Start with my 'wave equation in spacetime' worksheet for an introduction to a scalar wave propagating in curved spacetime: https://thenumbernine.github.io/symmath/tests/output/wave%20equation%20in%20spacetime.html
TODO fix the source terms - take them from the above mentioned worksheet.
Let the source function $f = \frac{dV}{d|\Phi|^2}$
$(\square - \frac{dV}{d|\Phi|^2}) \Phi = 0$
Let $V = \mu^2 |\Phi|^2 + \frac{1}{2} \lambda |\Phi|^4$
$\frac{dV}{d|\Phi|^2} = \mu^2 + \lambda |\Phi|^2$
Let $\Lambda = \frac{\lambda}{4 \pi \mu^2}$, so $\lambda = 4 \pi \Lambda \mu^2$
$\frac{dV}{d|\Phi|^2} = \mu^2 (1 + 4 \pi \Lambda |\Phi|^2)$
Hyperbolic conservation system:
Only using $\Phi, \Psi_i, \Pi$:
$\Phi_{,t}
- \beta^i \Phi_{,i}
= \alpha \Pi$
$\Psi_{i,t}
- \alpha \Pi_{,i}
- \beta^j \Psi_{i,j}
- {\beta^j}_{,i} \Phi_{,j}
=
\alpha_{,i} \Pi
$
$\Pi_{,t}
- \alpha \gamma^{ij} \Psi_{i,j}
- \beta^i \Pi_{,i}
=
- \frac{1}{\alpha^2} \alpha_{,t} \Pi
+ \frac{1}{\alpha} \alpha_{,t} \Pi
+ \alpha K \Pi
+ \Psi_i \alpha_{,j} \gamma^{ij}
- \Psi_i \alpha {}^{(3)} \Gamma^i
- \alpha \frac{dV}{d|\Phi|^2} \Phi
$
$\left[\begin{matrix}
\Phi \\
\Psi_i \\
\Pi
\end{matrix}\right]_{,t}
+ \left[\begin{matrix}
- \beta^j \Phi_{,j} \\
- \alpha \Pi_{,i}
- \beta^j \Psi_{i,j}
- {\beta^j}_{,i} \Phi_{,j} \\
- \alpha \gamma^{kj} \Psi_{k,j}
- \beta^j \Pi_{,j}
\end{matrix}\right]
= \left[\begin{matrix}
\alpha \Pi \\
\alpha_{,i} \Pi \\
- \frac{1}{\alpha^2} \alpha_{,t} \Pi
+ \frac{1}{\alpha} \alpha_{,t} \Pi
+ \alpha K \Pi
+ \Psi_i \alpha_{,j} \gamma^{ij}
- \Psi_i \alpha {}^{(3)} \Gamma^i
- \alpha \frac{dV}{d|\Phi|^2} \Phi
\end{matrix}\right]$
$\left[\begin{matrix}
\Phi \\
\Psi_i \\
\Pi
\end{matrix}\right]_{,t}
+ \left[\begin{matrix}
-\beta^j & 0 & 0 \\
- {\beta^j}_{,i} &
- \beta^j \delta^k_i &
- \alpha \delta^j_i
\\
0 & - \alpha \gamma^{jk} & - \beta^j
\end{matrix}\right] \left[\begin{matrix}
\Phi \\
\Psi_k \\
\Pi
\end{matrix}\right]_{,j}
= \left[\begin{matrix}
\alpha \Pi \\
\alpha_{,i} \Pi \\
- \frac{1}{\alpha^2} \alpha_{,t} \Pi
+ \frac{1}{\alpha} \alpha_{,t} \Pi
+ \alpha K \Pi
+ \Psi_i \alpha_{,j} \gamma^{ij}
- \Psi_i \alpha {}^{(3)} \Gamma^i
- \alpha \frac{dV}{d|\Phi|^2} \Phi
\end{matrix}\right]$
Only using $\Pi, \Psi_i$:
$\left[\begin{matrix}
\Pi \\
\Psi_i
\end{matrix}\right]_{,t}
+ \left[\begin{matrix}
- \beta^k &
- \alpha \gamma^{jk}
\\
- \alpha \delta^k_i &
- \beta^k \delta^j_i
\end{matrix}\right] \left[\begin{matrix}
\Pi \\
\Psi_j
\end{matrix}\right]_{,k}
= \left[\begin{matrix}
\Pi (
\frac{1}{\alpha} \alpha_{,t} (1 - \frac{1}{\alpha})
+ \alpha K
)
+ \Psi_i (
\alpha_{,j} \gamma^{ij}
- \alpha {}^{(3)} \Gamma^i
)
- \alpha \frac{dV}{d|\Phi|^2} \Phi
\\
\alpha_{,i} \Pi
+ {\beta^k}_{,i} \Psi_k
\end{matrix}\right]$
Using $\alpha, \beta^k, \gamma_{ij}, \Phi, \Psi_k, \Pi$:
$\Phi_{,t}
- \beta^i \Phi_{,i}
= \alpha \Pi$
$\Psi_{i,t}
- \alpha_{,i} \Pi
- \alpha \Pi_{,i}
- \beta^j \Psi_{i,j}
- {\beta^j}_{,i} \Phi_{,j}
= 0
$
$\Pi_{,t}
- \frac{1}{\alpha} \beta^i \alpha_{,i} \Pi
- \alpha \gamma^{ij} \Psi_{i,j}
- \beta^i \Pi_{,i}
- \frac{1}{\alpha} \beta^i {\beta^j}_{,i} \Psi_j
+ \frac{1}{\alpha^2} \beta^m \alpha_{,m} \beta^i \Psi_i
+ \alpha \Psi_i (
{}^{(3)} \Gamma^i
- \frac{1}{\alpha^3} \beta^i \beta^m \alpha_{,m}
)
=
- \frac{1}{\alpha^2} \alpha_{,t} \Pi
+ \frac{1}{\alpha^2} \alpha_{,t} \beta^i \Psi_i
+ \frac{1}{\alpha} \alpha_{,t} \Pi
- \frac{1}{\alpha} {\beta^i}_{,t} \Psi_i
- \alpha \Psi_i (
\frac{1}{\alpha^3} \beta^i \alpha_{,t}
- \frac{1}{\alpha^2} {\beta^i}_{,t}
)
- \frac{1}{\alpha} \beta^m \alpha_{,m} \Pi
+ \alpha K \Pi
+ K \beta^i \Psi_i
- \frac{1}{\alpha^2} \Psi_i \beta^i \alpha^2 K
- \alpha \frac{dV}{d|\Phi|^2} \Phi
- \Psi_i \frac{1}{\alpha} (\beta^m {\beta^i}_{,m} + \alpha \alpha_{,j} \gamma^{ij})
$
Using
$\alpha_{,t} = -\alpha^2 f(\alpha) K + \alpha_{,j} \beta^j$
${\beta^i}_{,t} = {\beta^i}_{,j} \beta^j + B^i$
$\gamma_{ij,t} = -2 \alpha K_{ij} + \gamma_{ij,k} \beta^k + 2 \gamma_{k(i} {\beta^k}_{,j)}$
$\Pi_{,t}
+ \frac{1}{\alpha^2} \alpha_{,j} \beta^j \Pi
- \frac{1}{\alpha} \alpha_{,j} \beta^j \Pi
- \alpha \gamma^{ij} \Psi_{i,j}
- \beta^i \Pi_{,i}
- \frac{1}{\alpha} \beta^i {\beta^j}_{,i} \Psi_j
+ \alpha \Psi_i {}^{(3)} \Gamma^i
=
(1 - \alpha) f(\alpha) K \Pi
+ \alpha K \Pi
- \alpha \frac{dV}{d|\Phi|^2} \Phi
- \Psi_i \gamma^{ij} \beta^k {\beta^i}_{,k} \alpha_{,j}
$
$\left[\begin{matrix}
\alpha \\
\beta^k \\
\gamma_{ij} \\
\Phi \\
\Psi_k \\
\Pi
\end{matrix}\right]_{,t}
+ \left[\begin{matrix}
- \alpha_{,j} \beta^j \\
- {\beta^k}_{,j} \beta^j \\
- \gamma_{ij,k} \beta^k
- \gamma_{ki} {\beta^k}_{,j}
- \gamma_{kj} {\beta^k}_{,i} \\
- \beta^i \Phi_{,i} \\
- \alpha_{,k} \Pi
- \alpha \Pi_{,k}
- \beta^j \Psi_{k,j}
- {\beta^j}_{,k} \Phi_{,j} \\
+ \frac{1}{\alpha^2} \alpha_{,j} \beta^j \Pi
- \frac{1}{\alpha} \alpha_{,j} \beta^j \Pi
- \alpha \gamma^{ij} \Psi_{i,j}
- \beta^i \Pi_{,i}
- \frac{1}{\alpha} \beta^i {\beta^j}_{,i} \Psi_j
+ \alpha \Psi_i {}^{(3)} \Gamma^i
\end{matrix}\right]
= \left[\begin{matrix}
- \alpha^2 f(\alpha) K \\
B^k \\
-2 \alpha K_{ij} \\
\alpha \Pi \\
0 \\
(f - \alpha f + \alpha) K \Pi
- \alpha \frac{dV}{d|\Phi|^2} \Phi
- \Psi_i \gamma^{ij} \beta^k {\beta^i}_{,k} \alpha_{,j}
\end{matrix}\right]
$
$\left[\begin{matrix}
\alpha \\
\beta^k \\
\gamma_{ij} \\
\Phi \\
\Psi_k \\
\Pi
\end{matrix}\right]_{,t}
+ \left[\begin{matrix}
-\beta^r & 0 & 0 & 0 & 0 & 0 \\
0 & -\delta^k_m \beta^r & 0 & 0 & 0 & 0 \\
0 & -\gamma_{mi} \delta^r_j - \gamma_{mj} \delta^r_i &
- \delta^p_i \delta^q_j \beta^r & 0 & 0 & 0 \\
0 & 0 & 0 & -\delta^m_k \beta^r & 0 & 0 \\
- \delta^r_k \Pi &
- \delta^r_k \Psi_m &
0 &
0 &
- \delta^m_k \beta^r &
- \alpha \delta^r_k \\
(\frac{1}{\alpha^2} - \frac{1}{\alpha}) \beta^r \Pi &
- \frac{1}{\alpha} \beta^r \Psi_m &
+ \frac{1}{2} \alpha (2 \Psi^p \gamma^{qr} - \Psi^r \gamma^{pq}) &
0 &
- \alpha \gamma^{rm} &
- \beta^r
\end{matrix}\right]
\left[\begin{matrix}
\alpha \\
\beta^m \\
\gamma_{pq} \\
\Phi \\
\Psi_m \\
\Pi
\end{matrix}\right]_{,r}
= \left[\begin{matrix}
- \alpha^2 f(\alpha) K \\
B^k \\
-2 \alpha K_{ij} \\
\alpha \Pi \\
0 \\
(f - \alpha f + \alpha) K \Pi
- \alpha \frac{dV}{d|\Phi|^2} \Phi
- \Psi^j \beta^k {\beta^i}_{,k} \alpha_{,j}
\end{matrix}\right]
$
Stress-energy:
2017 Escorihuela-Tomàs et al, eqn 2:
$T_{ab} = \frac{1}{2} (
(\Phi_{,a})^* \Phi_{,b}
+ \Phi_{,a} (\Phi_{,b})^*
)
- \frac{1}{2} g_{ab} (
g^{cd} (\Phi_{,c})^* \Phi_{,d}
+ V
)$
$\rho = n^a n^b T_{ab}$
$= n^a n^b (
\frac{1}{2} (
(\Phi_{,a})^* \Phi_{,b}
+ \Phi_{,a} (\Phi_{,b})^*
)
- \frac{1}{2} g_{ab} (
g^{cd} (\Phi_{,c})^* \Phi_{,d}
+ V
)
)$
$= \frac{1}{2} (
n^a (\Phi_{,a})^* n^b \Phi_{,b}
+ n^a \Phi_{,a} n^b (\Phi_{,b})^*
+ g^{cd} (\Phi_{,c})^* \Phi_{,d}
+ V
)$
$= \frac{1}{2} (
\Pi^* \cdot \Pi
+ \Pi \cdot \Pi^*
+ (\gamma^{cd} - n^c n^d) (\Phi_{,c})^* \Phi_{,d}
+ V
)$
$=
\frac{1}{2} (
\Pi^* \cdot \Pi
+ \Pi \cdot \Pi^*
)
+ \frac{1}{2} (\gamma^{cd} (\Phi_{,c})^* \Phi_{,d} - \Pi^* \cdot \Pi)
+ \frac{1}{2} V
$
$= \frac{1}{2} (
\Pi \cdot \Pi^*
+ (\Psi^c)^* \Psi_c
+ V
)$
(2010 Alcubierre, Mendez has $= \frac{1}{2} (\Pi^2 + \frac{1}{exp(-4 \phi)} \Psi^2)$)
$S^u = -\gamma^{ua} n^b T_{ab}$
$S^u = -\gamma^{ua} n^b (
\frac{1}{2} (
(\Phi_{,a})^* \Phi_{,b}
+ \Phi_{,a} (\Phi_{,b})^*
)
- \frac{1}{2} g_{ab} (
g^{cd} (\Phi_{,c})^* \Phi_{,d}
+ V
)
)$
$S^u = -\frac{1}{2} \gamma^{ua} ((\Phi_{,a})^* \Pi + \Phi_{,a} \Pi^*)$
(2010 Alcubierre, Mendez has $ = -\Pi \Psi$)
$S_{uv} = {\gamma_u}^a {\gamma_v}^b T_{ab}$
$S_{uv} = \frac{1}{2} {\gamma_u}^a {\gamma_v}^b (
(\Phi_{,a})^* \Phi_{,b}
+ \Phi_{,a} (\Phi_{,b})^*
- g_{ab} (
g^{cd} (\Phi_{,c})^* \Phi_{,d}
+ V
)
)$
$S_{uv} = \frac{1}{2} (
{\gamma_u}^a {\gamma_v}^b (\Phi_{,a})^* \Phi_{,b}
+ {\gamma_u}^a {\gamma_v}^b \Phi_{,a} (\Phi_{,b})^*
- \gamma_{uv} (
(\gamma^{cd} - n^c n^d) (\Phi_{,c})^* \Phi_{,d}
+ V
)
)$
$S_{uv} = \frac{1}{2} (
{\gamma_u}^a {\gamma_v}^b (
(\Phi_{,a})^* \Phi_{,b}
+ \Phi_{,a} (\Phi_{,b})^*
)
- \gamma_{uv} (
\gamma^{cd} (\Phi_{,c})^* \Phi_{,d}
- (\Pi)^* \Pi
+ V
)
)$
$S_{uv} = \frac{1}{2} (
(\Phi_{,u})^* \Phi_{,v}
+ \Phi_{,u} (\Phi_{,v})^*
- \gamma_{uv} (
\gamma^{cd} (\Phi_{,c})^* \Phi_{,d}
- (\Pi)^* \Pi
+ V
)
)$
(2010 Alcubierre, Mendez has $\frac{1}{2} (\Pi^2 \pm \frac{\Psi^2}{exp(-4 \phi)} )$ for ${T^r}_r$ and ${T^\theta}_\theta$)
Weighting the internal partial derivative using the volume element:
manifold:
$g^{ab} \nabla_a \nabla_b \Phi = S$
$g^{ab} \Phi_{,ab} - g^{ab} {\Gamma^c}_{ab} \Phi_{,c} = S$
$g^{ab} \Phi_{,ab} - \Gamma^c \Phi_{,c} = S$
submanifold:
$\gamma^{ij} {}^{(3)} \nabla_i {}^{(3)} \nabla_j \Phi = S$
$\gamma^{ij} \Phi_{,ij} - {}^{(3)} \Gamma^k \Phi_{,k} = S$
back to the manifold:
$
g^{tt} \Phi_{,tt}
+ 2 g^{ti} \Phi_{,ti}
+ g^{ij} \Phi_{,ij}
- \Gamma^t \Phi_{,t}
- \Gamma^k \Phi_{,k}
= S$
$ g^{tt} \Phi_{,tt}
+ 2 g^{ti} \Phi_{,ti}
+ g^{ij} \Phi_{,ij}
- \Phi_{,i} (
{}^{(3)} \Gamma^i
+ \frac{1}{\alpha^3} \beta^i (\alpha_{,t} - \beta^m \alpha_{,m} + \alpha^2 K)
- \frac{1}{\alpha^2} ({\beta^i}_{,t} - \beta^m {\beta^i}_{,m} + \alpha \alpha_{,j} \gamma^{ij})
) = S + \Gamma^t \Phi_{,t}
$
...
Let $\Pi = \Phi_{,t}, \Psi_i = \Phi_{,i}$
$
g^{tt} \Pi_{,t}
+ 2 g^{ti} \Pi_{,i}
+ g^{ij} \Psi_{i,j}
=
\Gamma^t \Pi
+ \Gamma^i \Psi_i
+ S
$