp)_b = -(v^\perp)_b \nabla_a n^b$...
$=
\nabla^\perp_u (v^\perp)_v
+ {\gamma_u}^a n_v (v^\perp)^b \nabla_a n_b
- v^\top K_{uv}
+ n_v \nabla^\perp_u v^\top
$
$=
\nabla^\perp_u (v^\perp)_v
+ {\gamma_u}^a n_v (v^\perp)^b (-K_{ab} - n_a a_b)
- v^\top K_{uv}
+ n_v \nabla^\perp_u v^\top
$
...using ${\gamma_u}^a n_a = 0$...
$=
\nabla^\perp_u (v^\perp)_v
- K_{ua} (v^\perp)^a n_v
+ n_v \nabla^\perp_u v^\top
- K_{uv} v^\top
$
... for a spatial vector, set $v^\top = 0$:
${\gamma_u}^a \nabla_a (v^\perp)_v
= \nabla^\perp_u (v^\perp)_v - K_{ua} (v^\perp)^a n_v$
Normal contraction vs covariant derivative of projection:
$n^a \nabla_a v_v$
$= n^a (
\nabla_a (v^\perp)_v
- v^\top K_{av}
- v^\top n_a a_v
+ n_v \nabla_a v^\top
)$
$=
n^a \nabla_a (v^\perp)_v
- v^\top n^a K_{av}
- v^\top n^a n_a a_v
+ n_v n^a \nabla_a v^\top
$
$=
n^a \nabla_a (v^\perp)_v
+ v^\top a_v
+ n_v n^a \nabla_a v^\top
$
Projection and normal components of covariant derivative of vector:
$\nabla_u v_v$
$= \delta_u^a \nabla_a v_v$
$= ({\gamma_u}^a - n_u n^a) \nabla_a v_v$
$=
{\gamma_u}^a \nabla_a v_v
- n_u n^a \nabla_a v_v
$
$=
(
\nabla^\perp_u (v^\perp)_v
- K_{ub} (v^\perp)^b n_v
+ n_v \nabla^\perp_u v^\top
- K_{uv} v^\top
)
- n_u (
n^a \nabla_a (v^\perp)_v
+ v^\top a_v
+ n_v n^a \nabla_a v^\top
)
$
$=
\nabla^\perp_u (v^\perp)_v
- K_{ub} (v^\perp)^b n_v
+ n_v \nabla^\perp_u v^\top
- K_{uv} v^\top
- n_u n^a \nabla_a (v^\perp)_v
- n_u v^\top a_v
- n_u n_v n^a \nabla_a v^\top
$
$=
\nabla^\perp_u (v^\perp)_v
- n_u n^a \nabla_a (v^\perp)_v
- K_{ua} (v^\perp)^a n_v
- (K_{uv} + n_u a_v) v^\top
+ n_v \nabla^\perp_u v^\top
- n_u n_v n^a \nabla_a v^\top
$
Vector covariant derivative, contract both indexes with normal:
$n^a n^b \nabla_a v_b$
$= n^b (
n^a \nabla_a (v^\perp)_b
+ v^\top a_b
+ n_b n^a \nabla_a v^\top
)$
$=
n^a n^b \nabla_a (v^\perp)_b
+ v^\top n^b a_b
+ n^b n_b n^a \nabla_a v^\top
$
$=
n^b n^a \nabla_a (v^\perp)_b
- n^a \nabla_a v^\top
$
$=
-n^a (v^\perp)^b \nabla_a n_b
- n^a \nabla_a v^\top
$
$=
-n^a (v^\perp)^b (-K_{ab} - n_a a_b)
- n^a \nabla_a v^\top
$
$=
n^a (v^\perp)^b K_{ab}
+ n^a (v^\perp)^b n_a a_b
- n^a \nabla_a v^\top
$
$=
-(v^\perp)^a a_a
- n^a \nabla_a v^\top
$
Vector covariant derivative, projection of vector index:
${\gamma_v}^b \nabla_u v_b$
$= {\gamma_v}^b \nabla_u v_b$
$= (\delta_v^b + n_v n^b) (
\nabla^\perp_u (v^\perp)_b
- n_u n^a \nabla_a (v^\perp)_b
- K_{ua} (v^\perp)^a n_b
- (K_{ub} + n_u a_b) v^\top
+ n_b \nabla^\perp_u v^\top
- n_u n_b n^a \nabla_a v^\top
)$
$=
\delta_v^b \nabla^\perp_u (v^\perp)_b
- \delta_v^b n_u n^a \nabla_a (v^\perp)_b
- \delta_v^b K_{ua} (v^\perp)^a n_b
- \delta_v^b (K_{ub} + n_u a_b) v^\top
+ \delta_v^b n_b \nabla^\perp_u v^\top
- \delta_v^b n_u n_b n^a \nabla_a v^\top
+ n_v n^b \nabla^\perp_u (v^\perp)_b
- n_v n^b n_u n^a \nabla_a (v^\perp)_b
- n_v n^b K_{ua} (v^\perp)^a n_b
- n_v n^b (K_{ub} + n_u a_b) v^\top
+ n_v n^b n_b \nabla^\perp_u v^\top
- n_v n^b n_u n_b n^a \nabla_a v^\top
$
$=
\nabla^\perp_u (v^\perp)_v
- n_u n^a \nabla_a (v^\perp)_v
- n_v n_u n^a n^b \nabla_a (v^\perp)_b
- (K_{uv} + n_u a_v) v^\top
$
$=
\nabla^\perp_u (v^\perp)_v
- n_u n^a \nabla_a (v^\perp)_v
- n_v n_u n^a (v^\perp)^b (K_{ab} + n_a a_b)
- (K_{uv} + n_u a_v) v^\top
$
$=
\nabla^\perp_u (v^\perp)_v
- n_u n^a \nabla_a (v^\perp)_v
+ n_u n_v (v^\perp)^a a_a
- (K_{uv} + n_u a_v) v^\top
$
Vector covariant derivative, contract normal with covariant derivative index, project vector index:
$n^a {\gamma_v}^b \nabla_a v_b$
$= n^a (
\nabla^\perp_a (v^\perp)_v
- n_a n^b \nabla_b (v^\perp)_v
+ n_a n_v (v^\perp)^b a_b
- (K_{av} + n_a a_v) v^\top
)$
$=
n^a \nabla^\perp_a (v^\perp)_v
- n^a n_a n^b \nabla_b (v^\perp)_v
+ n^a n_a n_v (v^\perp)^b a_b
- n^a (K_{av} + n_a a_v) v^\top
$
$=
n^a \nabla_a (v^\perp)_v
- n_v (v^\perp)^a a_a
+ a_v v^\top
$
Vector covariant derivative, project covariant derivative index, contract normal with vector index:
$n^b {\gamma_u}^a \nabla_a v_b$
$= n^b (
\nabla^\perp_u (v^\perp)_b
- K_{ua} (v^\perp)^a n_b
+ n_b \nabla^\perp_u v^\top
- K_{ub} v^\top
)$
$=
n^b \nabla^\perp_u (v^\perp)_b
- n^b K_{ua} (v^\perp)^a n_b
+ n^b n_b \nabla^\perp_u v^\top
- n^b K_{ub} v^\top
$
$=
K_{ua} (v^\perp)^a
- \nabla^\perp_u v^\top
$
Vector covariant derivative, project both indexes:
${\gamma_u}^a {\gamma_v}^b \nabla_a v_b$
$= {\gamma_u}^a (
\nabla^\perp_a (v^\perp)_v
- n_a n^b \nabla_b (v^\perp)_v
+ n_a n_v (v^\perp)^b a_b
- (K_{av} + n_a a_v) v^\top
)$
$=
{\gamma_u}^a \nabla^\perp_a (v^\perp)_v
- {\gamma_u}^a n_a n^b \nabla_b (v^\perp)_v
+ {\gamma_u}^a n_a n_v (v^\perp)^b a_b
- {\gamma_u}^a (K_{av} + n_a a_v) v^\top
$
$=
\nabla^\perp_u (v^\perp)_v
- K_{uv} v^\top
$
Put them all together to verify it reconstructs the original.
$\nabla_u v_v$
$= \delta_u^a \delta_v^b \nabla_a v_b$
$= ({\gamma_u}^a - n_u n^a) ({\gamma_v}^b - n_v n^b) \nabla_a v_b$
$=
{\gamma_u}^a {\gamma_v}^b \nabla_a v_b
- n_u n^a {\gamma_v}^b \nabla_a v_b
- n_v {\gamma_u}^a n^b \nabla_a v_b
+ n_u n_v n^a n^b \nabla_a v_b
$
$=
(
\nabla^\perp_u (v^\perp)_v
- K_{uv} v^\top
)
- n_u (
n^a \nabla_a (v^\perp)_v
- n_v (v^\perp)^a a_a
+ a_v v^\top
)
- n_v (
K_{ua} (v^\perp)^a
- \nabla^\perp_u v^\top
)
+ n_u n_v (
-(v^\perp)^a a_a
- n^a \nabla_a v^\top
)
$
$=
\nabla^\perp_u (v^\perp)_v
- n_u n^a \nabla_a (v^\perp)_v
- K_{ua} (v^\perp)^a n_v
- (K_{uv} + n_u a_v) v^\top
+ n_v \nabla^\perp_u v^\top
- n_u n_v n^a \nabla_a v^\top
$
...should equal our previous definition of the covariant derivative in projected terms:
$=
\nabla^\perp_u (v^\perp)_v
- n_u n^a \nabla_a (v^\perp)_v
- K_{ua} (v^\perp)^a n_v
- (K_{uv} + n_u a_v) v^\top
+ n_v \nabla^\perp_u v^\top
- n_u n_v n^a \nabla_a v^\top
$
Vector divergence in projection decomposition:
$\nabla_a v^a$
$= g^{ab} \nabla_a v_b$
$= (\gamma^{ab} - n^a n^b) \nabla_a v_b$
$= (\gamma^{ab} - n^a n^b) (
\nabla^\perp_a (v^\perp)_b
- n_a n^c \nabla_c (v^\perp)_b
- K_{ac} (v^\perp)^c n_b
- (K_{ab} + n_a a_b) v^\top
+ \nabla^\perp_a v^\top n_b
- n_a n_b n^c \nabla_c v^\top
)$
$=
\gamma^{ab} \nabla^\perp_a (v^\perp)_b
- \gamma^{ab} n_a n^c \nabla_c (v^\perp)_b
- \gamma^{ab} K_{ac} (v^\perp)^c n_b
- \gamma^{ab} K_{ab} v^\top
- \gamma^{ab} n_a a_b v^\top
+ \gamma^{ab} \nabla^\perp_a v^\top n_b
- \gamma^{ab} n_a n_b n^c \nabla_c v^\top
- n^a n^b \nabla^\perp_a (v^\perp)_b
+ n^a n^b n_a n^c \nabla_c (v^\perp)_b
+ n^a n^b K_{ac} (v^\perp)^c n_b
+ n^a n^b K_{ab} v^\top
+ n^a n^b n_a a_b v^\top
- n^a n^b \nabla^\perp_a v^\top n_b
+ n^a n^b n_a n_b n^c \nabla_c v^\top
$
$=
\nabla^\perp_a (v^\perp)^a
- K v^\top
- n^c n^b \nabla_c (v^\perp)_b
+ n^a \nabla_a v^\top
$
$=
\nabla^\perp_a (v^\perp)^a
- K v^\top
+ n^c (v^\perp)^b (-K_{cb} - n_c a_b)
+ n^a \nabla_a v^\top
$
$=
\nabla^\perp_a (v^\perp)^a
+ a_a (v^\perp)^a
+ n^a \nabla_a v^\top
- K v^\top
$
Projection vs covariant derivative of a (p q) tensor:
$\nabla_u {T^A}_B$
$= \nabla_u (
\overset{p}{\underset{i=1}{\Pi}} \delta_{c_i}^{a_i}
\cdot
\overset{q}{\underset{i=1}{\Pi}} \delta_{b_i}^{d_i}
\cdot
{T^C}_D
)$
$= \nabla_u (
\overset{p}{\underset{i=1}{\Pi}} ({\gamma_{c_i}}^{a_i} - n_{c_i} n^{a_i})
\cdot
\overset{q}{\underset{i=1}{\Pi}} ({\gamma_{b_i}}^{d_i} - n_{b_i} n^{d_i})
\cdot
{T^C}_D
)$
$= \nabla_u \left(
\left(
\overset{p}{\underset{i=1}{\Pi}} {\gamma_{c_i}}^{a_i}
\cdot \overset{q}{\underset{i=1}{\Pi}} {\gamma_{b_i}}^{d_i}
- n_{c_k} n^{a_k}
\cdot \overset{p}{\underset{i \ne k}{\underset{i=1}{\Pi}}} {\gamma_{c_i}}^{a_i}
\cdot \overset{q}{\underset{i=1}{\Pi}} {\gamma_{b_i}}^{d_i}
- n_{b_k} n^{d_k}
\cdot \overset{p}{\underset{i=1}{\Pi}} {\gamma_{c_i}}^{a_i}
\cdot \overset{q}{\underset{i \ne k}{\underset{i=1}{\Pi}}} {\gamma_{b_i}}^{d_i}
+ ...
+ (-1)^{p+q}
\overset{p}{\underset{i=1}{\Pi}} n_{c_i} n^{a_i}
\cdot \overset{q}{\underset{i=1}{\Pi}} n_{b_i} n^{d_i}
\right)
\cdot {T^C}_D
\right)$
$= \nabla_u \left(
\perp {T^A}_B
+ \left(
- n_{c_k} n^{a_k}
\cdot \overset{p}{\underset{i \ne k}{\underset{i=1}{\Pi}}} {\gamma_{c_i}}^{a_i}
\cdot \overset{q}{\underset{i=1}{\Pi}} {\gamma_{b_i}}^{d_i}
- n_{b_k} n^{d_k}
\cdot \overset{p}{\underset{i=1}{\Pi}} {\gamma_{c_i}}^{a_i}
\cdot \overset{q}{\underset{i \ne k}{\underset{i=1}{\Pi}}} {\gamma_{b_i}}^{d_i}
+ ...
+ (-1)^{p+q}
\overset{p}{\underset{i=1}{\Pi}} n_{c_i} n^{a_i}
\cdot \overset{q}{\underset{i=1}{\Pi}} n_{b_i} n^{d_i}
\right)
\cdot {T^C}_D
\right)$
...next distribute that covariant derivative...
...next replace the individual $\nabla_a n_b = -K_{ab} - n_a a_b$...
...next do the same delta-to-gamma-plus-n's trick on the outside...
... and finally you will have a general (p q) relation between a tensor's covariant derivative, its projection's projected covariant derivative, and a bunch of normal and acceleration and extrinsic curvature vectors.
$\nabla^\perp_a \nabla^\perp_b v^c$
$= \nabla^\perp_a ({\gamma_b}^d {\gamma_e}^c \nabla_d v^e)$
$= {\gamma_a}^p {\gamma_b}^q {\gamma_r}^c \nabla_p ({\gamma_q}^d {\gamma_e}^r \nabla_d v^e)$
$= {\gamma_a}^p {\gamma_b}^q {\gamma_r}^c ({\gamma_e}^r \nabla_p {\gamma_q}^d \nabla_d v^e
+ {\gamma_q}^d \nabla_p {\gamma_e}^r \nabla_d v^e
+ {\gamma_q}^d {\gamma_e}^r \nabla_p \nabla_d v^e)$
$= {\gamma_a}^p {\gamma_b}^q {\gamma_r}^c {\gamma_e}^r \nabla_p {\gamma_q}^d \nabla_d v^e
+ {\gamma_a}^p {\gamma_b}^q {\gamma_r}^c {\gamma_q}^d \nabla_p {\gamma_e}^r \nabla_d v^e
+ {\gamma_a}^p {\gamma_b}^q {\gamma_r}^c {\gamma_q}^d {\gamma_e}^r \nabla_p \nabla_d v^e$
$= {\gamma_a}^p {\gamma_b}^q {\gamma_r}^c (\nabla_p {\gamma_q}^d \nabla_d v^r + \nabla_p {\gamma_e}^r \nabla_q v^e + \nabla_p \nabla_q v^r)$
$= {\gamma_a}^p {\gamma_b}^q {\gamma_r}^c \nabla_p \nabla_q v^r
+ {\gamma_a}^p {\gamma_b}^q {\gamma_r}^c \nabla_p (n_q n^d) \nabla_d v^r
+ {\gamma_a}^p {\gamma_b}^q {\gamma_r}^c \nabla_p (n_e n^r) \nabla_q v^e$
$= {\gamma_a}^p {\gamma_b}^q {\gamma_r}^c \nabla_p \nabla_q v^r
+ {\gamma_a}^p {\gamma_b}^q {\gamma_r}^c (n^d \nabla_p n_q + n_q \nabla_p n^d) \nabla_d v^r
+ {\gamma_a}^p {\gamma_b}^q {\gamma_r}^c (n^r \nabla_p n_e + n_e \nabla_p n^r) \nabla_q v^e$
$= {\gamma_a}^p {\gamma_b}^q {\gamma_r}^c \nabla_p \nabla_q v^r
+ {\gamma_a}^p {\gamma_b}^q {\gamma_r}^c n^d \nabla_p n_q \nabla_d v^r
+ {\gamma_a}^p {\gamma_b}^q {\gamma_r}^c n_e \nabla_p n^r \nabla_q v^e$
$= {\gamma_a}^p {\gamma_b}^q {\gamma_r}^c \nabla_p \nabla_q v^r
+ {\gamma_r}^c ({\gamma_a}^p {\gamma_b}^q \nabla_p n_q) n^d \nabla_d v^r
+ {\gamma_b}^q ({\gamma_a}^p {\gamma_r}^c \nabla_p n^r) n_e \nabla_q v^e$
$= {\gamma_a}^p {\gamma_b}^q {\gamma_r}^c \nabla_p \nabla_q v^r
- K_{ab} {\gamma_r}^c n^p \nabla_p v^r
+ {\gamma_b}^q g^{cs} ({\gamma_a}^p {\gamma_s}^r \nabla_p n_r) n_e \nabla_q v^e$
$= {\gamma_a}^p {\gamma_b}^q {\gamma_r}^c \nabla_p \nabla_q v^r
- K_{ab} {\gamma_r}^c n^p \nabla_p v^r
- {K_a}^c n_e {\gamma_b}^q \nabla_q v^e$
$\nabla^\perp_a \nabla^\perp_b v^c = {\gamma_a}^p {\gamma_b}^q {\gamma_r}^c \nabla_p \nabla_q v^r
- K_{ab} {\gamma_r}^c n^p \nabla_p v^r
- {K_a}^c K_{bp} v^p$
Riemann metric tensor
$R_{abcd}$
add identity to prepare for substitution with projection matrix
$=g^p_a g^q_b g^r_c g^s_d R_{pqrs}$
...and substitute...
$=({\gamma_a}^p-n^p n_a) ({\gamma_b}^q - n^q n_b) ({\gamma_c}^r - n^r n_c) ({\gamma_d}^s-n^s n_d) R_{pqrs}$
$= (
{\gamma_a}^p {\gamma_b}^q {\gamma_c}^r {\gamma_d}^s
- n^p n_a {\gamma_b}^q {\gamma_c}^r {\gamma_d}^s
- {\gamma_a}^p n^q n_b {\gamma_c}^r {\gamma_d}^s
- {\gamma_a}^p {\gamma_b}^q n^r n_c {\gamma_d}^s
- {\gamma_a}^p {\gamma_b}^q {\gamma_c}^r n^s n_d
+ n^p n_a n^q n_b {\gamma_c}^r {\gamma_d}^s
+ n^p n_a {\gamma_b}^q n^r n_c {\gamma_d}^s
+ n^p n_a {\gamma_b}^q {\gamma_c}^r n^s n_d
+ {\gamma_a}^p n^q n_b n^r n_c {\gamma_d}^s
+ {\gamma_a}^p n^q n_b {\gamma_c}^r n^s n_d
+ {\gamma_a}^p {\gamma_b}^q n^r n_c n^s n_d
- n^p n_a n^q n_b n^r n_c {\gamma_d}^s
- n^p n_a n^q n_b {\gamma_c}^r n^s n_d
- n^p n_a {\gamma_b}^q n^r n_c n^s n_d
- {\gamma_a}^p n^q n_b n^r n_c n^s n_d
+ n^p n_a n^q n_b n^r n_c n^s n_d ) R_{pqrs}$
remove contraction of antisymmetric indices...
$= ({\gamma_a}^p {\gamma_b}^q {\gamma_c}^r {\gamma_d}^s
- {\gamma_b}^q {\gamma_c}^r {\gamma_d}^s n^p n_a
- {\gamma_a}^p {\gamma_c}^r {\gamma_d}^s n^q n_b
- {\gamma_a}^p {\gamma_b}^q {\gamma_d}^s n^r n_c
- {\gamma_a}^p {\gamma_b}^q {\gamma_c}^r n^s n_d
+ {\gamma_b}^q {\gamma_d}^s n^p n_a n^r n_c
+ {\gamma_b}^q {\gamma_c}^r n^p n_a n^s n_d
+ {\gamma_a}^p {\gamma_d}^s n^q n_b n^r n_c
+ {\gamma_a}^p {\gamma_c}^r n^q n_b n^s n_d) R_{pqrs}$
use antisymmetry to switch indices, and combine ...
$R_{abcd} = ({\gamma_a}^p {\gamma_b}^q {\gamma_c}^r {\gamma_d}^s
- 2 {\gamma_a}^p {\gamma_b}^q {\gamma_{[c}}^r n_{d]} n^s
- 2 {\gamma_c}^p {\gamma_d}^q {\gamma_{[a}}^r n_{b]} n^s
+ 2 {\gamma_a}^p {\gamma_{[c}}^r n_{d]} n_b n^q n^s
- 2 {\gamma_b}^p {\gamma_{[c}}^r n_{d]} n_a n^q n^s) R_{pqrs}$ ... (2010 Baumgarte & Shapiro, eqn 2.61)
Riemann tensor of spatial covariant derivative:
${(R^\perp)^c}_{dab} v^d$
$= \nabla^\perp_a \nabla^\perp_b v^c
- \nabla^\perp_b \nabla^\perp_a v^c
$
$= {\gamma_a}^p {\gamma_b}^q {\gamma_r}^c \nabla_p \nabla_q v^r
- K_{ab} {\gamma_r}^c n^p \nabla_p v^r
- {K_a}^c K_{bp} v^p
- {\gamma_b}^p {\gamma_a}^q {\gamma_r}^c \nabla_p \nabla_q v^r
+ K_{ba} {\gamma_r}^c n^p \nabla_p v^r
+ {K_b}^c K_{ap} v^p
$
$= 2 {\gamma_a}^p {\gamma_b}^q {\gamma_r}^c \nabla_{[p} \nabla_{q]} v^r
- 2 K_{[ab]} {\gamma_r}^c n^p \nabla_p v^r
- 2 {K_{[a}}^c K_{b]p} v^p
$
$= {\gamma_a}^p {\gamma_b}^q {\gamma_r}^c {R^r}_{dpq} v^d
- {K_a}^c K_{bd} v^d + {K_b}^c K_{ad} v^d
- 2 K_{[ab]} {\gamma_r}^c n^p \nabla_p v^r
$
${(R^\perp)^c}_{dab} v^d =
{\gamma_a}^p {\gamma_b}^q {\gamma_r}^c {R^r}_{dpq} v^d
- {K_a}^c K_{bd} v^d
+ {K_b}^c K_{ad} v^d
- 2 K_{[ab]} {\gamma_r}^c n^p \nabla_p v^r
$
...if $v^d$ is spatial...
${(R^\perp)^c}_{dab} v^d =
{\gamma_a}^p {\gamma_b}^q {\gamma_r}^c {\gamma_d}^s {R^r}_{spq} v^d
- {K_a}^c K_{bd} v^d
+ {K_b}^c K_{ad} v^d
- 2 K_{[ab]} {\gamma_r}^c n^p \nabla_p v^r
$
...if $K_{[ab]} = 0$... But this is only true for projected-commutation-free basii (prove me)...
${(R^\perp)^c}_{dab} v^d =
({\gamma_a}^p {\gamma_b}^q {\gamma_r}^c {\gamma_d}^s {R^r}_{spq}
- {K_a}^c K_{bd}
+ {K_b}^c K_{ad}) v^d
$
${R^\perp}_{abcd} + K_{ac} K_{bd} - K_{ad} K_{bc} = {\gamma_a}^p {\gamma_b}^q {\gamma_c}^r {\gamma_d}^s R_{pqrs}$
Gauss' equation. (2008 Alcubierre, eqn 2.41; 2010 Baumgarte & Shapiro, eqn 2.68)
Gauss' equation, contracting once with the spatial metric:
$ \gamma^{ac} (
{R^\perp}_{abcd}
+ K_{ac} K_{bd}
- K_{ad} K_{bc}
) = \gamma^{ac} {\gamma_a}^p {\gamma_b}^q {\gamma_c}^r {\gamma_d}^s R_{pqrs}
$
$ \gamma^{ac} {R^\perp}_{abcd}
+ \gamma^{ac} K_{ac} K_{bd}
- \gamma^{ac} K_{ad} K_{bc}
= \gamma^{pr} {\gamma_b}^q {\gamma_d}^s R_{pqrs}
$
$ {R^\perp}_{bd}
+ K K_{bd}
- K_{bc} {K^c}_d
= \gamma^{pr} {\gamma_b}^q {\gamma_d}^s R_{pqrs}
$ (Baumgarte & Shapiro 2010, eqn. 2.84)
Gauss' equation contracting twice with the spatial metric:
$ \gamma^{bd} ({R^\perp}_{bd}
+ K K_{bd}
- K_{bc} {K^c}_d
) = \gamma^{bd} \gamma^{pr} {\gamma_b}^q {\gamma_d}^s R_{pqrs}
$
$ \gamma^{bd} {R^\perp}_{bd}
+ \gamma^{bd} K K_{bd}
- \gamma^{bd} K_{bc} {K^c}_d
= \gamma^{pr} \gamma^{qs} R_{pqrs}
$
$ R^\perp
+ K^2
- K_{ab} K^{ab}
= \gamma^{pr} \gamma^{qs} R_{pqrs}
$ (2008 Alcubierre, eqn 2.4.4, 2010 Baumgarte & Shapiro, eqn 2.85)
spatial Riemann is orthogonal to normal:
${R^\perp}_{abcd} n^d$
$={\gamma_a}^p {\gamma_b}^q {\gamma_c}^r {\gamma_d}^s R_{pqrs} n^d - K_{ac} K_{bd} n^d + K_{ad} K_{bc} n^d$
$={\gamma_a}^p {\gamma_b}^q {\gamma_c}^r (n^d {\gamma_d}^s) R_{pqrs} - K_{ac} (K_{bd} n^d) + (K_{ad} n^d) K_{bc}$
$={\gamma_a}^p {\gamma_b}^q {\gamma_c}^r \cdot 0 \cdot R_{pqrs} - K_{ac} \cdot 0 + 0 \cdot K_{bc}$
${R^\perp}_{abcd} n^d = 0$
spatial covariant derivative of extrinsic curvature:
$\nabla^\perp_a K_{bc}$
$=-{\gamma_a}^p {\gamma_b}^q {\gamma_c}^r \nabla_p K_{qr}$
$=-{\gamma_a}^p {\gamma_b}^q {\gamma_c}^r \nabla_p (\nabla_q n_r + n_q a_r)$
$=-{\gamma_a}^p {\gamma_b}^q {\gamma_c}^r (\nabla_p \nabla_q n_r + \nabla_p n_q a_r + n_q \nabla_p a_r)$
$=-{\gamma_a}^p {\gamma_b}^q {\gamma_c}^r (\nabla_p \nabla_q n_r + \nabla_p n_q a_r)$
$=-{\gamma_a}^p {\gamma_b}^q {\gamma_c}^r (\nabla_p \nabla_q n_r + (\nabla_p n_q + n_p a_q) a_r)$
$=-{\gamma_a}^p {\gamma_b}^q {\gamma_c}^r (\nabla_p \nabla_q n_r + K_{pq} a_r)$
$=-{\gamma_a}^p {\gamma_b}^q {\gamma_c}^r \nabla_p \nabla_q n_r + {\gamma_a}^p {\gamma_b}^q {\gamma_c}^r K_{pq} a_r$
$\nabla^\perp_a K_{bc}=-{\gamma_a}^p {\gamma_b}^q {\gamma_c}^r \nabla_p \nabla_q n_r + a_c K_{ab}$
therefore
$\nabla^\perp_{[a} K_{b]c} = -{\gamma_a}^p {\gamma_b}^q {\gamma_c}^r \nabla_{[p} \nabla_{q]} n_r$
Codazzi equation:
$\nabla^\perp_b K_{ac} - \nabla^\perp_b K_{bc} = {\gamma_a}^p {\gamma_b}^q {\gamma_c}^r n^s R_{pqrs}$
Decomposition of the Einstein curvature tensor:
${\gamma_a}^u n^v G_{uv}$
$= {\gamma_a}^u n^v (R_{uv} - \frac{1}{2} g_{uv} R)$
$= {\gamma_a}^u n^v R_{uv} - \frac{1}{2} {\gamma_a}^u n^v g_{uv} R$
$= {\gamma_a}^u n^v R_{uv} - \frac{1}{2} {\gamma_a}^u n_u R$
$= {\gamma_a}^u n^v R_{uv}$ (2008 Alcubierre, eqn 2.4.7)
$= {\gamma_a}^q g^{pr} n^s R_{pqrs}$
${\gamma_a}^u {\gamma_b}^v G_{uv}$
$= {\gamma_a}^u {\gamma_b}^v (R_{uv} - \frac{1}{2} g_{uv} R)$
$= {\gamma_a}^u {\gamma_b}^v R_{uv} - \frac{1}{2} {\gamma_a}^u {\gamma_b}^v g_{uv} R$
$= {\gamma_a}^u {\gamma_b}^v R_{uv} - \frac{1}{2} \gamma_{ab} R$
$= g^{pr} ({\gamma_a}^q {\gamma_b}^s - \frac{1}{2} \gamma_{ab} g^{qs}) R_{pqrs}$
$n^u n^v G_{uv}$
$= n^u n^v (R_{uv} - \frac{1}{2} R g_{uv})$
$= n^u n^v R_{uv} - \frac{1}{2} R n^u n_u$
$= n^u n^v R_{uv} + \frac{1}{2} R$
$= n^u n^v R_{uv} + \frac{1}{2} g^{uv} R_{uv}$
$= \frac{1}{2} (\gamma^{uv} + n^u n^v) R_{uv}$ ... this says the trace-reversal of the Ricci, i.e. the Einstein tensor, is the Ricci contracted with the trace-reversal of the projection.
$= \frac{1}{2} R + R_{uv} n^u n^v$
Therefore:
$n^u n^v G_{uv} = n^u n^v R_{uv} + \frac{1}{2} R$ (2008 Alcubierre, eqn. 2.7.2)
Keep going:
$n^u n^v G_{uv} = \frac{1}{2} (g^{pr} g^{qs} R_{pqrs} + g^{pr} n^q n^s R_{pqrs} + n^p n^r g^{qs} R_{pqrs})$
...using $R_{abcd} n^c n^d = 0$
$= \frac{1}{2} (g^{pr} g^{qs} R_{pqrs} + g^{pr} n^q n^s R_{pqrs} + n^p n^r g^{qs} R_{pqrs} + n^p n^r n^q n^s R_{pqrs})$
...un-distribute...
$= \frac{1}{2} (g^{pr} + n^p n^r) (g^{qs} + n^q n^s) R_{pqrs}$
...use definition of projection...
$= \frac{1}{2} \gamma^{pr} \gamma^{qs} R_{pqrs}$
Therefore:
$n^u n^v G_{uv} = \frac{1}{2} \gamma^{pr} \gamma^{qs} R_{pqrs}$ (2008 Alcubierre, eqn 2.4.3)
Also:
$\gamma^{pr} \gamma^{qs} R_{pqrs} = R + 2 R_{uv} n^u n^v$ (2010 Baumgarte & Shapiro, eqn. 2.86)
Using
$\gamma^{pr} \gamma^{qs} R_{pqrs} = R^\perp + K^2 - K^{ab} K_{ab}$
and
$n^u n^v G_{uv} = \frac{1}{2} \gamma^{pr} \gamma^{qs} R_{pqrs}$
we get:
$2 n^u n^v G_{uv} = R^\perp + K^2 - K^{ab} K_{ab}$ (2008 Alcubierre, eqn 2.4.5, 2010 Baumgarte & Shapiro, eqn. 2.88)
start with Codazzi equation:
$ \nabla^\perp_b K_{ac}
- \nabla^\perp_b K_{bc}
= {\gamma_a}^p {\gamma_c}^r {\gamma_b}^q n^s R_{pqrs}
$
multiply by $\gamma^{ac}$:
$ \gamma^{ac} \nabla^\perp_b K_{ac}
- \gamma^{ac} \nabla^\perp_b K_{bc}
= \gamma^{ac} {\gamma_a}^p {\gamma_c}^r {\gamma_b}^q n^s R_{pqrs}
$
$ \gamma^{ac} \nabla^\perp_b K_{ac}
- \gamma^{ac} \nabla^\perp_b K_{bc}
= \gamma^{pr} {\gamma_b}^q n^s R_{pqrs}
= g^{pr} {\gamma_b}^q n^s R_{pqrs} + n^p n^r n^s {\gamma_b}^q R_{pqrs}
$
... using $R_{pqrs} n^r n^s = 0$ by antisymmetry of $R_{pqrs}$ and symmetry of $n^r n^s$:
... and using $\gamma_{ab} \nabla^\perp_c = \nabla^\perp_c \gamma_{ab}$
$ \nabla^\perp_b (\gamma^{ac} K_{ac})
- \nabla^\perp_b (K_{bc} \gamma^{ac})
= g^{pr} {\gamma_b}^q n^s R_{pqrs}
$
...using $g^{pr} {\gamma_b}^q n^s R_{pqrs} = {\gamma_a}^u n^v G_{uv}$
${\gamma_a}^u n^v G_{uv} = \nabla^\perp_a K - \nabla^\perp_b {K_a}^b$ (2008 Alcubierre eqn 2.4.8)
Therefore...
${\gamma_a}^u n^v (R_{uv} - \frac{1}{2} g_{uv} R) = \nabla^\perp_a K - \nabla^\perp_b {K_a}^b$
${\gamma_a}^u n^v R_{uv} - \frac{1}{2} {\gamma_a}^u n_u R = \nabla^\perp_a K - \nabla^\perp_b {K_a}^b$
${\gamma_a}^u n^v R_{uv} = \nabla^\perp_a K - \nabla^\perp_b {K_a}^b$
...and the trace-reversal part cancels anyways.
${\gamma_a}^u {\gamma_b}^v R_{uv}$
$= {\gamma_a}^u {\gamma_b}^v g^{cd} R_{ucvd}$
$= {\gamma_a}^u {\gamma_b}^v (\gamma^{cd} - n^c n^d) R_{ucvd}$
$= {\gamma_a}^u {\gamma_b}^v \gamma^{cd} R_{ucvd}
- {\gamma_a}^u {\gamma_b}^v n^c n^d R_{ucvd}
$
using ${R^\perp}_{abcd} + K_{ac} K_{bd} - K_{ad} K_{bc} = {\gamma_a}^p {\gamma_b}^q {\gamma_c}^r {\gamma_d}^s R_{pqrs}$
so $\gamma^{bd} {R^\perp}_{abcd} + \gamma^{bd} K_{ac} K_{bd} - \gamma^{bd} K_{ad} K_{bc} = {\gamma_a}^p {\gamma_c}^r \gamma^{bd} {\gamma_b}^q {\gamma_d}^s R_{pqrs}$
so ${R^\perp}_{ac} + K K_{ac} - {K_a}^b K_{bc} = {\gamma_a}^p {\gamma_c}^r \gamma^{qs} R_{pqrs}$
$=
{R^\perp}_{ab} + K K_{ab} - {K_a}^c K_{cb}
- {\gamma_a}^u {\gamma_b}^v n^c n^d R_{ucvd}
$
... using ${\gamma_a}^b = \delta_a^b + n_a n^b$
$=
{R^\perp}_{ab} + K K_{ab} - {K_a}^c K_{cb}
- (\delta_a^u + n_a n^u) (\delta_b^v + n_b n^v) n^c n^d R_{ucvd}
$
... using $n^a n^b R_{abcd} = 0$
$=
{R^\perp}_{ab} + K K_{ab} - {K_a}^c K_{cb}
- n^c n^d R_{acbd}
$
$=
{R^\perp}_{ab} + K K_{ab} - {K_a}^c K_{cb}
- {\gamma_a}^p {\gamma_b}^r n^q n^s R_{pqrs}
$
contracting Gauss' equation:
$R^\perp_{abcd} + K_{ac} K_{bd} - K_{ad} K_{bc} = {\gamma_a}^p {\gamma_b}^q {\gamma_c}^r {\gamma_d}^s R_{pqrs}$
$g^{ac} (R^\perp_{abcd} + K_{ac} K_{bd} - K_{ad} K_{bc}) = g^{ac} {\gamma_a}^p {\gamma_b}^q {\gamma_c}^r {\gamma_d}^s R_{pqrs}$
${R^{\perp a}}_{bad} + K K_{bd} - {K^a}_d K_{ba} = \gamma^{pr} {\gamma_b}^q {\gamma_d}^s R_{pqrs}$
$g^{bd}({R^\perp}_{bd} + K K_{bd} - {K^a}_d K_{ba}) = g^{bd} \gamma^{pr} {\gamma_b}^q {\gamma_d}^s R_{pqrs}$
${R^\perp} + K^2 - K^{ab} K_{ab} = \gamma^{pr} \gamma^{qs} R_{pqrs}$
...substitute projected Riemann...
$R^\perp+K^2-K_{ab}K^{ab}=R + 2n^a n^b R_{ab}$
...solve for R...
$R = R^\perp - K_{ab} K^{ab} + K^2 - 2 n^a n^b R_{ab}$
Back to the once-contracted Codazzi equations:
$ {R^\perp}_{bd}
+ K K_{bd}
- K_{bc} {K^c}_d
= \gamma^{pr} {\gamma_b}^q {\gamma_d}^s R_{pqrs}
$
now use $\gamma^{pr} = g^{pr} + n^p n^r$
$ {R^\perp}_{bd}
+ K K_{bd}
- K_{bc} {K^c}_d
= (g^{pr} + n^p n^r) {\gamma_b}^q {\gamma_d}^s R_{pqrs}
$
$ {R^\perp}_{bd}
+ K K_{bd}
- K_{bc} {K^c}_d
=
g^{pr} {\gamma_b}^q {\gamma_d}^s R_{pqrs}
+ n^p n^r {\gamma_b}^q {\gamma_d}^s R_{pqrs}
$
using $g^{pr} R_{pqrs} = R_{qs}$
$ {R^\perp}_{bd}
+ K K_{bd}
- K_{bc} {K^c}_d
=
{\gamma_b}^q {\gamma_d}^s R_{qs}
+ n^p n^r {\gamma_b}^q {\gamma_d}^s R_{pqrs}
$
using $R_{ab} = G_{ab} + \frac{1}{2} g_{ab} R$
Another route is to use the trace-reversed EFE here: $R_{ab} = 8 \pi (T_{ab} - \frac{1}{2} g_{ab} T)$, which lets you arrive at 2010 Baumgarte & Shapiro, eqn. 2.103. Maybe I'll do that later in the "ADM Constraints" worksheet where I introduce the EFE and the projected stress-energy tensor to these normal projected equations.
$ {R^\perp}_{bd}
+ K K_{bd}
- K_{bc} {K^c}_d
=
{\gamma_b}^q {\gamma_d}^s (G_{qs} + \frac{1}{2} g_{qs} R)
+ n^p n^r {\gamma_b}^q {\gamma_d}^s R_{pqrs}
$
$ {\gamma_b}^q {\gamma_d}^s G_{qs}
=
{R^\perp}_{bd}
+ K K_{bd}
- K_{bc} {K^c}_d
- \frac{1}{2} \gamma_{bd} R
- n^p n^r {\gamma_b}^q {\gamma_d}^s R_{pqrs}
$
... using our previous definition of projected Einstein curvature:
${\gamma_a}^u {\gamma_b}^v G_{uv} = {\gamma_a}^u {\gamma_b}^v R_{uv} - \frac{1}{2} \gamma_{ab} R$
$
{\gamma_b}^u {\gamma_d}^v R_{uv}
- \frac{1}{2} \gamma_{bd} R
=
{R^\perp}_{bd}
+ K K_{bd}
- K_{bc} {K^c}_d
- \frac{1}{2} \gamma_{bd} R
- n^p n^r {\gamma_b}^q {\gamma_d}^s R_{pqrs}
$
$
{\gamma_b}^u {\gamma_d}^v R_{uv}
=
{R^\perp}_{bd}
+ K K_{bd}
- K_{bc} {K^c}_d
- n^p n^r {\gamma_b}^q {\gamma_d}^s R_{pqrs}
$
... and we're back at our definition of projected Ricci curvature.
Start again with:
$ {\gamma_a}^u {\gamma_b}^v G_{uv}
=
{R^\perp}_{ab}
+ K K_{ab}
- K_{ac} {K^c}_b
- \frac{1}{2} \gamma_{ab} R
- n^p n^r {\gamma_a}^q {\gamma_b}^s R_{pqrs}
$
Substitute: $R = R^\perp - K_{ab} K^{ab} + K^2 - 2 n^a n^b R_{ab}$
$ {\gamma_a}^u {\gamma_b}^v G_{uv}
=
{R^\perp}_{ab}
+ K K_{ab}
- K_{ac} {K^c}_b
- \frac{1}{2} \gamma_{ab} (
R^\perp - K_{cd} K^{cd} + K^2 - 2 n^c n^d R_{cd}
)
- n^p n^r {\gamma_a}^q {\gamma_b}^s R_{pqrs}
$
$
\perp G_{ab}
=
(G^\perp)_{ab}
+ K K_{ab} - \frac{1}{2} \gamma_{ab} K^2
- K_{ac} {K^c}_b + \frac{1}{2} \gamma_{ab} K_{cd} K^{cd}
+ \gamma_{ab} n^c n^d R_{cd}
- n^p n^r {\gamma_a}^q {\gamma_b}^s R_{pqrs}
$
So those four $K_{ab}$ products are really two trace-reversals. Reversed using the $\gamma_{ab}$ metric, but reversed with the spacetime weight of $\frac{2}{n} = \frac{2}{4} = \frac{1}{2}$ instead of the 3-metric's weight $\frac{2}{3}$
For entertainment, what do all those $K_{ab}$'s look like in terms of ${\gamma_a}^b$'s?
$
\perp G_{ab}
=
(G^\perp)_{ab}
+ (
{\gamma_a}^p {\gamma_b}^q \gamma^{rs}
- \frac{1}{2} \gamma_{ab} \gamma^{pq} \gamma^{rs}
- {\gamma_a}^p \gamma^{qr} {\gamma_b}^s
+ \frac{1}{2} \gamma_{ab} \gamma^{pr} \gamma^{qs}
) K_{pq} K_{rs}
+ \gamma_{ab} n^c n^d R_{cd}
- n^p n^r {\gamma_a}^q {\gamma_b}^s R_{pqrs}
$
$
\perp G_{ab}
=
(G^\perp)_{ab}
+ (
2 {\gamma_a}^p {\gamma_b}^{[q} \gamma^{s]r}
+ \gamma_{ab} \gamma^{p[r} \gamma^{q]s}
) K_{pq} K_{rs}
+ \gamma_{ab} n^c n^d R_{cd}
- n^p n^r {\gamma_a}^q {\gamma_b}^s R_{pqrs}
$
Lie derivative along hypersurface normal of spatial hypersurface, in terms of extrinsic curvature:
$\mathcal{L}_{\vec{n}} \gamma_{ab}
= n^c e_c(\gamma_{ab})
+ \gamma_{cb} e_a(n^c)
+ \gamma_{ac} e_b(n^c)
$
$\mathcal{L}_{\vec{n}} \gamma_{ab}
= n^c \nabla_c \gamma_{ab}
+ \gamma_{cb} \nabla_a n^c
+ \gamma_{ac} \nabla_b n^c
$
substitute definition of projection:
$\mathcal{L}_{\vec{n}} \gamma_{ab}
= n^c \nabla_c (g_{ab} + n_a n_b)
+ (g_{cb} + n_c n_b) \nabla_a n^c
+ (g_{ac} + n_a n_c) \nabla_b n^c
$
distribute parenthesis:
$\mathcal{L}_{\vec{n}} \gamma_{ab}
= n^c \nabla_c g_{ab}
+ n_b n^c \nabla_c n_a
+ n_a n^c \nabla_c n_b
+ g_{cb} \nabla_a n^c
+ n_c n_b \nabla_a n^c
+ g_{ac} \nabla_b n^c
+ n_a n_c \nabla_b n^c
$
use the metric-cancelling property of our covariant derivative $\nabla_c g_{ab} = 0$,
using acceleration vector definition $a_u = n^a \nabla_a n_u$,
using $n_b \nabla_a n^b = 0$:
$ \mathcal{L}_{\vec{n}} \gamma_{ab}
= n_b a_a
+ n_a a_b
+ \nabla_a n_b
+ \nabla_b n_a
$
using $K_{ab} = -\nabla_a n_b - n_a a_b$:
$ \mathcal{L}_{\vec{n}} \gamma_{ab} = -K_{ab} - K_{ba}$
using $K_{ab} = K_{(ab)}$:
$ \mathcal{L}_{\vec{n}} \gamma_{ab} = -2 K_{ab}$ (2008 Alcubierre, eqn. 2.3.7, 2010 Baumgarte & Shapiro, eqn. 2.58)
Is the projection of the trace-reversal the same as the trace-reversal of the projection?
$\perp TR(T_{ab})$
$= {\gamma_a}^u {\gamma_b}^v (T_{uv} - \frac{1}{2} g_{ab} T)$
$= {\gamma_a}^u {\gamma_b}^v T_{uv} - \frac{1}{2} {\gamma_a}^u {\gamma_b}^v g_{ab} T$
$= \perp T_{ab} - \frac{1}{2} \gamma_{ab} T$
$= \perp T_{ab}
- \frac{1}{2} g_{ab} g^{uv} T_{uv}
- \frac{1}{2} n_a n_b g^{uv} T_{uv}
$
$= \perp T_{ab}
- \frac{1}{2} g_{ab} g^{uv} T_{uv}
- \frac{1}{2} g_{ab} n^u n^v T_{uv}
+ \frac{1}{2} g_{ab} n^u n^v T_{uv}
- \frac{1}{2} n_a n_b g^{uv} T_{uv}
$
$= {\gamma_a}^u {\gamma_b}^v T_{uv}
- \frac{1}{2} g_{ab} (
\gamma^{uv} T_{uv}
)
+ \frac{1}{2} g_{ab} n^u n^v T_{uv}
- \frac{1}{2} n_a n_b g^{uv} T_{uv}
$
$= {\gamma_a}^u {\gamma_b}^v T_{uv}
- \frac{1}{2} g_{ab} g^{cd} (
{\gamma_c}^u {\gamma_d}^v T_{uv}
)
+ \frac{1}{2} g_{ab} n^u n^v T_{uv}
- \frac{1}{2} n_a n_b g^{uv} T_{uv}
$
$= TR({\gamma_a}^u {\gamma_b}^v T_{uv})
+ \frac{1}{2} g_{ab} n^u n^v T_{uv}
- \frac{1}{2} n_a n_b g^{uv} T_{uv}
$
$= TR(\perp T_{ab})
+ \frac{1}{2} (g_{ab} n^u n^v - n_a n_b g^{uv}) T_{uv}
$
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