ADM formalism of metric tensor:
$g_{ab} = g_{ab}(\alpha, \beta^i, \gamma_{ij})$
$[g_{uv}] = u(i)\downarrow \overset{v(j)\rightarrow}{\left[ \matrix{-\alpha^2 + \beta_k \beta^k & \beta_j \\ \beta_i & \gamma_{ij}} \right]}$
$[g^{uv}] = u(i)\downarrow \overset{v(j)\rightarrow}{\left[ \matrix{-1/\alpha^2 & \beta^j / \alpha^2 \\ \beta^i / \alpha^2 & \gamma^{ij} - \beta^i \beta^j / \alpha^2} \right]}$
Derivatives:
$\frac{\partial}{\partial \alpha} [g_{uv}] = u(i)\downarrow\overset{v(j)\rightarrow}{\left[ \matrix{-2 \alpha & 0 \\ 0 & 0} \right]}$
$\frac{\partial}{\partial \beta^m} [g_{uv}] = u(i)\downarrow \overset{v(j)\rightarrow}{\left[ \matrix{ 2 \beta_m & \gamma_{mj} \\ \gamma_{mi} & 0} \right]}$
$\frac{\partial}{\partial \gamma_{mn}} [g_{uv}] = u(i)\downarrow \overset{v(j)\rightarrow}{\left[ \matrix{ \beta^m \beta^n & \beta^m \delta^j_n \\ \beta^m \delta^n_i & \delta^m_i \delta^n_j} \right]}$
Derivative of metric wrt metric:
$\frac{\partial}{\partial g_{uv}(x')} g_{ab}(x)$
$= \delta_a^u \delta_b^v \delta(x - x')$
...by symmetry of metric:
$= \delta_a^v \delta_b^u \delta(x - x')$
Derivative of inverse wrt metric:
$\delta^b_a = g^{bc} g_{ca}$
$0 = \partial \delta^b_a = \partial g^{bc} g_{ca} + g^{bc} \partial g_{ca}$
$g^{bc} \partial g_{ca} = -\partial g^{bc} g_{ca}$
$\partial g^{ab} = -g^{ac} \partial g_{cd} g^{db}$
$\frac{\partial g^{ab}(x)}{\partial g_{uv}(x')} = -g^{au}(x) g^{vb}(x) \delta(x - x')$
The index is correct for asymmetric $g_{ab}$.
Derivative of determinant wrt metric, using Jacobi's formula:
$\partial det A = tr(adj(A) \partial A) = det A \cdot tr(A^{-T} \partial A)$
$\frac{\partial g(x)}{\partial g_{uv}(x')} = g(x) g^{ba}(x) \frac{\partial g_{ab}(x)}{\partial g_{uv}(x')}$
$= g(x) g^{fe}(x) \delta(x - x')$
Levi-Civita tensor:
$\bar\epsilon_{abcd} = $ permutation tensor of $abcd$ with weight of 1, raised/lowered by $\eta_{ab}$.
$\epsilon_{abcd} = \sqrt{|det(g_{uv})|} \bar\epsilon_{abcd} = |det({e^a}_b)| \bar\epsilon_{abcd}$
...where ${e^a}_b$ is the transformation from a locally-Minkowski basis to our manifold, such that $g_{ab} = \eta_{uv} {e^u}_a {e^v}_b$.
$\epsilon^{abcd} = \frac{-1}{\sqrt{-g}} \bar\epsilon_{abcd}$
Derivatives:
$\frac{\epsilon_{abcd}}{\partial g_{ef}} = \frac{\partial \sqrt{-g}}{\partial g_{ef}} \bar\epsilon_{abcd}$
$ = -\frac{1}{2 \sqrt{-g}} \frac{\partial det g}{\partial g_{ef}} \bar\epsilon_{abcd}$
$ = -\frac{1}{2 \sqrt{-g}} g g^{fe} \bar\epsilon_{abcd}$
$ = \frac{1}{2} \sqrt{-g} g^{fe} \bar\epsilon_{abcd}$
Metric derivative wrt metric:
$\frac{\partial}{\partial g_{uv}} g_{ab,c}$
$= \partial_c ( \frac{\partial}{\partial g_{uv}} g_{ab} )$
$= \partial_c ( \delta^u_a \delta^b_v )$
$= 0$
Metric derivative, finite-difference:
$\frac{\partial}{\partial x^j} g_{ab}(x) \approx D^1_j[g_{ab}(x)] = \frac{1}{h} \Sigma_k \phi(k) g_{ab}(x + k \Delta \hat{x}^j)$
https://en.wikipedia.org/wiki/Finite_difference_coefficient:
... 1st-order: $\phi(\pm 1) = \pm \frac{1}{2}$
... 2nd-order: $\phi(\pm 1) = \pm \frac{8}{12}, \phi(\pm 2) = \mp \frac{1}{12}$
... 3rd-order: $\phi(\pm 1) = \pm \frac{45}{60}, \phi(\pm 2) = \mp \frac{9}{60}, \phi(\pm 3) = \pm \frac{1}{60}$
... 4rd-order: $\phi(\pm 1) = \pm \frac{672}{840}, \phi(\pm 2) = \mp \frac{168}{840}, \phi(\pm 3) = \pm \frac{32}{840}, \phi(\pm 4) = \mp \frac{3}{840}$
$\frac{\partial}{\partial g_{ef}(x + k \Delta \hat{x}^j)} \frac{\partial g_{ab}(x)}{\partial x} \approx \phi(k) \delta^e_a \delta^f_b$
$\frac{\partial}{\partial g_{pq}(x')} \frac{\partial g_{ab}(x)}{\partial x^k} \approx \frac{\partial}{\partial g_{pq}(x')} D^1_k[g_{ab}(x)]$
$= \delta(x'^{k'} - x^{k'}) \phi \left( {\frac{x'^k - x^k}{h}} \right) \delta^p_a \delta^q_b$
...for index k' spanning all but index k.
For nth-order spatial derivatives, $\frac{\partial}{\partial x} g_{ab}(x)$ references n cells away.
Christoffel symbols of the first kind:
$\Gamma_{abc} = \frac{1}{2} (
\frac{\partial}{\partial x^c} g_{ab}
+ \frac{\partial}{\partial x^b} g_{ac}
- \frac{\partial}{\partial x^a} g_{bc})$
Derivatives:
$\frac{\partial}{\partial g_{ef}} \Gamma_{abc} =
\frac{1}{2} (
\frac{\partial}{\partial x^c} \frac{\partial}{\partial g_{ef}} g_{ab}
+ \frac{\partial}{\partial x^b} \frac{\partial}{\partial g_{ef}} g_{ac}
- \frac{\partial}{\partial x^a} \frac{\partial}{\partial g_{ef}} g_{bc}
)$
$= \frac{1}{2} (
\frac{\partial}{\partial x^c} (\delta^e_a \delta^f_b)
+ \frac{\partial}{\partial x^b} (\delta^e_a \delta^f_c)
- \frac{\partial}{\partial x^a} (\delta^e_b \delta^f_c)
)$
$= 0$
For nth order spatial derivatives, $\Gamma_{abc}(x)$ references n cells away.
Useful identity:
$ \Gamma_{(ab)c} = {1 \over 2} (\Gamma_{abc} + \Gamma_{bac})$
$ = {1 \over 4} (\frac{\partial}{\partial x^c} g_{ab} + \partial_b g_{ac} - \partial_a g_{bc} + \frac{\partial}{\partial x^c} g_{ba} + \partial_a g_{bc} - \partial_b g_{ac})$
$ = {1 \over 4} (2 \frac{\partial}{\partial x^c} g_{ab})$
$ = {1 \over 2} \frac{\partial}{\partial x^c} g_{ab} $
so $\frac{\partial}{\partial x^c} g_{ab} = 2 \Gamma_{(ab)c}$
Discretized, for 1st order finite-difference approximation:
The spatial derivative sampling is only used when the indexes are spatial. Otherwise, for time derivatives it falls back to $\frac{\partial}{\partial t}g_{uv}(x)$
$\Gamma_{ttt}(x)
= \frac{1}{2} \frac{\partial}{\partial t} g_{tt}(x)$
$\Gamma_{itt}(x)
= \frac{\partial}{\partial t} g_{it}(x) - \frac{1}{2} \frac{\partial}{\partial x^i} g_{tt}(x)
\approx \frac{\partial}{\partial t} g_{ti}(x) - \frac{1}{4 h} ( g_{tt}(x + \Delta \hat{x}^i) - g_{tt}(x - \Delta \hat{x}^i) )$
$\Gamma_{tit}(x) = \Gamma_{tti}(x)
= \frac{1}{2} ( g_{ti,t}(x) + g_{tt,i}(x) - g_{it,t}(x))
= \frac{1}{2} g_{tt,i}(x)
\approx \frac{1}{4 h} ( g_{tt}(x + \Delta \hat{x}^i) - g_{tt}(x - \Delta \hat{x}^i) )$
$\Gamma_{ijt}(x) = \Gamma_{itj}(x)
= \frac{1}{2} ( g_{ij,t}(x) + g_{it,j}(x) - g_{jt,i}(x) )
\approx \frac{1}{2} \frac{\partial}{\partial t} g_{ij}(x)
+ \frac{1}{4 h} \left(
g_{ti}(x + \Delta \hat{x}^j) - g_{ti}(x - \Delta \hat{x}^j)
- g_{tj}(x + \Delta \hat{x}^i) + g_{tj}(x - \Delta \hat{x}^i)
\right)$
$\Gamma_{tij}(x) = \frac{1}{2} (g_{ti,j}(x) + g_{tj,i}(x) - g_{ij,t}(x))
\approx -\frac{1}{2} \frac{\partial}{\partial t} g_{ij}(x)
+ \frac{1}{4 h} \left(
g_{ti}(x + \Delta \hat{x}^j) - g_{ti}(x - \Delta \hat{x}^j)
+ g_{tj}(x + \Delta \hat{x}^i) - g_{tj}(x - \Delta \hat{x}^i)
\right)$
$\Gamma_{ijk}(x) = \frac{1}{2} (g_{ij,k}(x) + g_{ik,j}(x) - g_{jk,i}(x))
\approx \frac{1}{4 h} \left(
g_{ij}(x + \Delta\hat{x}^k) - g_{ij}(x - \Delta\hat{x}^k)
+ g_{ik}(x + \Delta\hat{x}^j) - g_{ik}(x - \Delta\hat{x}^j)
- g_{jk}(x + \Delta\hat{x}^i) - g_{jk}(x - \Delta\hat{x}^i)
\right)$
Discretized, for arbitrary kernel finite-difference approximation:
$\Gamma_{itt}(x)
\approx \frac{\partial}{\partial t} g_{ti}(x) - \frac{1}{2 h} \Sigma_k \phi(k) g_{tt}(x + k \Delta \hat{x}^i)$
$\Gamma_{tit}(x) = \Gamma_{tti}(x)
\approx \frac{1}{2 h} \Sigma_k \phi(k) g_{tt}(x + k \Delta \hat{x}^i)$
$\Gamma_{ijt}(x) = \Gamma_{itj}(x)
\approx \frac{1}{2} \frac{\partial}{\partial t} g_{ij}(x)
+ \frac{1}{2 h} \Sigma_k \phi(k) \left( g_{ti}(x + k \Delta \hat{x}^j) - g_{tj}(x + k \Delta \hat{x}^i) \right)$
$\Gamma_{tij}(x)
\approx -\frac{1}{2} \frac{\partial}{\partial t} g_{ij}(x)
+ \frac{1}{2 h} \Sigma_k \phi(k) \left( g_{ti}(x + k \Delta \hat{x}^j) + g_{tj}(x + k \Delta \hat{x}^i) \right)$
$\Gamma_{ijk}(x)
\approx \frac{1}{2 h} \Sigma_m \phi_m \left(
g_{ij}(x + m \Delta\hat{x}^k)
+ g_{ik}(x + m \Delta\hat{x}^j)
- g_{jk}(x + m \Delta\hat{x}^i)
\right)$
TODO decide from here on whether to work around the approx partial discretization spatial vs temporal, or to actually define each case.
1st-order discretized 1st-derivatives:
$\frac{\partial \Gamma_{abc}(x)}{\partial g_{ef}(x)} = 0$
$\frac{\partial \Gamma_{abc}(x)}{\partial g_{ef}(x + \Delta \hat{x}^j)} = \frac{1}{4} (\delta^e_a \delta^f_b \delta^j_c + \delta^e_a \delta^f_c \delta^j_b - \delta^e_b \delta^f_c \delta^j_a)$
$\frac{\partial \Gamma_{abc}(x)}{\partial g_{ef}(x - \Delta \hat{x}^j)} = -\frac{1}{4} (\delta^e_a \delta^f_b \delta^j_c + \delta^e_a \delta^f_c \delta^j_b - \delta^e_b \delta^f_c \delta^j_a)$
For 1st order spatial derivatives, $\frac{\partial}{\partial g_{ef}} \Gamma_{abc}(x)$ references cells 1 cell away.
Christoffel symbols of the second kind:
${\Gamma^a}_{bc} = g^{ad} \Gamma_{dbc}$
Derivatives:
$\frac{\partial}{\partial g_{ef}} {\Gamma^a}_{bc}
= \frac{\partial g^{ad}}{\partial g_{ef}} \Gamma_{dbc}
+ g^{ad} \frac{\partial}{\partial g_{ef}} \Gamma_{dbc}$
$ = -g^{ae} {\Gamma^f}_{bc}
+ g^{ad} \frac{\partial}{\partial g_{ef}} \Gamma_{dbc}$
$ = -g^{ae} {\Gamma^f}_{bc}$
Discretized derivatives:
$\frac{\partial}{\partial g_{ef}(x)} {\Gamma^a}_{bc}(x) = -g^{ae}(x) {\Gamma^f}_{bc}(x)$
$\frac{\partial}{\partial g_{ef}(x + \Delta \hat{x}^j)} {\Gamma^a}_{bc}(x)
= g^{ad}(x) \frac{\partial}{\partial g_{ef}(x + \Delta \hat{x}^j)} \Gamma_{dbc}(x)$
$ = \frac{1}{4} g^{ad}(x) (\delta^e_d \delta^f_b \delta^j_c + \delta^e_d \delta^f_c \delta^j_b - \delta^e_b \delta^f_c \delta^j_d)$
$ = \frac{1}{4} (g^{ae}(x) (\delta^f_b \delta^j_c + \delta^f_c \delta^j_b) - g^{aj}(x) \delta^e_b \delta^f_c)$
$\frac{\partial}{\partial g_{ef}(x - \Delta \hat{x}^j)} {\Gamma^a}_{bc}(x)
= -\frac{1}{4} (g^{ae}(x) (\delta^f_b \delta^j_c + \delta^f_c \delta^j_b) - g^{aj}(x) \delta^e_b \delta^f_c)$
so:
$\frac{\partial}{\partial g_{ef}(x)} {\Gamma^a}_{bc}(x) = -g^{ae}(x) {\Gamma^f}_{bc}(x)$
$\frac{\partial}{\partial g_{ef}(x)} {\Gamma^a}_{bc}(x - \Delta \hat{x}^j)
= \frac{1}{4} (g^{ae}(x - \Delta \hat{x}^j) (\delta^f_b \delta^j_c + \delta^f_c \delta^j_b) - g^{aj}(x - \Delta \hat{x}^j) \delta^e_b \delta^f_c)$
$\frac{\partial}{\partial g_{ef}(x)} {\Gamma^a}_{bc}(x + \Delta \hat{x}^j)
= -\frac{1}{4} (g^{ae}(x + \Delta \hat{x}^j) (\delta^f_b \delta^j_c + \delta^f_c \delta^j_b) - g^{aj}(x + \Delta \hat{x}^j) \delta^e_b \delta^f_c)$
so, for contracting only the spatial terms of $c$:
$\frac{\partial}{\partial g_{ef}(x + \Delta \hat{x}^j)} {\Gamma^a}_{bj}(x)
= \frac{1}{4} (g^{ae}(x) (\delta^f_b \delta^j_j + \delta^f_j \delta^j_b) - g^{aj}(x) \delta^e_b \delta^f_j)$
$ = \frac{1}{4} (g^{ae}(x) (3 \delta^f_b + \delta^f_b) - g^{aj}(x) \delta^e_b \delta^f_j)$
$ = g^{ae}(x) \delta^f_b - \frac{1}{4} g^{af}(x) \delta^e_b$
$\frac{\partial}{\partial g_{ef}(x - \Delta \hat{x}^j)} {\Gamma^a}_{bj}(x)
= -\frac{1}{4} (g^{ae}(x) (\delta^f_b \delta^j_j + \delta^f_j \delta^j_b) - g^{aj}(x) \delta^e_b \delta^f_j)$
$= -g^{ae}(x) \delta^f_b + \frac{1}{4} g^{af}(x) \delta^e_b$
then the full discretization of ${\Gamma^a}_{bc,c}(x)$ is...
${\Gamma^a}_{bt,t}(x) + {\Gamma^a}_{bj,j}(x)$
$={\Gamma^a}_{bt,t}(x) + \frac{1}{2 h} (
{\Gamma^a}_{bj}(x + \Delta \hat{x}^j)
- {\Gamma^a}_{bj}(x - \Delta \hat{x}^j)
)$
For 1st order spatial derivatives, $\frac{\partial}{\partial g_{ef}} {\Gamma^a}_{bc}(x)$ references cells 1 cell away.
Riemann curvature:
${R^a}_{bcd} = 2 (\frac{\partial}{\partial x^{[c}} {\Gamma^a}_{d]b} + {\Gamma^a}_{e[c} {\Gamma^e}_{d]b})$
Expanding the antisymmetric indexes:
${R^a}_{bcd} = {\Gamma^a}_{bd,c} - {\Gamma^a}_{bc,d} + {\Gamma^a}_{fc} {\Gamma^f}_{bd} - {\Gamma^a}_{fd} {\Gamma^f}_{bc}$
$= (g^{ae} \Gamma_{ebd})_{,c} - (g^{ae} \Gamma_{ebc})_{,d} + g^{ae} g^{fg} \Gamma_{efc} \Gamma_{gbd} - g^{ae} g^{fg} \Gamma_{efd} \Gamma_{gbc}$
$= {g^{ae}}_{,c} \Gamma_{ebd} + g^{ae} \Gamma_{ebd,c}
- {g^{ae}}_{,d} \Gamma_{ebc} - g^{ae} \Gamma_{ebc,d}
+ g^{ae} g^{fg} \Gamma_{efc} \Gamma_{gbd}
- g^{ae} g^{fg} \Gamma_{efd} \Gamma_{gbc}$
Replacing only $\Gamma_{abc,d}$'s in terms of $g_{ab,cd}$:
$=
\frac{1}{2} g^{ae} (g_{eb,d} + g_{ed,b} - g_{bd,e})_{,c}
- \frac{1}{2} g^{ae} (g_{eb,c} + g_{ec,b} - g_{bc,e})_{,d}
+ {g^{ae}}_{,c} \Gamma_{ebd}
- {g^{ae}}_{,d} \Gamma_{ebc}
+ g^{ae} g^{fg} \Gamma_{efc} \Gamma_{gbd}
- g^{ae} g^{fg} \Gamma_{efd} \Gamma_{gbc}$
$=
\frac{1}{2} g^{ae} (
g_{ed,cb} - g_{bd,ce} - g_{ec,db} + g_{bc,de}
)
+ {g^{ae}}_{,c} \Gamma_{ebd}
- {g^{ae}}_{,d} \Gamma_{ebc}
+ g^{ae} g^{fg} \Gamma_{efc} \Gamma_{gbd}
- g^{ae} g^{fg} \Gamma_{efd} \Gamma_{gbc}$
...then replacing the ${g^{ab}}_{,c}$ terms with $g_{ab,c}$ terms:
$=
\frac{1}{2} g^{ae} (
g_{ed,cb} - g_{bd,ce} - g_{ec,db} + g_{bc,de}
)
- g^{au} g_{uv,c} g^{ve} \Gamma_{ebd}
+ g^{au} g_{uv,d} g^{ve} \Gamma_{ebc}
+ g^{ae} g^{fg} \Gamma_{efc} \Gamma_{gbd}
- g^{ae} g^{fg} \Gamma_{efd} \Gamma_{gbc}$
... then replace those with $\Gamma_{abc}$...
$=
\frac{1}{2} g^{ae} (
g_{ed,cb} - g_{bd,ce} - g_{ec,db} + g_{bc,de}
)
- g^{au} (\Gamma_{uvc} + \Gamma_{vuc}) g^{ve} \Gamma_{ebd}
+ g^{au} (\Gamma_{uvd} + \Gamma_{vud}) g^{ve} \Gamma_{ebc}
+ g^{ae} g^{fg} \Gamma_{efc} \Gamma_{gbd}
- g^{ae} g^{fg} \Gamma_{efd} \Gamma_{gbc}$
$=
\frac{1}{2} g^{ae} (
g_{ed,cb} - g_{bd,ce} - g_{ec,db} + g_{bc,de}
)
+ g^{ae} g^{fg} \Gamma_{fed} \Gamma_{gbc}
- g^{ae} g^{fg} \Gamma_{fec} \Gamma_{gbd}
$
in lower form it looks a bit cleaner:
$R_{abcd} =
\frac{1}{2} (
g_{ad,bc} - g_{bd,ac} - g_{ac,bd} + g_{bc,ad}
)
+ g^{fg} \Gamma_{fad} \Gamma_{gbc}
- g^{fg} \Gamma_{fac} \Gamma_{gbd}
$
$R_{abcd} =
\frac{1}{2} (
g_{ad,bc} + g^{fg} \Gamma_{fad} \Gamma_{gbc}
+ g_{bc,ad} + g^{fg} \Gamma_{fbc} \Gamma_{gad}
- g_{bd,ac} - g^{fg} \Gamma_{fbd} \Gamma_{gac}
- g_{ac,bd} - g^{fg} \Gamma_{fac} \Gamma_{gbd}
)
$
$R_{abcd} =
g_{a[d,c]b} + g^{fg} \Gamma_{fa[d} \Gamma_{g|c]b}
- g_{b[d,c]a} - g^{fg} \Gamma_{fb[d} \Gamma_{g|c]a}
$
$R_{abcd} = 2 (g_{[a|[d,c]|b]} + g^{fg} \Gamma_{f[a|[d} \Gamma_{g|c]|b]})$
... or if you really need to not mix valence of antisymmetric indexes ...
$R_{abcd} = 2 \delta^{[u}_a \delta^{v]}_b (g_{a[d,c]b} + g^{fg} \Gamma_{fa[d} \Gamma_{g|c]b})$
... or if you really want to see indexes next to one another ...
$R_{abcd} = 2 \delta^{[u}_a \delta^{v]}_b (g_{u[d,c]v} + \Gamma_{fu[d} {\Gamma^f}_{c]v})$
Discretized derivative in lower form:
$\frac{\partial}{\partial g_{pq}(x')} R_{abcd}(x) =
\frac{1}{2} \frac{\partial}{\partial g_{pq}(x')} (
D^2_{bc}[g_{ad}(x)]
- D^2_{ac}[g_{bd}(x)]
- D^2_{bd}[g_{ac}(x)]
+ D^2_{ad}[g_{bc}(x)]
)
+ \frac{\partial}{\partial g_{pq}(x')} (
g^{fg}(x) (
\Gamma_{fad}(x) \Gamma_{gbc}(x) - \Gamma_{fac}(x) \Gamma_{gbd}(x)
)
)
$
$=
\frac{1}{2} \frac{\partial}{\partial g_{pq}(x')} (
D^2_{bc}[g_{ad}(x)]
- D^2_{ac}[g_{bd}(x)]
- D^2_{bd}[g_{ac}(x)]
+ D^2_{ad}[g_{bc}(x)]
)
+ (\frac{\partial}{\partial g_{pq}(x')} g^{fg}(x))
(
\Gamma_{fad}(x) \Gamma_{gbc}(x) - \Gamma_{fac}(x) \Gamma_{gbd}(x)
)
+ g^{fg}(x) (
\frac{\partial}{\partial g_{pq}(x')}\Gamma_{fad}(x) \Gamma_{gbc}(x)
+ \Gamma_{fad}(x) \frac{\partial}{\partial g_{pq}(x')}\Gamma_{gbc}(x)
- \frac{\partial}{\partial g_{pq}(x')}\Gamma_{fac}(x) \Gamma_{gbd}(x)
- \Gamma_{fac}(x) \frac{\partial}{\partial g_{pq}(x')}\Gamma_{gbd}(x)
)
$
$=
\frac{1}{2} \frac{\partial}{\partial g_{pq}(x')} (
D^2_{bc}[g_{ad}(x)]
- D^2_{ac}[g_{bd}(x)]
- D^2_{bd}[g_{ac}(x)]
+ D^2_{ad}[g_{bc}(x)]
)
- \delta(x - x') g^{pf}(x) g^{gq}(x)
(
\Gamma_{fad}(x) \Gamma_{gbc}(x) - \Gamma_{fac}(x) \Gamma_{gbd}(x)
)
+ g^{fg}(x) (
\frac{\partial}{\partial g_{pq}(x')}\Gamma_{fad}(x) \Gamma_{gbc}(x)
+ \Gamma_{fad}(x) \frac{\partial}{\partial g_{pq}(x')}\Gamma_{gbc}(x)
- \frac{\partial}{\partial g_{pq}(x')}\Gamma_{fac}(x) \Gamma_{gbd}(x)
- \Gamma_{fac}(x) \frac{\partial}{\partial g_{pq}(x')}\Gamma_{gbd}(x)
)
$
$=
\frac{1}{2} \frac{\partial}{\partial g_{pq}(x')} (
D^2_{bc}[g_{ad}(x)]
- D^2_{ac}[g_{bd}(x)]
- D^2_{bd}[g_{ac}(x)]
+ D^2_{ad}[g_{bc}(x)]
)
- \delta(x - x') g^{pf}(x) g^{gq}(x)
(
\Gamma_{fad}(x) \Gamma_{gbc}(x) - \Gamma_{fac}(x) \Gamma_{gbd}(x)
)
+ g^{fg}(x) (
\Gamma_{fbc}(x) \frac{\partial}{\partial g_{pq}(x')}\Gamma_{gad}(x)
+ \Gamma_{fad}(x) \frac{\partial}{\partial g_{pq}(x')}\Gamma_{gbc}(x)
- \Gamma_{fbd}(x) \frac{\partial}{\partial g_{pq}(x')}\Gamma_{gac}(x)
- \Gamma_{fac}(x) \frac{\partial}{\partial g_{pq}(x')}\Gamma_{gbd}(x)
)
$
...replacing the 2nd-derivatives with their stencils ...
$=
\frac{1}{2} \frac{\partial}{\partial g_{pq}(x')} (
\delta^p_a \delta^q_d \phi^2_{bc}(x - x')
- \delta^p_b \delta^q_d \phi^2_{ac}(x - x')
- \delta^p_a \delta^q_c \phi^2_{bd}(x - x')
+ \delta^p_b \delta^q_c \phi^2_{ad}(x - x')
)
- \delta(x - x') g^{pf}(x) g^{gq}(x)
(
\Gamma_{fad}(x) \Gamma_{gbc}(x) - \Gamma_{fac}(x) \Gamma_{gbd}(x)
)
+ g^{fg}(x) (
\Gamma_{fbc}(x) \frac{\partial}{\partial g_{pq}(x')}\Gamma_{gad}(x)
+ \Gamma_{fad}(x) \frac{\partial}{\partial g_{pq}(x')}\Gamma_{gbc}(x)
- \Gamma_{fbd}(x) \frac{\partial}{\partial g_{pq}(x')}\Gamma_{gac}(x)
- \Gamma_{fac}(x) \frac{\partial}{\partial g_{pq}(x')}\Gamma_{gbd}(x)
)
$
Ricci curvature:
$R_{ab} = {R^c}_{acd}$
Ricci curvature based on ${\Gamma^a}_{bc,d}$ and ${\Gamma^a}_{ec} {\Gamma^e}_{bd}$:
$R_{ab} = 2 ( \frac{\partial}{\partial x^{[c}} {\Gamma^c}_{b]a} + {\Gamma^c}_{d[c} {\Gamma^d}_{b]a} )$
Ricci curvature based on $g_{ab,cd}$ and ${\Gamma^u}_{ab} \Gamma_{ucd}$:
$R_{ab} =
\frac{1}{2} g^{ce} (
g_{eb,ca} - g_{ab,ce} - g_{ec,ba} + g_{ac,be}
)
+ g^{ce} g^{fg} \Gamma_{feb} \Gamma_{gac}
- g^{ce} g^{fg} \Gamma_{fec} \Gamma_{gab}
$
$=
\frac{1}{2} g^{uv} (
g_{au,bv}
+ g_{bv,au}
- g_{ab,uv}
- g_{uv,ab}
)
+ g^{ce} g^{fg} (\Gamma_{feb} \Gamma_{gac} - \Gamma_{fec} \Gamma_{gab})
$
$=
\frac{1}{2} g^{uv} (
g_{au,bv}
+ g_{bv,au}
- g_{ab,uv}
- g_{uv,ab}
)
+ \Gamma_{uva} {\Gamma^{uv}}_b
- {\Gamma^{uv}}_v \Gamma_{uab}
$
Derivatives of $R_{ab}$ based on $\partial \Gamma$ and $\Gamma^2$:
$\frac{\partial R_{ab}}{\partial g_{ef}} =
2 ( \frac{\partial}{\partial x^{[c}} (\frac{\partial}{\partial g_{ef}} {\Gamma^c}_{b]a})
+ \frac{\partial}{\partial g_{ef}} {\Gamma^c}_{d[c} {\Gamma^d}_{b]a}
+ {\Gamma^c}_{d[c} \frac{\partial}{\partial g_{ef}} {\Gamma^d}_{b]a}
)$
$ = -2 ( \frac{\partial}{\partial x^{[c}} (g^{ce} {\Gamma^f}_{b]a})
+ g^{ce} {\Gamma^f}_{d[c} {\Gamma^d}_{b]a}
+ {\Gamma^c}_{d[c} g^{de} {\Gamma^f}_{b]a}
)$
$ = -2 ( \frac{\partial}{\partial x^{[c}} g^{ce} {\Gamma^f}_{b]a}
+ g^{ce} \frac{\partial}{\partial x^{[c}} {\Gamma^f}_{b]a}
+ g^{ce} {\Gamma^f}_{d[c} {\Gamma^d}_{b]a}
+ {\Gamma^c}_{d[c} g^{de} {\Gamma^f}_{b]a}
)$
$ = 2 ( g^{cg} \frac{\partial}{\partial x^{[c}} g_{gh} g^{he} {\Gamma^f}_{b]a}
- g^{ce} \frac{\partial}{\partial x^{[c}} {\Gamma^f}_{b]a}
- g^{ce} {\Gamma^f}_{d[c} {\Gamma^d}_{b]a}
- {\Gamma^c}_{d[c} g^{de} {\Gamma^f}_{b]a}
)$
getting rid of the $\frac{\partial}{\partial x^c} g_{ab}$...
$ = 2 ( g^{cg} 2 \Gamma_{(gh)[c} g^{he} {\Gamma^f}_{b]a}
- g^{ce} (
\frac{\partial}{\partial x^{[c}} {\Gamma^f}_{b]a}
+ {\Gamma^f}_{d[c} {\Gamma^d}_{b]a}
)
- {\Gamma^{ce}}_{[c} {\Gamma^f}_{b]a}
)$
$ = 2 ( 2 {\Gamma^{(ce)}}_{[c} {\Gamma^f}_{b]a}
- g^{ce} (
\frac{\partial}{\partial x^{[c}} {\Gamma^f}_{b]a}
+ {\Gamma^f}_{d[c} {\Gamma^d}_{b]a}
)
- {\Gamma^{ce}}_{[c} {\Gamma^f}_{b]a}
)$
$ = 2 ( {\Gamma^{ec}}_{[c} {\Gamma^f}_{b]a}
- g^{ce} (
\frac{\partial}{\partial x^{[c}} {\Gamma^f}_{b]a}
+ {\Gamma^f}_{d[c} {\Gamma^d}_{b]a}
)
)$
getting rid of the $\frac{\partial}{\partial x^d} {\Gamma^a}_{bc}$...
$ = 2 {\Gamma^{ec}}_{[c} {\Gamma^f}_{b]a} - g^{ce} {R^f}_{acb}$
Discretized derivative of $R_{ab}$ based on $R_{abcd}$:
$\frac{\partial R_{ab}(x)}{g_{pq}(x')}$
$= (\frac{\partial}{g_{pq}(x')} g^{uv}(x)) R_{uavb}(x)
+ g^{uv}(x) \frac{\partial}{g_{pq}(x')} R_{uavb}(x)
$
$= -\delta(x - x') g^{pu}(x) g^{qv}(x) R_{uavb}(x)
+ g^{uv}(x) \frac{\partial}{g_{pq}(x')} R_{uavb}(x)
$
$= -\delta(x - x') {{{R^p}_a}^q}_b(x)
+ g^{uv}(x) \frac{\partial}{g_{pq}(x')} R_{uavb}(x)
$
Discretized, starting with continuous definition:
$R_{ab}(x) = 2 ( \frac{\partial}{\partial x^{[c}} {\Gamma^c}_{b]a}(x) + {\Gamma^c}_{d[c}(x) {\Gamma^d}_{b]a}(x) )$
$ =
\frac{\partial}{\partial x^c} {\Gamma^c}_{ab}(x)
- \frac{\partial}{\partial x^b} {\Gamma^c}_{ac}(x)
+ {\Gamma^c}_{dc}(x) {\Gamma^d}_{ab}(x)
- {\Gamma^c}_{db}(x) {\Gamma^d}_{ac}(x)
$
separate out time derivatives (because I'm only discretizing in space):
$=
+ \frac{\partial}{\partial x^t} {\Gamma^t}_{ab}(x)
+ \frac{\partial}{\partial x^k} {\Gamma^k}_{ab}(x)
- \frac{\partial}{\partial x^b} {\Gamma^t}_{at}(x)
- \frac{\partial}{\partial x^b} {\Gamma^k}_{ak}(x)
+ {\Gamma^c}_{dc}(x) {\Gamma^d}_{ab}(x)
- {\Gamma^c}_{db}(x) {\Gamma^d}_{ac}(x)
$
...but then shouldn't I also be separating out only the time derivatives of the discretized derivatives of the connection coefficients?...
...in fact, I never defined the time derivatives of the discretized derivatives of the connection coefficients. This is the first they've come into play...
Since the indexes of the Ricci show up in the derivatives of the connections, I think I'll define them separately...
$R_{tt}(x) =
\frac{\partial}{\partial x^t} {\Gamma^t}_{tt}(x)
+ \frac{\partial}{\partial x^k} {\Gamma^k}_{tt}(x)
- \frac{\partial}{\partial x^t} {\Gamma^t}_{tt}(x)
- \frac{\partial}{\partial x^t} {\Gamma^k}_{tk}(x)
+ {\Gamma^c}_{dc}(x) {\Gamma^d}_{tt}(x)
- {\Gamma^c}_{dt}(x) {\Gamma^d}_{tc}(x)
$
$R_{it}(x) = R_{ti}(x) =
\frac{\partial}{\partial x^t} {\Gamma^t}_{ti}(x)
+ \frac{\partial}{\partial x^k} {\Gamma^k}_{ti}(x)
- \frac{\partial}{\partial x^i} {\Gamma^t}_{tt}(x)
- \frac{\partial}{\partial x^i} {\Gamma^k}_{tk}(x)
+ {\Gamma^c}_{dc}(x) {\Gamma^d}_{ti}(x)
- {\Gamma^c}_{di}(x) {\Gamma^d}_{tc}(x)
$
$R_{ij}(x) =
\frac{\partial}{\partial x^t} {\Gamma^t}_{ij}(x)
+ \frac{\partial}{\partial x^k} {\Gamma^k}_{ij}(x)
- \frac{\partial}{\partial x^j} {\Gamma^t}_{it}(x)
- \frac{\partial}{\partial x^j} {\Gamma^k}_{ik}(x)
+ {\Gamma^c}_{dc}(x) {\Gamma^d}_{ij}(x)
- {\Gamma^c}_{dj}(x) {\Gamma^d}_{ic}(x)
$
...
$=
\frac{\partial}{\partial x^t}{\Gamma^t}_{ab}(x)
+ \frac{ {\Gamma^j}_{ba}(x + \Delta \hat{x}^j) - {\Gamma^j}_{ba}(x - \Delta \hat{x}^j)}{ 2 h }
- \frac{\partial}{\partial x^b} {\Gamma^t}_{at}(x)
- \frac{ {\Gamma^c}_{ca}(x + \Delta \hat{x}^b) - {\Gamma^c}_{ca}(x - \Delta \hat{x}^b)}{ 2 h }
+ 2 {\Gamma^c}_{d[c} (x) {\Gamma^d}_{b]a} (x)
$
$= \frac{1}{2 h} \left( {\Gamma^c}_{ba}(x + \Delta \hat{x}^c) - {\Gamma^c}_{ba}(x - \Delta \hat{x}^c)
- {\Gamma^c}_{ca}(x + \Delta \hat{x}^b) + {\Gamma^c}_{ca}(x - \Delta \hat{x}^b) \right)
+ 2 {\Gamma^c}_{d[c} (x) {\Gamma^d}_{b]a} (x)
$
Discretized 1st-derivatives, using $R_{ab}$ based on ${\Gamma^a}_{b[d,c]}$ and ${\Gamma^a}_{e[c} {\Gamma^e}_{d]b}$:
$\frac{\partial R_{ab}(x)}{\partial g_{ef}(x)}
= \frac{\partial}{\partial g_{ef}(x)} \frac{1}{2 h} \left( {\Gamma^c}_{ba}(x + \Delta \hat{x}^c) - {\Gamma^c}_{ba}(x - \Delta \hat{x}^c)
- {\Gamma^c}_{ca}(x + \Delta \hat{x}^b) + {\Gamma^c}_{ca}(x - \Delta \hat{x}^b) \right)
+ {\Gamma^c}_{dc} (x) {\Gamma^d}_{ab} (x)
- {\Gamma^c}_{db} (x) {\Gamma^d}_{ac} (x)
$
$= \frac{1}{2 h} \left(
\frac{\partial}{\partial g_{ef}(x)} {\Gamma^c}_{ba}(x + \Delta \hat{x}^c)
- \frac{\partial}{\partial g_{ef}(x)} {\Gamma^c}_{ba}(x - \Delta \hat{x}^c)
- \frac{\partial}{\partial g_{ef}(x)} {\Gamma^c}_{ca}(x + \Delta \hat{x}^b)
+ \frac{\partial}{\partial g_{ef}(x)} {\Gamma^c}_{ca}(x - \Delta \hat{x}^b) \right)
+ \frac{\partial}{\partial g_{ef}(x)} {\Gamma^c}_{dc} (x) {\Gamma^d}_{ab} (x)
+ {\Gamma^c}_{dc} (x) \frac{\partial}{\partial g_{ef}(x)} {\Gamma^d}_{ab} (x)
- \frac{\partial}{\partial g_{ef}(x)} {\Gamma^c}_{db} (x) {\Gamma^d}_{ac} (x)
- {\Gamma^c}_{db} (x) \frac{\partial}{\partial g_{ef}(x)} {\Gamma^d}_{ac} (x)
$
$= \frac{1}{2 h}
\left(
( -\frac{1}{4} (
g^{ce}(x + \Delta \hat{x}^c) (\delta^f_b \delta^c_a + \delta^f_a \delta^c_b)
- g^{cc}(x + \Delta \hat{x}^c) \delta^e_b \delta^f_a
)
)
- (
\frac{1}{4} (
g^{ce}(x - \Delta \hat{x}^c) (\delta^f_b \delta^c_a + \delta^f_a \delta^c_b)
- g^{cc}(x - \Delta \hat{x}^c) \delta^e_b \delta^f_a
)
) - (
-\frac{1}{4} (
g^{ce}(x + \Delta \hat{x}^b) (\delta^f_c \delta^b_a + \delta^f_a \delta^b_c)
- g^{cb}(x + \Delta \hat{x}^b) \delta^e_c \delta^f_a
)
) + (
\frac{1}{4} (
g^{ce}(x - \Delta \hat{x}^b) (\delta^f_c \delta^b_a + \delta^f_a \delta^b_c)
- g^{cb}(x - \Delta \hat{x}^b) \delta^e_c \delta^f_a
)
)
\right)
+ {\Gamma^d}_{ab} (x) (
-g^{ce}(x) {\Gamma^f}_{dc}(x)
)
+ {\Gamma^c}_{dc} (x) (
-g^{de}(x) {\Gamma^f}_{ab}(x)
)
- {\Gamma^d}_{ac} (x) (
-g^{ce}(x) {\Gamma^f}_{db}(x)
)
- {\Gamma^c}_{db} (x) (
-g^{de}(x) {\Gamma^f}_{ac}(x)
)
$
$= \frac{1}{8 \Delta}
\left(
- g^{ce}(x + \Delta \hat{x}^c) (\delta^f_b \delta^c_a + \delta^f_a \delta^c_b) + g^{cc}(x + \Delta \hat{x}^c) \delta^e_b \delta^f_a
- g^{ce}(x - \Delta \hat{x}^c) (\delta^f_b \delta^c_a + \delta^f_a \delta^c_b) + g^{cc}(x - \Delta \hat{x}^c) \delta^e_b \delta^f_a
+ g^{ce}(x + \Delta \hat{x}^b) (\delta^f_c \delta^b_a + \delta^f_a \delta^b_c) - g^{cb}(x + \Delta \hat{x}^b) \delta^e_c \delta^f_a
+ g^{ce}(x - \Delta \hat{x}^b) (\delta^f_c \delta^b_a + \delta^f_a \delta^b_c) - g^{cb}(x - \Delta \hat{x}^b) \delta^e_c \delta^f_a
\right)
- {\Gamma^d}_{ab} (x) g^{ce}(x) {\Gamma^f}_{dc}(x) - {\Gamma^c}_{dc} (x) g^{de}(x) {\Gamma^f}_{ab}(x)
+ {\Gamma^d}_{ac} (x) g^{ce}(x) {\Gamma^f}_{db}(x) + {\Gamma^c}_{db} (x) g^{de}(x) {\Gamma^f}_{ac}(x)
$
$= \frac{1}{8 \Delta}
\left(
- g^{ce}(x + \Delta \hat{x}^c) \delta^f_b \delta^c_a
- g^{ce}(x + \Delta \hat{x}^c) \delta^f_a \delta^c_b
+ g^{cc}(x + \Delta \hat{x}^c) \delta^e_b \delta^f_a
- g^{ce}(x - \Delta \hat{x}^c) \delta^f_b \delta^c_a
- g^{ce}(x - \Delta \hat{x}^c) \delta^f_a \delta^c_b
+ g^{cc}(x - \Delta \hat{x}^c) \delta^e_b \delta^f_a
+ g^{ce}(x + \Delta \hat{x}^b) \delta^f_c \delta^b_a
+ g^{ce}(x + \Delta \hat{x}^b) \delta^f_a \delta^b_c
- g^{cb}(x + \Delta \hat{x}^b) \delta^e_c \delta^f_a
+ g^{ce}(x - \Delta \hat{x}^b) \delta^f_c \delta^b_a
+ g^{ce}(x - \Delta \hat{x}^b) \delta^f_a \delta^b_c
- g^{cb}(x - \Delta \hat{x}^b) \delta^e_c \delta^f_a
\right)
- {\Gamma^d}_{ab} (x) g^{ce}(x) {\Gamma^f}_{dc}(x)
- {\Gamma^c}_{dc} (x) g^{de}(x) {\Gamma^f}_{ab}(x)
+ {\Gamma^d}_{ac} (x) g^{ce}(x) {\Gamma^f}_{db}(x)
+ {\Gamma^c}_{db} (x) g^{de}(x) {\Gamma^f}_{ac}(x)
$
Gaussian curvature:
$R = g^{ab} R_{ab}$
Einstein tensor / trace reversal of Ricci tensor:
$G_{ab} = G_{ab}(g_{ab})$
$G_{ab} = R_{ab} - \frac{1}{2} g_{ab} g^{cd} R_{cd}$
Derivatives:
$\frac{\partial G_{ab}}{\partial g_{ef}} =
\frac{\partial R_{ab}}{\partial g_{ef}}
- \frac{1}{2} (
\frac{\partial g_{ab}}{\partial g_{ef}} g^{cd} R_{cd}
+ g_{ab} \frac{\partial g^{cd}}{\partial g_{ef}} R_{cd}
+ g_{ab} g^{cd} \frac{\partial R_{cd}}{\partial g_{ef}}
)$
$= \frac{\partial R_{ab}}{\partial g_{ef}}
- \frac{1}{2} (
\delta^e_a \delta^f_b R
- g_{ab} R^{ef}
+ g_{ab} g^{cd} \frac{\partial R_{cd}}{\partial g_{ef}}
)$
$= - \frac{1}{2} ( \delta^e_a \delta^f_b R - g_{ab} R^{ef} )
+ (\delta^c_a \delta^d_b - \frac{1}{2} g_{ab} g^{cd}) \frac{\partial R_{cd}}{\partial g_{ef}}
$
Discrete derivatives in terms of $\frac{\partial}{\partial g_{pq}(x')} R_{cd}$:
$\frac{\partial}{\partial g_{pq}(x')} G_{ab}(x) = \frac{\partial}{\partial g_{pq}(x')} (R_{ab}(x) - \frac{1}{2} g_{ab}(x) g^{cd}(x) R_{cd}(x))$
$= \frac{\partial}{\partial g_{pq}(x')}R_{ab}(x) - \frac{1}{2} (\frac{\partial}{\partial g_{pq}(x')} g_{ab}(x) g^{cd}(x) R_{cd}(x) + g_{ab}(x) \frac{\partial}{\partial g_{pq}(x')} g^{cd}(x) R_{cd}(x) + g_{ab}(x) g^{cd}(x) \frac{\partial}{\partial g_{pq}(x')} R_{cd}(x))$
$= \frac{\partial}{\partial g_{pq}(x')}R_{ab}(x) - \frac{1}{2} (\delta^a_p \delta^b_q \delta(x-x') R(x) - g_{ab}(x) g^{ce}(x) \frac{\partial}{\partial g_{pq}(x')} g_{ef}(x) g^{fd}(x) R_{cd}(x) + g_{ab}(x) g^{cd}(x) \frac{\partial}{\partial g_{pq}(x')} R_{cd}(x))$
$= \frac{\partial}{\partial g_{pq}(x')}R_{ab}(x) - \frac{1}{2}\delta^p_a \delta^q_b \delta(x-x') R(x) + \frac{1}{2} g_{ab}(x) g^{ce}(x) \delta^p_e \delta^q_f \delta(x-x') g^{fd}(x) R_{cd}(x) - \frac{1}{2} g_{ab}(x) g^{cd}(x) \frac{\partial}{\partial g_{pq}(x')} R_{cd}(x)$
$= \frac{\partial}{\partial g_{pq}(x')}R_{ab}(x)
- \frac{1}{2}\delta^p_a \delta^q_b \delta(x-x') R(x)
+ \frac{1}{2} \delta(x-x') g_{ab}(x) g^{pc}(x) R_{cd}(x) g^{dq}(x)
- \frac{1}{2} g_{ab}(x) g^{cd}(x) \frac{\partial}{\partial g_{pq}(x')} R_{cd}(x)
$
$=
\frac{1}{2} \delta(x-x') (
g_{ab}(x) R^{pq}(x)
- \delta^p_a \delta^q_b R(x)
)
+ (
\delta^u_a \delta^v_b
- \frac{1}{2} g_{ab}(x) g^{uv}(x)
) \frac{\partial}{\partial g_{pq}(x')} R_{uv}(x)
$
Discrete derivatives in terms of $\frac{\partial}{\partial g_{pq}(x')} R_{ucvd}(x)$:
$=
\frac{\partial}{\partial g_{pq}(x')} G_{ab} (x)
$
$=
\frac{\partial}{\partial g_{pq}(x')} (
R_{ab}(x)
- \frac{1}{2} R(x) g_{ab}(x)
)
$
$=
\frac{\partial}{\partial g_{pq}(x')} (
g^{uv}(x) \delta^c_a \delta^d_b R_{ucvd}(x)
- \frac{1}{2} g^{uv}(x) g^{cd}(x) R_{ucvd}(x) g_{ab}(x)
)
$
$=
\frac{\partial}{\partial g_{pq}(x')} (
(
g^{uv}(x) \delta^c_a \delta^d_b
- \frac{1}{2} g^{cd}(x) g^{uv}(x) g_{ab}(x)
) R_{ucvd}(x)
)
$
$=
\frac{\partial}{\partial g_{pq}(x')} (
g^{uv}(x) \delta^c_a \delta^d_b
- \frac{1}{2} g^{cd}(x) g^{uv}(x) g_{ab}(x)
)
R_{ucvd}(x)
+ (
g^{uv}(x) \delta^c_a \delta^d_b
- \frac{1}{2} g^{cd}(x) g^{uv}(x) g_{ab}(x)
)
\frac{\partial}{\partial g_{pq}(x')} R_{ucvd}(x)
$
$=
(
\frac{\partial}{\partial g_{pq}(x')} g^{uv}(x) \delta^c_a \delta^d_b
- \frac{1}{2} \frac{\partial}{\partial g_{pq}(x')} g^{cd}(x) g^{uv}(x) g_{ab}(x)
- \frac{1}{2} g^{cd}(x) \frac{\partial}{\partial g_{pq}(x')} g^{uv}(x) g_{ab}(x)
- \frac{1}{2} g^{cd}(x) g^{uv}(x) \frac{\partial}{\partial g_{pq}(x')} g_{ab}(x)
)
R_{ucvd}(x)
+ g^{uv}(x) (
\delta^c_a \delta^d_b
- \frac{1}{2} g^{cd}(x) g_{ab}(x)
)
\frac{\partial}{\partial g_{pq}(x')} R_{ucvd}(x)
$
$=
\delta(x - x') (
- g^{pu}(x) g^{qv}(x) \delta^c_a \delta^d_b
+ \frac{1}{2} g^{pc}(x) g^{qd}(x) g^{uv}(x) g_{ab}(x)
+ \frac{1}{2} g^{cd}(x) g^{pu}(x) g^{qv}(x) g_{ab}(x)
- \frac{1}{2} g^{cd}(x) g^{uv}(x) \delta^p_a \delta^q_b
)
R_{ucvd}(x)
+ g^{uv}(x) (
\delta^c_a \delta^d_b
- \frac{1}{2} g^{cd}(x) g_{ab}(x)
)
\frac{\partial}{\partial g_{pq}(x')} R_{ucvd}(x)
$
$=
\delta(x - x') (
- {{{R^p}_a}^q}_b(x)
+ g_{ab}(x) R^{pq}(x)
- \frac{1}{2} \delta^p_a \delta^q_b R(x)
)
+ g^{uv}(x) (
\delta^c_a \delta^d_b
- \frac{1}{2} g^{cd}(x) g_{ab}(x)
)
\frac{\partial}{\partial g_{pq}(x')} R_{ucvd}(x)
$
Stress-energy tensor:
$T_{ab} = T_{ab}(g_{ab}, \rho, P, e_{int}, v^i, E^i, B^i, ...)$
$T_{ab} = T^{EM}_{ab} + T^{matter}_{ab} + ...$
Electromagnetism stress-energy:
$T^{EM}_{ab} = \frac{1}{8\pi} \downarrow a(i) \overset{\rightarrow b(j)}{ \left[\matrix{
g_{mn} (E^m E^n + B^m B^n) & -2 \epsilon_{tjmn} E^m B^n \\
-2 \epsilon_{timn} E^m B^n & (g_{ij} g_{mn} - 2 g_{im} g_{jn}) (E^m E^n + B^m B^n)
}\right]}$
Derivatives:
$\frac{\partial}{\partial g_{ef}} T^{EM}_{ab}
= \frac{1}{8\pi} \downarrow a(i) \overset{\rightarrow b(j)}{ \left[\matrix{
E^e E^f + B^e B^f & -2 \frac{\partial \epsilon_{tjmn}}{\partial g_{ef}} E^m B^n \\
-2 \frac{\partial \epsilon_{timn}}{\partial g_{ef}} E^m B^n &
(\delta^e_i \delta^f_j g_{mn}
+ g_{ij} \delta^e_m \delta^f_n
- 2 \delta^e_i \delta^f_m g_{jn}
- 2 g_{im} \delta^e_j \delta^f_n)
(E^m E^n + B^m B^n)
}\right]}$
$ = \frac{1}{8\pi} \downarrow a(i) \overset{\rightarrow b(j)}{ \left[\matrix{
E^e E^f + B^e B^f & -2 \frac{\partial \epsilon_{tjmn}}{\partial g_{ef}} E^m B^n \\
-2 \frac{\partial \epsilon_{timn}}{\partial g_{ef}} E^m B^n &
\delta^e_i \delta^f_j g_{mn} (E^m E^n + B^m B^n)
+ g_{ij} (E^e E^f + B^e B^f)
- 2 \delta^e_i g_{jn} (E^f E^n + B^f B^n)
- 2 g_{im} \delta^e_j (E^m E^f + B^m B^f)
}\right]}$
$ = \frac{1}{8\pi} \downarrow a(i) \overset{\rightarrow b(j)}{ \left[\matrix{
E^e E^f + B^e B^f & -\sqrt{-g} g^{fe} \bar\epsilon_{tjmn} E^m B^n \\
-\sqrt{-g} g^{fe} \bar\epsilon_{timn} E^m B^n &
\delta^e_i \delta^f_j g_{mn} (E^m E^n + B^m B^n)
+ g_{ij} (E^e E^f + B^e B^f)
- 2 \delta^e_i g_{jn} (E^f E^n + B^f B^n)
- 2 g_{im} \delta^e_j (E^m E^f + B^m B^f)
}\right]}$
Matter stress-energy:
$T^{matter}_{ab} = u_a u_b (\rho (1 + e_{int}) + P) + g_{ab} P$
Derivatives:
$\frac{\partial}{\partial g_{ef}} T^{matter}_{ab}
= (
\frac{\partial}{\partial g_{ef}} u_a u_b
+ u_a \frac{\partial}{\partial g_{ef}} u_b
) (\rho (1 + e_{int}) + P)
+ \frac{\partial}{\partial g_{ef}} g_{ab} P
$
$= (\delta^e_a u_b + \delta^e_b u_a ) u^f (\rho (1 + e_{int}) + P)
+ \delta^e_a \delta^f_b P
$
Einstein Field Equations:
$G_{ab} = 8 \pi T_{ab}$
Einstein Field Equation constraint tensor:
$EFE_{ab} = G_{ab} - 8 \pi T_{ab}$
Derivatives:
$\frac{\partial EFE_{ab}}{\partial g_{pq}}$
$ = \frac{\partial G_{ab}}{\partial g_{pq}} - 8 \pi \frac{\partial T_{ab}}{\partial g_{pq}}$
Discretized:
Constraint function:
$\Phi = \frac{1}{2} \underset{ab}\Sigma (EFE_{ab})^2$
$\Phi = \frac{1}{2} \underset{ab}\Sigma (G_{ab} - 8 \pi T_{ab})^2$
Derivatives:
$\frac{\partial \Phi}{\partial g_{pq}} = \frac{1}{2} \underset{ab}\Sigma \left[ 2 EFE_{ab} \right] \cdot \frac{\partial EFE_{ab}}{\partial g_{pq}}$
$ = \underset{ab}\Sigma \left[ EFE_{ab} \right] \cdot (\frac{\partial G_{ab}}{\partial g_{pq}} - 8 \pi \frac{\partial T_{ab}}{\partial g_{pq}})$
Discretized:
$\Phi = \frac{1}{2} \underset{a,b,x}\Sigma (EFE_{ab}(x))^2$
Discrete derivatives of $\Phi$:
$\frac{\partial \Phi}{\partial g_{pq}(x')}
= \frac{\partial}{\partial g_{pq}(x')} \frac{1}{2} \underset{a,b,x}\Sigma EFE_{ab}(x)^2
$
...substitute $\partial EFE_{ab}(x)$...
$= \underset{a,b,x}\Sigma EFE_{ab}(x)
\left(
\frac{\partial}{\partial g_{pq}(x')} G_{ab}(x)
- 8 \pi \frac{\partial}{\partial g_{pq}(x')} T_{ab}(x)
\right)
$
...substitute $G_{ab}(x)$...
$= \underset{a,b,x}\Sigma EFE_{ab}(x) \cdot
\left(
\frac{\partial}{\partial g_{pq}(x')}
\left(
R_{ab}(x) - \frac{1}{2} g_{ab}(x) g^{uv}(x) R_{uv}(x)
\right)
- 8 \pi \frac{\partial}{\partial g_{pq}(x')} T_{ab}(x)
\right)
$
...distribute $\frac{\partial}{\partial g_{pq}}$...
$= \underset{a,b,x}\Sigma EFE_{ab}(x) \cdot
\left(
\frac{\partial}{\partial g_{pq}(x')} R_{ab}(x)
- \frac{1}{2} \frac{\partial}{\partial g_{pq}(x')} g_{ab}(x) g^{uv}(x) R_{uv}(x)
- \frac{1}{2} g_{ab}(x) \frac{\partial}{\partial g_{pq}(x')} g^{uv}(x) R_{uv}(x)
- \frac{1}{2} g_{ab}(x) g^{uv}(x) \frac{\partial}{\partial g_{pq}(x')} R_{uv}(x)
- 8 \pi \frac{\partial}{\partial g_{pq}(x')} T_{ab}(x)
\right)
$
...substitute $\frac{\partial g^{uv}}{\partial g_{pq}}$, assume $\frac{\partial T_{ab}}{\partial g_{pq}}$ is only a function of x'...
$= \underset{a,b,x}\Sigma EFE_{ab}(x) \cdot
\left(
\frac{\partial}{\partial g_{pq}(x')} R_{ab}(x)
- \frac{1}{2} \delta^p_a \delta^q_b g^{uv}(x) R_{uv}(x) \delta(x' - x)
+ \frac{1}{2} g_{ab}(x) g^{pu}(x) g^{qv}(x) R_{uv}(x) \delta(x' - x)
- \frac{1}{2} g_{ab}(x) g^{uv}(x) \frac{\partial}{\partial g_{pq}(x')} R_{uv}(x)
- 8 \pi \frac{\partial}{\partial g_{pq}(x')} T_{ab}(x) \delta(x' - x)
\right)
$
...separate $\delta(x' - x)$'s and $\delta^p_a \delta^q_b$'s...
$=
- \frac{1}{2} EFE_{pq}(x') \cdot g^{uv}(x') R_{uv}(x')
\\
+ \underset{a,b}\Sigma EFE_{ab}(x') \cdot
\left(
\frac{1}{2} g_{ab}(x') g^{pu}(x') g^{qv}(x') R_{uv}(x')
- 8 \pi \frac{\partial}{\partial g_{pq}(x')} T_{ab}(x')
\right)
\\
+ \underset{a,b,x}\Sigma EFE_{ab}(x) \cdot
\left(
\frac{\partial}{\partial g_{pq}(x')} R_{ab}(x)
- \frac{1}{2} g_{ab}(x) g^{uv}(x) \frac{\partial}{\partial g_{pq}(x')} R_{uv}(x)
\right)
$
...simplify and separate $\frac{\partial R_{uv}(x)}{\partial g_{pq}}$'s...
$=
-\frac{1}{2} EFE_{pq}(x') R(x')
\\
+ \underset{a,b}\Sigma EFE_{ab}(x') \cdot
\left(
\frac{1}{2} g_{ab}(x') R^{pq}(x')
- 8 \pi \frac{\partial}{\partial g_{pq}(x')} T_{ab}(x')
\right)
\\
+ \underset{a,b,x}\Sigma EFE_{ab}(x) \cdot
\left(
\delta^u_a \delta^v_b
- \frac{1}{2} g_{ab}(x) g^{uv}(x)
\right) \frac{\partial}{\partial g_{pq}(x')} R_{uv}(x)
$
...substitute $R_{cudv}(x)$...
$=
-\frac{1}{2} EFE_{pq}(x') R(x')
\\
+ \underset{a,b}\Sigma EFE_{ab}(x') \cdot
\left(
\frac{1}{2} g_{ab}(x') R^{pq}(x')
- 8 \pi \frac{\partial}{\partial g_{pq}(x')} T_{ab}(x')
\right)
\\
+ \underset{a,b,x}\Sigma EFE_{ab}(x) \cdot
\left(
\delta^u_a \delta^v_b
- \frac{1}{2} g_{ab}(x) g^{uv}(x)
\right) \frac{\partial}{\partial g_{pq}(x')} \left(
g^{cd}(x) R_{cudv}(x)
\right)
$
...distribute...
$=
-\frac{1}{2} EFE_{pq}(x') R(x')
\\
+ \underset{a,b}\Sigma EFE_{ab}(x') \cdot
\left(
\frac{1}{2} g_{ab}(x') R^{pq}(x')
- 8 \pi \frac{\partial}{\partial g_{pq}(x')} T_{ab}(x')
\right)
\\
+ \underset{a,b,x}\Sigma EFE_{ab}(x) \cdot
\left(
\delta^u_a \delta^v_b
- \frac{1}{2} g_{ab}(x) g^{uv}(x)
\right) \left(
\frac{\partial}{\partial g_{pq}(x')} g^{cd}(x) R_{cudv}(x)
+ g^{cd}(x) \frac{\partial}{\partial g_{pq}(x')} R_{cudv}(x)
\right)
$
...simplify $\frac{\partial g^{cd}(x)}{\partial g_{pq}(x')}$...
$=
-\frac{1}{2} EFE_{pq}(x') R(x')
\\
+ \underset{a,b}\Sigma EFE_{ab}(x') \cdot
\left(
\frac{1}{2} g_{ab}(x') R^{pq}(x')
- 8 \pi \frac{\partial}{\partial g_{pq}(x')} T_{ab}(x')
\right)
\\
+ \underset{a,b,x}\Sigma EFE_{ab}(x) \cdot
\left(
\delta^u_a \delta^v_b
- \frac{1}{2} g_{ab}(x) g^{uv}(x)
\right) \left(
-g^{pc}(x) g^{qd}(x) \delta(x - x') R_{cudv}(x)
+ g^{cd}(x) \frac{\partial}{\partial g_{pq}(x')} R_{cudv}(x)
\right)
$
...distribute deltas...
$=
-\frac{1}{2} EFE_{pq}(x') R(x')
\\
+ \underset{a,b}\Sigma EFE_{ab}(x') \cdot
\left(
\frac{1}{2} g_{ab}(x') R^{pq}(x')
- 8 \pi \frac{\partial}{\partial g_{pq}(x')} T_{ab}(x')
- \left(
\delta^u_a \delta^v_b
- \frac{1}{2} g_{ab}(x') g^{uv}(x')
\right) {{{R^p}_u}^q}_v(x')
\right)
\\
+ \underset{a,b,x}\Sigma EFE_{ab}(x) \cdot
\left(
\delta^u_a \delta^v_b
- \frac{1}{2} g_{ab}(x) g^{uv}(x)
\right)
g^{cd}(x) \frac{\partial}{\partial g_{pq}(x')} R_{cudv}(x)
$
...substitute finite-difference definition of $R_{cudv}(x)$...
$=
-\frac{1}{2} EFE_{pq}(x') R(x')
\\
+ \underset{a,b}\Sigma EFE_{ab}(x') \cdot
\left(
\frac{1}{2} g_{ab}(x') R^{pq}(x')
- 8 \pi \frac{\partial}{\partial g_{pq}(x')} T_{ab}(x')
- \left(
\delta^u_a \delta^v_b
- \frac{1}{2} g_{ab}(x') g^{uv}(x')
\right) {{{R^p}_u}^q}_v(x')
\right)
\\
+ \underset{a,b,x}\Sigma EFE_{ab}(x) \cdot
\left(
\delta^u_a \delta^v_b
- \frac{1}{2} g_{ab}(x) g^{uv}(x)
\right)
g^{cd}(x) \frac{\partial}{\partial g_{pq}(x')} \left(
\frac{1}{2} \left(
D^2_{ud}[g_{cv}(x)]
+ D^2_{cv}[g_{ud}(x)]
- D^2_{cd}[g_{uv}(x)]
- D^2_{uv}[g_{cd}(x)]
\right)
+ g^{ef}(x) \left(
\Gamma_{ecv}(x) \Gamma_{fud}(x)
- \Gamma_{ecd}(x) \Gamma_{fuv}(x)
\right)
\right)
$
...distribute partial...
$=
-\frac{1}{2} EFE_{pq}(x') R(x')
\\
+ \underset{a,b}\Sigma EFE_{ab}(x') \cdot
\left(
\frac{1}{2} g_{ab}(x') R^{pq}(x')
- 8 \pi \frac{\partial}{\partial g_{pq}(x')} T_{ab}(x')
- \left(
\delta^u_a \delta^v_b
- \frac{1}{2} g_{ab}(x') g^{uv}(x')
\right) {{{R^p}_u}^q}_v(x')
\right)
\\
+ \underset{a,b,x}\Sigma EFE_{ab}(x) \cdot
\left(
\delta^u_a \delta^v_b
- \frac{1}{2} g_{ab}(x) g^{uv}(x)
\right)
g^{cd}(x) \left(
\frac{1}{2} \frac{\partial}{\partial g_{pq}(x')} \left(
D^2_{ud}[g_{cv}(x)]
+ D^2_{cv}[g_{ud}(x)]
- D^2_{cd}[g_{uv}(x)]
- D^2_{uv}[g_{cd}(x)]
\right)
+ \frac{\partial}{\partial g_{pq}(x')} g^{ef}(x) \left(
\Gamma_{ecv}(x) \Gamma_{fud}(x)
- \Gamma_{ecd}(x) \Gamma_{fuv}(x)
\right)
+ g^{ef}(x) \frac{\partial}{\partial g_{pq}(x')} \left(
\Gamma_{ecv}(x) \Gamma_{fud}(x)
- \Gamma_{ecd}(x) \Gamma_{fuv}(x)
\right)
\right)
$
...simplify $\frac{\partial g^{ef}(x)}{\partial g_{pq}(x')}$...
$=
-\frac{1}{2} EFE_{pq}(x') R(x')
\\
+ \underset{a,b}\Sigma EFE_{ab}(x') \cdot
\left(
\frac{1}{2} g_{ab}(x') R^{pq}(x')
- 8 \pi \frac{\partial}{\partial g_{pq}(x')} T_{ab}(x')
- \left(
\delta^u_a \delta^v_b
- \frac{1}{2} g_{ab}(x') g^{uv}(x')
\right) {{{R^p}_u}^q}_v(x')
\right)
\\
+ \underset{a,b,x}\Sigma EFE_{ab}(x) \cdot
\left(
\delta^u_a \delta^v_b
- \frac{1}{2} g_{ab}(x) g^{uv}(x)
\right)
g^{cd}(x) \left(
\frac{1}{2} \frac{\partial}{\partial g_{pq}(x')} \left(
D^2_{ud}[g_{cv}(x)]
+ D^2_{cv}[g_{ud}(x)]
- D^2_{cd}[g_{uv}(x)]
- D^2_{uv}[g_{cd}(x)]
\right)
- g^{pe}(x) g^{qf}(x) \delta(x' - x) \left(
\Gamma_{ecv}(x) \Gamma_{fud}(x)
- \Gamma_{ecd}(x) \Gamma_{fuv}(x)
\right)
+ g^{ef}(x) \frac{\partial}{\partial g_{pq}(x')} \left(
\Gamma_{ecv}(x) \Gamma_{fud}(x)
- \Gamma_{ecd}(x) \Gamma_{fuv}(x)
\right)
\right)
$
...simplify deltas...
$=
-\frac{1}{2} EFE_{pq}(x') R(x')
\\
+ \underset{a,b}\Sigma EFE_{ab}(x') \cdot
\left(
\frac{1}{2} g_{ab}(x') R^{pq}(x')
- 8 \pi \frac{\partial}{\partial g_{pq}(x')} T_{ab}(x')
+ \left(
\delta^u_a \delta^v_b
- \frac{1}{2} g_{ab}(x') g^{uv}(x')
\right) \left(
- {{{R^p}_u}^q}_v(x')
- {\Gamma^{pc}}_v(x') {\Gamma^q}_{cu}(x')
+ {\Gamma^{pc}}_c(x') {\Gamma^q}_{uv}(x')
\right)
\right)
\\
+ \underset{a,b,x}\Sigma EFE_{ab}(x) \cdot
\left(
\delta^u_a \delta^v_b
- \frac{1}{2} g_{ab}(x) g^{uv}(x)
\right)
g^{cd}(x) \left(
\frac{1}{2} \frac{\partial}{\partial g_{pq}(x')} \left(
D^2_{ud}[g_{cv}(x)]
+ D^2_{cv}[g_{ud}(x)]
- D^2_{cd}[g_{uv}(x)]
- D^2_{uv}[g_{cd}(x)]
\right)
+ g^{ef}(x) \frac{\partial}{\partial g_{pq}(x')} \left(
\Gamma_{ecv}(x) \Gamma_{fud}(x)
- \Gamma_{ecd}(x) \Gamma_{fuv}(x)
\right)
\right)
$
...distribute around ${{{R^p}_u}^q}_v(x')$ and combine...
$=
-\frac{1}{2} EFE_{pq}(x') R(x')
\\
+ \underset{a,b}\Sigma EFE_{ab}(x') \cdot
\left(
g_{ab}(x') R^{pq}(x')
- 8 \pi \frac{\partial}{\partial g_{pq}(x')} T_{ab}(x')
- {{{R^p}_a}^q}_b(x')
+ \left(
\delta^u_a \delta^v_b
- \frac{1}{2} g_{ab}(x') g^{uv}(x')
\right)
\left(
- {\Gamma^{pc}}_v(x') {\Gamma^q}_{cu}(x')
+ {\Gamma^{pc}}_c(x') {\Gamma^q}_{uv}(x')
\right)
\right)
\\
+ \underset{a,b,x}\Sigma EFE_{ab}(x) \cdot
\left(
\delta^u_a \delta^v_b
- \frac{1}{2} g_{ab}(x) g^{uv}(x)
\right)
g^{cd}(x) \left(
\frac{1}{2} \frac{\partial}{\partial g_{pq}(x')} \left(
D^2_{ud}[g_{cv}(x)]
+ D^2_{cv}[g_{ud}(x)]
- D^2_{cd}[g_{uv}(x)]
- D^2_{uv}[g_{cd}(x)]
\right)
+ g^{ef}(x) \frac{\partial}{\partial g_{pq}(x')} \left(
\Gamma_{ecv}(x) \Gamma_{fud}(x)
- \Gamma_{ecd}(x) \Gamma_{fuv}(x)
\right)
\right)
$
...substitute $\Gamma_{abc}(x)$ definition...
$=
-\frac{1}{2} EFE_{pq}(x') R(x')
\\
\quad + \underset{a,b}\Sigma EFE_{ab}(x') \cdot
\left(
g_{ab}(x') R^{pq}(x')
- 8 \pi \frac{\partial}{\partial g_{pq}(x')} T_{ab}(x')
- {{{R^p}_a}^q}_b(x')
+ \left(
\delta^u_a \delta^v_b
- \frac{1}{2} g_{ab}(x') g^{uv}(x')
\right)
\left(
- {\Gamma^{pc}}_v(x') {\Gamma^q}_{cu}(x')
+ {\Gamma^{pc}}_c(x') {\Gamma^q}_{uv}(x')
\right)
\right)
\\
\quad + \underset{a,b,x}\Sigma EFE_{ab}(x) \cdot
\left(
\delta^u_a \delta^v_b
- \frac{1}{2} g_{ab}(x) g^{uv}(x)
\right)
g^{cd}(x) \left(
\\
\quad\quad \frac{1}{2} \frac{\partial}{\partial g_{pq}(x')} \left(
D^2_{ud}[g_{cv}(x)]
+ D^2_{cv}[g_{ud}(x)]
- D^2_{cd}[g_{uv}(x)]
- D^2_{uv}[g_{cd}(x)]
\right)
\\
\quad\quad + g^{ef}(x) \frac{\partial}{\partial g_{pq}(x')} \left(
\\
\quad\quad\quad \frac{1}{4}
\left(
D_v[g_{ec}(x)]
+ D_c[g_{ev}(x)]
- D_e[g_{cv}(x)]
\right)
\left(
D_d[g_{fu}(x)]
+ D_u[g_{fd}(x)]
- D_f[g_{ud}(x)]
\right)
\\
\quad\quad\quad
- \frac{1}{4}
\left(
D_d[g_{ec}(x)]
+ D_c[g_{ed}(x)]
- D_e[g_{cd}(x)]
\right)
\left(
D_v[g_{fu}(x)]
+ D_u[g_{fv}(x)]
- D_f[g_{uv}(x)]
\right)
\\
\quad\quad
\right)
\\
\quad
\right)
$
...distribute multiplication, distribute $g^{cd}$, symmetrize cd, ef, uv, sometimes swap cd and ef...
$=
-\frac{1}{2} EFE_{pq}(x') R(x')
\\
\quad + \underset{a,b}\Sigma EFE_{ab}(x') \cdot
\left(
g_{ab}(x') R^{pq}(x')
- 8 \pi \frac{\partial}{\partial g_{pq}(x')} T_{ab}(x')
- {{{R^p}_a}^q}_b(x')
+ \left(
\delta^u_a \delta^v_b
- \frac{1}{2} g_{ab}(x') g^{uv}(x')
\right)
\left(
- {\Gamma^{pc}}_v(x') {\Gamma^q}_{cu}(x')
+ {\Gamma^{pc}}_c(x') {\Gamma^q}_{uv}(x')
\right)
\right)
\\
\quad + \underset{a,b,x}\Sigma EFE_{ab}(x) \cdot
\left(
\delta^u_a \delta^v_b
- \frac{1}{2} g_{ab}(x) g^{uv}(x)
\right)
\left(
\\ \quad\quad
\frac{1}{2} g^{cd}(x) \frac{\partial}{\partial g_{pq}(x')} \left(
2 D^2_{cu}[g_{dv}(x)]
- D^2_{cd}[g_{uv}(x)]
- D^2_{uv}[g_{cd}(x)]
\right)
\\ \quad\quad
+ \frac{1}{4} g^{cd}(x) g^{ef}(x) \frac{\partial}{\partial g_{pq}(x')} \left(
\\ \quad\quad\quad
D_u[g_{ce}(x)] D_v[g_{df}(x)]
\\ \quad\quad\quad
+ 2 D_c[g_{eu}(x)] D_d[g_{fv}(x)]
\\ \quad\quad\quad
- D_c[g_{ef}(x)] D_d[g_{uv}(x)]
\\ \quad\quad\quad
- 2 D_c[g_{eu}(x)] D_f[g_{dv}(x)]
\\ \quad\quad\quad
+ 2 D_c[g_{de}(x)] D_f[g_{uv}(x)]
\\ \quad\quad\quad
+ 2 D_c[g_{ef}(x)] D_u[g_{dv}(x)]
\\ \quad\quad\quad
- 4 D_c[g_{de}(x)] D_u[g_{fv}(x)]
\\ \quad\quad
\right)
\\
\quad
\right)
$
Derivatives of continuous $\Phi$ wrt $\alpha, \beta^i, \gamma_{ij}$:
$\frac{\partial \Phi}{\partial \alpha}
= \frac{\partial \Phi}{\partial g_{pq}} \frac{\partial g_{pq}}{\partial \alpha}
= -2 \alpha \frac{\partial \Phi}{\partial g_{tt}}
$
$\frac{\partial \Phi}{\partial \beta^m}
= \frac{\partial \Phi}{\partial g_{pq}} \frac{\partial g_{pq}}{\partial \beta^m}
$
$ = p(i)\downarrow \overset{q(j)\rightarrow}{\left[\matrix{
\frac{\partial \Phi}{\partial g_{tt}} &
\frac{\partial \Phi}{\partial g_{tj}} \\
\frac{\partial \Phi}{\partial g_{it}} &
\frac{\partial \Phi}{\partial g_{ij}}
}\right]}
\cdot p(i)\downarrow \overset{q(j)\rightarrow}{\left[ \matrix{
2 \beta_m & \gamma_{mj} \\ \gamma_{mi} & 0
} \right]}$
$ = 2 \frac{\partial \Phi}{\partial g_{tt}} \beta_m
+ 2 \frac{\partial \Phi}{\partial g_{ti}} \gamma_{im}
$
$\frac{\partial \Phi}{\partial \gamma_{mn}}
= \frac{\partial \Phi}{\partial g_{pq}} \frac{\partial g_{pq}}{\partial \gamma_{mn}}
$
$ = p(i)\downarrow \overset{q(j)\rightarrow}{\left[\matrix{
\frac{\partial \Phi}{\partial g_{tt}} &
\frac{\partial \Phi}{\partial g_{tj}} \\
\frac{\partial \Phi}{\partial g_{it}} &
\frac{\partial \Phi}{\partial g_{ij}}
}\right]}
\cdot p(i)\downarrow \overset{q(j)\rightarrow}{\left[ \matrix{
\beta^m \beta^n & \beta^m \delta^j_n \\ \beta^m \delta^n_i & \delta^m_i \delta^n_j
} \right]}$
$ = \frac{\partial \Phi}{\partial g_{tt}} \beta^m \beta^n
+ 2 \frac{\partial \Phi}{\partial g_{tn}} \beta^m
+ \frac{\partial \Phi}{\partial g_{mn}}
$