Let $\eta_{ab} = diag(-1,1,1,1)$ (someday I'll switch)
Indexes denoted $ijklmn$ are raised/lowered with the spatial metric $\gamma_{ij}$ alone.
All others are raised/lowered by the spacetime metric $g_{ab}$.
$\partial = (\frac{1}{c} \partial_t, \partial_i) = (\partial_0, \partial_i)$ is in units of $\frac{1}{m}$
$dx = (c dt, dx^i) = (dx^0, dx^i)$ is in units $m$
$\partial_u T = T_{,u} = (\partial_0, \partial_i) T$
Vectors / matrices are displayed as $V = \left[ V_0, V_x, V_y, V_z \right] = \left[ c V_t, V_x, V_y, V_z \right]$.
Take note when referencing $V_t$ that
$V_t dt = V_t \frac{1}{c} dx^0 = V_0 dx^0$, therefore $V_t = \frac{1}{c} V_0$
In SI units (from my 'natural units' worksheet in my symmath library):
$\mu_0 = 1.25663706212 \cdot 10^{-6} \frac{kg \cdot m}{C^2} \approx \frac{4 \pi}{10^7} \frac{kg \cdot m}{C^2}$
$\epsilon_0 = \frac{1}{\mu_0 c^2}$ is in units $\frac{C^2 \cdot s^2}{kg \cdot m^3}$
$\frac{1}{c^2} = \mu_0 \epsilon_0$ is in units $\frac{s^2}{m^2}$
vector math rules:
$\vec\nabla \cdot \vec{V} = {V^i}_{,i}$ but this neglects the metric which isn't good.
$(\vec\nabla \times \vec{V})^i = \bar\epsilon^{ijk} V_{k,j}$ but this neglects the metric which isn't good, and I only use this symbol in the macroscopic equations in medium, possibly only in SR, so I should change this definition.
exterior algebra rules:
$a \wedge b = a \otimes b - b \otimes a$
$a^1 \wedge ... \wedge a^p = \delta^{1...p}_{i_1 ... i_p} a^{i_1} \otimes ... \otimes a^{i_p}$
$= p! \delta^{1...p}_{[i_1 ... i_p]} a^{i_1} \otimes ... \otimes a^{i_p}$
$= \epsilon_{i_1 ... i_p} a^{i_1} \otimes ... \otimes a^{i_p}$
k-form on n-dimensional manifold:
$T = T_{\alpha_1 ... \alpha_k} dx^{\alpha_1} \otimes ... \otimes dx^{\alpha_k}$
$= \frac{1}{k!} \cdot T_{\alpha_1 ... \alpha_k} dx^{\alpha_1} \wedge ... \wedge dx^{\alpha_k}$
Exterior derivative:
$dT = (dT)_{\alpha_1 ... \alpha_{k+1}} dx^{\alpha_1} \otimes ... \otimes dx^{\alpha_{k+1}}$
$= \partial_{\beta_1} T_{\beta_2 ... \beta_{k+1}} \delta^{\beta_1 ... \beta_{k+1}}_{\alpha_1 ... \alpha_{k+1}} dx^{\alpha_1} \otimes ... \otimes dx^{\alpha_{k+1}}$
$= (k+1)! \cdot \partial_{[\alpha_1} T_{\alpha_2 ... \alpha_{k+1}]} dx^{\alpha_1} \otimes ... \otimes dx^{\alpha_{k+1}}$
$= \partial_{\alpha_1} T_{\alpha_2 ... \alpha_{k+1}} dx^{\alpha_1} \wedge ... \wedge dx^{\alpha_{k+1}}$
$= dx^{\alpha_1} \wedge \partial_{\alpha_1} (T_{\alpha_2 ... \alpha_{k+1}} dx^{\alpha_2} \wedge ... \wedge dx^{\alpha_{k+1}})$
$= dx^\mu \wedge \partial_\mu T$
Recap from my ADM metric worksheet in the Differential Geometry section:
(In terms of units, there might be a $c$ somewhere in here...)
$[g_{uv}] = \downarrow u(i) \overset{\rightarrow v(j)}{
\left[ \begin{matrix}
-\alpha^2 + \beta_k \beta^k & \beta_j \\
\beta_i & \gamma_{ij}
\end{matrix} \right] }$
$[g^{uv}] = \downarrow u(i) \overset{\rightarrow v(j)}{ \left[ \begin{matrix}
-1/\alpha^2 & \beta^j / \alpha^2 \\
\beta^i / \alpha^2 & \gamma^{ij} - \beta^i \beta^j / \alpha^2
\end{matrix} \right] }$
For $\alpha$ the lapse between $\Sigma_t$ and $\Sigma_{t + dt}$
And $\beta$ the shift in coordinates, $\beta \in \Sigma$. Note that, for any vector $u \in \Sigma$ that $u^0 = 0$.
And $\gamma_{ij}$ the metric of $\Sigma$
Where $ \beta_j = \gamma_{ij} \beta^i = \gamma_{u j} \beta^u = g_{u j} \beta^j $ courtesy of the fact that $ g_{ij} = \gamma_{ij} $ and $ \beta^0 = 0 $.
But this also means, for some $u \in \Sigma$, $u^0 = 0$, however $u_t = u^j \beta_j$, while $u_i = \gamma_{ij} u^j$.
ADM metric determinant: $g = det[g_{ab}] = \alpha \sqrt{\gamma}$
ADM spatial metric determinant: $\gamma = det[\gamma_{ij}]$
normal in the ADM metric:
$n_u = \overset{\rightarrow u(i)}{ \left[ \begin{matrix} -\alpha & 0 \end{matrix} \right] } $
$n^u = \downarrow u(i) \left[ \begin{matrix} 1/\alpha \\ -\beta^i/\alpha \end{matrix} \right]$
notice: $n_u n^u = -1$
Recap from my Levi-Civita tensor worksheet in the Differential Geometry section, with specifics for the ADM metric:
Let $[abcd] = \bar\epsilon_{abcd} = $ permutation symbol for $abcd$
Same with $[ijk] = \bar\epsilon_{ijk} = $ spatial permutation symbol for $ijk$
I'm going to raise and lower $\bar\epsilon_{ijk}$ by $\eta_{ij}$
Therefore $\bar\epsilon_{0ijk} = \bar\epsilon_{ijk} = \bar\epsilon^{ijk} = -\bar\epsilon_{ijk0} = -\bar\epsilon^{0ijk}$.
If I was to start describing things in terms of a diagonalized basis at each point, and if I was to give them a separate coordinate system, then I might replace $\bar\epsilon_{abcd}$ with the Levi-Civita permutation tensor associated with the indexes associated with the diagonalized metric.
TODO put the rest of this in the Differential Geometry ADM formalism worksheet:
TODO spatial indexes of vectors ... should raised vectors be raised using the 3 or 4 metric? How to distinguish?
Let $\epsilon_{abcd} = \sqrt{|g|} [abcd] = \sqrt{|g|} \bar\epsilon_{abcd} $ is the spacetime Levi-Civita permutation tensor
So $\epsilon_{0ijk} = \alpha \sqrt\gamma \bar\epsilon_{0ijk} = \alpha \sqrt\gamma \bar\epsilon_{ijk}$ (see the ADM section for more details)
$\epsilon_{ijk} = \sqrt\gamma [ijk] = \sqrt\gamma \bar\epsilon_{ijk}$ is the spatial Levi-Civita permutation tensor
So $\epsilon_{0ijk} = \alpha \epsilon_{ijk}$
And ${\epsilon^0}_{ijk} = g^{0a} \epsilon_{aijk} = g^{00} \epsilon_{0ijk} + g^{0\Sigma} \epsilon_{\Sigma ijk}$, but since $ijk \in \Sigma$ we know $\epsilon_{\Sigma ijk} = 0$
So ${\epsilon^0}_{ijk} = -\frac{1}{\alpha^2} \epsilon_{0ijk} = -\frac{1}{\alpha} \epsilon_{ijk} = -\frac{1}{\alpha} \sqrt\gamma \bar\epsilon_{0ijk} = -\frac{1}{\alpha} \sqrt\gamma \bar\epsilon_{ijk}$
Notice that ${\epsilon^0}_{ijk}$ does not account for all terms of ${\epsilon^0}_{\alpha\beta\gamma}$.
${\epsilon^{0i}}_{jk}$
$= g^{0a} g^{ib} \epsilon_{abjk} $
$= (
g^{00} g^{i0} \epsilon_{00jk}
+ g^{0m} g^{i0} \epsilon_{m0jk}
+ g^{00} g^{in} \epsilon_{0njk}
+ g^{0m} g^{in} \epsilon_{mnjk}
) $
$= (
0
+ \alpha^{-4} \beta^m \beta^i \epsilon_{m0jk}
- \alpha^{-2} (\gamma^{in} - \alpha^{-2} \beta^i \beta^n) \epsilon_{0njk}
+ \alpha^{-2} \beta^m (\gamma^{in} - \alpha^{-2} \beta^i \beta^n) \epsilon_{mnjk}
)$
$= -\frac{1}{\alpha^2} \gamma^{im} \epsilon_{0mjk}$
$= -\frac{1}{\alpha} \gamma^{im} \epsilon_{mjk}$
$= -\frac{1}{\alpha} {\epsilon^i}_{jk}$ ... where the hypersurface Levi-Civita tensor is raised by the 3-metric.
Notice that, like above ${\epsilon^{0i}}_{jk}$ do not account for all terms of ${\epsilon^{0i}}_{\mu\nu}$.
In ADM formalism, $\epsilon^{0abc} u_a v_b w_c = \epsilon^{0ijk} u_i v_j w_k$ since $\epsilon^{00ab} = 0$ due to its antisymmetric. The associated $u_0, v_0, w_0$ components will be cancelled.
In ADM formalism:
$\star dx^0 = {\epsilon^0}_{\alpha\beta\gamma} dx^\alpha \otimes dx^\beta \otimes dx^\gamma$
$= g^{0\mu} \epsilon_{\mu\alpha\beta\gamma} dx^\alpha \otimes dx^\beta \otimes dx^\gamma$
$= (g^{00} \epsilon_{0 \alpha \beta \gamma} + g^{0i} \epsilon_{i \alpha \beta \gamma}) dx^\alpha \otimes dx^\beta \otimes dx^\gamma$
$= (-\alpha^{-2} \epsilon_{0\alpha\beta\gamma} + \alpha^{-2} \beta^i \epsilon_{i\alpha\beta\gamma}) dx^\alpha \otimes dx^\beta \otimes dx^\gamma$
$= -\alpha^{-2} \epsilon_{0ijk} dx^i \otimes dx^j \otimes dx^k
+ \alpha^{-2} \beta^i \epsilon_{i0jk} dx^0 \otimes dx^j \otimes dx^k
+ \alpha^{-2} \beta^i \epsilon_{ij0k} dx^j \otimes dx^0 \otimes dx^k
+ \alpha^{-2} \beta^i \epsilon_{ijk0} dx^j \otimes dx^k \otimes dx^0
$
$= -\alpha^{-2} \epsilon_{0ijk} dx^i \otimes dx^j \otimes dx^k
- \alpha^{-2} \beta^i \epsilon_{0ijk} dx^0 \otimes dx^j \otimes dx^k
+ \alpha^{-2} \beta^i \epsilon_{0ijk} dx^j \otimes dx^0 \otimes dx^k
- \alpha^{-2} \beta^i \epsilon_{0ijk} dx^j \otimes dx^k \otimes dx^0
$
Therefore, for $\star dx^0 = (\star dx^0)_{\alpha\beta\gamma} dx^\alpha \otimes dx^\beta \otimes dx^\gamma$, we find:
$(\star dx^0)_{ijk} = -\frac{1}{\alpha^2} \epsilon_{0ijk} = -\frac{1}{\alpha} \epsilon_{ijk} = -\frac{1}{\alpha} \sqrt\gamma \bar\epsilon_{ijk}$
$(\star dx^0)_{0jk} = -\frac{1}{\alpha^2} \beta^i \epsilon_{0ijk} = -\frac{1}{\alpha} \beta^i \epsilon_{ijk} = -\frac{1}{\alpha} \sqrt\gamma \beta^i \bar\epsilon_{ijk}$
$\star dx^i = {\epsilon^i}_{\alpha\beta\gamma} dx^\alpha \otimes dx^\beta \otimes dx^\gamma$
$= (g^{i0} \epsilon_{0\alpha\beta\gamma} + g^{ij} \epsilon_{j\alpha\beta\gamma}) dx^\alpha \otimes dx^\beta \otimes dx^\gamma$
$= (\alpha^{-2} \beta^i \epsilon_{0\alpha\beta\gamma} + (\gamma^{ij} - \alpha^{-2} \beta^i \beta^j) \epsilon_{j\alpha\beta\gamma}) dx^\alpha \otimes dx^\beta \otimes dx^\gamma$
$= \alpha^{-2} \beta^i \epsilon_{0\alpha\beta\gamma} dx^\alpha \otimes dx^\beta \otimes dx^\gamma
+ (\gamma^{ij} - \alpha^{-2} \beta^i \beta^j) \epsilon_{j\alpha\beta\gamma} dx^\alpha \otimes dx^\beta \otimes dx^\gamma
$
$= \alpha^{-2} \beta^i \epsilon_{0jkl} dx^j \otimes dx^k \otimes dx^l
+ (\gamma^{im} - \alpha^{-2} \beta^i \beta^m) \epsilon_{m0jk} dx^0 \otimes dx^j \otimes dx^k
+ (\gamma^{im} - \alpha^{-2} \beta^i \beta^m) \epsilon_{mj0k} dx^j \otimes dx^0 \otimes dx^k
+ (\gamma^{im} - \alpha^{-2} \beta^i \beta^m) \epsilon_{mjk0} dx^j \otimes dx^k \otimes dx^0
$
Therefore, for $\star dx^i = (\star dx^i)_{\alpha \beta \gamma} dx^\alpha \otimes dx^\beta \otimes dx^\gamma$, we find:
$(\star dx^i)_{jkl} = \frac{1}{\alpha^2} \beta^i \epsilon_{0jkl} = \frac{1}{\alpha} \beta^i \epsilon_{jkl}$
$(\star dx^i)_{0jk} = (\frac{1}{\alpha^2} \beta^i \beta^m - \gamma^{im}) \epsilon_{0mjk} = (\frac{1}{\alpha} \beta^i \beta^m - \alpha \gamma^{im}) \epsilon_{mjk} = \frac{1}{\alpha} \beta^i \beta^m \epsilon_{mjk} - \alpha {\epsilon^i}_{jk}$
$\star (dx^0 \otimes dx^i) = {\epsilon^{0i}}_{\alpha\beta} dx^\alpha \otimes dx^\beta$
$= {\epsilon^{0i}}_{\alpha\beta} dx^\alpha \otimes dx^\beta$
$= g^{0\mu} g^{i\nu} \epsilon_{\mu\nu\alpha\beta} dx^\alpha \otimes dx^\beta$
$=
g^{00} g^{i0} \epsilon_{00\alpha\beta} dx^\alpha \otimes dx^\beta
+ g^{00} g^{ij} \epsilon_{0j\alpha\beta} dx^\alpha \otimes dx^\beta
+ g^{0j} g^{i0} \epsilon_{j0\alpha\beta} dx^\alpha \otimes dx^\beta
+ g^{0j} g^{ik} \epsilon_{jk\alpha\beta} dx^\alpha \otimes dx^\beta
$
$=
(g^{00} g^{ij} - g^{0j} g^{i0}) \epsilon_{0jkl} dx^k \otimes dx^l
+ g^{0j} g^{ik} \epsilon_{jk0m} dx^0 \otimes dx^m
+ g^{0j} g^{ik} \epsilon_{jkm0} dx^m \otimes dx^0
$
$=
(-\frac{1}{\alpha^2} (\gamma^{ij} - \frac{1}{\alpha^2} \beta^i \beta^j) - \frac{1}{\alpha^4} \beta^i \beta^j) \epsilon_{0jkl} dx^k \otimes dx^l
+ \frac{1}{\alpha^2} \beta^j (\gamma^{ik} - \frac{1}{\alpha^2} \beta^i \beta^k) \epsilon_{0jkm} dx^0 \otimes dx^m
- \frac{1}{\alpha^2} \beta^j (\gamma^{ik} - \frac{1}{\alpha^2} \beta^i \beta^k) \epsilon_{0jkm} dx^m \otimes dx^0
$
$=
(
-\frac{1}{\alpha^2} (\gamma^{ij} - \frac{1}{\alpha^2} \beta^i \beta^j)
- \frac{1}{\alpha^4} \beta^i \beta^j
) \epsilon_{0jkl} dx^k \otimes dx^l
+ \frac{1}{\alpha^2} \beta^j (
\gamma^{ik}
- \frac{1}{\alpha^2} \beta^i \beta^k
) \epsilon_{0jkm} dx^0 \otimes dx^m
- \frac{1}{\alpha^2} \beta^j (
\gamma^{ik}
- \frac{1}{\alpha^2} \beta^i \beta^k
) \epsilon_{0jkm} dx^m \otimes dx^0
$
Therefore, for $\star (dx^0 \otimes dx^i) = (\star (dx^0 \otimes dx^i))_{\alpha\beta} dx^\alpha \otimes dx^\beta$
$(\star (dx^0 \otimes dx^i))_{kl} = -\frac{1}{\alpha^2} \gamma^{ij} \epsilon_{0jkl}
= -\frac{1}{\alpha} \gamma^{ij} \epsilon_{jkl}
= -\frac{1}{\alpha} {\epsilon^i}_{kl}$
$(\star (dx^0 \otimes dx^i))_{0m} = -(\star (dx^0 \otimes dx^i))_{m0}
= \frac{1}{\alpha^2} (
\beta^j \gamma^{ik}
- \frac{1}{\alpha^2} \beta^i \beta^j \beta^k
) \epsilon_{0jkm}
= \frac{1}{\alpha^2} \beta^j \gamma^{ik} \epsilon_{0jkm}
= \frac{1}{\alpha} \beta^j \gamma^{ik} \epsilon_{jkm}
= -\frac{1}{\alpha} \beta^j {\epsilon^i}_{jm}
$
...where the hypersurface Levi-Civita tensor is raised by the 3-metric.
So for spatial one-form $V = V_\mu dx^\mu = V_i dx^i$ with $V_0 = 0$, we get:
$\star (V \otimes dx^0) = \star (V_i dx^i \otimes dx^0) = -V_i \star (dx^0 \otimes dx^i)$
$\star (dx^j \otimes dx^k) = {\epsilon^{jk}}_{\alpha\beta} dx^\alpha \otimes dx^\beta$
$= g^{j\mu} g^{k\nu} \epsilon_{\mu\nu\alpha\beta} dx^\alpha \otimes dx^\beta$
$= (
g^{j0} g^{k0} \epsilon_{00\alpha\beta}
+ g^{jl} g^{k0} \epsilon_{l0\alpha\beta}
+ g^{j0} g^{km} \epsilon_{0m\alpha\beta}
+ g^{jl} g^{km} \epsilon_{lm\alpha\beta}
) dx^\alpha \otimes dx^\beta$
$=
(
\frac{1}{\alpha^2} (\gamma^{jl} - \frac{1}{\alpha^2} \beta^j \beta^l) \beta^k \epsilon_{l0pq}
+ \frac{1}{\alpha^2} \beta^j (\gamma^{km} - \frac{1}{\alpha^2} \beta^k \beta^m) \epsilon_{0mpq}
) dx^p \otimes dx^q
+ (\gamma^{jl} - \frac{1}{\alpha^2} \beta^j \beta^l) (\gamma^{km} - \frac{1}{\alpha^2} \beta^k \beta^m) \epsilon_{lm0p} dx^0 \otimes dx^p
+ (\gamma^{jl} - \frac{1}{\alpha^2} \beta^j \beta^l) (\gamma^{km} - \frac{1}{\alpha^2} \beta^k \beta^m) \epsilon_{lmp0} dx^p \otimes dx^0
$
$=
\frac{1}{\alpha^2} (\beta^j \gamma^{km} - \gamma^{jm} \beta^k) \epsilon_{0mpq} dx^p \otimes dx^q
+ (
\gamma^{jl} \gamma^{km}
- \frac{1}{\alpha^2} \beta^k \beta^m \gamma^{jl}
- \frac{1}{\alpha^2} \beta^j \beta^l \gamma^{km}
) \epsilon_{0lmp} dx^0 \otimes dx^p
- (
\gamma^{jl} \gamma^{km}
- \frac{1}{\alpha^2} \beta^k \beta^m \gamma^{jl}
- \frac{1}{\alpha^2} \beta^j \beta^l \gamma^{km}
) \epsilon_{0lmp} dx^p \otimes dx^0
$
$(\star (dx^j \otimes dx^k))_{pq}
= \frac{1}{\alpha^2} (\beta^j \gamma^{km} - \gamma^{jm} \beta^k) \epsilon_{0mpq}
= \frac{1}{\alpha} (\beta^j \gamma^{km} - \gamma^{jm} \beta^k) \epsilon_{mpq}
= 2 \frac{1}{\alpha} \beta^{[j} {\epsilon^{k]}}_{pq}
$
$(\star (dx^j \otimes dx^k))_{0p} = -(\star (dx^j \otimes dx^k))_{p0}
= (
\gamma^{jl} \gamma^{km}
- \frac{1}{\alpha^2} \beta^k \beta^m \gamma^{jl}
- \frac{1}{\alpha^2} \beta^j \beta^l \gamma^{km}
) \epsilon_{0lmp}
= (
\alpha \gamma^{jl} \gamma^{km}
- \frac{1}{\alpha} \beta^k \beta^m \gamma^{jl}
- \frac{1}{\alpha} \beta^j \beta^l \gamma^{km}
) \epsilon_{lmp}
$
Double-dual:
$\star F_{ab} dx^a \otimes dx^b = \frac{1}{2} F_{ab} {\epsilon^{ab}}_{cd} dx^c \otimes dx^d$
$\star^2 F_{ab} dx^a \otimes dx^b = \frac{1}{4} F_{ab} {\epsilon^{ab}}_{uv} {\epsilon^{uv}}_{cd} dx^c \otimes dx^d$
$= \frac{1}{4} F_{ab} {\epsilon^{ab}}_{uv} {\epsilon_{cd}}^{uv} dx^c \otimes dx^d$
$= -\frac{1}{2} F_{ab} \delta^{ab}_{cd} dx^c \otimes dx^d$
$= -F_{ab} (\delta^a_{[c} \delta^b_{d]}) dx^c \otimes dx^d$
$= -F_{cd} dx^c \otimes dx^d$
Sources:
Misner, Throne, Wheeler 1973
Baumgarte & Shapiro, 2010
That one Alcubierre et al paper on charged black holes
Tajmar and DeMatos' paper
A few wiki pages:
https://en.wikipedia.org/wiki/Covariant_formulation_of_classical_electromagnetism
https://en.wikipedia.org/wiki/Electromagnetic_tensor
https://en.wikipedia.org/wiki/Maxwell%27s_equations
https://en.wikipedia.org/wiki/Magnetic_susceptibility
https://en.wikipedia.org/wiki/Electric_susceptibility
https://en.wikipedia.org/wiki/Maxwell%27s_equations_in_curved_spacetime