---

Eduard Larrañaga (ealarranaga@unal.edu.co)

---

# Propiedades de la Solución de  Schwarzschild
---

## Resumen

Este cuaderno utiliza [SageMath](https://www.sagemath.org) y [SageManifolds](https://sagemanifolds.obspm.fr/index.html) para calcular algunas propiedades geométricas del espacio-tiempo de Schwarzschild incluyendo  simbolos de Christoffel, el tensor de Riemann, el tensor de Ricci y el escalar de Kretschmann.

Nota: Este cuaderno fue creado utilizando SageMath >= 9.4

```python
version()
```

    'SageMath version 9.4, Release Date: 2021-08-22'

Activamos la visualización matemática via LaTeX,

```python
%display latex
```

## 1.  Definición del Espacio-Tiempo

El espaciotiempo se define como una variedad Lorentziana 4-dimensional,

```python
MU = Manifold(4, 'MU', structure='Lorentzian', latex_name='\mathcal{M}')
print(MU)
```

    4-dimensional Lorentzian manifold MU

### 1.1. Coordenadas Estandard

Se introducen las coordenadas esfericas (estandard) o **coordenadas de Schwarzschild-Droste**  utilizando el método `.chart()` del objeto variedad`MU`. El argumento del método `.chart()` es una cadena de texto que define cada una de las coordenadas y sus rangos, si son diferentes de $(-\infty, +\infty)$, al igual que el sìmbolo LaTeX que las representará.

Note que las variables de `Python` para cada coordenada son declaradas dentro del operador `<...>`en el lado izquierdo. La variable `X` denotará la carta coordenada.

```python
X.<t,r,th,ph> = MU.chart(r"t r:(0,+oo) th:(0,pi):\theta ph:(0,2*pi):\phi")
X
```

\[\newcommand{\Bold}[1]{\mathbf{#1}}\left(\mathcal{M},(t, r, {\theta}, {\phi})\right)\]

## 2. Tensor Metrico

Para definir la métrica de Schwarzschild, es necesario declarar el parámetro de masa $M$ como una variable simbólica utilizando al función `var()`. Adicionalmente asumiremos que este parámetro toma valores positivos.

```python
M = var('M')
assume(M>=0)
a = var('a', domain='real')
Sigma = var('Sigma', latex_name='\\Sigma')
Delta = var('Delta', latex_name='\\Delta')
F = 1 - 2*M/r
Sigma = r**2 + a**2*cos(th)**2
Delta = r**2 - 2*M*r + a**2
#Delta = F*Sigma + a**2*sin(th)**2
```

Ahora definimos el tensor métrico

```python
g = MU.metric()
g[0,0] = -(1 - 2*M*r/Sigma)
g[1,1] = Sigma/Delta
g[2,2] = Sigma
g[3,3] = (r**2 + a**2 + 2*M*r*(a*sin(th))**2/Sigma)*sin(th)^2
g[0,3] = -2*M*r*a*(sin(th))**2/Sigma
g.display()
```

\[\newcommand{\Bold}[1]{\mathbf{#1}}g = \left( \frac{2 \, M r}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} - 1 \right) \mathrm{d} t\otimes \mathrm{d} t + \left( -\frac{2 \, M a r \sin\left({\theta}\right)^{2}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \right) \mathrm{d} t\otimes \mathrm{d} {\phi} + \left( \frac{a^{2} \cos\left({\theta}\right)^{2} + r^{2}}{a^{2} - 2 \, M r + r^{2}} \right) \mathrm{d} r\otimes \mathrm{d} r + \left( a^{2} \cos\left({\theta}\right)^{2} + r^{2} \right) \mathrm{d} {\theta}\otimes \mathrm{d} {\theta} + \left( -\frac{2 \, M a r \sin\left({\theta}\right)^{2}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \right) \mathrm{d} {\phi}\otimes \mathrm{d} t + {\left(\frac{2 \, M a^{2} r \sin\left({\theta}\right)^{2}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} + a^{2} + r^{2}\right)} \sin\left({\theta}\right)^{2} \mathrm{d} {\phi}\otimes \mathrm{d} {\phi}\]

```python
g = MU.metric()
g[0,0] = (Delta-a**2*sin(th)**2)/Sigma
g[1,1] = -Sigma/Delta
g[2,2] = -Sigma
g[3,3] = -((r**2 + a**2)**2 - Delta*a**2*sin(th)**2)*sin(th)^2/Sigma
g[0,3] = (r**2 + a**2 - Delta)*a*sin(th)**2/Sigma
g.display()
```

\[\newcommand{\Bold}[1]{\mathbf{#1}}g = \left( -\frac{2 \, M}{r} + 1 \right) \mathrm{d} t\otimes \mathrm{d} t + \left( -\frac{{\left(a^{2} \sin\left({\theta}\right)^{2} - a^{2} - r^{2} - {\left(a^{2} \cos\left({\theta}\right)^{2} + r^{2}\right)} {\left(\frac{2 \, M}{r} - 1\right)}\right)} a \sin\left({\theta}\right)^{2}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \right) \mathrm{d} t\otimes \mathrm{d} {\phi} + \left( -\frac{a^{2} \cos\left({\theta}\right)^{2} + r^{2}}{a^{2} \sin\left({\theta}\right)^{2} - {\left(a^{2} \cos\left({\theta}\right)^{2} + r^{2}\right)} {\left(\frac{2 \, M}{r} - 1\right)}} \right) \mathrm{d} r\otimes \mathrm{d} r + \left( -a^{2} \cos\left({\theta}\right)^{2} - r^{2} \right) \mathrm{d} {\theta}\otimes \mathrm{d} {\theta} + \left( -\frac{{\left(a^{2} \sin\left({\theta}\right)^{2} - a^{2} - r^{2} - {\left(a^{2} \cos\left({\theta}\right)^{2} + r^{2}\right)} {\left(\frac{2 \, M}{r} - 1\right)}\right)} a \sin\left({\theta}\right)^{2}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \right) \mathrm{d} {\phi}\otimes \mathrm{d} t + \left( \frac{{\left({\left(a^{2} \sin\left({\theta}\right)^{2} - {\left(a^{2} \cos\left({\theta}\right)^{2} + r^{2}\right)} {\left(\frac{2 \, M}{r} - 1\right)}\right)} a^{2} \sin\left({\theta}\right)^{2} - {\left(a^{2} + r^{2}\right)}^{2}\right)} \sin\left({\theta}\right)^{2}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \right) \mathrm{d} {\phi}\otimes \mathrm{d} {\phi}\]

```python
g[:]
```

\[\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrr}
\frac{2 \, M r}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} - 1 & 0 & 0 & -\frac{2 \, M a r \sin\left({\theta}\right)^{2}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \\
0 & \frac{a^{2} \cos\left({\theta}\right)^{2} + r^{2}}{a^{2} - 2 \, M r + r^{2}} & 0 & 0 \\
0 & 0 & a^{2} \cos\left({\theta}\right)^{2} + r^{2} & 0 \\
-\frac{2 \, M a r \sin\left({\theta}\right)^{2}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} & 0 & 0 & {\left(\frac{2 \, M a^{2} r \sin\left({\theta}\right)^{2}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} + a^{2} + r^{2}\right)} \sin\left({\theta}\right)^{2}
\end{array}\right)\]

## 3. Simbolos de Christoffel (Conexiones)

Los simbolos de Christoffel (de segunda especie) para la métrica $g$ en las coordenadas de Schwarzschild-Droste se definen como

\begin{equation}
\Gamma^\alpha_{\mu \nu} = \frac{1}{2} g^{\alpha \sigma} \left[ \partial_\mu g_{\sigma \nu} + \partial_\nu g_{\mu \sigma } - \partial_\sigma g_{\mu \nu} \right]
\end{equation}

Estas componentes se pueden obtener mediante el método `.christoffel_symbols_display()` del objeto métrica, `g`. Por defecto, solo se mostrarán las componentes no-nulas y no-redundantes (i.e. se tiene en cuenta la propiedad de simetría $\Gamma^\alpha_{\mu \nu} = \Gamma^\alpha_{\nu \mu}$).

De igual forma, utilizando el símbolo `?` al final del comando, i.e. `g.christoffel_symbols_display?` se visualiza la información y opciones disponibles para el método.

```python
g.christoffel_symbols_display?
```

```python
g.christoffel_symbols_display()
```

\[\newcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{lcl} \Gamma_{ \phantom{\, t} \, t \, r }^{ \, t \phantom{\, t} \phantom{\, r} } & = & -\frac{M}{2 \, M r - r^{2}} \\ \Gamma_{ \phantom{\, r} \, t \, t }^{ \, r \phantom{\, t} \phantom{\, t} } & = & -\frac{2 \, M^{2} - M r}{r^{3}} \\ \Gamma_{ \phantom{\, r} \, r \, r }^{ \, r \phantom{\, r} \phantom{\, r} } & = & \frac{M}{2 \, M r - r^{2}} \\ \Gamma_{ \phantom{\, r} \, {\theta} \, {\theta} }^{ \, r \phantom{\, {\theta}} \phantom{\, {\theta}} } & = & 2 \, M - r \\ \Gamma_{ \phantom{\, r} \, {\phi} \, {\phi} }^{ \, r \phantom{\, {\phi}} \phantom{\, {\phi}} } & = & {\left(2 \, M - r\right)} \sin\left({\theta}\right)^{2} \\ \Gamma_{ \phantom{\, {\theta}} \, r \, {\theta} }^{ \, {\theta} \phantom{\, r} \phantom{\, {\theta}} } & = & \frac{1}{r} \\ \Gamma_{ \phantom{\, {\theta}} \, {\phi} \, {\phi} }^{ \, {\theta} \phantom{\, {\phi}} \phantom{\, {\phi}} } & = & -\cos\left({\theta}\right) \sin\left({\theta}\right) \\ \Gamma_{ \phantom{\, {\phi}} \, r \, {\phi} }^{ \, {\phi} \phantom{\, r} \phantom{\, {\phi}} } & = & \frac{1}{r} \\ \Gamma_{ \phantom{\, {\phi}} \, {\theta} \, {\phi} }^{ \, {\phi} \phantom{\, {\theta}} \phantom{\, {\phi}} } & = & \frac{\cos\left({\theta}\right)}{\sin\left({\theta}\right)} \end{array}\]

También es posible acceder a cada componente particular,

```python
g.christoffel_symbols()[0,1,0]
```

\[\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{M}{2 \, M r - r^{2}}\]

Podemos comprobar que la simetría en los indices inferiores se satisface,

```python
g.christoffel_symbols()[0,0,1] == g.christoffel_symbols()[0,1,0]
```

\[\newcommand{\Bold}[1]{\mathbf{#1}}\mathrm{True}\]

## 4. Tensor de Curvatura

El tensor de curvatura de Riemann se define como

\begin{equation}
R^\mu_{\,\, \nu \rho \sigma} = \frac{\partial \Gamma^\mu_{ \nu \sigma}}{\partial x^{\rho}} - \frac{\partial \Gamma^\mu_{ \nu \rho}}{\partial x^{\sigma}} + \Gamma^{\alpha}_{\nu \sigma}\Gamma^\mu_{\rho \alpha} - \Gamma^{\alpha}_{\ \nu \rho}\Gamma^\mu_{\sigma \alpha}
\end{equation}

Este puede calculares mediante el método `.riemann()` de la métrica,

```python
Riemann = g.riemann()
print(Riemann)
```

    Tensor field Riem(g) of type (1,3) on the 4-dimensional Lorentzian manifold MU

**Es importante notar que este es un tensor de orden (1,3)**.

```python
Riemann
```

\[\newcommand{\Bold}[1]{\mathbf{#1}}\mathrm{Riem}\left(g\right)\]

Para visualizar sus componentes utilizamos el método `.display()`,

```python
Riemann.display()
```

\[\newcommand{\Bold}[1]{\mathbf{#1}}\mathrm{Riem}\left(g\right) = \left( -\frac{2 \, M}{2 \, M r^{2} - r^{3}} \right) \frac{\partial}{\partial t }\otimes \mathrm{d} r\otimes \mathrm{d} t\otimes \mathrm{d} r + \left( \frac{2 \, M}{2 \, M r^{2} - r^{3}} \right) \frac{\partial}{\partial t }\otimes \mathrm{d} r\otimes \mathrm{d} r\otimes \mathrm{d} t -\frac{M}{r} \frac{\partial}{\partial t }\otimes \mathrm{d} {\theta}\otimes \mathrm{d} t\otimes \mathrm{d} {\theta} + \frac{M}{r} \frac{\partial}{\partial t }\otimes \mathrm{d} {\theta}\otimes \mathrm{d} {\theta}\otimes \mathrm{d} t -\frac{M \sin\left({\theta}\right)^{2}}{r} \frac{\partial}{\partial t }\otimes \mathrm{d} {\phi}\otimes \mathrm{d} t\otimes \mathrm{d} {\phi} + \frac{M \sin\left({\theta}\right)^{2}}{r} \frac{\partial}{\partial t }\otimes \mathrm{d} {\phi}\otimes \mathrm{d} {\phi}\otimes \mathrm{d} t -\frac{2 \, {\left(2 \, M^{2} - M r\right)}}{r^{4}} \frac{\partial}{\partial r }\otimes \mathrm{d} t\otimes \mathrm{d} t\otimes \mathrm{d} r + \frac{2 \, {\left(2 \, M^{2} - M r\right)}}{r^{4}} \frac{\partial}{\partial r }\otimes \mathrm{d} t\otimes \mathrm{d} r\otimes \mathrm{d} t -\frac{M}{r} \frac{\partial}{\partial r }\otimes \mathrm{d} {\theta}\otimes \mathrm{d} r\otimes \mathrm{d} {\theta} + \frac{M}{r} \frac{\partial}{\partial r }\otimes \mathrm{d} {\theta}\otimes \mathrm{d} {\theta}\otimes \mathrm{d} r -\frac{M \sin\left({\theta}\right)^{2}}{r} \frac{\partial}{\partial r }\otimes \mathrm{d} {\phi}\otimes \mathrm{d} r\otimes \mathrm{d} {\phi} + \frac{M \sin\left({\theta}\right)^{2}}{r} \frac{\partial}{\partial r }\otimes \mathrm{d} {\phi}\otimes \mathrm{d} {\phi}\otimes \mathrm{d} r + \left( \frac{2 \, M^{2} - M r}{r^{4}} \right) \frac{\partial}{\partial {\theta} }\otimes \mathrm{d} t\otimes \mathrm{d} t\otimes \mathrm{d} {\theta} + \left( -\frac{2 \, M^{2} - M r}{r^{4}} \right) \frac{\partial}{\partial {\theta} }\otimes \mathrm{d} t\otimes \mathrm{d} {\theta}\otimes \mathrm{d} t + \left( -\frac{M}{2 \, M r^{2} - r^{3}} \right) \frac{\partial}{\partial {\theta} }\otimes \mathrm{d} r\otimes \mathrm{d} r\otimes \mathrm{d} {\theta} + \left( \frac{M}{2 \, M r^{2} - r^{3}} \right) \frac{\partial}{\partial {\theta} }\otimes \mathrm{d} r\otimes \mathrm{d} {\theta}\otimes \mathrm{d} r + \frac{2 \, M \sin\left({\theta}\right)^{2}}{r} \frac{\partial}{\partial {\theta} }\otimes \mathrm{d} {\phi}\otimes \mathrm{d} {\theta}\otimes \mathrm{d} {\phi} -\frac{2 \, M \sin\left({\theta}\right)^{2}}{r} \frac{\partial}{\partial {\theta} }\otimes \mathrm{d} {\phi}\otimes \mathrm{d} {\phi}\otimes \mathrm{d} {\theta} + \left( \frac{2 \, M^{2} - M r}{r^{4}} \right) \frac{\partial}{\partial {\phi} }\otimes \mathrm{d} t\otimes \mathrm{d} t\otimes \mathrm{d} {\phi} + \left( -\frac{2 \, M^{2} - M r}{r^{4}} \right) \frac{\partial}{\partial {\phi} }\otimes \mathrm{d} t\otimes \mathrm{d} {\phi}\otimes \mathrm{d} t + \left( -\frac{M}{2 \, M r^{2} - r^{3}} \right) \frac{\partial}{\partial {\phi} }\otimes \mathrm{d} r\otimes \mathrm{d} r\otimes \mathrm{d} {\phi} + \left( \frac{M}{2 \, M r^{2} - r^{3}} \right) \frac{\partial}{\partial {\phi} }\otimes \mathrm{d} r\otimes \mathrm{d} {\phi}\otimes \mathrm{d} r -\frac{2 \, M}{r} \frac{\partial}{\partial {\phi} }\otimes \mathrm{d} {\theta}\otimes \mathrm{d} {\theta}\otimes \mathrm{d} {\phi} + \frac{2 \, M}{r} \frac{\partial}{\partial {\phi} }\otimes \mathrm{d} {\theta}\otimes \mathrm{d} {\phi}\otimes \mathrm{d} {\theta}\]

```python
Riemann[:]
```

\[\newcommand{\Bold}[1]{\mathbf{#1}}\left[\left[\left[\left[0, 0, 0, 0\right], \left[0, 0, 0, 0\right], \left[0, 0, 0, 0\right], \left[0, 0, 0, 0\right]\right], \left[\left[0, -\frac{2 \, M}{2 \, M r^{2} - r^{3}}, 0, 0\right], \left[\frac{2 \, M}{2 \, M r^{2} - r^{3}}, 0, 0, 0\right], \left[0, 0, 0, 0\right], \left[0, 0, 0, 0\right]\right], \left[\left[0, 0, -\frac{M}{r}, 0\right], \left[0, 0, 0, 0\right], \left[\frac{M}{r}, 0, 0, 0\right], \left[0, 0, 0, 0\right]\right], \left[\left[0, 0, 0, -\frac{M \sin\left({\theta}\right)^{2}}{r}\right], \left[0, 0, 0, 0\right], \left[0, 0, 0, 0\right], \left[\frac{M \sin\left({\theta}\right)^{2}}{r}, 0, 0, 0\right]\right]\right], \left[\left[\left[0, -\frac{2 \, {\left(2 \, M^{2} - M r\right)}}{r^{4}}, 0, 0\right], \left[\frac{2 \, {\left(2 \, M^{2} - M r\right)}}{r^{4}}, 0, 0, 0\right], \left[0, 0, 0, 0\right], \left[0, 0, 0, 0\right]\right], \left[\left[0, 0, 0, 0\right], \left[0, 0, 0, 0\right], \left[0, 0, 0, 0\right], \left[0, 0, 0, 0\right]\right], \left[\left[0, 0, 0, 0\right], \left[0, 0, -\frac{M}{r}, 0\right], \left[0, \frac{M}{r}, 0, 0\right], \left[0, 0, 0, 0\right]\right], \left[\left[0, 0, 0, 0\right], \left[0, 0, 0, -\frac{M \sin\left({\theta}\right)^{2}}{r}\right], \left[0, 0, 0, 0\right], \left[0, \frac{M \sin\left({\theta}\right)^{2}}{r}, 0, 0\right]\right]\right], \left[\left[\left[0, 0, \frac{2 \, M^{2} - M r}{r^{4}}, 0\right], \left[0, 0, 0, 0\right], \left[-\frac{2 \, M^{2} - M r}{r^{4}}, 0, 0, 0\right], \left[0, 0, 0, 0\right]\right], \left[\left[0, 0, 0, 0\right], \left[0, 0, -\frac{M}{2 \, M r^{2} - r^{3}}, 0\right], \left[0, \frac{M}{2 \, M r^{2} - r^{3}}, 0, 0\right], \left[0, 0, 0, 0\right]\right], \left[\left[0, 0, 0, 0\right], \left[0, 0, 0, 0\right], \left[0, 0, 0, 0\right], \left[0, 0, 0, 0\right]\right], \left[\left[0, 0, 0, 0\right], \left[0, 0, 0, 0\right], \left[0, 0, 0, \frac{2 \, M \sin\left({\theta}\right)^{2}}{r}\right], \left[0, 0, -\frac{2 \, M \sin\left({\theta}\right)^{2}}{r}, 0\right]\right]\right], \left[\left[\left[0, 0, 0, \frac{2 \, M^{2} - M r}{r^{4}}\right], \left[0, 0, 0, 0\right], \left[0, 0, 0, 0\right], \left[-\frac{2 \, M^{2} - M r}{r^{4}}, 0, 0, 0\right]\right], \left[\left[0, 0, 0, 0\right], \left[0, 0, 0, -\frac{M}{2 \, M r^{2} - r^{3}}\right], \left[0, 0, 0, 0\right], \left[0, \frac{M}{2 \, M r^{2} - r^{3}}, 0, 0\right]\right], \left[\left[0, 0, 0, 0\right], \left[0, 0, 0, 0\right], \left[0, 0, 0, -\frac{2 \, M}{r}\right], \left[0, 0, \frac{2 \, M}{r}, 0\right]\right], \left[\left[0, 0, 0, 0\right], \left[0, 0, 0, 0\right], \left[0, 0, 0, 0\right], \left[0, 0, 0, 0\right]\right]\right]\right]\]

Cada componente particular se visualiza utilizando los índices correspondientes. Por ejemplo, la componente $R^0_{\ \,101} = R^t_{\ \,trt}$ será

```python
Riemann[0,1,1,0]
```

\[\newcommand{\Bold}[1]{\mathbf{#1}}\frac{2 \, M}{2 \, M r^{2} - r^{3}}\]

```python
Riemann[0,1,0,1]
```

Para obtener el tensor de curvatura del tipo (0,4) se utiliza el método `.down()`:

```python
Riemann_dddd = Riemann.down(g)
Riemann_dddd
```

\[\newcommand{\Bold}[1]{\mathbf{#1}}\mbox{Tensor field of type (0,4) on the 4-dimensional Lorentzian manifold MU}\]

```python
Riemann_dddd.display()
```

\[\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{2 \, M}{r^{3}} \mathrm{d} t\otimes \mathrm{d} r\otimes \mathrm{d} t\otimes \mathrm{d} r + \frac{2 \, M}{r^{3}} \mathrm{d} t\otimes \mathrm{d} r\otimes \mathrm{d} r\otimes \mathrm{d} t + \left( -\frac{2 \, M^{2} - M r}{r^{2}} \right) \mathrm{d} t\otimes \mathrm{d} {\theta}\otimes \mathrm{d} t\otimes \mathrm{d} {\theta} + \left( \frac{2 \, M^{2} - M r}{r^{2}} \right) \mathrm{d} t\otimes \mathrm{d} {\theta}\otimes \mathrm{d} {\theta}\otimes \mathrm{d} t -\frac{{\left(2 \, M^{2} - M r\right)} \sin\left({\theta}\right)^{2}}{r^{2}} \mathrm{d} t\otimes \mathrm{d} {\phi}\otimes \mathrm{d} t\otimes \mathrm{d} {\phi} + \frac{{\left(2 \, M^{2} - M r\right)} \sin\left({\theta}\right)^{2}}{r^{2}} \mathrm{d} t\otimes \mathrm{d} {\phi}\otimes \mathrm{d} {\phi}\otimes \mathrm{d} t + \frac{2 \, M}{r^{3}} \mathrm{d} r\otimes \mathrm{d} t\otimes \mathrm{d} t\otimes \mathrm{d} r -\frac{2 \, M}{r^{3}} \mathrm{d} r\otimes \mathrm{d} t\otimes \mathrm{d} r\otimes \mathrm{d} t + \left( \frac{M}{2 \, M - r} \right) \mathrm{d} r\otimes \mathrm{d} {\theta}\otimes \mathrm{d} r\otimes \mathrm{d} {\theta} + \left( -\frac{M}{2 \, M - r} \right) \mathrm{d} r\otimes \mathrm{d} {\theta}\otimes \mathrm{d} {\theta}\otimes \mathrm{d} r + \left( \frac{M \sin\left({\theta}\right)^{2}}{2 \, M - r} \right) \mathrm{d} r\otimes \mathrm{d} {\phi}\otimes \mathrm{d} r\otimes \mathrm{d} {\phi} + \left( -\frac{M \sin\left({\theta}\right)^{2}}{2 \, M - r} \right) \mathrm{d} r\otimes \mathrm{d} {\phi}\otimes \mathrm{d} {\phi}\otimes \mathrm{d} r + \left( \frac{2 \, M^{2} - M r}{r^{2}} \right) \mathrm{d} {\theta}\otimes \mathrm{d} t\otimes \mathrm{d} t\otimes \mathrm{d} {\theta} + \left( -\frac{2 \, M^{2} - M r}{r^{2}} \right) \mathrm{d} {\theta}\otimes \mathrm{d} t\otimes \mathrm{d} {\theta}\otimes \mathrm{d} t + \left( -\frac{M}{2 \, M - r} \right) \mathrm{d} {\theta}\otimes \mathrm{d} r\otimes \mathrm{d} r\otimes \mathrm{d} {\theta} + \left( \frac{M}{2 \, M - r} \right) \mathrm{d} {\theta}\otimes \mathrm{d} r\otimes \mathrm{d} {\theta}\otimes \mathrm{d} r + 2 \, M r \sin\left({\theta}\right)^{2} \mathrm{d} {\theta}\otimes \mathrm{d} {\phi}\otimes \mathrm{d} {\theta}\otimes \mathrm{d} {\phi} -2 \, M r \sin\left({\theta}\right)^{2} \mathrm{d} {\theta}\otimes \mathrm{d} {\phi}\otimes \mathrm{d} {\phi}\otimes \mathrm{d} {\theta} + \frac{{\left(2 \, M^{2} - M r\right)} \sin\left({\theta}\right)^{2}}{r^{2}} \mathrm{d} {\phi}\otimes \mathrm{d} t\otimes \mathrm{d} t\otimes \mathrm{d} {\phi} -\frac{{\left(2 \, M^{2} - M r\right)} \sin\left({\theta}\right)^{2}}{r^{2}} \mathrm{d} {\phi}\otimes \mathrm{d} t\otimes \mathrm{d} {\phi}\otimes \mathrm{d} t + \left( -\frac{M \sin\left({\theta}\right)^{2}}{2 \, M - r} \right) \mathrm{d} {\phi}\otimes \mathrm{d} r\otimes \mathrm{d} r\otimes \mathrm{d} {\phi} + \left( \frac{M \sin\left({\theta}\right)^{2}}{2 \, M - r} \right) \mathrm{d} {\phi}\otimes \mathrm{d} r\otimes \mathrm{d} {\phi}\otimes \mathrm{d} r -2 \, M r \sin\left({\theta}\right)^{2} \mathrm{d} {\phi}\otimes \mathrm{d} {\theta}\otimes \mathrm{d} {\theta}\otimes \mathrm{d} {\phi} + 2 \, M r \sin\left({\theta}\right)^{2} \mathrm{d} {\phi}\otimes \mathrm{d} {\theta}\otimes \mathrm{d} {\phi}\otimes \mathrm{d} {\theta}\]

```python
Riemann_dddd.display_comp()
```

\[\newcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{lcl} X_{ \, t \, r \, t \, r }^{ \phantom{\, t}\phantom{\, r}\phantom{\, t}\phantom{\, r} } & = & -\frac{2 \, M}{r^{3}} \\ X_{ \, t \, r \, r \, t }^{ \phantom{\, t}\phantom{\, r}\phantom{\, r}\phantom{\, t} } & = & \frac{2 \, M}{r^{3}} \\ X_{ \, t \, {\theta} \, t \, {\theta} }^{ \phantom{\, t}\phantom{\, {\theta}}\phantom{\, t}\phantom{\, {\theta}} } & = & -\frac{2 \, M^{2} - M r}{r^{2}} \\ X_{ \, t \, {\theta} \, {\theta} \, t }^{ \phantom{\, t}\phantom{\, {\theta}}\phantom{\, {\theta}}\phantom{\, t} } & = & \frac{2 \, M^{2} - M r}{r^{2}} \\ X_{ \, t \, {\phi} \, t \, {\phi} }^{ \phantom{\, t}\phantom{\, {\phi}}\phantom{\, t}\phantom{\, {\phi}} } & = & -\frac{{\left(2 \, M^{2} - M r\right)} \sin\left({\theta}\right)^{2}}{r^{2}} \\ X_{ \, t \, {\phi} \, {\phi} \, t }^{ \phantom{\, t}\phantom{\, {\phi}}\phantom{\, {\phi}}\phantom{\, t} } & = & \frac{{\left(2 \, M^{2} - M r\right)} \sin\left({\theta}\right)^{2}}{r^{2}} \\ X_{ \, r \, t \, t \, r }^{ \phantom{\, r}\phantom{\, t}\phantom{\, t}\phantom{\, r} } & = & \frac{2 \, M}{r^{3}} \\ X_{ \, r \, t \, r \, t }^{ \phantom{\, r}\phantom{\, t}\phantom{\, r}\phantom{\, t} } & = & -\frac{2 \, M}{r^{3}} \\ X_{ \, r \, {\theta} \, r \, {\theta} }^{ \phantom{\, r}\phantom{\, {\theta}}\phantom{\, r}\phantom{\, {\theta}} } & = & \frac{M}{2 \, M - r} \\ X_{ \, r \, {\theta} \, {\theta} \, r }^{ \phantom{\, r}\phantom{\, {\theta}}\phantom{\, {\theta}}\phantom{\, r} } & = & -\frac{M}{2 \, M - r} \\ X_{ \, r \, {\phi} \, r \, {\phi} }^{ \phantom{\, r}\phantom{\, {\phi}}\phantom{\, r}\phantom{\, {\phi}} } & = & \frac{M \sin\left({\theta}\right)^{2}}{2 \, M - r} \\ X_{ \, r \, {\phi} \, {\phi} \, r }^{ \phantom{\, r}\phantom{\, {\phi}}\phantom{\, {\phi}}\phantom{\, r} } & = & -\frac{M \sin\left({\theta}\right)^{2}}{2 \, M - r} \\ X_{ \, {\theta} \, t \, t \, {\theta} }^{ \phantom{\, {\theta}}\phantom{\, t}\phantom{\, t}\phantom{\, {\theta}} } & = & \frac{2 \, M^{2} - M r}{r^{2}} \\ X_{ \, {\theta} \, t \, {\theta} \, t }^{ \phantom{\, {\theta}}\phantom{\, t}\phantom{\, {\theta}}\phantom{\, t} } & = & -\frac{2 \, M^{2} - M r}{r^{2}} \\ X_{ \, {\theta} \, r \, r \, {\theta} }^{ \phantom{\, {\theta}}\phantom{\, r}\phantom{\, r}\phantom{\, {\theta}} } & = & -\frac{M}{2 \, M - r} \\ X_{ \, {\theta} \, r \, {\theta} \, r }^{ \phantom{\, {\theta}}\phantom{\, r}\phantom{\, {\theta}}\phantom{\, r} } & = & \frac{M}{2 \, M - r} \\ X_{ \, {\theta} \, {\phi} \, {\theta} \, {\phi} }^{ \phantom{\, {\theta}}\phantom{\, {\phi}}\phantom{\, {\theta}}\phantom{\, {\phi}} } & = & 2 \, M r \sin\left({\theta}\right)^{2} \\ X_{ \, {\theta} \, {\phi} \, {\phi} \, {\theta} }^{ \phantom{\, {\theta}}\phantom{\, {\phi}}\phantom{\, {\phi}}\phantom{\, {\theta}} } & = & -2 \, M r \sin\left({\theta}\right)^{2} \\ X_{ \, {\phi} \, t \, t \, {\phi} }^{ \phantom{\, {\phi}}\phantom{\, t}\phantom{\, t}\phantom{\, {\phi}} } & = & \frac{{\left(2 \, M^{2} - M r\right)} \sin\left({\theta}\right)^{2}}{r^{2}} \\ X_{ \, {\phi} \, t \, {\phi} \, t }^{ \phantom{\, {\phi}}\phantom{\, t}\phantom{\, {\phi}}\phantom{\, t} } & = & -\frac{{\left(2 \, M^{2} - M r\right)} \sin\left({\theta}\right)^{2}}{r^{2}} \\ X_{ \, {\phi} \, r \, r \, {\phi} }^{ \phantom{\, {\phi}}\phantom{\, r}\phantom{\, r}\phantom{\, {\phi}} } & = & -\frac{M \sin\left({\theta}\right)^{2}}{2 \, M - r} \\ X_{ \, {\phi} \, r \, {\phi} \, r }^{ \phantom{\, {\phi}}\phantom{\, r}\phantom{\, {\phi}}\phantom{\, r} } & = & \frac{M \sin\left({\theta}\right)^{2}}{2 \, M - r} \\ X_{ \, {\phi} \, {\theta} \, {\theta} \, {\phi} }^{ \phantom{\, {\phi}}\phantom{\, {\theta}}\phantom{\, {\theta}}\phantom{\, {\phi}} } & = & -2 \, M r \sin\left({\theta}\right)^{2} \\ X_{ \, {\phi} \, {\theta} \, {\phi} \, {\theta} }^{ \phantom{\, {\phi}}\phantom{\, {\theta}}\phantom{\, {\phi}}\phantom{\, {\theta}} } & = & 2 \, M r \sin\left({\theta}\right)^{2} \end{array}\]

```python
Riemann_dddd[0,1,0,1]
```

```python
Riemann_dddd[0,1,1,0]
```

```python
Riemann_dddd[1,0,0,1]
```

```python
Riemann_dddd[1,0,0,1]
```

## 5. Tensor de Ricci

El tensor de Ricci se calcula con el método `.ricci()` de la metrica,

```python
Ricci = g.ricci()
Ricci[:]
```

\[\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrr}
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0
\end{array}\right)\]

Como era de esperar, todas sus componentes son cero en este caso,

```python
Ricci.display()
```

\[\newcommand{\Bold}[1]{\mathbf{#1}}\mathrm{Ric}\left(g\right) = 0\]

```python
Ricci[:]
```

\[\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrr}
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0
\end{array}\right)\]

### Tensor de Ricci Tensor (Version Alternativa)

Otra forma de calcular este tensor es contrayendo el trensor de Riemann

\begin{equation}
R_{\mu \nu} = R^\alpha_{\,\, \mu \alpha \nu} = g ^{\alpha \beta} R_{\alpha \mu \beta \nu}
\end{equation}

```python
Ric = g.up(g)['^{ab}'] * Riemann.down(g)['_{acbd}']
Ricci[:]
```

\[\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrr}
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0
\end{array}\right)\]

### 5.1. Escalar de Curvatura de Ricci

El escalar de curvatura puede obtenerse contrayendo la métrica con el tesnor de Ricci,

```python
R = g['_{ab}'] * Ricci.up(g)['^{ab}']
print(R)
```

    Scalar field dzero(∂/∂th) on the 4-dimensional Lorentzian manifold MU

```python
R.display()
```

\[\newcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{llcl} \mathrm{d}0\left( \frac{\partial}{\partial {\theta} } \right) : & MU & \longrightarrow & \mathbb{R} \\ & \left(t, r, {\theta}, {\phi}\right) & \longmapsto & 0 \end{array}\]

La expresión simbólica de un campo escalar se realiza utilizando el método `.expr()`:

```python
R.expr()
```

\[\newcommand{\Bold}[1]{\mathbf{#1}}0\]

También es posible utilizar el método `.ricci_scalar()`

```python
R = g.ricci_scalar()
R.expr()
```

\[\newcommand{\Bold}[1]{\mathbf{#1}}\frac{2 \, {\left(11 \, {\left(M^{2} a^{4} - M^{2} a^{2}\right)} r^{12} - 44 \, {\left(M^{3} a^{4} - M^{3} a^{2}\right)} r^{11} + 2 \, {\left(5 \, M^{2} a^{6} - 24 \, M^{4} a^{2} + {\left(24 \, M^{4} - 5 \, M^{2}\right)} a^{4}\right)} r^{10} - {\left(3 \, M^{2} a^{16} - 3 \, M^{2} a^{14} - {\left(M^{2} a^{12} - M^{2} a^{10}\right)} r^{4} - 2 \, {\left(M^{2} a^{14} + 4 \, M^{4} a^{10} - {\left(4 \, M^{4} + M^{2}\right)} a^{12}\right)} r^{2} - 8 \, {\left(M^{3} a^{14} - M^{3} a^{12}\right)} r\right)} \cos\left({\theta}\right)^{10} - 28 \, {\left(M^{3} a^{6} - M^{3} a^{4}\right)} r^{9} - {\left({\left(20 \, M^{4} - 3 \, M^{2}\right)} a^{8} + 20 \, M^{4} a^{4} - {\left(40 \, M^{4} - 3 \, M^{2}\right)} a^{6}\right)} r^{8} + {\left(3 \, M^{2} a^{16} - 3 \, M^{2} a^{14} - 44 \, {\left(M^{3} a^{10} - M^{3} a^{8}\right)} r^{5} + {\left({\left(12 \, M^{4} + 11 \, M^{2}\right)} a^{12} - 60 \, M^{4} a^{8} + {\left(48 \, M^{4} - 11 \, M^{2}\right)} a^{10}\right)} r^{4} - 4 \, {\left(2 \, M^{5} a^{8} + {\left(2 \, M^{5} + 9 \, M^{3}\right)} a^{12} - {\left(4 \, M^{5} + 9 \, M^{3}\right)} a^{10}\right)} r^{3} - 2 \, {\left({\left(2 \, M^{4} + 3 \, M^{2}\right)} a^{14} + 6 \, M^{4} a^{10} - {\left(8 \, M^{4} + 3 \, M^{2}\right)} a^{12}\right)} r^{2} - 8 \, {\left(M^{3} a^{14} - M^{3} a^{12}\right)} r\right)} \cos\left({\theta}\right)^{8} - 24 \, {\left(M^{5} a^{8} - 2 \, M^{5} a^{6} + M^{5} a^{4}\right)} r^{7} - 4 \, {\left(M^{4} a^{10} - 2 \, M^{4} a^{8} + M^{4} a^{6}\right)} r^{6} - 2 \, {\left(3 \, {\left(M^{2} a^{8} - M^{2} a^{6}\right)} r^{8} + 30 \, {\left(M^{3} a^{8} - M^{3} a^{6}\right)} r^{7} + 4 \, {\left({\left(3 \, M^{4} - 4 \, M^{2}\right)} a^{10} + 18 \, M^{4} a^{6} - {\left(21 \, M^{4} - 4 \, M^{2}\right)} a^{8}\right)} r^{6} - 4 \, {\left(16 \, M^{5} a^{6} + {\left(16 \, M^{5} - 7 \, M^{3}\right)} a^{10} - {\left(32 \, M^{5} - 7 \, M^{3}\right)} a^{8}\right)} r^{5} + {\left({\left(24 \, M^{4} - 5 \, M^{2}\right)} a^{12} - 12 \, M^{4} a^{8} - {\left(12 \, M^{4} - 5 \, M^{2}\right)} a^{10}\right)} r^{4} - 2 \, {\left(4 \, M^{5} a^{8} + {\left(4 \, M^{5} + 5 \, M^{3}\right)} a^{12} - {\left(8 \, M^{5} + 5 \, M^{3}\right)} a^{10}\right)} r^{3} - 2 \, {\left({\left(2 \, M^{4} + 3 \, M^{2}\right)} a^{14} + 2 \, M^{4} a^{10} - {\left(4 \, M^{4} + 3 \, M^{2}\right)} a^{12}\right)} r^{2}\right)} \cos\left({\theta}\right)^{6} - 2 \, {\left(4 \, {\left(M^{2} a^{6} - M^{2} a^{4}\right)} r^{10} - 6 \, {\left(M^{3} a^{6} - M^{3} a^{4}\right)} r^{9} + {\left({\left(18 \, M^{4} - 25 \, M^{2}\right)} a^{8} + 14 \, M^{4} a^{4} - {\left(32 \, M^{4} - 25 \, M^{2}\right)} a^{6}\right)} r^{8} + 4 \, {\left(3 \, M^{5} a^{4} + {\left(3 \, M^{5} + 2 \, M^{3}\right)} a^{8} - 2 \, {\left(3 \, M^{5} + M^{3}\right)} a^{6}\right)} r^{7} - 6 \, {\left({\left(M^{4} + 3 \, M^{2}\right)} a^{10} + 11 \, M^{4} a^{6} - 3 \, {\left(4 \, M^{4} + M^{2}\right)} a^{8}\right)} r^{6} + 2 \, {\left(64 \, M^{5} a^{6} + {\left(64 \, M^{5} - 9 \, M^{3}\right)} a^{10} - {\left(128 \, M^{5} - 9 \, M^{3}\right)} a^{8}\right)} r^{5} - {\left({\left(22 \, M^{4} + 9 \, M^{2}\right)} a^{12} + 22 \, M^{4} a^{8} - {\left(44 \, M^{4} + 9 \, M^{2}\right)} a^{10}\right)} r^{4} + 4 \, {\left(M^{5} a^{12} - 2 \, M^{5} a^{10} + M^{5} a^{8}\right)} r^{3} + 2 \, {\left(M^{4} a^{14} - 2 \, M^{4} a^{12} + M^{4} a^{10}\right)} r^{2}\right)} \cos\left({\theta}\right)^{4} - {\left(3 \, {\left(M^{2} a^{4} - M^{2} a^{2}\right)} r^{12} - 28 \, {\left(M^{3} a^{4} - M^{3} a^{2}\right)} r^{11} - 2 \, {\left(19 \, M^{2} a^{6} + 24 \, M^{4} a^{2} - {\left(24 \, M^{4} + 19 \, M^{2}\right)} a^{4}\right)} r^{10} + 48 \, {\left(M^{3} a^{6} - M^{3} a^{4}\right)} r^{9} - {\left({\left(56 \, M^{4} + 33 \, M^{2}\right)} a^{8} + 48 \, M^{4} a^{4} - {\left(104 \, M^{4} + 33 \, M^{2}\right)} a^{6}\right)} r^{8} - 4 \, {\left(12 \, M^{5} a^{4} + {\left(12 \, M^{5} - 5 \, M^{3}\right)} a^{8} - {\left(24 \, M^{5} - 5 \, M^{3}\right)} a^{6}\right)} r^{7} - 4 \, {\left({\left(4 \, M^{4} + 3 \, M^{2}\right)} a^{10} + 4 \, M^{4} a^{6} - {\left(8 \, M^{4} + 3 \, M^{2}\right)} a^{8}\right)} r^{6} - 128 \, {\left(M^{5} a^{10} - 2 \, M^{5} a^{8} + M^{5} a^{6}\right)} r^{5} + 8 \, {\left(M^{4} a^{12} - 2 \, M^{4} a^{10} + M^{4} a^{8}\right)} r^{4}\right)} \cos\left({\theta}\right)^{2}\right)}}{4 \, M a^{2} r^{15} + 4 \, M r^{17} - r^{18} - 2 \, {\left(2 \, M^{2} + a^{2}\right)} r^{16} - {\left({\left(8 \, M^{2} + 1\right)} a^{4} - 8 \, M^{2} a^{2}\right)} r^{14} - {\left(a^{18} - 4 \, M a^{16} r - 4 \, M a^{14} r^{3} + a^{14} r^{4} + 2 \, {\left(2 \, M^{2} a^{14} + a^{16}\right)} r^{2}\right)} \cos\left({\theta}\right)^{14} + 16 \, {\left(M^{3} a^{4} - M^{3} a^{2}\right)} r^{13} - 8 \, {\left(M^{2} a^{6} - M^{2} a^{4}\right)} r^{12} + {\left(28 \, M a^{12} r^{5} - 7 \, a^{12} r^{6} + 2 \, {\left({\left(4 \, M^{2} - 7\right)} a^{14} - 18 \, M^{2} a^{12}\right)} r^{4} + 4 \, {\left(4 \, M^{3} a^{12} - {\left(4 \, M^{3} - 7 \, M\right)} a^{14}\right)} r^{3} + {\left({\left(8 \, M^{2} - 7\right)} a^{16} - 8 \, M^{2} a^{14}\right)} r^{2}\right)} \cos\left({\theta}\right)^{12} - 16 \, {\left(M^{4} a^{8} - 2 \, M^{4} a^{6} + M^{4} a^{4}\right)} r^{10} + {\left(84 \, M a^{10} r^{7} - 21 \, a^{10} r^{8} + 2 \, {\left({\left(20 \, M^{2} - 21\right)} a^{12} - 62 \, M^{2} a^{10}\right)} r^{6} + 4 \, {\left(20 \, M^{3} a^{10} - {\left(20 \, M^{3} - 21 \, M\right)} a^{12}\right)} r^{5} - {\left({\left(16 \, M^{4} - 32 \, M^{2} + 21\right)} a^{14} + 16 \, M^{4} a^{10} - 32 \, {\left(M^{4} - M^{2}\right)} a^{12}\right)} r^{4} + 16 \, {\left(M^{3} a^{14} - M^{3} a^{12}\right)} r^{3} - 8 \, {\left(M^{2} a^{16} - M^{2} a^{14}\right)} r^{2}\right)} \cos\left({\theta}\right)^{10} + {\left(140 \, M a^{8} r^{9} - 35 \, a^{8} r^{10} + 10 \, {\left({\left(8 \, M^{2} - 7\right)} a^{10} - 22 \, M^{2} a^{8}\right)} r^{8} + 20 \, {\left(8 \, M^{3} a^{8} - {\left(8 \, M^{3} - 7 \, M\right)} a^{10}\right)} r^{7} - {\left({\left(48 \, M^{4} - 40 \, M^{2} + 35\right)} a^{12} + 48 \, M^{4} a^{8} - 8 \, {\left(12 \, M^{4} - 5 \, M^{2}\right)} a^{10}\right)} r^{6} + 80 \, {\left(M^{3} a^{12} - M^{3} a^{10}\right)} r^{5} + 8 \, {\left({\left(4 \, M^{4} - 5 \, M^{2}\right)} a^{14} + 4 \, M^{4} a^{10} - {\left(8 \, M^{4} - 5 \, M^{2}\right)} a^{12}\right)} r^{4}\right)} \cos\left({\theta}\right)^{8} + {\left(140 \, M a^{6} r^{11} - 35 \, a^{6} r^{12} + 10 \, {\left({\left(8 \, M^{2} - 7\right)} a^{8} - 22 \, M^{2} a^{6}\right)} r^{10} + 20 \, {\left(8 \, M^{3} a^{6} - {\left(8 \, M^{3} - 7 \, M\right)} a^{8}\right)} r^{9} + {\left(96 \, M^{4} a^{8} - {\left(48 \, M^{4} + 35\right)} a^{10} - 48 \, M^{4} a^{6}\right)} r^{8} + 160 \, {\left(M^{3} a^{10} - M^{3} a^{8}\right)} r^{7} + 16 \, {\left({\left(6 \, M^{4} - 5 \, M^{2}\right)} a^{12} + 6 \, M^{4} a^{8} - {\left(12 \, M^{4} - 5 \, M^{2}\right)} a^{10}\right)} r^{6} - 16 \, {\left(M^{4} a^{14} - 2 \, M^{4} a^{12} + M^{4} a^{10}\right)} r^{4}\right)} \cos\left({\theta}\right)^{6} + {\left(84 \, M a^{4} r^{13} - 21 \, a^{4} r^{14} + 2 \, {\left({\left(20 \, M^{2} - 21\right)} a^{6} - 62 \, M^{2} a^{4}\right)} r^{12} + 4 \, {\left(20 \, M^{3} a^{4} - {\left(20 \, M^{3} - 21 \, M\right)} a^{6}\right)} r^{11} - {\left({\left(16 \, M^{4} + 40 \, M^{2} + 21\right)} a^{8} + 16 \, M^{4} a^{4} - 8 \, {\left(4 \, M^{4} + 5 \, M^{2}\right)} a^{6}\right)} r^{10} + 160 \, {\left(M^{3} a^{8} - M^{3} a^{6}\right)} r^{9} + 16 \, {\left({\left(6 \, M^{4} - 5 \, M^{2}\right)} a^{10} + 6 \, M^{4} a^{6} - {\left(12 \, M^{4} - 5 \, M^{2}\right)} a^{8}\right)} r^{8} - 48 \, {\left(M^{4} a^{12} - 2 \, M^{4} a^{10} + M^{4} a^{8}\right)} r^{6}\right)} \cos\left({\theta}\right)^{4} + {\left(28 \, M a^{2} r^{15} - 7 \, a^{2} r^{16} + 2 \, {\left({\left(4 \, M^{2} - 7\right)} a^{4} - 18 \, M^{2} a^{2}\right)} r^{14} + 4 \, {\left(4 \, M^{3} a^{2} - {\left(4 \, M^{3} - 7 \, M\right)} a^{4}\right)} r^{13} - {\left({\left(32 \, M^{2} + 7\right)} a^{6} - 32 \, M^{2} a^{4}\right)} r^{12} + 80 \, {\left(M^{3} a^{6} - M^{3} a^{4}\right)} r^{11} + 8 \, {\left({\left(4 \, M^{4} - 5 \, M^{2}\right)} a^{8} + 4 \, M^{4} a^{4} - {\left(8 \, M^{4} - 5 \, M^{2}\right)} a^{6}\right)} r^{10} - 48 \, {\left(M^{4} a^{10} - 2 \, M^{4} a^{8} + M^{4} a^{6}\right)} r^{8}\right)} \cos\left({\theta}\right)^{2}}\]

## 6. Escalar de Kretschmann

El escalar de Kretschmann es el "cuadrado" del tensor de Riemann,

\begin{equation}
K = R_{\mu \nu \rho \sigma} R^{\mu \nu \rho \sigma}
\end{equation}

Para calcularlo, se necesita el tensor de curvtura con todos los indices abajo y con todos los indices arriba,  i.e. $R_{\mu \nu \rho \sigma}$ y
$R^{\mu \nu \rho \sigma}$. Estos se evaluan utilizando los métodos`.down()` y `.up()`. Luego se realiza la contracción de los índices en el orden establecido,

```python
K = Riemann.down(g)['_{abcd}'] * Riemann.up(g)['^{abcd}']
K
```

\[\newcommand{\Bold}[1]{\mathbf{#1}}\mbox{Scalar field on the 4-dimensional Lorentzian manifold MU}\]

```python
K.display()
```

\[\newcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{llcl} & MU & \longrightarrow & \mathbb{R} \\ & \left(t, r, {\theta}, {\phi}\right) & \longmapsto & \frac{48 \, M^{2}}{r^{6}} \end{array}\]

```python
K.expr()
```

\[\newcommand{\Bold}[1]{\mathbf{#1}}\frac{48 \, M^{2}}{r^{6}}\]

### Summary of methods

```
Manifold.chart()
Manifold.metric()
```

```
metric.christoffel_symbols_display()
metric.christoffel_symbols()[]
metric.riemann()
metric.ricci()
```

```
tensor.display()
tensor[:]
tensor.expr()
tensor.up(metric)
tensor.down(metric)
```

```python
g.display?
```

```python
Einstein = Ricci - R*g/2
Einstein[:]
```

\[\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrr}
\frac{9 \, {\left(M^{2} a^{4} - M^{2} a^{2}\right)} r^{14} - 54 \, {\left(M^{3} a^{4} - M^{3} a^{2}\right)} r^{13} + 6 \, {\left(M^{2} a^{6} - 18 \, M^{4} a^{2} + {\left(18 \, M^{4} - M^{2}\right)} a^{4}\right)} r^{12} - {\left(M^{2} a^{18} - M^{2} a^{16} - 3 \, {\left(M^{2} a^{14} - M^{2} a^{12}\right)} r^{4} + 8 \, {\left(M^{3} a^{14} - M^{3} a^{12}\right)} r^{3} - 6 \, {\left(M^{2} a^{16} - M^{2} a^{14}\right)} r^{2}\right)} \cos\left({\theta}\right)^{12} - 36 \, {\left(M^{3} a^{6} - 2 \, M^{5} a^{2} + {\left(2 \, M^{5} - M^{3}\right)} a^{4}\right)} r^{11} - {\left({\left(28 \, M^{4} - M^{2}\right)} a^{8} + 72 \, M^{4} a^{4} - {\left(100 \, M^{4} - M^{2}\right)} a^{6}\right)} r^{10} + {\left(M^{2} a^{18} - M^{2} a^{16} + 3 \, {\left(M^{2} a^{12} - M^{2} a^{10}\right)} r^{6} - 42 \, {\left(M^{3} a^{12} - M^{3} a^{10}\right)} r^{5} + {\left({\left(4 \, M^{4} + 15 \, M^{2}\right)} a^{14} - 48 \, M^{4} a^{10} + {\left(44 \, M^{4} - 15 \, M^{2}\right)} a^{12}\right)} r^{4} + 4 \, {\left({\left(2 \, M^{5} - 5 \, M^{3}\right)} a^{14} - {\left(2 \, M^{5} - 5 \, M^{3}\right)} a^{12}\right)} r^{3} - {\left({\left(12 \, M^{4} + 11 \, M^{2}\right)} a^{16} - {\left(12 \, M^{4} + 11 \, M^{2}\right)} a^{14}\right)} r^{2} + 6 \, {\left(M^{3} a^{16} - M^{3} a^{14}\right)} r\right)} \cos\left({\theta}\right)^{10} + 6 \, {\left(8 \, M^{5} a^{4} + {\left(8 \, M^{5} - M^{3}\right)} a^{8} - {\left(16 \, M^{5} - M^{3}\right)} a^{6}\right)} r^{9} - 12 \, {\left(M^{4} a^{10} - 2 \, M^{4} a^{8} + M^{4} a^{6}\right)} r^{8} - {\left(18 \, {\left(M^{2} a^{10} - M^{2} a^{8}\right)} r^{8} + 24 \, {\left(M^{3} a^{10} - M^{3} a^{8}\right)} r^{7} + {\left({\left(4 \, M^{4} - 9 \, M^{2}\right)} a^{12} + 132 \, M^{4} a^{8} - {\left(136 \, M^{4} - 9 \, M^{2}\right)} a^{10}\right)} r^{6} - 2 \, {\left(48 \, M^{5} a^{8} + {\left(16 \, M^{5} - 3 \, M^{3}\right)} a^{12} - {\left(64 \, M^{5} - 3 \, M^{3}\right)} a^{10}\right)} r^{5} + 4 \, {\left(7 \, {\left(M^{4} + M^{2}\right)} a^{14} - 6 \, M^{4} a^{10} - {\left(M^{4} + 7 \, M^{2}\right)} a^{12}\right)} r^{4} + 4 \, {\left({\left(2 \, M^{5} - 13 \, M^{3}\right)} a^{14} - {\left(2 \, M^{5} - 13 \, M^{3}\right)} a^{12}\right)} r^{3} - {\left({\left(24 \, M^{4} + 5 \, M^{2}\right)} a^{16} + 12 \, M^{4} a^{12} - {\left(36 \, M^{4} + 5 \, M^{2}\right)} a^{14}\right)} r^{2} + 6 \, {\left(M^{3} a^{16} - M^{3} a^{14}\right)} r\right)} \cos\left({\theta}\right)^{8} + 8 \, {\left(M^{5} a^{10} - 2 \, M^{5} a^{8} + M^{5} a^{6}\right)} r^{7} - 2 \, {\left(21 \, {\left(M^{2} a^{8} - M^{2} a^{6}\right)} r^{10} - 50 \, {\left(M^{3} a^{8} - M^{3} a^{6}\right)} r^{9} + {\left({\left(10 \, M^{4} - 3 \, M^{2}\right)} a^{10} + 6 \, M^{4} a^{6} - {\left(16 \, M^{4} - 3 \, M^{2}\right)} a^{8}\right)} r^{8} - 4 \, {\left(15 \, M^{5} a^{6} + {\left(5 \, M^{5} + 6 \, M^{3}\right)} a^{10} - 2 \, {\left(10 \, M^{5} + 3 \, M^{3}\right)} a^{8}\right)} r^{7} - {\left({\left(10 \, M^{4} - 11 \, M^{2}\right)} a^{12} - 32 \, M^{6} a^{6} - {\left(32 \, M^{6} + 88 \, M^{4} - 11 \, M^{2}\right)} a^{10} + 2 \, {\left(32 \, M^{6} + 39 \, M^{4}\right)} a^{8}\right)} r^{6} + 2 \, {\left(36 \, M^{5} a^{8} + {\left(20 \, M^{5} - 21 \, M^{3}\right)} a^{12} - 7 \, {\left(8 \, M^{5} - 3 \, M^{3}\right)} a^{10}\right)} r^{5} - 5 \, {\left({\left(6 \, M^{4} + M^{2}\right)} a^{14} + 6 \, M^{4} a^{10} - {\left(12 \, M^{4} + M^{2}\right)} a^{12}\right)} r^{4} + 4 \, {\left(M^{5} a^{10} + {\left(M^{5} + 3 \, M^{3}\right)} a^{14} - {\left(2 \, M^{5} + 3 \, M^{3}\right)} a^{12}\right)} r^{3} + 6 \, {\left(M^{4} a^{16} - 2 \, M^{4} a^{14} + M^{4} a^{12}\right)} r^{2}\right)} \cos\left({\theta}\right)^{6} - {\left(33 \, {\left(M^{2} a^{6} - M^{2} a^{4}\right)} r^{12} - 144 \, {\left(M^{3} a^{6} - M^{3} a^{4}\right)} r^{11} + 12 \, {\left({\left(M^{4} - 2 \, M^{2}\right)} a^{8} - 15 \, M^{4} a^{4} + 2 \, {\left(7 \, M^{4} + M^{2}\right)} a^{6}\right)} r^{10} + 4 \, {\left(12 \, M^{5} a^{4} - {\left(4 \, M^{5} - 5 \, M^{3}\right)} a^{8} - {\left(8 \, M^{5} + 5 \, M^{3}\right)} a^{6}\right)} r^{9} - {\left({\left(76 \, M^{4} + 7 \, M^{2}\right)} a^{10} + 132 \, M^{4} a^{6} - {\left(208 \, M^{4} + 7 \, M^{2}\right)} a^{8}\right)} r^{8} + 8 \, {\left(17 \, M^{5} a^{10} - 44 \, M^{5} a^{8} + 27 \, M^{5} a^{6}\right)} r^{7} - 2 \, {\left(5 \, {\left(2 \, M^{4} + M^{2}\right)} a^{12} + 64 \, M^{6} a^{6} + {\left(64 \, M^{6} - 16 \, M^{4} - 5 \, M^{2}\right)} a^{10} - 2 \, {\left(64 \, M^{6} - 3 \, M^{4}\right)} a^{8}\right)} r^{6} - 4 \, {\left(8 \, M^{5} a^{8} + {\left(8 \, M^{5} - 9 \, M^{3}\right)} a^{12} - {\left(16 \, M^{5} - 9 \, M^{3}\right)} a^{10}\right)} r^{5} + 36 \, {\left(M^{4} a^{14} - 2 \, M^{4} a^{12} + M^{4} a^{10}\right)} r^{4} - 8 \, {\left(M^{5} a^{14} - 2 \, M^{5} a^{12} + M^{5} a^{10}\right)} r^{3}\right)} \cos\left({\theta}\right)^{4} - {\left(9 \, {\left(M^{2} a^{4} - M^{2} a^{2}\right)} r^{14} - 54 \, {\left(M^{3} a^{4} - M^{3} a^{2}\right)} r^{13} - 27 \, {\left(M^{2} a^{6} + 4 \, M^{4} a^{2} - {\left(4 \, M^{4} + M^{2}\right)} a^{4}\right)} r^{12} + 36 \, {\left(3 \, M^{3} a^{6} + 2 \, M^{5} a^{2} - {\left(2 \, M^{5} + 3 \, M^{3}\right)} a^{4}\right)} r^{11} - {\left({\left(40 \, M^{4} + 17 \, M^{2}\right)} a^{8} - 108 \, M^{4} a^{4} + 17 \, {\left(4 \, M^{4} - M^{2}\right)} a^{6}\right)} r^{10} + 2 \, {\left({\left(32 \, M^{5} + 37 \, M^{3}\right)} a^{8} - {\left(32 \, M^{5} + 37 \, M^{3}\right)} a^{6}\right)} r^{9} + {\left({\left(44 \, M^{4} - 5 \, M^{2}\right)} a^{10} + 108 \, M^{4} a^{6} - {\left(152 \, M^{4} - 5 \, M^{2}\right)} a^{8}\right)} r^{8} - 8 \, {\left(11 \, M^{5} a^{6} + {\left(11 \, M^{5} - 3 \, M^{3}\right)} a^{10} - {\left(22 \, M^{5} - 3 \, M^{3}\right)} a^{8}\right)} r^{7} + 4 \, {\left(9 \, M^{4} a^{12} + 16 \, M^{6} a^{6} + 2 \, {\left(8 \, M^{6} - 9 \, M^{4}\right)} a^{10} - {\left(32 \, M^{6} - 9 \, M^{4}\right)} a^{8}\right)} r^{6} - 16 \, {\left(M^{5} a^{12} - 2 \, M^{5} a^{10} + M^{5} a^{8}\right)} r^{5}\right)} \cos\left({\theta}\right)^{2}}{4 \, M a^{2} r^{17} + 4 \, M r^{19} - r^{20} - 2 \, {\left(2 \, M^{2} + a^{2}\right)} r^{18} - {\left({\left(8 \, M^{2} + 1\right)} a^{4} - 8 \, M^{2} a^{2}\right)} r^{16} - {\left(a^{20} - 4 \, M a^{18} r - 4 \, M a^{16} r^{3} + a^{16} r^{4} + 2 \, {\left(2 \, M^{2} a^{16} + a^{18}\right)} r^{2}\right)} \cos\left({\theta}\right)^{16} + 16 \, {\left(M^{3} a^{4} - M^{3} a^{2}\right)} r^{15} - 8 \, {\left(M^{2} a^{6} - M^{2} a^{4}\right)} r^{14} + 8 \, {\left(4 \, M a^{14} r^{5} - a^{14} r^{6} + {\left({\left(M^{2} - 2\right)} a^{16} - 5 \, M^{2} a^{14}\right)} r^{4} + 2 \, {\left(M^{3} a^{14} - {\left(M^{3} - 2 \, M\right)} a^{16}\right)} r^{3} + {\left({\left(M^{2} - 1\right)} a^{18} - M^{2} a^{16}\right)} r^{2}\right)} \cos\left({\theta}\right)^{14} - 16 \, {\left(M^{4} a^{8} - 2 \, M^{4} a^{6} + M^{4} a^{4}\right)} r^{12} + 4 \, {\left(28 \, M a^{12} r^{7} - 7 \, a^{12} r^{8} + 2 \, {\left({\left(6 \, M^{2} - 7\right)} a^{14} - 20 \, M^{2} a^{12}\right)} r^{6} + 4 \, {\left(6 \, M^{3} a^{12} - {\left(6 \, M^{3} - 7 \, M\right)} a^{14}\right)} r^{5} - {\left({\left(4 \, M^{4} - 10 \, M^{2} + 7\right)} a^{16} + 4 \, M^{4} a^{12} - 2 \, {\left(4 \, M^{4} - 5 \, M^{2}\right)} a^{14}\right)} r^{4} + 4 \, {\left(M^{3} a^{16} - M^{3} a^{14}\right)} r^{3} - 2 \, {\left(M^{2} a^{18} - M^{2} a^{16}\right)} r^{2}\right)} \cos\left({\theta}\right)^{12} + 8 \, {\left(28 \, M a^{10} r^{9} - 7 \, a^{10} r^{10} + {\left({\left(15 \, M^{2} - 14\right)} a^{12} - 43 \, M^{2} a^{10}\right)} r^{8} + 2 \, {\left(15 \, M^{3} a^{10} - {\left(15 \, M^{3} - 14 \, M\right)} a^{12}\right)} r^{7} - {\left({\left(8 \, M^{4} - 9 \, M^{2} + 7\right)} a^{14} + 8 \, M^{4} a^{10} - {\left(16 \, M^{4} - 9 \, M^{2}\right)} a^{12}\right)} r^{6} + 12 \, {\left(M^{3} a^{14} - M^{3} a^{12}\right)} r^{5} + 2 \, {\left({\left(2 \, M^{4} - 3 \, M^{2}\right)} a^{16} + 2 \, M^{4} a^{12} - {\left(4 \, M^{4} - 3 \, M^{2}\right)} a^{14}\right)} r^{4}\right)} \cos\left({\theta}\right)^{10} + 2 \, {\left(140 \, M a^{8} r^{11} - 35 \, a^{8} r^{12} + 10 \, {\left({\left(8 \, M^{2} - 7\right)} a^{10} - 22 \, M^{2} a^{8}\right)} r^{10} + 20 \, {\left(8 \, M^{3} a^{8} - {\left(8 \, M^{3} - 7 \, M\right)} a^{10}\right)} r^{9} - {\left({\left(48 \, M^{4} - 20 \, M^{2} + 35\right)} a^{12} + 48 \, M^{4} a^{8} - 4 \, {\left(24 \, M^{4} - 5 \, M^{2}\right)} a^{10}\right)} r^{8} + 120 \, {\left(M^{3} a^{12} - M^{3} a^{10}\right)} r^{7} + 4 \, {\left({\left(16 \, M^{4} - 15 \, M^{2}\right)} a^{14} + 16 \, M^{4} a^{10} - {\left(32 \, M^{4} - 15 \, M^{2}\right)} a^{12}\right)} r^{6} - 8 \, {\left(M^{4} a^{16} - 2 \, M^{4} a^{14} + M^{4} a^{12}\right)} r^{4}\right)} \cos\left({\theta}\right)^{8} + 8 \, {\left(28 \, M a^{6} r^{13} - 7 \, a^{6} r^{14} + {\left({\left(15 \, M^{2} - 14\right)} a^{8} - 43 \, M^{2} a^{6}\right)} r^{12} + 2 \, {\left(15 \, M^{3} a^{6} - {\left(15 \, M^{3} - 14 \, M\right)} a^{8}\right)} r^{11} - {\left({\left(8 \, M^{4} + 5 \, M^{2} + 7\right)} a^{10} + 8 \, M^{4} a^{6} - {\left(16 \, M^{4} + 5 \, M^{2}\right)} a^{8}\right)} r^{10} + 40 \, {\left(M^{3} a^{10} - M^{3} a^{8}\right)} r^{9} + 4 \, {\left({\left(6 \, M^{4} - 5 \, M^{2}\right)} a^{12} + 6 \, M^{4} a^{8} - {\left(12 \, M^{4} - 5 \, M^{2}\right)} a^{10}\right)} r^{8} - 8 \, {\left(M^{4} a^{14} - 2 \, M^{4} a^{12} + M^{4} a^{10}\right)} r^{6}\right)} \cos\left({\theta}\right)^{6} + 4 \, {\left(28 \, M a^{4} r^{15} - 7 \, a^{4} r^{16} + 2 \, {\left({\left(6 \, M^{2} - 7\right)} a^{6} - 20 \, M^{2} a^{4}\right)} r^{14} + 4 \, {\left(6 \, M^{3} a^{4} - {\left(6 \, M^{3} - 7 \, M\right)} a^{6}\right)} r^{13} - {\left({\left(4 \, M^{4} + 18 \, M^{2} + 7\right)} a^{8} + 4 \, M^{4} a^{4} - 2 \, {\left(4 \, M^{4} + 9 \, M^{2}\right)} a^{6}\right)} r^{12} + 60 \, {\left(M^{3} a^{8} - M^{3} a^{6}\right)} r^{11} + 2 \, {\left({\left(16 \, M^{4} - 15 \, M^{2}\right)} a^{10} + 16 \, M^{4} a^{6} - {\left(32 \, M^{4} - 15 \, M^{2}\right)} a^{8}\right)} r^{10} - 24 \, {\left(M^{4} a^{12} - 2 \, M^{4} a^{10} + M^{4} a^{8}\right)} r^{8}\right)} \cos\left({\theta}\right)^{4} + 8 \, {\left(4 \, M a^{2} r^{17} - a^{2} r^{18} + {\left({\left(M^{2} - 2\right)} a^{4} - 5 \, M^{2} a^{2}\right)} r^{16} + 2 \, {\left(M^{3} a^{2} - {\left(M^{3} - 2 \, M\right)} a^{4}\right)} r^{15} - {\left({\left(5 \, M^{2} + 1\right)} a^{6} - 5 \, M^{2} a^{4}\right)} r^{14} + 12 \, {\left(M^{3} a^{6} - M^{3} a^{4}\right)} r^{13} + 2 \, {\left({\left(2 \, M^{4} - 3 \, M^{2}\right)} a^{8} + 2 \, M^{4} a^{4} - {\left(4 \, M^{4} - 3 \, M^{2}\right)} a^{6}\right)} r^{12} - 8 \, {\left(M^{4} a^{10} - 2 \, M^{4} a^{8} + M^{4} a^{6}\right)} r^{10}\right)} \cos\left({\theta}\right)^{2}} & 0 & 0 & \frac{2 \, {\left(13 \, {\left(M^{3} a^{6} - M^{3} a^{4}\right)} r^{13} - 42 \, {\left(M^{4} a^{6} - M^{4} a^{4}\right)} r^{12} - {\left(3 \, {\left(M^{3} a^{14} - M^{3} a^{12}\right)} r^{5} - 6 \, {\left(M^{4} a^{14} - M^{4} a^{12}\right)} r^{4} + 4 \, {\left(M^{3} a^{16} + M^{5} a^{12} - {\left(M^{5} + M^{3}\right)} a^{14}\right)} r^{3} + 6 \, {\left(M^{4} a^{16} - M^{4} a^{14}\right)} r^{2} - 3 \, {\left(M^{3} a^{18} - M^{3} a^{16}\right)} r\right)} \cos\left({\theta}\right)^{12} + 12 \, {\left(M^{3} a^{8} - 3 \, M^{5} a^{4} + {\left(3 \, M^{5} - M^{3}\right)} a^{6}\right)} r^{11} - 22 \, {\left(M^{4} a^{8} - M^{4} a^{6}\right)} r^{10} + 2 \, {\left(12 \, {\left(M^{4} a^{12} - M^{4} a^{10}\right)} r^{6} + {\left(14 \, M^{5} a^{10} - {\left(2 \, M^{5} + M^{3}\right)} a^{14} - {\left(12 \, M^{5} - M^{3}\right)} a^{12}\right)} r^{5} - 2 \, {\left(M^{4} a^{14} - M^{4} a^{12}\right)} r^{4} + 2 \, {\left(3 \, M^{5} a^{12} + {\left(M^{5} + 4 \, M^{3}\right)} a^{16} - 4 \, {\left(M^{5} + M^{3}\right)} a^{14}\right)} r^{3} + 6 \, {\left(M^{4} a^{16} - M^{4} a^{14}\right)} r^{2} - 3 \, {\left(M^{3} a^{18} - M^{3} a^{16}\right)} r\right)} \cos\left({\theta}\right)^{10} - 3 \, {\left(4 \, M^{5} a^{6} + {\left(4 \, M^{5} - M^{3}\right)} a^{10} - {\left(8 \, M^{5} - M^{3}\right)} a^{8}\right)} r^{9} + {\left(18 \, {\left(M^{3} a^{10} - M^{3} a^{8}\right)} r^{9} - 4 \, {\left(M^{4} a^{10} - M^{4} a^{8}\right)} r^{8} + 4 \, {\left(12 \, M^{5} a^{8} + {\left(2 \, M^{5} - M^{3}\right)} a^{12} - {\left(14 \, M^{5} - M^{3}\right)} a^{10}\right)} r^{7} - 4 \, {\left(8 \, M^{6} a^{8} + {\left(8 \, M^{6} + 5 \, M^{4}\right)} a^{12} - {\left(16 \, M^{6} + 5 \, M^{4}\right)} a^{10}\right)} r^{6} - {\left(44 \, M^{5} a^{10} - {\left(20 \, M^{5} + 11 \, M^{3}\right)} a^{14} - {\left(24 \, M^{5} - 11 \, M^{3}\right)} a^{12}\right)} r^{5} - 2 \, {\left(M^{4} a^{14} - M^{4} a^{12}\right)} r^{4} - 4 \, {\left(4 \, M^{5} a^{12} + 3 \, {\left(M^{5} + 2 \, M^{3}\right)} a^{16} - {\left(7 \, M^{5} + 6 \, M^{3}\right)} a^{14}\right)} r^{3} - 6 \, {\left(M^{4} a^{16} - M^{4} a^{14}\right)} r^{2} + 3 \, {\left(M^{3} a^{18} - M^{3} a^{16}\right)} r\right)} \cos\left({\theta}\right)^{8} - 4 \, {\left(M^{5} a^{12} - 2 \, M^{5} a^{10} + M^{5} a^{8}\right)} r^{7} + 4 \, {\left(6 \, {\left(M^{3} a^{8} - M^{3} a^{6}\right)} r^{11} - 14 \, {\left(M^{4} a^{8} - M^{4} a^{6}\right)} r^{10} - {\left(5 \, M^{5} a^{6} - {\left(3 \, M^{5} - 10 \, M^{3}\right)} a^{10} - 2 \, {\left(M^{5} + 5 \, M^{3}\right)} a^{8}\right)} r^{9} + 6 \, {\left(M^{4} a^{10} - M^{4} a^{8}\right)} r^{8} - 5 \, {\left(5 \, M^{5} a^{8} + {\left(M^{5} + M^{3}\right)} a^{12} - {\left(6 \, M^{5} + M^{3}\right)} a^{10}\right)} r^{7} + 24 \, {\left(M^{6} a^{12} - 2 \, M^{6} a^{10} + M^{6} a^{8}\right)} r^{6} - {\left(M^{5} a^{10} + 3 \, {\left(3 \, M^{5} + 2 \, M^{3}\right)} a^{14} - 2 \, {\left(5 \, M^{5} + 3 \, M^{3}\right)} a^{12}\right)} r^{5} + 3 \, {\left(M^{5} a^{12} + {\left(M^{5} + M^{3}\right)} a^{16} - {\left(2 \, M^{5} + M^{3}\right)} a^{14}\right)} r^{3}\right)} \cos\left({\theta}\right)^{6} + {\left(9 \, {\left(M^{3} a^{6} - M^{3} a^{4}\right)} r^{13} - 34 \, {\left(M^{4} a^{6} - M^{4} a^{4}\right)} r^{12} - 4 \, {\left(14 \, M^{3} a^{8} + 9 \, M^{5} a^{4} - {\left(9 \, M^{5} + 14 \, M^{3}\right)} a^{6}\right)} r^{11} + 122 \, {\left(M^{4} a^{8} - M^{4} a^{6}\right)} r^{10} + {\left(28 \, M^{5} a^{6} - {\left(36 \, M^{5} + 11 \, M^{3}\right)} a^{10} + {\left(8 \, M^{5} + 11 \, M^{3}\right)} a^{8}\right)} r^{9} + 12 \, {\left(M^{4} a^{10} - M^{4} a^{8}\right)} r^{8} + 4 \, {\left(13 \, M^{5} a^{8} + 3 \, {\left(M^{5} + M^{3}\right)} a^{12} - {\left(16 \, M^{5} + 3 \, M^{3}\right)} a^{10}\right)} r^{7} - 4 \, {\left(24 \, M^{6} a^{8} + {\left(24 \, M^{6} + M^{4}\right)} a^{12} - {\left(48 \, M^{6} + M^{4}\right)} a^{10}\right)} r^{6} + 2 \, {\left(14 \, M^{5} a^{10} + {\left(14 \, M^{5} + 9 \, M^{3}\right)} a^{14} - {\left(28 \, M^{5} + 9 \, M^{3}\right)} a^{12}\right)} r^{5} - 4 \, {\left(M^{5} a^{16} - 2 \, M^{5} a^{14} + M^{5} a^{12}\right)} r^{3}\right)} \cos\left({\theta}\right)^{4} - 2 \, {\left(11 \, {\left(M^{3} a^{6} - M^{3} a^{4}\right)} r^{13} - 38 \, {\left(M^{4} a^{6} - M^{4} a^{4}\right)} r^{12} - 2 \, {\left(5 \, M^{3} a^{8} + 18 \, M^{5} a^{4} - {\left(18 \, M^{5} + 5 \, M^{3}\right)} a^{6}\right)} r^{11} + 22 \, {\left(M^{4} a^{8} - M^{4} a^{6}\right)} r^{10} - {\left(2 \, M^{5} a^{6} + 3 \, {\left(6 \, M^{5} + 5 \, M^{3}\right)} a^{10} - 5 \, {\left(4 \, M^{5} + 3 \, M^{3}\right)} a^{8}\right)} r^{9} + 16 \, {\left(M^{4} a^{10} - M^{4} a^{8}\right)} r^{8} - 2 \, {\left(M^{5} a^{8} + {\left(M^{5} + 3 \, M^{3}\right)} a^{12} - {\left(2 \, M^{5} + 3 \, M^{3}\right)} a^{10}\right)} r^{7} - 16 \, {\left(M^{6} a^{12} - 2 \, M^{6} a^{10} + M^{6} a^{8}\right)} r^{6} + 4 \, {\left(M^{5} a^{14} - 2 \, M^{5} a^{12} + M^{5} a^{10}\right)} r^{5}\right)} \cos\left({\theta}\right)^{2}\right)}}{4 \, M a^{2} r^{17} + 4 \, M r^{19} - r^{20} - 2 \, {\left(2 \, M^{2} + a^{2}\right)} r^{18} - {\left({\left(8 \, M^{2} + 1\right)} a^{4} - 8 \, M^{2} a^{2}\right)} r^{16} - {\left(a^{20} - 4 \, M a^{18} r - 4 \, M a^{16} r^{3} + a^{16} r^{4} + 2 \, {\left(2 \, M^{2} a^{16} + a^{18}\right)} r^{2}\right)} \cos\left({\theta}\right)^{16} + 16 \, {\left(M^{3} a^{4} - M^{3} a^{2}\right)} r^{15} - 8 \, {\left(M^{2} a^{6} - M^{2} a^{4}\right)} r^{14} + 8 \, {\left(4 \, M a^{14} r^{5} - a^{14} r^{6} + {\left({\left(M^{2} - 2\right)} a^{16} - 5 \, M^{2} a^{14}\right)} r^{4} + 2 \, {\left(M^{3} a^{14} - {\left(M^{3} - 2 \, M\right)} a^{16}\right)} r^{3} + {\left({\left(M^{2} - 1\right)} a^{18} - M^{2} a^{16}\right)} r^{2}\right)} \cos\left({\theta}\right)^{14} - 16 \, {\left(M^{4} a^{8} - 2 \, M^{4} a^{6} + M^{4} a^{4}\right)} r^{12} + 4 \, {\left(28 \, M a^{12} r^{7} - 7 \, a^{12} r^{8} + 2 \, {\left({\left(6 \, M^{2} - 7\right)} a^{14} - 20 \, M^{2} a^{12}\right)} r^{6} + 4 \, {\left(6 \, M^{3} a^{12} - {\left(6 \, M^{3} - 7 \, M\right)} a^{14}\right)} r^{5} - {\left({\left(4 \, M^{4} - 10 \, M^{2} + 7\right)} a^{16} + 4 \, M^{4} a^{12} - 2 \, {\left(4 \, M^{4} - 5 \, M^{2}\right)} a^{14}\right)} r^{4} + 4 \, {\left(M^{3} a^{16} - M^{3} a^{14}\right)} r^{3} - 2 \, {\left(M^{2} a^{18} - M^{2} a^{16}\right)} r^{2}\right)} \cos\left({\theta}\right)^{12} + 8 \, {\left(28 \, M a^{10} r^{9} - 7 \, a^{10} r^{10} + {\left({\left(15 \, M^{2} - 14\right)} a^{12} - 43 \, M^{2} a^{10}\right)} r^{8} + 2 \, {\left(15 \, M^{3} a^{10} - {\left(15 \, M^{3} - 14 \, M\right)} a^{12}\right)} r^{7} - {\left({\left(8 \, M^{4} - 9 \, M^{2} + 7\right)} a^{14} + 8 \, M^{4} a^{10} - {\left(16 \, M^{4} - 9 \, M^{2}\right)} a^{12}\right)} r^{6} + 12 \, {\left(M^{3} a^{14} - M^{3} a^{12}\right)} r^{5} + 2 \, {\left({\left(2 \, M^{4} - 3 \, M^{2}\right)} a^{16} + 2 \, M^{4} a^{12} - {\left(4 \, M^{4} - 3 \, M^{2}\right)} a^{14}\right)} r^{4}\right)} \cos\left({\theta}\right)^{10} + 2 \, {\left(140 \, M a^{8} r^{11} - 35 \, a^{8} r^{12} + 10 \, {\left({\left(8 \, M^{2} - 7\right)} a^{10} - 22 \, M^{2} a^{8}\right)} r^{10} + 20 \, {\left(8 \, M^{3} a^{8} - {\left(8 \, M^{3} - 7 \, M\right)} a^{10}\right)} r^{9} - {\left({\left(48 \, M^{4} - 20 \, M^{2} + 35\right)} a^{12} + 48 \, M^{4} a^{8} - 4 \, {\left(24 \, M^{4} - 5 \, M^{2}\right)} a^{10}\right)} r^{8} + 120 \, {\left(M^{3} a^{12} - M^{3} a^{10}\right)} r^{7} + 4 \, {\left({\left(16 \, M^{4} - 15 \, M^{2}\right)} a^{14} + 16 \, M^{4} a^{10} - {\left(32 \, M^{4} - 15 \, M^{2}\right)} a^{12}\right)} r^{6} - 8 \, {\left(M^{4} a^{16} - 2 \, M^{4} a^{14} + M^{4} a^{12}\right)} r^{4}\right)} \cos\left({\theta}\right)^{8} + 8 \, {\left(28 \, M a^{6} r^{13} - 7 \, a^{6} r^{14} + {\left({\left(15 \, M^{2} - 14\right)} a^{8} - 43 \, M^{2} a^{6}\right)} r^{12} + 2 \, {\left(15 \, M^{3} a^{6} - {\left(15 \, M^{3} - 14 \, M\right)} a^{8}\right)} r^{11} - {\left({\left(8 \, M^{4} + 5 \, M^{2} + 7\right)} a^{10} + 8 \, M^{4} a^{6} - {\left(16 \, M^{4} + 5 \, M^{2}\right)} a^{8}\right)} r^{10} + 40 \, {\left(M^{3} a^{10} - M^{3} a^{8}\right)} r^{9} + 4 \, {\left({\left(6 \, M^{4} - 5 \, M^{2}\right)} a^{12} + 6 \, M^{4} a^{8} - {\left(12 \, M^{4} - 5 \, M^{2}\right)} a^{10}\right)} r^{8} - 8 \, {\left(M^{4} a^{14} - 2 \, M^{4} a^{12} + M^{4} a^{10}\right)} r^{6}\right)} \cos\left({\theta}\right)^{6} + 4 \, {\left(28 \, M a^{4} r^{15} - 7 \, a^{4} r^{16} + 2 \, {\left({\left(6 \, M^{2} - 7\right)} a^{6} - 20 \, M^{2} a^{4}\right)} r^{14} + 4 \, {\left(6 \, M^{3} a^{4} - {\left(6 \, M^{3} - 7 \, M\right)} a^{6}\right)} r^{13} - {\left({\left(4 \, M^{4} + 18 \, M^{2} + 7\right)} a^{8} + 4 \, M^{4} a^{4} - 2 \, {\left(4 \, M^{4} + 9 \, M^{2}\right)} a^{6}\right)} r^{12} + 60 \, {\left(M^{3} a^{8} - M^{3} a^{6}\right)} r^{11} + 2 \, {\left({\left(16 \, M^{4} - 15 \, M^{2}\right)} a^{10} + 16 \, M^{4} a^{6} - {\left(32 \, M^{4} - 15 \, M^{2}\right)} a^{8}\right)} r^{10} - 24 \, {\left(M^{4} a^{12} - 2 \, M^{4} a^{10} + M^{4} a^{8}\right)} r^{8}\right)} \cos\left({\theta}\right)^{4} + 8 \, {\left(4 \, M a^{2} r^{17} - a^{2} r^{18} + {\left({\left(M^{2} - 2\right)} a^{4} - 5 \, M^{2} a^{2}\right)} r^{16} + 2 \, {\left(M^{3} a^{2} - {\left(M^{3} - 2 \, M\right)} a^{4}\right)} r^{15} - {\left({\left(5 \, M^{2} + 1\right)} a^{6} - 5 \, M^{2} a^{4}\right)} r^{14} + 12 \, {\left(M^{3} a^{6} - M^{3} a^{4}\right)} r^{13} + 2 \, {\left({\left(2 \, M^{4} - 3 \, M^{2}\right)} a^{8} + 2 \, M^{4} a^{4} - {\left(4 \, M^{4} - 3 \, M^{2}\right)} a^{6}\right)} r^{12} - 8 \, {\left(M^{4} a^{10} - 2 \, M^{4} a^{8} + M^{4} a^{6}\right)} r^{10}\right)} \cos\left({\theta}\right)^{2}} \\
0 & \frac{11 \, {\left(M^{2} a^{4} - M^{2} a^{2}\right)} r^{12} - 28 \, {\left(M^{3} a^{4} - M^{3} a^{2}\right)} r^{11} + 2 \, {\left(11 \, M^{2} a^{6} - 6 \, M^{4} a^{2} + {\left(6 \, M^{4} - 11 \, M^{2}\right)} a^{4}\right)} r^{10} + {\left(M^{2} a^{16} - M^{2} a^{14} - 3 \, {\left(M^{2} a^{12} - M^{2} a^{10}\right)} r^{4} + 4 \, {\left(M^{3} a^{12} - M^{3} a^{10}\right)} r^{3} - 2 \, {\left(M^{2} a^{14} + 2 \, M^{4} a^{10} - {\left(2 \, M^{4} + M^{2}\right)} a^{12}\right)} r^{2} - 4 \, {\left(M^{3} a^{14} - M^{3} a^{12}\right)} r\right)} \cos\left({\theta}\right)^{10} - 28 \, {\left(M^{3} a^{6} - M^{3} a^{4}\right)} r^{9} + 11 \, {\left({\left(4 \, M^{4} + M^{2}\right)} a^{8} + 4 \, M^{4} a^{4} - {\left(8 \, M^{4} + M^{2}\right)} a^{6}\right)} r^{8} - {\left(M^{2} a^{16} - M^{2} a^{14} - 28 \, {\left(M^{2} a^{10} - M^{2} a^{8}\right)} r^{6} + 64 \, {\left(M^{3} a^{10} - M^{3} a^{8}\right)} r^{5} + {\left({\left(4 \, M^{4} - 71 \, M^{2}\right)} a^{12} + 20 \, M^{4} a^{8} - {\left(24 \, M^{4} - 71 \, M^{2}\right)} a^{10}\right)} r^{4} - 4 \, {\left(2 \, M^{5} a^{8} + {\left(2 \, M^{5} - 23 \, M^{3}\right)} a^{12} - {\left(4 \, M^{5} - 23 \, M^{3}\right)} a^{10}\right)} r^{3} + 2 \, {\left({\left(2 \, M^{4} - 21 \, M^{2}\right)} a^{14} - {\left(2 \, M^{4} - 21 \, M^{2}\right)} a^{12}\right)} r^{2} - 4 \, {\left(M^{3} a^{14} - M^{3} a^{12}\right)} r\right)} \cos\left({\theta}\right)^{8} - 24 \, {\left(M^{5} a^{8} - 2 \, M^{5} a^{6} + M^{5} a^{4}\right)} r^{7} + 44 \, {\left(M^{4} a^{10} - 2 \, M^{4} a^{8} + M^{4} a^{6}\right)} r^{6} + 2 \, {\left(23 \, {\left(M^{2} a^{8} - M^{2} a^{6}\right)} r^{8} - 48 \, {\left(M^{3} a^{8} - M^{3} a^{6}\right)} r^{7} - 2 \, {\left({\left(26 \, M^{4} - 17 \, M^{2}\right)} a^{10} + 28 \, M^{4} a^{6} - {\left(54 \, M^{4} - 17 \, M^{2}\right)} a^{8}\right)} r^{6} + 4 \, {\left(2 \, M^{5} a^{6} + {\left(2 \, M^{5} - 5 \, M^{3}\right)} a^{10} - {\left(4 \, M^{5} - 5 \, M^{3}\right)} a^{8}\right)} r^{5} - {\left({\left(48 \, M^{4} + 13 \, M^{2}\right)} a^{12} + 40 \, M^{4} a^{8} - {\left(88 \, M^{4} + 13 \, M^{2}\right)} a^{10}\right)} r^{4} - 4 \, {\left(2 \, M^{5} a^{8} + {\left(2 \, M^{5} - 13 \, M^{3}\right)} a^{12} - {\left(4 \, M^{5} - 13 \, M^{3}\right)} a^{10}\right)} r^{3} + 4 \, {\left({\left(M^{4} - 6 \, M^{2}\right)} a^{14} + M^{4} a^{10} - 2 \, {\left(M^{4} - 3 \, M^{2}\right)} a^{12}\right)} r^{2}\right)} \cos\left({\theta}\right)^{6} - 2 \, {\left(2 \, {\left(M^{2} a^{6} - M^{2} a^{4}\right)} r^{10} - 8 \, {\left(M^{3} a^{6} - M^{3} a^{4}\right)} r^{9} - {\left({\left(30 \, M^{4} - 37 \, M^{2}\right)} a^{8} + 38 \, M^{4} a^{4} - {\left(68 \, M^{4} - 37 \, M^{2}\right)} a^{6}\right)} r^{8} + 4 \, {\left(3 \, M^{5} a^{4} + {\left(3 \, M^{5} - 19 \, M^{3}\right)} a^{8} - {\left(6 \, M^{5} - 19 \, M^{3}\right)} a^{6}\right)} r^{7} - 2 \, {\left({\left(71 \, M^{4} - 38 \, M^{2}\right)} a^{10} + 73 \, M^{4} a^{6} - 2 \, {\left(72 \, M^{4} - 19 \, M^{2}\right)} a^{8}\right)} r^{6} + 4 \, {\left(4 \, M^{5} a^{6} + {\left(4 \, M^{5} - 21 \, M^{3}\right)} a^{10} - {\left(8 \, M^{5} - 21 \, M^{3}\right)} a^{8}\right)} r^{5} - {\left({\left(110 \, M^{4} - 41 \, M^{2}\right)} a^{12} + 110 \, M^{4} a^{8} - {\left(220 \, M^{4} - 41 \, M^{2}\right)} a^{10}\right)} r^{4} - 4 \, {\left(M^{5} a^{12} - 2 \, M^{5} a^{10} + M^{5} a^{8}\right)} r^{3} + 2 \, {\left(M^{4} a^{14} - 2 \, M^{4} a^{12} + M^{4} a^{10}\right)} r^{2}\right)} \cos\left({\theta}\right)^{4} - {\left(19 \, {\left(M^{2} a^{4} - M^{2} a^{2}\right)} r^{12} - 44 \, {\left(M^{3} a^{4} - M^{3} a^{2}\right)} r^{11} + 2 \, {\left(29 \, M^{2} a^{6} - 6 \, M^{4} a^{2} + {\left(6 \, M^{4} - 29 \, M^{2}\right)} a^{4}\right)} r^{10} - 76 \, {\left(M^{3} a^{6} - M^{3} a^{4}\right)} r^{9} + {\left({\left(104 \, M^{4} + 63 \, M^{2}\right)} a^{8} + 120 \, M^{4} a^{4} - 7 \, {\left(32 \, M^{4} + 9 \, M^{2}\right)} a^{6}\right)} r^{8} - 8 \, {\left(6 \, M^{5} a^{4} + {\left(6 \, M^{5} + 5 \, M^{3}\right)} a^{8} - {\left(12 \, M^{5} + 5 \, M^{3}\right)} a^{6}\right)} r^{7} + 8 \, {\left({\left(28 \, M^{4} + 3 \, M^{2}\right)} a^{10} + 28 \, M^{4} a^{6} - {\left(56 \, M^{4} + 3 \, M^{2}\right)} a^{8}\right)} r^{6} - 16 \, {\left(M^{5} a^{10} - 2 \, M^{5} a^{8} + M^{5} a^{6}\right)} r^{5} + 120 \, {\left(M^{4} a^{12} - 2 \, M^{4} a^{10} + M^{4} a^{8}\right)} r^{4}\right)} \cos\left({\theta}\right)^{2}}{6 \, M r^{17} - r^{18} - 3 \, {\left(4 \, M^{2} + a^{2}\right)} r^{16} + 4 \, {\left(2 \, M^{3} + 3 \, M a^{2}\right)} r^{15} - {\left({\left(8 \, M^{2} + 3\right)} a^{4} + 4 \, M^{2} a^{2}\right)} r^{14} - 2 \, {\left(16 \, M^{3} a^{2} - {\left(16 \, M^{3} + 3 \, M\right)} a^{4}\right)} r^{13} - {\left({\left(16 \, M^{2} + 1\right)} a^{6} - 32 \, M^{4} a^{2} + 16 \, {\left(2 \, M^{4} - M^{2}\right)} a^{4}\right)} r^{12} - {\left(a^{18} - 6 \, M a^{16} r - 6 \, M a^{12} r^{5} + a^{12} r^{6} + 3 \, {\left(4 \, M^{2} a^{12} + a^{14}\right)} r^{4} - 4 \, {\left(2 \, M^{3} a^{12} + 3 \, M a^{14}\right)} r^{3} + 3 \, {\left(4 \, M^{2} a^{14} + a^{16}\right)} r^{2}\right)} \cos\left({\theta}\right)^{12} + 32 \, {\left(M^{3} a^{6} - M^{3} a^{4}\right)} r^{11} - 8 \, {\left({\left(2 \, M^{4} + M^{2}\right)} a^{8} + 2 \, M^{4} a^{4} - {\left(4 \, M^{4} + M^{2}\right)} a^{6}\right)} r^{10} + 2 \, {\left(18 \, M a^{10} r^{7} - 3 \, a^{10} r^{8} + {\left({\left(4 \, M^{2} - 9\right)} a^{12} - 40 \, M^{2} a^{10}\right)} r^{6} + 4 \, {\left(10 \, M^{3} a^{10} - {\left(4 \, M^{3} - 9 \, M\right)} a^{12}\right)} r^{5} + {\left({\left(8 \, M^{2} - 9\right)} a^{14} - 16 \, M^{4} a^{10} + 4 \, {\left(4 \, M^{4} - 11 \, M^{2}\right)} a^{12}\right)} r^{4} + 2 \, {\left(8 \, M^{3} a^{12} - {\left(8 \, M^{3} - 9 \, M\right)} a^{14}\right)} r^{3} + {\left({\left(4 \, M^{2} - 3\right)} a^{16} - 4 \, M^{2} a^{14}\right)} r^{2}\right)} \cos\left({\theta}\right)^{10} + 32 \, {\left(M^{5} a^{8} - 2 \, M^{5} a^{6} + M^{5} a^{4}\right)} r^{9} - 16 \, {\left(M^{4} a^{10} - 2 \, M^{4} a^{8} + M^{4} a^{6}\right)} r^{8} + {\left(90 \, M a^{8} r^{9} - 15 \, a^{8} r^{10} + {\left({\left(32 \, M^{2} - 45\right)} a^{10} - 212 \, M^{2} a^{8}\right)} r^{8} + 4 \, {\left(62 \, M^{3} a^{8} - {\left(32 \, M^{3} - 45 \, M\right)} a^{10}\right)} r^{7} - {\left({\left(16 \, M^{4} - 56 \, M^{2} + 45\right)} a^{12} + 144 \, M^{4} a^{8} - 4 \, {\left(40 \, M^{4} - 59 \, M^{2}\right)} a^{10}\right)} r^{6} + 2 \, {\left(16 \, M^{5} a^{8} + {\left(16 \, M^{5} - 48 \, M^{3} + 45 \, M\right)} a^{12} - 16 \, {\left(2 \, M^{5} - 3 \, M^{3}\right)} a^{10}\right)} r^{5} - {\left({\left(16 \, M^{4} - 16 \, M^{2} + 15\right)} a^{14} - 16 \, M^{4} a^{10} + 16 \, M^{2} a^{12}\right)} r^{4} + 32 \, {\left(M^{3} a^{14} - M^{3} a^{12}\right)} r^{3} - 8 \, {\left(M^{2} a^{16} - M^{2} a^{14}\right)} r^{2}\right)} \cos\left({\theta}\right)^{8} + 4 \, {\left(30 \, M a^{6} r^{11} - 5 \, a^{6} r^{12} + 3 \, {\left({\left(4 \, M^{2} - 5\right)} a^{8} - 24 \, M^{2} a^{6}\right)} r^{10} + 4 \, {\left(22 \, M^{3} a^{6} - 3 \, {\left(4 \, M^{3} - 5 \, M\right)} a^{8}\right)} r^{9} - {\left({\left(8 \, M^{4} - 16 \, M^{2} + 15\right)} a^{10} + 56 \, M^{4} a^{6} - 4 \, {\left(16 \, M^{4} - 19 \, M^{2}\right)} a^{8}\right)} r^{8} + 2 \, {\left(8 \, M^{5} a^{6} + {\left(8 \, M^{5} - 8 \, M^{3} + 15 \, M\right)} a^{10} - 8 \, {\left(2 \, M^{5} - M^{3}\right)} a^{8}\right)} r^{7} - {\left({\left(4 \, M^{2} + 5\right)} a^{12} - 32 \, M^{4} a^{8} + 4 \, {\left(8 \, M^{4} - M^{2}\right)} a^{10}\right)} r^{6} - 16 \, {\left(M^{5} a^{8} + {\left(M^{5} - 2 \, M^{3}\right)} a^{12} - 2 \, {\left(M^{5} - M^{3}\right)} a^{10}\right)} r^{5} + 8 \, {\left({\left(M^{4} - M^{2}\right)} a^{14} + M^{4} a^{10} - {\left(2 \, M^{4} - M^{2}\right)} a^{12}\right)} r^{4}\right)} \cos\left({\theta}\right)^{6} + {\left(90 \, M a^{4} r^{13} - 15 \, a^{4} r^{14} + {\left({\left(32 \, M^{2} - 45\right)} a^{6} - 212 \, M^{2} a^{4}\right)} r^{12} + 4 \, {\left(62 \, M^{3} a^{4} - {\left(32 \, M^{3} - 45 \, M\right)} a^{6}\right)} r^{11} - {\left({\left(16 \, M^{4} - 16 \, M^{2} + 45\right)} a^{8} + 144 \, M^{4} a^{4} - 4 \, {\left(40 \, M^{4} - 49 \, M^{2}\right)} a^{6}\right)} r^{10} + 2 \, {\left(16 \, M^{5} a^{4} + {\left(16 \, M^{5} + 32 \, M^{3} + 45 \, M\right)} a^{8} - 32 \, {\left(M^{5} + M^{3}\right)} a^{6}\right)} r^{9} + {\left({\left(48 \, M^{4} - 64 \, M^{2} - 15\right)} a^{10} + 240 \, M^{4} a^{6} - 32 \, {\left(9 \, M^{4} - 2 \, M^{2}\right)} a^{8}\right)} r^{8} - 64 \, {\left(2 \, M^{5} a^{6} + {\left(2 \, M^{5} - 3 \, M^{3}\right)} a^{10} - {\left(4 \, M^{5} - 3 \, M^{3}\right)} a^{8}\right)} r^{7} + 48 \, {\left({\left(M^{4} - M^{2}\right)} a^{12} + M^{4} a^{8} - {\left(2 \, M^{4} - M^{2}\right)} a^{10}\right)} r^{6} + 32 \, {\left(M^{5} a^{12} - 2 \, M^{5} a^{10} + M^{5} a^{8}\right)} r^{5} - 16 \, {\left(M^{4} a^{14} - 2 \, M^{4} a^{12} + M^{4} a^{10}\right)} r^{4}\right)} \cos\left({\theta}\right)^{4} + 2 \, {\left(18 \, M a^{2} r^{15} - 3 \, a^{2} r^{16} + {\left({\left(4 \, M^{2} - 9\right)} a^{4} - 40 \, M^{2} a^{2}\right)} r^{14} + 4 \, {\left(10 \, M^{3} a^{2} - {\left(4 \, M^{3} - 9 \, M\right)} a^{4}\right)} r^{13} - {\left({\left(8 \, M^{2} + 9\right)} a^{6} + 16 \, M^{4} a^{2} - 4 \, {\left(4 \, M^{4} - 7 \, M^{2}\right)} a^{4}\right)} r^{12} - 6 \, {\left(8 \, M^{3} a^{4} - {\left(8 \, M^{3} + 3 \, M\right)} a^{6}\right)} r^{11} + {\left({\left(16 \, M^{4} - 28 \, M^{2} - 3\right)} a^{8} + 80 \, M^{4} a^{4} - 4 \, {\left(24 \, M^{4} - 7 \, M^{2}\right)} a^{6}\right)} r^{10} - 32 \, {\left(M^{5} a^{4} + {\left(M^{5} - 2 \, M^{3}\right)} a^{8} - 2 \, {\left(M^{5} - M^{3}\right)} a^{6}\right)} r^{9} - 16 \, {\left(M^{2} a^{10} - M^{2} a^{8}\right)} r^{8} + 32 \, {\left(M^{5} a^{10} - 2 \, M^{5} a^{8} + M^{5} a^{6}\right)} r^{7} - 16 \, {\left(M^{4} a^{12} - 2 \, M^{4} a^{10} + M^{4} a^{8}\right)} r^{6}\right)} \cos\left({\theta}\right)^{2}} & -\frac{4 \, {\left({\left({\left(M^{2} a^{12} - M^{2} a^{10}\right)} r^{3} - 3 \, {\left(M^{3} a^{12} - M^{3} a^{10}\right)} r^{2} + 2 \, {\left(M^{2} a^{14} - M^{2} a^{12}\right)} r\right)} \cos\left({\theta}\right)^{9} - 2 \, {\left(3 \, {\left(M^{2} a^{10} - M^{2} a^{8}\right)} r^{5} - 5 \, {\left(M^{3} a^{10} - M^{3} a^{8}\right)} r^{4} + {\left({\left(2 \, M^{4} + 3 \, M^{2}\right)} a^{12} + 2 \, M^{4} a^{8} - {\left(4 \, M^{4} + 3 \, M^{2}\right)} a^{10}\right)} r^{3} - 3 \, {\left(M^{3} a^{12} - M^{3} a^{10}\right)} r^{2} + 2 \, {\left(M^{2} a^{14} - M^{2} a^{12}\right)} r\right)} \cos\left({\theta}\right)^{7} - 2 \, {\left(5 \, {\left(M^{2} a^{8} - M^{2} a^{6}\right)} r^{7} - 10 \, {\left(M^{3} a^{8} - M^{3} a^{6}\right)} r^{6} - {\left({\left(12 \, M^{4} - M^{2}\right)} a^{10} + 12 \, M^{4} a^{6} - {\left(24 \, M^{4} - M^{2}\right)} a^{8}\right)} r^{5} + 5 \, {\left(M^{3} a^{10} - M^{3} a^{8}\right)} r^{4} - {\left({\left(6 \, M^{4} + M^{2}\right)} a^{12} + 6 \, M^{4} a^{8} - {\left(12 \, M^{4} + M^{2}\right)} a^{10}\right)} r^{3}\right)} \cos\left({\theta}\right)^{5} + 2 \, {\left({\left(M^{2} a^{6} - M^{2} a^{4}\right)} r^{9} - {\left(M^{3} a^{6} - M^{3} a^{4}\right)} r^{8} - {\left({\left(6 \, M^{4} - 11 \, M^{2}\right)} a^{8} + 6 \, M^{4} a^{4} - {\left(12 \, M^{4} - 11 \, M^{2}\right)} a^{6}\right)} r^{7} - 19 \, {\left(M^{3} a^{8} - M^{3} a^{6}\right)} r^{6} - 4 \, {\left({\left(7 \, M^{4} - 2 \, M^{2}\right)} a^{10} + 7 \, M^{4} a^{6} - 2 \, {\left(7 \, M^{4} - M^{2}\right)} a^{8}\right)} r^{5} - 4 \, {\left(M^{4} a^{12} - 2 \, M^{4} a^{10} + M^{4} a^{8}\right)} r^{3}\right)} \cos\left({\theta}\right)^{3} + {\left(5 \, {\left(M^{2} a^{4} - M^{2} a^{2}\right)} r^{11} - 9 \, {\left(M^{3} a^{4} - M^{3} a^{2}\right)} r^{10} + 16 \, {\left(M^{2} a^{6} - M^{2} a^{4}\right)} r^{9} - 22 \, {\left(M^{3} a^{6} - M^{3} a^{4}\right)} r^{8} + 2 \, {\left({\left(6 \, M^{4} + 5 \, M^{2}\right)} a^{8} + 6 \, M^{4} a^{4} - {\left(12 \, M^{4} + 5 \, M^{2}\right)} a^{6}\right)} r^{7} + 32 \, {\left(M^{4} a^{10} - 2 \, M^{4} a^{8} + M^{4} a^{6}\right)} r^{5}\right)} \cos\left({\theta}\right)\right)} \sin\left({\theta}\right)}{4 \, M a^{2} r^{13} + 4 \, M r^{15} - r^{16} - 2 \, {\left(2 \, M^{2} + a^{2}\right)} r^{14} - {\left({\left(8 \, M^{2} + 1\right)} a^{4} - 8 \, M^{2} a^{2}\right)} r^{12} - {\left(a^{16} - 4 \, M a^{14} r - 4 \, M a^{12} r^{3} + a^{12} r^{4} + 2 \, {\left(2 \, M^{2} a^{12} + a^{14}\right)} r^{2}\right)} \cos\left({\theta}\right)^{12} + 16 \, {\left(M^{3} a^{4} - M^{3} a^{2}\right)} r^{11} - 8 \, {\left(M^{2} a^{6} - M^{2} a^{4}\right)} r^{10} + 2 \, {\left(12 \, M a^{10} r^{5} - 3 \, a^{10} r^{6} + 2 \, {\left({\left(2 \, M^{2} - 3\right)} a^{12} - 8 \, M^{2} a^{10}\right)} r^{4} + 4 \, {\left(2 \, M^{3} a^{10} - {\left(2 \, M^{3} - 3 \, M\right)} a^{12}\right)} r^{3} + {\left({\left(4 \, M^{2} - 3\right)} a^{14} - 4 \, M^{2} a^{12}\right)} r^{2}\right)} \cos\left({\theta}\right)^{10} - 16 \, {\left(M^{4} a^{8} - 2 \, M^{4} a^{6} + M^{4} a^{4}\right)} r^{8} + {\left(60 \, M a^{8} r^{7} - 15 \, a^{8} r^{8} + 2 \, {\left({\left(16 \, M^{2} - 15\right)} a^{10} - 46 \, M^{2} a^{8}\right)} r^{6} + 4 \, {\left(16 \, M^{3} a^{8} - {\left(16 \, M^{3} - 15 \, M\right)} a^{10}\right)} r^{5} - {\left({\left(16 \, M^{4} - 24 \, M^{2} + 15\right)} a^{12} + 16 \, M^{4} a^{8} - 8 \, {\left(4 \, M^{4} - 3 \, M^{2}\right)} a^{10}\right)} r^{4} + 16 \, {\left(M^{3} a^{12} - M^{3} a^{10}\right)} r^{3} - 8 \, {\left(M^{2} a^{14} - M^{2} a^{12}\right)} r^{2}\right)} \cos\left({\theta}\right)^{8} + 4 \, {\left(20 \, M a^{6} r^{9} - 5 \, a^{6} r^{10} + 2 \, {\left({\left(6 \, M^{2} - 5\right)} a^{8} - 16 \, M^{2} a^{6}\right)} r^{8} + 4 \, {\left(6 \, M^{3} a^{6} - {\left(6 \, M^{3} - 5 \, M\right)} a^{8}\right)} r^{7} - {\left({\left(8 \, M^{4} - 4 \, M^{2} + 5\right)} a^{10} + 8 \, M^{4} a^{6} - 4 \, {\left(4 \, M^{4} - M^{2}\right)} a^{8}\right)} r^{6} + 16 \, {\left(M^{3} a^{10} - M^{3} a^{8}\right)} r^{5} + 8 \, {\left({\left(M^{4} - M^{2}\right)} a^{12} + M^{4} a^{8} - {\left(2 \, M^{4} - M^{2}\right)} a^{10}\right)} r^{4}\right)} \cos\left({\theta}\right)^{6} + {\left(60 \, M a^{4} r^{11} - 15 \, a^{4} r^{12} + 2 \, {\left({\left(16 \, M^{2} - 15\right)} a^{6} - 46 \, M^{2} a^{4}\right)} r^{10} + 4 \, {\left(16 \, M^{3} a^{4} - {\left(16 \, M^{3} - 15 \, M\right)} a^{6}\right)} r^{9} - {\left({\left(16 \, M^{4} + 16 \, M^{2} + 15\right)} a^{8} + 16 \, M^{4} a^{4} - 16 \, {\left(2 \, M^{4} + M^{2}\right)} a^{6}\right)} r^{8} + 96 \, {\left(M^{3} a^{8} - M^{3} a^{6}\right)} r^{7} + 16 \, {\left({\left(4 \, M^{4} - 3 \, M^{2}\right)} a^{10} + 4 \, M^{4} a^{6} - {\left(8 \, M^{4} - 3 \, M^{2}\right)} a^{8}\right)} r^{6} - 16 \, {\left(M^{4} a^{12} - 2 \, M^{4} a^{10} + M^{4} a^{8}\right)} r^{4}\right)} \cos\left({\theta}\right)^{4} + 2 \, {\left(12 \, M a^{2} r^{13} - 3 \, a^{2} r^{14} + 2 \, {\left({\left(2 \, M^{2} - 3\right)} a^{4} - 8 \, M^{2} a^{2}\right)} r^{12} + 4 \, {\left(2 \, M^{3} a^{2} - {\left(2 \, M^{3} - 3 \, M\right)} a^{4}\right)} r^{11} - 3 \, {\left({\left(4 \, M^{2} + 1\right)} a^{6} - 4 \, M^{2} a^{4}\right)} r^{10} + 32 \, {\left(M^{3} a^{6} - M^{3} a^{4}\right)} r^{9} + 16 \, {\left({\left(M^{4} - M^{2}\right)} a^{8} + M^{4} a^{4} - {\left(2 \, M^{4} - M^{2}\right)} a^{6}\right)} r^{8} - 16 \, {\left(M^{4} a^{10} - 2 \, M^{4} a^{8} + M^{4} a^{6}\right)} r^{6}\right)} \cos\left({\theta}\right)^{2}} & 0 \\
0 & -\frac{4 \, {\left({\left({\left(M^{2} a^{12} - M^{2} a^{10}\right)} r^{3} - 3 \, {\left(M^{3} a^{12} - M^{3} a^{10}\right)} r^{2} + 2 \, {\left(M^{2} a^{14} - M^{2} a^{12}\right)} r\right)} \cos\left({\theta}\right)^{9} - 2 \, {\left(3 \, {\left(M^{2} a^{10} - M^{2} a^{8}\right)} r^{5} - 5 \, {\left(M^{3} a^{10} - M^{3} a^{8}\right)} r^{4} + {\left({\left(2 \, M^{4} + 3 \, M^{2}\right)} a^{12} + 2 \, M^{4} a^{8} - {\left(4 \, M^{4} + 3 \, M^{2}\right)} a^{10}\right)} r^{3} - 3 \, {\left(M^{3} a^{12} - M^{3} a^{10}\right)} r^{2} + 2 \, {\left(M^{2} a^{14} - M^{2} a^{12}\right)} r\right)} \cos\left({\theta}\right)^{7} - 2 \, {\left(5 \, {\left(M^{2} a^{8} - M^{2} a^{6}\right)} r^{7} - 10 \, {\left(M^{3} a^{8} - M^{3} a^{6}\right)} r^{6} - {\left({\left(12 \, M^{4} - M^{2}\right)} a^{10} + 12 \, M^{4} a^{6} - {\left(24 \, M^{4} - M^{2}\right)} a^{8}\right)} r^{5} + 5 \, {\left(M^{3} a^{10} - M^{3} a^{8}\right)} r^{4} - {\left({\left(6 \, M^{4} + M^{2}\right)} a^{12} + 6 \, M^{4} a^{8} - {\left(12 \, M^{4} + M^{2}\right)} a^{10}\right)} r^{3}\right)} \cos\left({\theta}\right)^{5} + 2 \, {\left({\left(M^{2} a^{6} - M^{2} a^{4}\right)} r^{9} - {\left(M^{3} a^{6} - M^{3} a^{4}\right)} r^{8} - {\left({\left(6 \, M^{4} - 11 \, M^{2}\right)} a^{8} + 6 \, M^{4} a^{4} - {\left(12 \, M^{4} - 11 \, M^{2}\right)} a^{6}\right)} r^{7} - 19 \, {\left(M^{3} a^{8} - M^{3} a^{6}\right)} r^{6} - 4 \, {\left({\left(7 \, M^{4} - 2 \, M^{2}\right)} a^{10} + 7 \, M^{4} a^{6} - 2 \, {\left(7 \, M^{4} - M^{2}\right)} a^{8}\right)} r^{5} - 4 \, {\left(M^{4} a^{12} - 2 \, M^{4} a^{10} + M^{4} a^{8}\right)} r^{3}\right)} \cos\left({\theta}\right)^{3} + {\left(5 \, {\left(M^{2} a^{4} - M^{2} a^{2}\right)} r^{11} - 9 \, {\left(M^{3} a^{4} - M^{3} a^{2}\right)} r^{10} + 16 \, {\left(M^{2} a^{6} - M^{2} a^{4}\right)} r^{9} - 22 \, {\left(M^{3} a^{6} - M^{3} a^{4}\right)} r^{8} + 2 \, {\left({\left(6 \, M^{4} + 5 \, M^{2}\right)} a^{8} + 6 \, M^{4} a^{4} - {\left(12 \, M^{4} + 5 \, M^{2}\right)} a^{6}\right)} r^{7} + 32 \, {\left(M^{4} a^{10} - 2 \, M^{4} a^{8} + M^{4} a^{6}\right)} r^{5}\right)} \cos\left({\theta}\right)\right)} \sin\left({\theta}\right)}{4 \, M a^{2} r^{13} + 4 \, M r^{15} - r^{16} - 2 \, {\left(2 \, M^{2} + a^{2}\right)} r^{14} - {\left({\left(8 \, M^{2} + 1\right)} a^{4} - 8 \, M^{2} a^{2}\right)} r^{12} - {\left(a^{16} - 4 \, M a^{14} r - 4 \, M a^{12} r^{3} + a^{12} r^{4} + 2 \, {\left(2 \, M^{2} a^{12} + a^{14}\right)} r^{2}\right)} \cos\left({\theta}\right)^{12} + 16 \, {\left(M^{3} a^{4} - M^{3} a^{2}\right)} r^{11} - 8 \, {\left(M^{2} a^{6} - M^{2} a^{4}\right)} r^{10} + 2 \, {\left(12 \, M a^{10} r^{5} - 3 \, a^{10} r^{6} + 2 \, {\left({\left(2 \, M^{2} - 3\right)} a^{12} - 8 \, M^{2} a^{10}\right)} r^{4} + 4 \, {\left(2 \, M^{3} a^{10} - {\left(2 \, M^{3} - 3 \, M\right)} a^{12}\right)} r^{3} + {\left({\left(4 \, M^{2} - 3\right)} a^{14} - 4 \, M^{2} a^{12}\right)} r^{2}\right)} \cos\left({\theta}\right)^{10} - 16 \, {\left(M^{4} a^{8} - 2 \, M^{4} a^{6} + M^{4} a^{4}\right)} r^{8} + {\left(60 \, M a^{8} r^{7} - 15 \, a^{8} r^{8} + 2 \, {\left({\left(16 \, M^{2} - 15\right)} a^{10} - 46 \, M^{2} a^{8}\right)} r^{6} + 4 \, {\left(16 \, M^{3} a^{8} - {\left(16 \, M^{3} - 15 \, M\right)} a^{10}\right)} r^{5} - {\left({\left(16 \, M^{4} - 24 \, M^{2} + 15\right)} a^{12} + 16 \, M^{4} a^{8} - 8 \, {\left(4 \, M^{4} - 3 \, M^{2}\right)} a^{10}\right)} r^{4} + 16 \, {\left(M^{3} a^{12} - M^{3} a^{10}\right)} r^{3} - 8 \, {\left(M^{2} a^{14} - M^{2} a^{12}\right)} r^{2}\right)} \cos\left({\theta}\right)^{8} + 4 \, {\left(20 \, M a^{6} r^{9} - 5 \, a^{6} r^{10} + 2 \, {\left({\left(6 \, M^{2} - 5\right)} a^{8} - 16 \, M^{2} a^{6}\right)} r^{8} + 4 \, {\left(6 \, M^{3} a^{6} - {\left(6 \, M^{3} - 5 \, M\right)} a^{8}\right)} r^{7} - {\left({\left(8 \, M^{4} - 4 \, M^{2} + 5\right)} a^{10} + 8 \, M^{4} a^{6} - 4 \, {\left(4 \, M^{4} - M^{2}\right)} a^{8}\right)} r^{6} + 16 \, {\left(M^{3} a^{10} - M^{3} a^{8}\right)} r^{5} + 8 \, {\left({\left(M^{4} - M^{2}\right)} a^{12} + M^{4} a^{8} - {\left(2 \, M^{4} - M^{2}\right)} a^{10}\right)} r^{4}\right)} \cos\left({\theta}\right)^{6} + {\left(60 \, M a^{4} r^{11} - 15 \, a^{4} r^{12} + 2 \, {\left({\left(16 \, M^{2} - 15\right)} a^{6} - 46 \, M^{2} a^{4}\right)} r^{10} + 4 \, {\left(16 \, M^{3} a^{4} - {\left(16 \, M^{3} - 15 \, M\right)} a^{6}\right)} r^{9} - {\left({\left(16 \, M^{4} + 16 \, M^{2} + 15\right)} a^{8} + 16 \, M^{4} a^{4} - 16 \, {\left(2 \, M^{4} + M^{2}\right)} a^{6}\right)} r^{8} + 96 \, {\left(M^{3} a^{8} - M^{3} a^{6}\right)} r^{7} + 16 \, {\left({\left(4 \, M^{4} - 3 \, M^{2}\right)} a^{10} + 4 \, M^{4} a^{6} - {\left(8 \, M^{4} - 3 \, M^{2}\right)} a^{8}\right)} r^{6} - 16 \, {\left(M^{4} a^{12} - 2 \, M^{4} a^{10} + M^{4} a^{8}\right)} r^{4}\right)} \cos\left({\theta}\right)^{4} + 2 \, {\left(12 \, M a^{2} r^{13} - 3 \, a^{2} r^{14} + 2 \, {\left({\left(2 \, M^{2} - 3\right)} a^{4} - 8 \, M^{2} a^{2}\right)} r^{12} + 4 \, {\left(2 \, M^{3} a^{2} - {\left(2 \, M^{3} - 3 \, M\right)} a^{4}\right)} r^{11} - 3 \, {\left({\left(4 \, M^{2} + 1\right)} a^{6} - 4 \, M^{2} a^{4}\right)} r^{10} + 32 \, {\left(M^{3} a^{6} - M^{3} a^{4}\right)} r^{9} + 16 \, {\left({\left(M^{4} - M^{2}\right)} a^{8} + M^{4} a^{4} - {\left(2 \, M^{4} - M^{2}\right)} a^{6}\right)} r^{8} - 16 \, {\left(M^{4} a^{10} - 2 \, M^{4} a^{8} + M^{4} a^{6}\right)} r^{6}\right)} \cos\left({\theta}\right)^{2}} & -\frac{23 \, {\left(M^{2} a^{4} - M^{2} a^{2}\right)} r^{12} - 80 \, {\left(M^{3} a^{4} - M^{3} a^{2}\right)} r^{11} + 2 \, {\left(17 \, M^{2} a^{6} - 36 \, M^{4} a^{2} + {\left(36 \, M^{4} - 17 \, M^{2}\right)} a^{4}\right)} r^{10} - {\left(3 \, M^{2} a^{16} - 3 \, M^{2} a^{14} + 3 \, {\left(M^{2} a^{12} - M^{2} a^{10}\right)} r^{4} - 8 \, {\left(M^{3} a^{12} - M^{3} a^{10}\right)} r^{3} + 2 \, {\left(M^{2} a^{14} - 4 \, M^{4} a^{10} + {\left(4 \, M^{4} - M^{2}\right)} a^{12}\right)} r^{2} - 8 \, {\left(M^{3} a^{14} - M^{3} a^{12}\right)} r\right)} \cos\left({\theta}\right)^{10} - 64 \, {\left(M^{3} a^{6} - M^{3} a^{4}\right)} r^{9} + {\left({\left(28 \, M^{4} + 15 \, M^{2}\right)} a^{8} + 28 \, M^{4} a^{4} - {\left(56 \, M^{4} + 15 \, M^{2}\right)} a^{6}\right)} r^{8} + {\left(3 \, M^{2} a^{16} - 3 \, M^{2} a^{14} + 28 \, {\left(M^{2} a^{10} - M^{2} a^{8}\right)} r^{6} - 120 \, {\left(M^{3} a^{10} - M^{3} a^{8}\right)} r^{5} + {\left({\left(12 \, M^{4} + 83 \, M^{2}\right)} a^{12} - 100 \, M^{4} a^{8} + {\left(88 \, M^{4} - 83 \, M^{2}\right)} a^{10}\right)} r^{4} - 8 \, {\left(M^{5} a^{8} + {\left(M^{5} + 18 \, M^{3}\right)} a^{12} - 2 \, {\left(M^{5} + 9 \, M^{3}\right)} a^{10}\right)} r^{3} - 2 \, {\left({\left(2 \, M^{4} - 19 \, M^{2}\right)} a^{14} + 6 \, M^{4} a^{10} - {\left(8 \, M^{4} - 19 \, M^{2}\right)} a^{12}\right)} r^{2} - 8 \, {\left(M^{3} a^{14} - M^{3} a^{12}\right)} r\right)} \cos\left({\theta}\right)^{8} - 72 \, {\left(M^{5} a^{8} - 2 \, M^{5} a^{6} + M^{5} a^{4}\right)} r^{7} + 44 \, {\left(M^{4} a^{10} - 2 \, M^{4} a^{8} + M^{4} a^{6}\right)} r^{6} + 2 \, {\left(23 \, {\left(M^{2} a^{8} - M^{2} a^{6}\right)} r^{8} - 96 \, {\left(M^{3} a^{8} - M^{3} a^{6}\right)} r^{7} - 2 \, {\left({\left(34 \, M^{4} - 29 \, M^{2}\right)} a^{10} + 78 \, M^{4} a^{6} - {\left(112 \, M^{4} - 29 \, M^{2}\right)} a^{8}\right)} r^{6} + 8 \, {\left(13 \, M^{5} a^{6} + {\left(13 \, M^{5} - 8 \, M^{3}\right)} a^{10} - 2 \, {\left(13 \, M^{5} - 4 \, M^{3}\right)} a^{8}\right)} r^{5} - {\left({\left(80 \, M^{4} + M^{2}\right)} a^{12} + 24 \, M^{4} a^{8} - {\left(104 \, M^{4} + M^{2}\right)} a^{10}\right)} r^{4} + 8 \, {\left(M^{5} a^{8} + {\left(M^{5} + 8 \, M^{3}\right)} a^{12} - 2 \, {\left(M^{5} + 4 \, M^{3}\right)} a^{10}\right)} r^{3} + 4 \, {\left({\left(M^{4} - 4 \, M^{2}\right)} a^{14} + M^{4} a^{10} - 2 \, {\left(M^{4} - 2 \, M^{2}\right)} a^{12}\right)} r^{2}\right)} \cos\left({\theta}\right)^{6} - 2 \, {\left(2 \, {\left(M^{2} a^{6} - M^{2} a^{4}\right)} r^{10} - 4 \, {\left(M^{3} a^{6} - M^{3} a^{4}\right)} r^{9} - {\left({\left(6 \, M^{4} - M^{2}\right)} a^{8} + 14 \, M^{4} a^{4} - {\left(20 \, M^{4} - M^{2}\right)} a^{6}\right)} r^{8} + 4 \, {\left(9 \, M^{5} a^{4} + {\left(9 \, M^{5} - 16 \, M^{3}\right)} a^{8} - 2 \, {\left(9 \, M^{5} - 8 \, M^{3}\right)} a^{6}\right)} r^{7} - 2 \, {\left({\left(71 \, M^{4} - 24 \, M^{2}\right)} a^{10} + 115 \, M^{4} a^{6} - 6 \, {\left(31 \, M^{4} - 4 \, M^{2}\right)} a^{8}\right)} r^{6} + 4 \, {\left(52 \, M^{5} a^{6} + {\left(52 \, M^{5} - 27 \, M^{3}\right)} a^{10} - {\left(104 \, M^{5} - 27 \, M^{3}\right)} a^{8}\right)} r^{5} - {\left({\left(134 \, M^{4} - 29 \, M^{2}\right)} a^{12} + 134 \, M^{4} a^{8} - {\left(268 \, M^{4} - 29 \, M^{2}\right)} a^{10}\right)} r^{4} + 4 \, {\left(M^{5} a^{12} - 2 \, M^{5} a^{10} + M^{5} a^{8}\right)} r^{3} + 2 \, {\left(M^{4} a^{14} - 2 \, M^{4} a^{12} + M^{4} a^{10}\right)} r^{2}\right)} \cos\left({\theta}\right)^{4} - {\left(19 \, {\left(M^{2} a^{4} - M^{2} a^{2}\right)} r^{12} - 72 \, {\left(M^{3} a^{4} - M^{3} a^{2}\right)} r^{11} + 2 \, {\left(5 \, M^{2} a^{6} - 36 \, M^{4} a^{2} + {\left(36 \, M^{4} - 5 \, M^{2}\right)} a^{4}\right)} r^{10} - 24 \, {\left(M^{3} a^{6} - M^{3} a^{4}\right)} r^{9} + {\left({\left(40 \, M^{4} + 19 \, M^{2}\right)} a^{8} + 56 \, M^{4} a^{4} - {\left(96 \, M^{4} + 19 \, M^{2}\right)} a^{6}\right)} r^{8} - 16 \, {\left(9 \, M^{5} a^{4} + {\left(9 \, M^{5} + M^{3}\right)} a^{8} - {\left(18 \, M^{5} + M^{3}\right)} a^{6}\right)} r^{7} + 8 \, {\left({\left(24 \, M^{4} + M^{2}\right)} a^{10} + 24 \, M^{4} a^{6} - {\left(48 \, M^{4} + M^{2}\right)} a^{8}\right)} r^{6} - 208 \, {\left(M^{5} a^{10} - 2 \, M^{5} a^{8} + M^{5} a^{6}\right)} r^{5} + 120 \, {\left(M^{4} a^{12} - 2 \, M^{4} a^{10} + M^{4} a^{8}\right)} r^{4}\right)} \cos\left({\theta}\right)^{2}}{4 \, M a^{2} r^{13} + 4 \, M r^{15} - r^{16} - 2 \, {\left(2 \, M^{2} + a^{2}\right)} r^{14} - {\left({\left(8 \, M^{2} + 1\right)} a^{4} - 8 \, M^{2} a^{2}\right)} r^{12} - {\left(a^{16} - 4 \, M a^{14} r - 4 \, M a^{12} r^{3} + a^{12} r^{4} + 2 \, {\left(2 \, M^{2} a^{12} + a^{14}\right)} r^{2}\right)} \cos\left({\theta}\right)^{12} + 16 \, {\left(M^{3} a^{4} - M^{3} a^{2}\right)} r^{11} - 8 \, {\left(M^{2} a^{6} - M^{2} a^{4}\right)} r^{10} + 2 \, {\left(12 \, M a^{10} r^{5} - 3 \, a^{10} r^{6} + 2 \, {\left({\left(2 \, M^{2} - 3\right)} a^{12} - 8 \, M^{2} a^{10}\right)} r^{4} + 4 \, {\left(2 \, M^{3} a^{10} - {\left(2 \, M^{3} - 3 \, M\right)} a^{12}\right)} r^{3} + {\left({\left(4 \, M^{2} - 3\right)} a^{14} - 4 \, M^{2} a^{12}\right)} r^{2}\right)} \cos\left({\theta}\right)^{10} - 16 \, {\left(M^{4} a^{8} - 2 \, M^{4} a^{6} + M^{4} a^{4}\right)} r^{8} + {\left(60 \, M a^{8} r^{7} - 15 \, a^{8} r^{8} + 2 \, {\left({\left(16 \, M^{2} - 15\right)} a^{10} - 46 \, M^{2} a^{8}\right)} r^{6} + 4 \, {\left(16 \, M^{3} a^{8} - {\left(16 \, M^{3} - 15 \, M\right)} a^{10}\right)} r^{5} - {\left({\left(16 \, M^{4} - 24 \, M^{2} + 15\right)} a^{12} + 16 \, M^{4} a^{8} - 8 \, {\left(4 \, M^{4} - 3 \, M^{2}\right)} a^{10}\right)} r^{4} + 16 \, {\left(M^{3} a^{12} - M^{3} a^{10}\right)} r^{3} - 8 \, {\left(M^{2} a^{14} - M^{2} a^{12}\right)} r^{2}\right)} \cos\left({\theta}\right)^{8} + 4 \, {\left(20 \, M a^{6} r^{9} - 5 \, a^{6} r^{10} + 2 \, {\left({\left(6 \, M^{2} - 5\right)} a^{8} - 16 \, M^{2} a^{6}\right)} r^{8} + 4 \, {\left(6 \, M^{3} a^{6} - {\left(6 \, M^{3} - 5 \, M\right)} a^{8}\right)} r^{7} - {\left({\left(8 \, M^{4} - 4 \, M^{2} + 5\right)} a^{10} + 8 \, M^{4} a^{6} - 4 \, {\left(4 \, M^{4} - M^{2}\right)} a^{8}\right)} r^{6} + 16 \, {\left(M^{3} a^{10} - M^{3} a^{8}\right)} r^{5} + 8 \, {\left({\left(M^{4} - M^{2}\right)} a^{12} + M^{4} a^{8} - {\left(2 \, M^{4} - M^{2}\right)} a^{10}\right)} r^{4}\right)} \cos\left({\theta}\right)^{6} + {\left(60 \, M a^{4} r^{11} - 15 \, a^{4} r^{12} + 2 \, {\left({\left(16 \, M^{2} - 15\right)} a^{6} - 46 \, M^{2} a^{4}\right)} r^{10} + 4 \, {\left(16 \, M^{3} a^{4} - {\left(16 \, M^{3} - 15 \, M\right)} a^{6}\right)} r^{9} - {\left({\left(16 \, M^{4} + 16 \, M^{2} + 15\right)} a^{8} + 16 \, M^{4} a^{4} - 16 \, {\left(2 \, M^{4} + M^{2}\right)} a^{6}\right)} r^{8} + 96 \, {\left(M^{3} a^{8} - M^{3} a^{6}\right)} r^{7} + 16 \, {\left({\left(4 \, M^{4} - 3 \, M^{2}\right)} a^{10} + 4 \, M^{4} a^{6} - {\left(8 \, M^{4} - 3 \, M^{2}\right)} a^{8}\right)} r^{6} - 16 \, {\left(M^{4} a^{12} - 2 \, M^{4} a^{10} + M^{4} a^{8}\right)} r^{4}\right)} \cos\left({\theta}\right)^{4} + 2 \, {\left(12 \, M a^{2} r^{13} - 3 \, a^{2} r^{14} + 2 \, {\left({\left(2 \, M^{2} - 3\right)} a^{4} - 8 \, M^{2} a^{2}\right)} r^{12} + 4 \, {\left(2 \, M^{3} a^{2} - {\left(2 \, M^{3} - 3 \, M\right)} a^{4}\right)} r^{11} - 3 \, {\left({\left(4 \, M^{2} + 1\right)} a^{6} - 4 \, M^{2} a^{4}\right)} r^{10} + 32 \, {\left(M^{3} a^{6} - M^{3} a^{4}\right)} r^{9} + 16 \, {\left({\left(M^{4} - M^{2}\right)} a^{8} + M^{4} a^{4} - {\left(2 \, M^{4} - M^{2}\right)} a^{6}\right)} r^{8} - 16 \, {\left(M^{4} a^{10} - 2 \, M^{4} a^{8} + M^{4} a^{6}\right)} r^{6}\right)} \cos\left({\theta}\right)^{2}} & 0 \\
\frac{2 \, {\left(13 \, {\left(M^{3} a^{6} - M^{3} a^{4}\right)} r^{13} - 42 \, {\left(M^{4} a^{6} - M^{4} a^{4}\right)} r^{12} - {\left(3 \, {\left(M^{3} a^{14} - M^{3} a^{12}\right)} r^{5} - 6 \, {\left(M^{4} a^{14} - M^{4} a^{12}\right)} r^{4} + 4 \, {\left(M^{3} a^{16} + M^{5} a^{12} - {\left(M^{5} + M^{3}\right)} a^{14}\right)} r^{3} + 6 \, {\left(M^{4} a^{16} - M^{4} a^{14}\right)} r^{2} - 3 \, {\left(M^{3} a^{18} - M^{3} a^{16}\right)} r\right)} \cos\left({\theta}\right)^{12} + 12 \, {\left(M^{3} a^{8} - 3 \, M^{5} a^{4} + {\left(3 \, M^{5} - M^{3}\right)} a^{6}\right)} r^{11} - 22 \, {\left(M^{4} a^{8} - M^{4} a^{6}\right)} r^{10} + 2 \, {\left(12 \, {\left(M^{4} a^{12} - M^{4} a^{10}\right)} r^{6} + {\left(14 \, M^{5} a^{10} - {\left(2 \, M^{5} + M^{3}\right)} a^{14} - {\left(12 \, M^{5} - M^{3}\right)} a^{12}\right)} r^{5} - 2 \, {\left(M^{4} a^{14} - M^{4} a^{12}\right)} r^{4} + 2 \, {\left(3 \, M^{5} a^{12} + {\left(M^{5} + 4 \, M^{3}\right)} a^{16} - 4 \, {\left(M^{5} + M^{3}\right)} a^{14}\right)} r^{3} + 6 \, {\left(M^{4} a^{16} - M^{4} a^{14}\right)} r^{2} - 3 \, {\left(M^{3} a^{18} - M^{3} a^{16}\right)} r\right)} \cos\left({\theta}\right)^{10} - 3 \, {\left(4 \, M^{5} a^{6} + {\left(4 \, M^{5} - M^{3}\right)} a^{10} - {\left(8 \, M^{5} - M^{3}\right)} a^{8}\right)} r^{9} + {\left(18 \, {\left(M^{3} a^{10} - M^{3} a^{8}\right)} r^{9} - 4 \, {\left(M^{4} a^{10} - M^{4} a^{8}\right)} r^{8} + 4 \, {\left(12 \, M^{5} a^{8} + {\left(2 \, M^{5} - M^{3}\right)} a^{12} - {\left(14 \, M^{5} - M^{3}\right)} a^{10}\right)} r^{7} - 4 \, {\left(8 \, M^{6} a^{8} + {\left(8 \, M^{6} + 5 \, M^{4}\right)} a^{12} - {\left(16 \, M^{6} + 5 \, M^{4}\right)} a^{10}\right)} r^{6} - {\left(44 \, M^{5} a^{10} - {\left(20 \, M^{5} + 11 \, M^{3}\right)} a^{14} - {\left(24 \, M^{5} - 11 \, M^{3}\right)} a^{12}\right)} r^{5} - 2 \, {\left(M^{4} a^{14} - M^{4} a^{12}\right)} r^{4} - 4 \, {\left(4 \, M^{5} a^{12} + 3 \, {\left(M^{5} + 2 \, M^{3}\right)} a^{16} - {\left(7 \, M^{5} + 6 \, M^{3}\right)} a^{14}\right)} r^{3} - 6 \, {\left(M^{4} a^{16} - M^{4} a^{14}\right)} r^{2} + 3 \, {\left(M^{3} a^{18} - M^{3} a^{16}\right)} r\right)} \cos\left({\theta}\right)^{8} - 4 \, {\left(M^{5} a^{12} - 2 \, M^{5} a^{10} + M^{5} a^{8}\right)} r^{7} + 4 \, {\left(6 \, {\left(M^{3} a^{8} - M^{3} a^{6}\right)} r^{11} - 14 \, {\left(M^{4} a^{8} - M^{4} a^{6}\right)} r^{10} - {\left(5 \, M^{5} a^{6} - {\left(3 \, M^{5} - 10 \, M^{3}\right)} a^{10} - 2 \, {\left(M^{5} + 5 \, M^{3}\right)} a^{8}\right)} r^{9} + 6 \, {\left(M^{4} a^{10} - M^{4} a^{8}\right)} r^{8} - 5 \, {\left(5 \, M^{5} a^{8} + {\left(M^{5} + M^{3}\right)} a^{12} - {\left(6 \, M^{5} + M^{3}\right)} a^{10}\right)} r^{7} + 24 \, {\left(M^{6} a^{12} - 2 \, M^{6} a^{10} + M^{6} a^{8}\right)} r^{6} - {\left(M^{5} a^{10} + 3 \, {\left(3 \, M^{5} + 2 \, M^{3}\right)} a^{14} - 2 \, {\left(5 \, M^{5} + 3 \, M^{3}\right)} a^{12}\right)} r^{5} + 3 \, {\left(M^{5} a^{12} + {\left(M^{5} + M^{3}\right)} a^{16} - {\left(2 \, M^{5} + M^{3}\right)} a^{14}\right)} r^{3}\right)} \cos\left({\theta}\right)^{6} + {\left(9 \, {\left(M^{3} a^{6} - M^{3} a^{4}\right)} r^{13} - 34 \, {\left(M^{4} a^{6} - M^{4} a^{4}\right)} r^{12} - 4 \, {\left(14 \, M^{3} a^{8} + 9 \, M^{5} a^{4} - {\left(9 \, M^{5} + 14 \, M^{3}\right)} a^{6}\right)} r^{11} + 122 \, {\left(M^{4} a^{8} - M^{4} a^{6}\right)} r^{10} + {\left(28 \, M^{5} a^{6} - {\left(36 \, M^{5} + 11 \, M^{3}\right)} a^{10} + {\left(8 \, M^{5} + 11 \, M^{3}\right)} a^{8}\right)} r^{9} + 12 \, {\left(M^{4} a^{10} - M^{4} a^{8}\right)} r^{8} + 4 \, {\left(13 \, M^{5} a^{8} + 3 \, {\left(M^{5} + M^{3}\right)} a^{12} - {\left(16 \, M^{5} + 3 \, M^{3}\right)} a^{10}\right)} r^{7} - 4 \, {\left(24 \, M^{6} a^{8} + {\left(24 \, M^{6} + M^{4}\right)} a^{12} - {\left(48 \, M^{6} + M^{4}\right)} a^{10}\right)} r^{6} + 2 \, {\left(14 \, M^{5} a^{10} + {\left(14 \, M^{5} + 9 \, M^{3}\right)} a^{14} - {\left(28 \, M^{5} + 9 \, M^{3}\right)} a^{12}\right)} r^{5} - 4 \, {\left(M^{5} a^{16} - 2 \, M^{5} a^{14} + M^{5} a^{12}\right)} r^{3}\right)} \cos\left({\theta}\right)^{4} - 2 \, {\left(11 \, {\left(M^{3} a^{6} - M^{3} a^{4}\right)} r^{13} - 38 \, {\left(M^{4} a^{6} - M^{4} a^{4}\right)} r^{12} - 2 \, {\left(5 \, M^{3} a^{8} + 18 \, M^{5} a^{4} - {\left(18 \, M^{5} + 5 \, M^{3}\right)} a^{6}\right)} r^{11} + 22 \, {\left(M^{4} a^{8} - M^{4} a^{6}\right)} r^{10} - {\left(2 \, M^{5} a^{6} + 3 \, {\left(6 \, M^{5} + 5 \, M^{3}\right)} a^{10} - 5 \, {\left(4 \, M^{5} + 3 \, M^{3}\right)} a^{8}\right)} r^{9} + 16 \, {\left(M^{4} a^{10} - M^{4} a^{8}\right)} r^{8} - 2 \, {\left(M^{5} a^{8} + {\left(M^{5} + 3 \, M^{3}\right)} a^{12} - {\left(2 \, M^{5} + 3 \, M^{3}\right)} a^{10}\right)} r^{7} - 16 \, {\left(M^{6} a^{12} - 2 \, M^{6} a^{10} + M^{6} a^{8}\right)} r^{6} + 4 \, {\left(M^{5} a^{14} - 2 \, M^{5} a^{12} + M^{5} a^{10}\right)} r^{5}\right)} \cos\left({\theta}\right)^{2}\right)}}{4 \, M a^{2} r^{17} + 4 \, M r^{19} - r^{20} - 2 \, {\left(2 \, M^{2} + a^{2}\right)} r^{18} - {\left({\left(8 \, M^{2} + 1\right)} a^{4} - 8 \, M^{2} a^{2}\right)} r^{16} - {\left(a^{20} - 4 \, M a^{18} r - 4 \, M a^{16} r^{3} + a^{16} r^{4} + 2 \, {\left(2 \, M^{2} a^{16} + a^{18}\right)} r^{2}\right)} \cos\left({\theta}\right)^{16} + 16 \, {\left(M^{3} a^{4} - M^{3} a^{2}\right)} r^{15} - 8 \, {\left(M^{2} a^{6} - M^{2} a^{4}\right)} r^{14} + 8 \, {\left(4 \, M a^{14} r^{5} - a^{14} r^{6} + {\left({\left(M^{2} - 2\right)} a^{16} - 5 \, M^{2} a^{14}\right)} r^{4} + 2 \, {\left(M^{3} a^{14} - {\left(M^{3} - 2 \, M\right)} a^{16}\right)} r^{3} + {\left({\left(M^{2} - 1\right)} a^{18} - M^{2} a^{16}\right)} r^{2}\right)} \cos\left({\theta}\right)^{14} - 16 \, {\left(M^{4} a^{8} - 2 \, M^{4} a^{6} + M^{4} a^{4}\right)} r^{12} + 4 \, {\left(28 \, M a^{12} r^{7} - 7 \, a^{12} r^{8} + 2 \, {\left({\left(6 \, M^{2} - 7\right)} a^{14} - 20 \, M^{2} a^{12}\right)} r^{6} + 4 \, {\left(6 \, M^{3} a^{12} - {\left(6 \, M^{3} - 7 \, M\right)} a^{14}\right)} r^{5} - {\left({\left(4 \, M^{4} - 10 \, M^{2} + 7\right)} a^{16} + 4 \, M^{4} a^{12} - 2 \, {\left(4 \, M^{4} - 5 \, M^{2}\right)} a^{14}\right)} r^{4} + 4 \, {\left(M^{3} a^{16} - M^{3} a^{14}\right)} r^{3} - 2 \, {\left(M^{2} a^{18} - M^{2} a^{16}\right)} r^{2}\right)} \cos\left({\theta}\right)^{12} + 8 \, {\left(28 \, M a^{10} r^{9} - 7 \, a^{10} r^{10} + {\left({\left(15 \, M^{2} - 14\right)} a^{12} - 43 \, M^{2} a^{10}\right)} r^{8} + 2 \, {\left(15 \, M^{3} a^{10} - {\left(15 \, M^{3} - 14 \, M\right)} a^{12}\right)} r^{7} - {\left({\left(8 \, M^{4} - 9 \, M^{2} + 7\right)} a^{14} + 8 \, M^{4} a^{10} - {\left(16 \, M^{4} - 9 \, M^{2}\right)} a^{12}\right)} r^{6} + 12 \, {\left(M^{3} a^{14} - M^{3} a^{12}\right)} r^{5} + 2 \, {\left({\left(2 \, M^{4} - 3 \, M^{2}\right)} a^{16} + 2 \, M^{4} a^{12} - {\left(4 \, M^{4} - 3 \, M^{2}\right)} a^{14}\right)} r^{4}\right)} \cos\left({\theta}\right)^{10} + 2 \, {\left(140 \, M a^{8} r^{11} - 35 \, a^{8} r^{12} + 10 \, {\left({\left(8 \, M^{2} - 7\right)} a^{10} - 22 \, M^{2} a^{8}\right)} r^{10} + 20 \, {\left(8 \, M^{3} a^{8} - {\left(8 \, M^{3} - 7 \, M\right)} a^{10}\right)} r^{9} - {\left({\left(48 \, M^{4} - 20 \, M^{2} + 35\right)} a^{12} + 48 \, M^{4} a^{8} - 4 \, {\left(24 \, M^{4} - 5 \, M^{2}\right)} a^{10}\right)} r^{8} + 120 \, {\left(M^{3} a^{12} - M^{3} a^{10}\right)} r^{7} + 4 \, {\left({\left(16 \, M^{4} - 15 \, M^{2}\right)} a^{14} + 16 \, M^{4} a^{10} - {\left(32 \, M^{4} - 15 \, M^{2}\right)} a^{12}\right)} r^{6} - 8 \, {\left(M^{4} a^{16} - 2 \, M^{4} a^{14} + M^{4} a^{12}\right)} r^{4}\right)} \cos\left({\theta}\right)^{8} + 8 \, {\left(28 \, M a^{6} r^{13} - 7 \, a^{6} r^{14} + {\left({\left(15 \, M^{2} - 14\right)} a^{8} - 43 \, M^{2} a^{6}\right)} r^{12} + 2 \, {\left(15 \, M^{3} a^{6} - {\left(15 \, M^{3} - 14 \, M\right)} a^{8}\right)} r^{11} - {\left({\left(8 \, M^{4} + 5 \, M^{2} + 7\right)} a^{10} + 8 \, M^{4} a^{6} - {\left(16 \, M^{4} + 5 \, M^{2}\right)} a^{8}\right)} r^{10} + 40 \, {\left(M^{3} a^{10} - M^{3} a^{8}\right)} r^{9} + 4 \, {\left({\left(6 \, M^{4} - 5 \, M^{2}\right)} a^{12} + 6 \, M^{4} a^{8} - {\left(12 \, M^{4} - 5 \, M^{2}\right)} a^{10}\right)} r^{8} - 8 \, {\left(M^{4} a^{14} - 2 \, M^{4} a^{12} + M^{4} a^{10}\right)} r^{6}\right)} \cos\left({\theta}\right)^{6} + 4 \, {\left(28 \, M a^{4} r^{15} - 7 \, a^{4} r^{16} + 2 \, {\left({\left(6 \, M^{2} - 7\right)} a^{6} - 20 \, M^{2} a^{4}\right)} r^{14} + 4 \, {\left(6 \, M^{3} a^{4} - {\left(6 \, M^{3} - 7 \, M\right)} a^{6}\right)} r^{13} - {\left({\left(4 \, M^{4} + 18 \, M^{2} + 7\right)} a^{8} + 4 \, M^{4} a^{4} - 2 \, {\left(4 \, M^{4} + 9 \, M^{2}\right)} a^{6}\right)} r^{12} + 60 \, {\left(M^{3} a^{8} - M^{3} a^{6}\right)} r^{11} + 2 \, {\left({\left(16 \, M^{4} - 15 \, M^{2}\right)} a^{10} + 16 \, M^{4} a^{6} - {\left(32 \, M^{4} - 15 \, M^{2}\right)} a^{8}\right)} r^{10} - 24 \, {\left(M^{4} a^{12} - 2 \, M^{4} a^{10} + M^{4} a^{8}\right)} r^{8}\right)} \cos\left({\theta}\right)^{4} + 8 \, {\left(4 \, M a^{2} r^{17} - a^{2} r^{18} + {\left({\left(M^{2} - 2\right)} a^{4} - 5 \, M^{2} a^{2}\right)} r^{16} + 2 \, {\left(M^{3} a^{2} - {\left(M^{3} - 2 \, M\right)} a^{4}\right)} r^{15} - {\left({\left(5 \, M^{2} + 1\right)} a^{6} - 5 \, M^{2} a^{4}\right)} r^{14} + 12 \, {\left(M^{3} a^{6} - M^{3} a^{4}\right)} r^{13} + 2 \, {\left({\left(2 \, M^{4} - 3 \, M^{2}\right)} a^{8} + 2 \, M^{4} a^{4} - {\left(4 \, M^{4} - 3 \, M^{2}\right)} a^{6}\right)} r^{12} - 8 \, {\left(M^{4} a^{10} - 2 \, M^{4} a^{8} + M^{4} a^{6}\right)} r^{10}\right)} \cos\left({\theta}\right)^{2}} & 0 & 0 & -\frac{{\left(M^{2} a^{4} - M^{2} a^{2}\right)} r^{16} + 3 \, {\left(M^{2} a^{6} - M^{2} a^{4}\right)} r^{14} + {\left(M^{2} a^{20} - M^{2} a^{18} + {\left(M^{2} a^{14} - M^{2} a^{12}\right)} r^{6} - 6 \, {\left(M^{3} a^{14} - M^{3} a^{12}\right)} r^{5} + 3 \, {\left(M^{2} a^{16} - 4 \, M^{4} a^{12} + {\left(4 \, M^{4} - M^{2}\right)} a^{14}\right)} r^{4} - 4 \, {\left(3 \, M^{3} a^{16} - 2 \, M^{5} a^{12} + {\left(2 \, M^{5} - 3 \, M^{3}\right)} a^{14}\right)} r^{3} + 3 \, {\left(M^{2} a^{18} - 4 \, M^{4} a^{14} + {\left(4 \, M^{4} - M^{2}\right)} a^{16}\right)} r^{2} - 6 \, {\left(M^{3} a^{18} - M^{3} a^{16}\right)} r\right)} \cos\left({\theta}\right)^{14} + 6 \, {\left(M^{3} a^{6} - M^{3} a^{4}\right)} r^{13} - 3 \, {\left({\left(20 \, M^{4} - M^{2}\right)} a^{8} - {\left(20 \, M^{4} - M^{2}\right)} a^{6}\right)} r^{12} - {\left(2 \, M^{2} a^{20} - 2 \, M^{2} a^{18} - {\left(M^{2} a^{12} - M^{2} a^{10}\right)} r^{8} + {\left({\left(20 \, M^{4} - M^{2}\right)} a^{14} - 12 \, M^{4} a^{10} - {\left(8 \, M^{4} - M^{2}\right)} a^{12}\right)} r^{6} + 2 \, {\left(8 \, M^{5} a^{10} - 3 \, {\left(8 \, M^{5} + 3 \, M^{3}\right)} a^{14} + {\left(16 \, M^{5} + 9 \, M^{3}\right)} a^{12}\right)} r^{5} + 3 \, {\left({\left(8 \, M^{4} + M^{2}\right)} a^{16} - 20 \, M^{4} a^{12} + {\left(12 \, M^{4} - M^{2}\right)} a^{14}\right)} r^{4} + 4 \, {\left(10 \, M^{5} a^{12} + {\left(4 \, M^{5} - 9 \, M^{3}\right)} a^{16} - {\left(14 \, M^{5} - 9 \, M^{3}\right)} a^{14}\right)} r^{3} - {\left({\left(12 \, M^{4} - 5 \, M^{2}\right)} a^{18} + 48 \, M^{4} a^{14} - 5 \, {\left(12 \, M^{4} - M^{2}\right)} a^{16}\right)} r^{2} - 18 \, {\left(M^{3} a^{18} - M^{3} a^{16}\right)} r\right)} \cos\left({\theta}\right)^{12} + 12 \, {\left({\left(6 \, M^{5} + M^{3}\right)} a^{8} - {\left(6 \, M^{5} + M^{3}\right)} a^{6}\right)} r^{11} - {\left({\left(56 \, M^{4} - M^{2}\right)} a^{10} + 12 \, M^{4} a^{6} - {\left(68 \, M^{4} - M^{2}\right)} a^{8}\right)} r^{10} + {\left(M^{2} a^{20} - M^{2} a^{18} - 6 \, {\left(M^{2} a^{10} - M^{2} a^{8}\right)} r^{10} + 44 \, {\left(M^{3} a^{10} - M^{3} a^{8}\right)} r^{9} + 4 \, {\left({\left(5 \, M^{4} - 6 \, M^{2}\right)} a^{12} + 27 \, M^{4} a^{8} - 2 \, {\left(16 \, M^{4} - 3 \, M^{2}\right)} a^{10}\right)} r^{8} - 40 \, {\left(3 \, M^{5} a^{8} + {\left(M^{5} - 3 \, M^{3}\right)} a^{12} - {\left(4 \, M^{5} - 3 \, M^{3}\right)} a^{10}\right)} r^{7} + {\left({\left(68 \, M^{4} - 35 \, M^{2}\right)} a^{14} + 64 \, M^{6} a^{8} + {\left(64 \, M^{6} - 224 \, M^{4} + 35 \, M^{2}\right)} a^{12} - 4 \, {\left(32 \, M^{6} - 39 \, M^{4}\right)} a^{10}\right)} r^{6} - 6 \, {\left(8 \, M^{5} a^{10} + 5 \, {\left(8 \, M^{5} - 3 \, M^{3}\right)} a^{14} - 3 \, {\left(16 \, M^{5} - 5 \, M^{3}\right)} a^{12}\right)} r^{5} + 3 \, {\left({\left(36 \, M^{4} - 7 \, M^{2}\right)} a^{16} - 8 \, M^{4} a^{12} - 7 \, {\left(4 \, M^{4} - M^{2}\right)} a^{14}\right)} r^{4} + 4 \, {\left(20 \, M^{5} a^{12} + {\left(14 \, M^{5} - M^{3}\right)} a^{16} - {\left(34 \, M^{5} - M^{3}\right)} a^{14}\right)} r^{3} - 3 \, {\left({\left(12 \, M^{4} + M^{2}\right)} a^{18} + 24 \, M^{4} a^{14} - {\left(36 \, M^{4} + M^{2}\right)} a^{16}\right)} r^{2} - 18 \, {\left(M^{3} a^{18} - M^{3} a^{16}\right)} r\right)} \cos\left({\theta}\right)^{10} + 6 \, {\left(M^{3} a^{10} - M^{3} a^{8}\right)} r^{9} - 12 \, {\left(M^{4} a^{12} - 2 \, M^{4} a^{10} + M^{4} a^{8}\right)} r^{8} - {\left(14 \, {\left(M^{2} a^{8} - M^{2} a^{6}\right)} r^{12} - 72 \, {\left(M^{3} a^{8} - M^{3} a^{6}\right)} r^{11} - 2 \, {\left({\left(50 \, M^{4} - 23 \, M^{2}\right)} a^{10} + 66 \, M^{4} a^{6} - {\left(116 \, M^{4} - 23 \, M^{2}\right)} a^{8}\right)} r^{10} + 4 \, {\left(24 \, M^{5} a^{6} + {\left(40 \, M^{5} - 31 \, M^{3}\right)} a^{10} - {\left(64 \, M^{5} - 31 \, M^{3}\right)} a^{8}\right)} r^{9} - {\left(7 \, {\left(4 \, M^{4} - 7 \, M^{2}\right)} a^{12} - 12 \, M^{4} a^{8} - {\left(16 \, M^{4} - 49 \, M^{2}\right)} a^{10}\right)} r^{8} - 8 \, {\left(35 \, M^{5} a^{8} + {\left(5 \, M^{5} - 3 \, M^{3}\right)} a^{12} - {\left(40 \, M^{5} - 3 \, M^{3}\right)} a^{10}\right)} r^{7} + {\left({\left(48 \, M^{4} + 11 \, M^{2}\right)} a^{14} + 256 \, M^{6} a^{8} + {\left(256 \, M^{6} - 360 \, M^{4} - 11 \, M^{2}\right)} a^{12} - 8 \, {\left(64 \, M^{6} - 39 \, M^{4}\right)} a^{10}\right)} r^{6} - 2 \, {\left(128 \, M^{5} a^{10} + 7 \, {\left(32 \, M^{5} - 9 \, M^{3}\right)} a^{14} - {\left(352 \, M^{5} - 63 \, M^{3}\right)} a^{12}\right)} r^{5} + {\left({\left(180 \, M^{4} - 11 \, M^{2}\right)} a^{16} + 120 \, M^{4} a^{12} - {\left(300 \, M^{4} - 11 \, M^{2}\right)} a^{14}\right)} r^{4} + 4 \, {\left(20 \, M^{5} a^{12} + {\left(18 \, M^{5} + 11 \, M^{3}\right)} a^{16} - {\left(38 \, M^{5} + 11 \, M^{3}\right)} a^{14}\right)} r^{3} - {\left({\left(36 \, M^{4} + 5 \, M^{2}\right)} a^{18} + 48 \, M^{4} a^{14} - {\left(84 \, M^{4} + 5 \, M^{2}\right)} a^{16}\right)} r^{2} - 6 \, {\left(M^{3} a^{18} - M^{3} a^{16}\right)} r\right)} \cos\left({\theta}\right)^{8} - 8 \, {\left(M^{5} a^{12} - 2 \, M^{5} a^{10} + M^{5} a^{8}\right)} r^{7} - {\left(11 \, {\left(M^{2} a^{6} - M^{2} a^{4}\right)} r^{14} - 42 \, {\left(M^{3} a^{6} - M^{3} a^{4}\right)} r^{13} - {\left({\left(60 \, M^{4} - 29 \, M^{2}\right)} a^{8} + 48 \, M^{4} a^{4} - {\left(108 \, M^{4} - 29 \, M^{2}\right)} a^{6}\right)} r^{12} + 12 \, {\left({\left(6 \, M^{5} - M^{3}\right)} a^{8} - {\left(6 \, M^{5} - M^{3}\right)} a^{6}\right)} r^{11} + {\left({\left(244 \, M^{4} + 11 \, M^{2}\right)} a^{10} + 192 \, M^{4} a^{6} - {\left(436 \, M^{4} + 11 \, M^{2}\right)} a^{8}\right)} r^{10} - 6 \, {\left(48 \, M^{5} a^{6} + {\left(80 \, M^{5} - 27 \, M^{3}\right)} a^{10} - {\left(128 \, M^{5} - 27 \, M^{3}\right)} a^{8}\right)} r^{9} + {\left({\left(192 \, M^{4} - 31 \, M^{2}\right)} a^{12} + 312 \, M^{4} a^{8} - {\left(504 \, M^{4} - 31 \, M^{2}\right)} a^{10}\right)} r^{8} + 8 \, {\left(14 \, M^{5} a^{8} - {\left(16 \, M^{5} - 21 \, M^{3}\right)} a^{12} + {\left(2 \, M^{5} - 21 \, M^{3}\right)} a^{10}\right)} r^{7} + 2 \, {\left({\left(32 \, M^{4} - 17 \, M^{2}\right)} a^{14} - 192 \, M^{6} a^{8} - {\left(192 \, M^{6} - 4 \, M^{4} - 17 \, M^{2}\right)} a^{12} + 12 \, {\left(32 \, M^{6} - 3 \, M^{4}\right)} a^{10}\right)} r^{6} + 4 \, {\left(80 \, M^{5} a^{10} + 3 \, {\left(32 \, M^{5} + M^{3}\right)} a^{14} - {\left(176 \, M^{5} + 3 \, M^{3}\right)} a^{12}\right)} r^{5} - 2 \, {\left({\left(66 \, M^{4} + 5 \, M^{2}\right)} a^{16} + 66 \, M^{4} a^{12} - {\left(132 \, M^{4} + 5 \, M^{2}\right)} a^{14}\right)} r^{4} - 8 \, {\left(5 \, M^{5} a^{12} + {\left(5 \, M^{5} + 3 \, M^{3}\right)} a^{16} - {\left(10 \, M^{5} + 3 \, M^{3}\right)} a^{14}\right)} r^{3} + 12 \, {\left(M^{4} a^{18} - 2 \, M^{4} a^{16} + M^{4} a^{14}\right)} r^{2}\right)} \cos\left({\theta}\right)^{6} - {\left(3 \, {\left(M^{2} a^{4} - M^{2} a^{2}\right)} r^{16} - 8 \, {\left(M^{3} a^{4} - M^{3} a^{2}\right)} r^{15} + 3 \, {\left(M^{2} a^{6} - M^{2} a^{4}\right)} r^{14} + 30 \, {\left(M^{3} a^{6} - M^{3} a^{4}\right)} r^{13} + {\left({\left(180 \, M^{4} - 19 \, M^{2}\right)} a^{8} + 96 \, M^{4} a^{4} - {\left(276 \, M^{4} - 19 \, M^{2}\right)} a^{6}\right)} r^{12} - 108 \, {\left({\left(2 \, M^{5} - M^{3}\right)} a^{8} - {\left(2 \, M^{5} - M^{3}\right)} a^{6}\right)} r^{11} - 3 \, {\left({\left(44 \, M^{4} + 15 \, M^{2}\right)} a^{10} - 8 \, M^{4} a^{6} - 3 \, {\left(12 \, M^{4} + 5 \, M^{2}\right)} a^{8}\right)} r^{10} + 2 \, {\left(144 \, M^{5} a^{6} + {\left(240 \, M^{5} + 29 \, M^{3}\right)} a^{10} - {\left(384 \, M^{5} + 29 \, M^{3}\right)} a^{8}\right)} r^{9} - 4 \, {\left({\left(52 \, M^{4} + 9 \, M^{2}\right)} a^{12} + 54 \, M^{4} a^{8} - {\left(106 \, M^{4} + 9 \, M^{2}\right)} a^{10}\right)} r^{8} + 16 \, {\left(9 \, M^{5} a^{8} + {\left(14 \, M^{5} - 3 \, M^{3}\right)} a^{12} - {\left(23 \, M^{5} - 3 \, M^{3}\right)} a^{10}\right)} r^{7} - 2 \, {\left(5 \, {\left(10 \, M^{4} + M^{2}\right)} a^{14} - 128 \, M^{6} a^{8} - {\left(128 \, M^{6} + 104 \, M^{4} + 5 \, M^{2}\right)} a^{12} + 2 \, {\left(128 \, M^{6} + 27 \, M^{4}\right)} a^{10}\right)} r^{6} - 36 \, {\left(4 \, M^{5} a^{10} + {\left(4 \, M^{5} + M^{3}\right)} a^{14} - {\left(8 \, M^{5} + M^{3}\right)} a^{12}\right)} r^{5} + 36 \, {\left(M^{4} a^{16} - 2 \, M^{4} a^{14} + M^{4} a^{12}\right)} r^{4} + 8 \, {\left(M^{5} a^{16} - 2 \, M^{5} a^{14} + M^{5} a^{12}\right)} r^{3}\right)} \cos\left({\theta}\right)^{4} + {\left(2 \, {\left(M^{2} a^{4} - M^{2} a^{2}\right)} r^{16} - 8 \, {\left(M^{3} a^{4} - M^{3} a^{2}\right)} r^{15} + 11 \, {\left(M^{2} a^{6} - M^{2} a^{4}\right)} r^{14} - 18 \, {\left(M^{3} a^{6} - M^{3} a^{4}\right)} r^{13} + 3 \, {\left({\left(60 \, M^{4} + 7 \, M^{2}\right)} a^{8} + 16 \, M^{4} a^{4} - {\left(76 \, M^{4} + 7 \, M^{2}\right)} a^{6}\right)} r^{12} - 12 \, {\left({\left(18 \, M^{5} - M^{3}\right)} a^{8} - {\left(18 \, M^{5} - M^{3}\right)} a^{6}\right)} r^{11} + {\left(17 \, {\left(4 \, M^{4} + M^{2}\right)} a^{10} + 96 \, M^{4} a^{6} - {\left(164 \, M^{4} + 17 \, M^{2}\right)} a^{8}\right)} r^{10} + 2 \, {\left(48 \, M^{5} a^{6} + {\left(80 \, M^{5} + 23 \, M^{3}\right)} a^{10} - {\left(128 \, M^{5} + 23 \, M^{3}\right)} a^{8}\right)} r^{9} - {\left({\left(52 \, M^{4} - 5 \, M^{2}\right)} a^{12} - 12 \, M^{4} a^{8} - 5 \, {\left(8 \, M^{4} - M^{2}\right)} a^{10}\right)} r^{8} + 8 \, {\left(13 \, M^{5} a^{8} + {\left(13 \, M^{5} + 3 \, M^{3}\right)} a^{12} - {\left(26 \, M^{5} + 3 \, M^{3}\right)} a^{10}\right)} r^{7} - 4 \, {\left(9 \, M^{4} a^{14} - 16 \, M^{6} a^{8} - 2 \, {\left(8 \, M^{6} + 9 \, M^{4}\right)} a^{12} + {\left(32 \, M^{6} + 9 \, M^{4}\right)} a^{10}\right)} r^{6} - 16 \, {\left(M^{5} a^{14} - 2 \, M^{5} a^{12} + M^{5} a^{10}\right)} r^{5}\right)} \cos\left({\theta}\right)^{2}}{4 \, M a^{2} r^{17} + 4 \, M r^{19} - r^{20} - 2 \, {\left(2 \, M^{2} + a^{2}\right)} r^{18} - {\left({\left(8 \, M^{2} + 1\right)} a^{4} - 8 \, M^{2} a^{2}\right)} r^{16} - {\left(a^{20} - 4 \, M a^{18} r - 4 \, M a^{16} r^{3} + a^{16} r^{4} + 2 \, {\left(2 \, M^{2} a^{16} + a^{18}\right)} r^{2}\right)} \cos\left({\theta}\right)^{16} + 16 \, {\left(M^{3} a^{4} - M^{3} a^{2}\right)} r^{15} - 8 \, {\left(M^{2} a^{6} - M^{2} a^{4}\right)} r^{14} + 8 \, {\left(4 \, M a^{14} r^{5} - a^{14} r^{6} + {\left({\left(M^{2} - 2\right)} a^{16} - 5 \, M^{2} a^{14}\right)} r^{4} + 2 \, {\left(M^{3} a^{14} - {\left(M^{3} - 2 \, M\right)} a^{16}\right)} r^{3} + {\left({\left(M^{2} - 1\right)} a^{18} - M^{2} a^{16}\right)} r^{2}\right)} \cos\left({\theta}\right)^{14} - 16 \, {\left(M^{4} a^{8} - 2 \, M^{4} a^{6} + M^{4} a^{4}\right)} r^{12} + 4 \, {\left(28 \, M a^{12} r^{7} - 7 \, a^{12} r^{8} + 2 \, {\left({\left(6 \, M^{2} - 7\right)} a^{14} - 20 \, M^{2} a^{12}\right)} r^{6} + 4 \, {\left(6 \, M^{3} a^{12} - {\left(6 \, M^{3} - 7 \, M\right)} a^{14}\right)} r^{5} - {\left({\left(4 \, M^{4} - 10 \, M^{2} + 7\right)} a^{16} + 4 \, M^{4} a^{12} - 2 \, {\left(4 \, M^{4} - 5 \, M^{2}\right)} a^{14}\right)} r^{4} + 4 \, {\left(M^{3} a^{16} - M^{3} a^{14}\right)} r^{3} - 2 \, {\left(M^{2} a^{18} - M^{2} a^{16}\right)} r^{2}\right)} \cos\left({\theta}\right)^{12} + 8 \, {\left(28 \, M a^{10} r^{9} - 7 \, a^{10} r^{10} + {\left({\left(15 \, M^{2} - 14\right)} a^{12} - 43 \, M^{2} a^{10}\right)} r^{8} + 2 \, {\left(15 \, M^{3} a^{10} - {\left(15 \, M^{3} - 14 \, M\right)} a^{12}\right)} r^{7} - {\left({\left(8 \, M^{4} - 9 \, M^{2} + 7\right)} a^{14} + 8 \, M^{4} a^{10} - {\left(16 \, M^{4} - 9 \, M^{2}\right)} a^{12}\right)} r^{6} + 12 \, {\left(M^{3} a^{14} - M^{3} a^{12}\right)} r^{5} + 2 \, {\left({\left(2 \, M^{4} - 3 \, M^{2}\right)} a^{16} + 2 \, M^{4} a^{12} - {\left(4 \, M^{4} - 3 \, M^{2}\right)} a^{14}\right)} r^{4}\right)} \cos\left({\theta}\right)^{10} + 2 \, {\left(140 \, M a^{8} r^{11} - 35 \, a^{8} r^{12} + 10 \, {\left({\left(8 \, M^{2} - 7\right)} a^{10} - 22 \, M^{2} a^{8}\right)} r^{10} + 20 \, {\left(8 \, M^{3} a^{8} - {\left(8 \, M^{3} - 7 \, M\right)} a^{10}\right)} r^{9} - {\left({\left(48 \, M^{4} - 20 \, M^{2} + 35\right)} a^{12} + 48 \, M^{4} a^{8} - 4 \, {\left(24 \, M^{4} - 5 \, M^{2}\right)} a^{10}\right)} r^{8} + 120 \, {\left(M^{3} a^{12} - M^{3} a^{10}\right)} r^{7} + 4 \, {\left({\left(16 \, M^{4} - 15 \, M^{2}\right)} a^{14} + 16 \, M^{4} a^{10} - {\left(32 \, M^{4} - 15 \, M^{2}\right)} a^{12}\right)} r^{6} - 8 \, {\left(M^{4} a^{16} - 2 \, M^{4} a^{14} + M^{4} a^{12}\right)} r^{4}\right)} \cos\left({\theta}\right)^{8} + 8 \, {\left(28 \, M a^{6} r^{13} - 7 \, a^{6} r^{14} + {\left({\left(15 \, M^{2} - 14\right)} a^{8} - 43 \, M^{2} a^{6}\right)} r^{12} + 2 \, {\left(15 \, M^{3} a^{6} - {\left(15 \, M^{3} - 14 \, M\right)} a^{8}\right)} r^{11} - {\left({\left(8 \, M^{4} + 5 \, M^{2} + 7\right)} a^{10} + 8 \, M^{4} a^{6} - {\left(16 \, M^{4} + 5 \, M^{2}\right)} a^{8}\right)} r^{10} + 40 \, {\left(M^{3} a^{10} - M^{3} a^{8}\right)} r^{9} + 4 \, {\left({\left(6 \, M^{4} - 5 \, M^{2}\right)} a^{12} + 6 \, M^{4} a^{8} - {\left(12 \, M^{4} - 5 \, M^{2}\right)} a^{10}\right)} r^{8} - 8 \, {\left(M^{4} a^{14} - 2 \, M^{4} a^{12} + M^{4} a^{10}\right)} r^{6}\right)} \cos\left({\theta}\right)^{6} + 4 \, {\left(28 \, M a^{4} r^{15} - 7 \, a^{4} r^{16} + 2 \, {\left({\left(6 \, M^{2} - 7\right)} a^{6} - 20 \, M^{2} a^{4}\right)} r^{14} + 4 \, {\left(6 \, M^{3} a^{4} - {\left(6 \, M^{3} - 7 \, M\right)} a^{6}\right)} r^{13} - {\left({\left(4 \, M^{4} + 18 \, M^{2} + 7\right)} a^{8} + 4 \, M^{4} a^{4} - 2 \, {\left(4 \, M^{4} + 9 \, M^{2}\right)} a^{6}\right)} r^{12} + 60 \, {\left(M^{3} a^{8} - M^{3} a^{6}\right)} r^{11} + 2 \, {\left({\left(16 \, M^{4} - 15 \, M^{2}\right)} a^{10} + 16 \, M^{4} a^{6} - {\left(32 \, M^{4} - 15 \, M^{2}\right)} a^{8}\right)} r^{10} - 24 \, {\left(M^{4} a^{12} - 2 \, M^{4} a^{10} + M^{4} a^{8}\right)} r^{8}\right)} \cos\left({\theta}\right)^{4} + 8 \, {\left(4 \, M a^{2} r^{17} - a^{2} r^{18} + {\left({\left(M^{2} - 2\right)} a^{4} - 5 \, M^{2} a^{2}\right)} r^{16} + 2 \, {\left(M^{3} a^{2} - {\left(M^{3} - 2 \, M\right)} a^{4}\right)} r^{15} - {\left({\left(5 \, M^{2} + 1\right)} a^{6} - 5 \, M^{2} a^{4}\right)} r^{14} + 12 \, {\left(M^{3} a^{6} - M^{3} a^{4}\right)} r^{13} + 2 \, {\left({\left(2 \, M^{4} - 3 \, M^{2}\right)} a^{8} + 2 \, M^{4} a^{4} - {\left(4 \, M^{4} - 3 \, M^{2}\right)} a^{6}\right)} r^{12} - 8 \, {\left(M^{4} a^{10} - 2 \, M^{4} a^{8} + M^{4} a^{6}\right)} r^{10}\right)} \cos\left({\theta}\right)^{2}}
\end{array}\right)\]

```python
Einstein_ud = Einstein.up(g,1)
Einstein_ud[:]
```

\[\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrr}
-\frac{4 \, M^{2} {\Sigma}^{2} r^{6} + {\Sigma}^{4} a^{4} - {\Delta} {\Sigma}^{4} a^{2} - 4 \, {\left(6 \, M^{3} {\Sigma} a^{2} + M {\Sigma}^{3}\right)} r^{5} + {\left(32 \, M^{2} {\Sigma}^{2} a^{2} + {\Sigma}^{4}\right)} r^{4} + {\left({\Delta} M^{2} {\Sigma}^{2} a^{4} + 16 \, {\left(M^{4} a^{8} - 5 \, M^{4} a^{6} + 4 \, M^{4} a^{4}\right)} r^{4} + 8 \, {\left(5 \, M^{3} {\Sigma} a^{6} - 8 \, M^{3} {\Sigma} a^{4}\right)} r^{3} + 4 \, {\left({\Delta} M^{4} a^{8} - 2 \, {\Delta} M^{4} a^{6} + {\left({\Delta} M^{4} + 4 \, M^{2} {\Sigma}^{2}\right)} a^{4}\right)} r^{2} + 4 \, {\left({\Delta} M^{3} {\Sigma} a^{6} - {\Delta} M^{3} {\Sigma} a^{4}\right)} r\right)} \sin\left({\theta}\right)^{4} - 2 \, {\left(12 \, M^{3} {\Sigma} a^{4} + 7 \, M {\Sigma}^{3} a^{2}\right)} r^{3} + 2 \, {\left(14 \, M^{2} {\Sigma}^{2} a^{4} - {\left(2 \, {\Delta} M^{2} {\Sigma}^{2} - {\Sigma}^{4}\right)} a^{2}\right)} r^{2} - {\left({\Delta} M^{2} {\Sigma}^{2} a^{6} - 12 \, M {\Sigma}^{3} a^{4} r + 16 \, {\left(M^{3} {\Sigma} a^{4} - 3 \, M^{3} {\Sigma} a^{2}\right)} r^{5} - 8 \, {\left(4 \, M^{4} a^{6} - 6 \, M^{2} {\Sigma}^{2} a^{2} - {\left(4 \, M^{4} - M^{2} {\Sigma}^{2}\right)} a^{4}\right)} r^{4} + 4 \, {\left(8 \, M^{3} {\Sigma} a^{6} - 20 \, M^{3} {\Sigma} a^{4} - 3 \, M {\Sigma}^{3} a^{2}\right)} r^{3} - {\left(8 \, M^{2} {\Sigma}^{2} a^{6} + {\left({\Delta} - 56\right)} M^{2} {\Sigma}^{2} a^{4}\right)} r^{2}\right)} \sin\left({\theta}\right)^{2} - 2 \, {\left(5 \, M {\Sigma}^{3} a^{4} - 2 \, {\Delta} M {\Sigma}^{3} a^{2}\right)} r}{4 \, M^{2} {\Sigma}^{3} r^{6} - 4 \, M {\Sigma}^{4} a^{4} r - 8 \, M {\Sigma}^{4} a^{2} r^{3} - 4 \, M {\Sigma}^{4} r^{5} + {\Sigma}^{5} a^{4} + {\left(8 \, M^{2} {\Sigma}^{3} a^{2} + {\Sigma}^{5}\right)} r^{4} + 4 \, {\left(M^{2} {\Sigma}^{3} a^{4} r^{2} + 4 \, {\left(M^{4} {\Sigma} a^{8} - 2 \, M^{4} {\Sigma} a^{6} + M^{4} {\Sigma} a^{4}\right)} r^{4} + 4 \, {\left(M^{3} {\Sigma}^{2} a^{6} - M^{3} {\Sigma}^{2} a^{4}\right)} r^{3}\right)} \sin\left({\theta}\right)^{4} + 2 \, {\left(2 \, M^{2} {\Sigma}^{3} a^{4} + {\Sigma}^{5} a^{2}\right)} r^{2} + 4 \, {\left(M {\Sigma}^{4} a^{4} r - 4 \, {\left(M^{3} {\Sigma}^{2} a^{4} - M^{3} {\Sigma}^{2} a^{2}\right)} r^{5} + 2 \, {\left(M^{2} {\Sigma}^{3} a^{4} - 2 \, M^{2} {\Sigma}^{3} a^{2}\right)} r^{4} - {\left(4 \, M^{3} {\Sigma}^{2} a^{6} - 4 \, M^{3} {\Sigma}^{2} a^{4} - M {\Sigma}^{4} a^{2}\right)} r^{3} + 2 \, {\left(M^{2} {\Sigma}^{3} a^{6} - 2 \, M^{2} {\Sigma}^{3} a^{4}\right)} r^{2}\right)} \sin\left({\theta}\right)^{2}} & 0 & 0 & -\frac{32 \, {\left(M^{3} {\Sigma} a^{6} r^{3} + 2 \, {\left(M^{4} a^{8} - M^{4} a^{6}\right)} r^{4}\right)} \sin\left({\theta}\right)^{6} - {\left(40 \, M^{3} {\Sigma} a^{4} r^{5} + {\Delta} M^{2} {\Sigma}^{2} a^{6} + 4 \, {\left(12 \, M^{4} a^{8} - 12 \, M^{4} a^{6} - 5 \, M^{2} {\Sigma}^{2} a^{4}\right)} r^{4} + 4 \, {\left({\left({\Delta} + 16\right)} M^{3} {\Sigma} a^{6} - {\Delta} M^{3} {\Sigma} a^{4}\right)} r^{3} - {\left(20 \, M^{2} {\Sigma}^{2} a^{6} - 3 \, {\Delta} M^{2} {\Sigma}^{2} a^{4}\right)} r^{2} + 4 \, {\left({\Delta} M^{3} {\Sigma} a^{8} - {\Delta} M^{3} {\Sigma} a^{6}\right)} r\right)} \sin\left({\theta}\right)^{4} + {\left(32 \, M^{3} {\Sigma} a^{4} r^{5} + {\Delta} M^{2} {\Sigma}^{2} a^{6} - 3 \, {\Delta} M {\Sigma}^{3} a^{4} r - {\left(16 \, M^{2} {\Sigma}^{2} a^{4} - {\Delta} M^{2} {\Sigma}^{2} a^{2}\right)} r^{4} + {\left(32 \, M^{3} {\Sigma} a^{6} - {\Delta} M {\Sigma}^{3} a^{2}\right)} r^{3} - 2 \, {\left(8 \, M^{2} {\Sigma}^{2} a^{6} - 3 \, {\Delta} M^{2} {\Sigma}^{2} a^{4}\right)} r^{2}\right)} \sin\left({\theta}\right)^{2}}{4 \, M^{2} {\Sigma}^{3} r^{6} - 4 \, M {\Sigma}^{4} a^{4} r - 8 \, M {\Sigma}^{4} a^{2} r^{3} - 4 \, M {\Sigma}^{4} r^{5} + {\Sigma}^{5} a^{4} + {\left(8 \, M^{2} {\Sigma}^{3} a^{2} + {\Sigma}^{5}\right)} r^{4} + 4 \, {\left(M^{2} {\Sigma}^{3} a^{4} r^{2} + 4 \, {\left(M^{4} {\Sigma} a^{8} - 2 \, M^{4} {\Sigma} a^{6} + M^{4} {\Sigma} a^{4}\right)} r^{4} + 4 \, {\left(M^{3} {\Sigma}^{2} a^{6} - M^{3} {\Sigma}^{2} a^{4}\right)} r^{3}\right)} \sin\left({\theta}\right)^{4} + 2 \, {\left(2 \, M^{2} {\Sigma}^{3} a^{4} + {\Sigma}^{5} a^{2}\right)} r^{2} + 4 \, {\left(M {\Sigma}^{4} a^{4} r - 4 \, {\left(M^{3} {\Sigma}^{2} a^{4} - M^{3} {\Sigma}^{2} a^{2}\right)} r^{5} + 2 \, {\left(M^{2} {\Sigma}^{3} a^{4} - 2 \, M^{2} {\Sigma}^{3} a^{2}\right)} r^{4} - {\left(4 \, M^{3} {\Sigma}^{2} a^{6} - 4 \, M^{3} {\Sigma}^{2} a^{4} - M {\Sigma}^{4} a^{2}\right)} r^{3} + 2 \, {\left(M^{2} {\Sigma}^{3} a^{6} - 2 \, M^{2} {\Sigma}^{3} a^{4}\right)} r^{2}\right)} \sin\left({\theta}\right)^{2}} \\
0 & \frac{{\Delta} M^{2} {\Sigma}^{2} a^{6} - 4 \, M^{2} {\Sigma}^{2} r^{6} + 4 \, {\left(6 \, M^{3} {\Sigma} a^{4} - 6 \, M^{3} {\Sigma} a^{2} + M {\Sigma}^{3}\right)} r^{5} - {\left({\Delta} M^{2} {\Sigma}^{2} + {\Sigma}^{4}\right)} a^{4} - {\left(32 \, M^{4} a^{8} - 64 \, M^{4} a^{6} - 16 \, M^{2} {\Sigma}^{2} a^{2} - 2 \, {\Delta} M^{2} {\Sigma}^{2} + 4 \, {\left(8 \, M^{4} + 3 \, M^{2} {\Sigma}^{2}\right)} a^{4} + {\Sigma}^{4}\right)} r^{4} - 2 \, {\left(8 \, {\left(3 \, M^{4} a^{8} - 7 \, M^{4} a^{6} + 4 \, M^{4} a^{4}\right)} r^{4} + 4 \, {\left(7 \, M^{3} {\Sigma} a^{6} - 8 \, M^{3} {\Sigma} a^{4}\right)} r^{3} - 2 \, {\left({\Delta} M^{4} a^{8} - 2 \, {\Delta} M^{4} a^{6} + {\left({\Delta} M^{4} - 4 \, M^{2} {\Sigma}^{2}\right)} a^{4}\right)} r^{2} - {\left({\Delta} M^{3} {\Sigma} a^{6} - {\Delta} M^{3} {\Sigma} a^{4}\right)} r\right)} \cos\left({\theta}\right)^{4} - {\left(8 \, M^{3} {\Sigma} a^{6} + 2 \, {\left(3 \, {\Delta} - 4\right)} M^{3} {\Sigma} a^{4} + {\Delta} M {\Sigma}^{3} - 2 \, {\left(3 \, {\Delta} M^{3} {\Sigma} + M {\Sigma}^{3}\right)} a^{2}\right)} r^{3} + {\left(4 \, {\Delta} M^{4} a^{8} - 4 \, {\left(2 \, {\Delta} M^{4} + 3 \, M^{2} {\Sigma}^{2}\right)} a^{6} + {\left(4 \, {\Delta} M^{4} + {\left({\Delta} + 12\right)} M^{2} {\Sigma}^{2}\right)} a^{4} - {\left({\Delta} M^{2} {\Sigma}^{2} + 2 \, {\Sigma}^{4}\right)} a^{2}\right)} r^{2} - {\left({\Delta} M^{2} {\Sigma}^{2} a^{6} - {\Delta} M^{2} {\Sigma}^{2} a^{4} + 8 \, {\left(5 \, M^{3} {\Sigma} a^{4} - 6 \, M^{3} {\Sigma} a^{2}\right)} r^{5} - 4 \, {\left(20 \, M^{4} a^{8} - 44 \, M^{4} a^{6} - 12 \, M^{2} {\Sigma}^{2} a^{2} + {\left(24 \, M^{4} + 5 \, M^{2} {\Sigma}^{2}\right)} a^{4}\right)} r^{4} - 6 \, {\left(8 \, M^{3} {\Sigma} a^{6} + {\left({\Delta} - 8\right)} M^{3} {\Sigma} a^{4} - {\left({\Delta} M^{3} {\Sigma} - 2 \, M {\Sigma}^{3}\right)} a^{2}\right)} r^{3} + {\left(8 \, {\Delta} M^{4} a^{8} - 3 \, {\Delta} M^{2} {\Sigma}^{2} a^{2} - 4 \, {\left(4 \, {\Delta} M^{4} + 5 \, M^{2} {\Sigma}^{2}\right)} a^{6} + {\left(8 \, {\Delta} M^{4} + {\left({\Delta} + 24\right)} M^{2} {\Sigma}^{2}\right)} a^{4}\right)} r^{2} + 2 \, {\left({\Delta} M^{3} {\Sigma} a^{6} - {\left({\Delta} M^{3} {\Sigma} + 6 \, M {\Sigma}^{3}\right)} a^{4}\right)} r\right)} \cos\left({\theta}\right)^{2} - {\left(2 \, M {\Sigma}^{3} a^{4} + {\Delta} M {\Sigma}^{3} a^{2}\right)} r}{4 \, M^{2} {\Sigma}^{3} r^{6} - 4 \, M {\Sigma}^{4} a^{4} r - 8 \, M {\Sigma}^{4} a^{2} r^{3} - 4 \, M {\Sigma}^{4} r^{5} + {\Sigma}^{5} a^{4} + {\left(8 \, M^{2} {\Sigma}^{3} a^{2} + {\Sigma}^{5}\right)} r^{4} + 4 \, {\left(M^{2} {\Sigma}^{3} a^{4} r^{2} + 4 \, {\left(M^{4} {\Sigma} a^{8} - 2 \, M^{4} {\Sigma} a^{6} + M^{4} {\Sigma} a^{4}\right)} r^{4} + 4 \, {\left(M^{3} {\Sigma}^{2} a^{6} - M^{3} {\Sigma}^{2} a^{4}\right)} r^{3}\right)} \sin\left({\theta}\right)^{4} + 2 \, {\left(2 \, M^{2} {\Sigma}^{3} a^{4} + {\Sigma}^{5} a^{2}\right)} r^{2} + 4 \, {\left(M {\Sigma}^{4} a^{4} r - 4 \, {\left(M^{3} {\Sigma}^{2} a^{4} - M^{3} {\Sigma}^{2} a^{2}\right)} r^{5} + 2 \, {\left(M^{2} {\Sigma}^{3} a^{4} - 2 \, M^{2} {\Sigma}^{3} a^{2}\right)} r^{4} - {\left(4 \, M^{3} {\Sigma}^{2} a^{6} - 4 \, M^{3} {\Sigma}^{2} a^{4} - M {\Sigma}^{4} a^{2}\right)} r^{3} + 2 \, {\left(M^{2} {\Sigma}^{3} a^{6} - 2 \, M^{2} {\Sigma}^{3} a^{4}\right)} r^{2}\right)} \sin\left({\theta}\right)^{2}} & -\frac{4 \, {\left({\Delta} M^{2} {\Sigma}^{2} a^{4} r + 4 \, {\left({\Delta} M^{4} a^{8} - 2 \, {\Delta} M^{4} a^{6} + {\Delta} M^{4} a^{4}\right)} r^{3} + 4 \, {\left({\Delta} M^{3} {\Sigma} a^{6} - {\Delta} M^{3} {\Sigma} a^{4}\right)} r^{2}\right)} \cos\left({\theta}\right) \sin\left({\theta}\right)^{4} + {\left(3 \, {\Delta} M {\Sigma}^{3} a^{4} - 12 \, {\left({\Delta} M^{3} {\Sigma} a^{4} - {\Delta} M^{3} {\Sigma} a^{2}\right)} r^{4} + 4 \, {\left(2 \, {\Delta} M^{2} {\Sigma}^{2} a^{4} - 3 \, {\Delta} M^{2} {\Sigma}^{2} a^{2}\right)} r^{3} - 3 \, {\left(4 \, {\Delta} M^{3} {\Sigma} a^{6} - 4 \, {\Delta} M^{3} {\Sigma} a^{4} - {\Delta} M {\Sigma}^{3} a^{2}\right)} r^{2} + 4 \, {\left(2 \, {\Delta} M^{2} {\Sigma}^{2} a^{6} - 3 \, {\Delta} M^{2} {\Sigma}^{2} a^{4}\right)} r\right)} \cos\left({\theta}\right) \sin\left({\theta}\right)^{2} + {\left(4 \, {\Delta} M^{2} {\Sigma}^{2} r^{5} - 4 \, {\Delta} M {\Sigma}^{3} a^{2} r^{2} - 4 \, {\Delta} M {\Sigma}^{3} r^{4} + {\Delta} {\Sigma}^{4} a^{2} r + {\left(4 \, {\Delta} M^{2} {\Sigma}^{2} a^{2} + {\Delta} {\Sigma}^{4}\right)} r^{3}\right)} \cos\left({\theta}\right)}{4 \, {\left(M^{2} {\Sigma}^{3} a^{4} r^{2} + 4 \, {\left(M^{4} {\Sigma} a^{8} - 2 \, M^{4} {\Sigma} a^{6} + M^{4} {\Sigma} a^{4}\right)} r^{4} + 4 \, {\left(M^{3} {\Sigma}^{2} a^{6} - M^{3} {\Sigma}^{2} a^{4}\right)} r^{3}\right)} \sin\left({\theta}\right)^{5} + 4 \, {\left(M {\Sigma}^{4} a^{4} r - 4 \, {\left(M^{3} {\Sigma}^{2} a^{4} - M^{3} {\Sigma}^{2} a^{2}\right)} r^{5} + 2 \, {\left(M^{2} {\Sigma}^{3} a^{4} - 2 \, M^{2} {\Sigma}^{3} a^{2}\right)} r^{4} - {\left(4 \, M^{3} {\Sigma}^{2} a^{6} - 4 \, M^{3} {\Sigma}^{2} a^{4} - M {\Sigma}^{4} a^{2}\right)} r^{3} + 2 \, {\left(M^{2} {\Sigma}^{3} a^{6} - 2 \, M^{2} {\Sigma}^{3} a^{4}\right)} r^{2}\right)} \sin\left({\theta}\right)^{3} + {\left(4 \, M^{2} {\Sigma}^{3} r^{6} - 4 \, M {\Sigma}^{4} a^{4} r - 8 \, M {\Sigma}^{4} a^{2} r^{3} - 4 \, M {\Sigma}^{4} r^{5} + {\Sigma}^{5} a^{4} + {\left(8 \, M^{2} {\Sigma}^{3} a^{2} + {\Sigma}^{5}\right)} r^{4} + 2 \, {\left(2 \, M^{2} {\Sigma}^{3} a^{4} + {\Sigma}^{5} a^{2}\right)} r^{2}\right)} \sin\left({\theta}\right)} & 0 \\
0 & -\frac{4 \, {\left(M^{2} {\Sigma}^{2} a^{4} r + 4 \, {\left(M^{4} a^{8} - 2 \, M^{4} a^{6} + M^{4} a^{4}\right)} r^{3} + 4 \, {\left(M^{3} {\Sigma} a^{6} - M^{3} {\Sigma} a^{4}\right)} r^{2}\right)} \cos\left({\theta}\right) \sin\left({\theta}\right)^{4} + {\left(3 \, M {\Sigma}^{3} a^{4} - 12 \, {\left(M^{3} {\Sigma} a^{4} - M^{3} {\Sigma} a^{2}\right)} r^{4} + 4 \, {\left(2 \, M^{2} {\Sigma}^{2} a^{4} - 3 \, M^{2} {\Sigma}^{2} a^{2}\right)} r^{3} - 3 \, {\left(4 \, M^{3} {\Sigma} a^{6} - 4 \, M^{3} {\Sigma} a^{4} - M {\Sigma}^{3} a^{2}\right)} r^{2} + 4 \, {\left(2 \, M^{2} {\Sigma}^{2} a^{6} - 3 \, M^{2} {\Sigma}^{2} a^{4}\right)} r\right)} \cos\left({\theta}\right) \sin\left({\theta}\right)^{2} + {\left(4 \, M^{2} {\Sigma}^{2} r^{5} - 4 \, M {\Sigma}^{3} a^{2} r^{2} - 4 \, M {\Sigma}^{3} r^{4} + {\Sigma}^{4} a^{2} r + {\left(4 \, M^{2} {\Sigma}^{2} a^{2} + {\Sigma}^{4}\right)} r^{3}\right)} \cos\left({\theta}\right)}{4 \, {\left(M^{2} {\Sigma}^{3} a^{4} r^{2} + 4 \, {\left(M^{4} {\Sigma} a^{8} - 2 \, M^{4} {\Sigma} a^{6} + M^{4} {\Sigma} a^{4}\right)} r^{4} + 4 \, {\left(M^{3} {\Sigma}^{2} a^{6} - M^{3} {\Sigma}^{2} a^{4}\right)} r^{3}\right)} \sin\left({\theta}\right)^{5} + 4 \, {\left(M {\Sigma}^{4} a^{4} r - 4 \, {\left(M^{3} {\Sigma}^{2} a^{4} - M^{3} {\Sigma}^{2} a^{2}\right)} r^{5} + 2 \, {\left(M^{2} {\Sigma}^{3} a^{4} - 2 \, M^{2} {\Sigma}^{3} a^{2}\right)} r^{4} - {\left(4 \, M^{3} {\Sigma}^{2} a^{6} - 4 \, M^{3} {\Sigma}^{2} a^{4} - M {\Sigma}^{4} a^{2}\right)} r^{3} + 2 \, {\left(M^{2} {\Sigma}^{3} a^{6} - 2 \, M^{2} {\Sigma}^{3} a^{4}\right)} r^{2}\right)} \sin\left({\theta}\right)^{3} + {\left(4 \, M^{2} {\Sigma}^{3} r^{6} - 4 \, M {\Sigma}^{4} a^{4} r - 8 \, M {\Sigma}^{4} a^{2} r^{3} - 4 \, M {\Sigma}^{4} r^{5} + {\Sigma}^{5} a^{4} + {\left(8 \, M^{2} {\Sigma}^{3} a^{2} + {\Sigma}^{5}\right)} r^{4} + 2 \, {\left(2 \, M^{2} {\Sigma}^{3} a^{4} + {\Sigma}^{5} a^{2}\right)} r^{2}\right)} \sin\left({\theta}\right)} & \frac{3 \, {\Delta} M^{2} {\Sigma}^{2} a^{6} - 3 \, {\Delta} M^{2} {\Sigma}^{2} a^{4} + {\Delta} M^{2} {\Sigma}^{2} r^{4} - 5 \, {\Delta} M {\Sigma}^{3} a^{2} r + {\Delta} {\Sigma}^{4} a^{2} - {\left(8 \, M^{3} {\Sigma} a^{6} r^{3} + {\Delta} M^{2} {\Sigma}^{2} a^{4} + 16 \, {\left(M^{4} a^{8} - M^{4} a^{6}\right)} r^{4} + 4 \, {\left({\Delta} M^{4} a^{8} - 2 \, {\Delta} M^{4} a^{6} + {\Delta} M^{4} a^{4}\right)} r^{2} + 2 \, {\left({\Delta} M^{3} {\Sigma} a^{6} - {\Delta} M^{3} {\Sigma} a^{4}\right)} r\right)} \cos\left({\theta}\right)^{4} - {\left(2 \, {\Delta} M^{3} {\Sigma} a^{4} - 2 \, {\Delta} M^{3} {\Sigma} a^{2} + {\Delta} M {\Sigma}^{3}\right)} r^{3} - {\left(4 \, {\Delta} M^{4} a^{8} - 8 \, {\Delta} M^{4} a^{6} - {\Delta} M^{2} {\Sigma}^{2} a^{2} + {\left(4 \, {\Delta} M^{4} + {\Delta} M^{2} {\Sigma}^{2}\right)} a^{4}\right)} r^{2} - {\left(8 \, M^{3} {\Sigma} a^{4} r^{5} + 3 \, {\Delta} M^{2} {\Sigma}^{2} a^{6} - 3 \, {\Delta} M^{2} {\Sigma}^{2} a^{4} - 4 \, {\left(4 \, M^{4} a^{8} - 4 \, M^{4} a^{6} + M^{2} {\Sigma}^{2} a^{4}\right)} r^{4} - 2 \, {\left({\Delta} M^{3} {\Sigma} a^{4} - {\Delta} M^{3} {\Sigma} a^{2}\right)} r^{3} - {\left(8 \, {\Delta} M^{4} a^{8} + 3 \, {\Delta} M^{2} {\Sigma}^{2} a^{2} - 4 \, {\left(4 \, {\Delta} M^{4} - M^{2} {\Sigma}^{2}\right)} a^{6} + {\left(8 \, {\Delta} M^{4} + {\Delta} M^{2} {\Sigma}^{2}\right)} a^{4}\right)} r^{2} - 2 \, {\left({\Delta} M^{3} {\Sigma} a^{6} - {\Delta} M^{3} {\Sigma} a^{4}\right)} r\right)} \cos\left({\theta}\right)^{2}}{4 \, M^{2} {\Sigma}^{3} r^{6} - 4 \, M {\Sigma}^{4} a^{4} r - 8 \, M {\Sigma}^{4} a^{2} r^{3} - 4 \, M {\Sigma}^{4} r^{5} + {\Sigma}^{5} a^{4} + {\left(8 \, M^{2} {\Sigma}^{3} a^{2} + {\Sigma}^{5}\right)} r^{4} + 4 \, {\left(M^{2} {\Sigma}^{3} a^{4} r^{2} + 4 \, {\left(M^{4} {\Sigma} a^{8} - 2 \, M^{4} {\Sigma} a^{6} + M^{4} {\Sigma} a^{4}\right)} r^{4} + 4 \, {\left(M^{3} {\Sigma}^{2} a^{6} - M^{3} {\Sigma}^{2} a^{4}\right)} r^{3}\right)} \sin\left({\theta}\right)^{4} + 2 \, {\left(2 \, M^{2} {\Sigma}^{3} a^{4} + {\Sigma}^{5} a^{2}\right)} r^{2} + 4 \, {\left(M {\Sigma}^{4} a^{4} r - 4 \, {\left(M^{3} {\Sigma}^{2} a^{4} - M^{3} {\Sigma}^{2} a^{2}\right)} r^{5} + 2 \, {\left(M^{2} {\Sigma}^{3} a^{4} - 2 \, M^{2} {\Sigma}^{3} a^{2}\right)} r^{4} - {\left(4 \, M^{3} {\Sigma}^{2} a^{6} - 4 \, M^{3} {\Sigma}^{2} a^{4} - M {\Sigma}^{4} a^{2}\right)} r^{3} + 2 \, {\left(M^{2} {\Sigma}^{3} a^{6} - 2 \, M^{2} {\Sigma}^{3} a^{4}\right)} r^{2}\right)} \sin\left({\theta}\right)^{2}} & 0 \\
-\frac{8 \, M^{3} {\Sigma} a^{2} r^{5} - {\Delta} M^{2} {\Sigma}^{2} a^{4} - 8 \, {\left(2 \, M^{4} a^{6} - 2 \, M^{4} a^{4} + M^{2} {\Sigma}^{2} a^{2}\right)} r^{4} + 2 \, {\left(4 \, M^{3} {\Sigma} a^{6} - 4 \, M^{3} {\Sigma} a^{4} + M {\Sigma}^{3} a^{2}\right)} r^{3} - {\left(4 \, M^{2} {\Sigma}^{2} a^{4} + 3 \, {\Delta} M^{2} {\Sigma}^{2} a^{2}\right)} r^{2} + {\left({\Delta} M^{2} {\Sigma}^{2} a^{4} + 4 \, {\left({\Delta} M^{3} {\Sigma} a^{6} - {\Delta} M^{3} {\Sigma} a^{4}\right)} r\right)} \sin\left({\theta}\right)^{2} + {\left(2 \, M {\Sigma}^{3} a^{4} + {\Delta} M {\Sigma}^{3} a^{2}\right)} r}{4 \, M^{2} {\Sigma}^{3} r^{6} - 4 \, M {\Sigma}^{4} a^{4} r - 8 \, M {\Sigma}^{4} a^{2} r^{3} - 4 \, M {\Sigma}^{4} r^{5} + {\Sigma}^{5} a^{4} + {\left(8 \, M^{2} {\Sigma}^{3} a^{2} + {\Sigma}^{5}\right)} r^{4} + 4 \, {\left(M^{2} {\Sigma}^{3} a^{4} r^{2} + 4 \, {\left(M^{4} {\Sigma} a^{8} - 2 \, M^{4} {\Sigma} a^{6} + M^{4} {\Sigma} a^{4}\right)} r^{4} + 4 \, {\left(M^{3} {\Sigma}^{2} a^{6} - M^{3} {\Sigma}^{2} a^{4}\right)} r^{3}\right)} \sin\left({\theta}\right)^{4} + 2 \, {\left(2 \, M^{2} {\Sigma}^{3} a^{4} + {\Sigma}^{5} a^{2}\right)} r^{2} + 4 \, {\left(M {\Sigma}^{4} a^{4} r - 4 \, {\left(M^{3} {\Sigma}^{2} a^{4} - M^{3} {\Sigma}^{2} a^{2}\right)} r^{5} + 2 \, {\left(M^{2} {\Sigma}^{3} a^{4} - 2 \, M^{2} {\Sigma}^{3} a^{2}\right)} r^{4} - {\left(4 \, M^{3} {\Sigma}^{2} a^{6} - 4 \, M^{3} {\Sigma}^{2} a^{4} - M {\Sigma}^{4} a^{2}\right)} r^{3} + 2 \, {\left(M^{2} {\Sigma}^{3} a^{6} - 2 \, M^{2} {\Sigma}^{3} a^{4}\right)} r^{2}\right)} \sin\left({\theta}\right)^{2}} & 0 & 0 & -\frac{8 \, M^{3} {\Sigma} a^{6} r^{3} + 8 \, M^{3} {\Sigma} a^{4} r^{5} + {\Delta} M^{2} {\Sigma}^{2} a^{4} - {\left(4 \, M^{2} {\Sigma}^{2} a^{4} - {\Delta} M^{2} {\Sigma}^{2}\right)} r^{4} + 4 \, {\left(2 \, M^{3} {\Sigma} a^{6} r^{3} + 4 \, {\left(M^{4} a^{8} - M^{4} a^{6}\right)} r^{4} + {\left({\Delta} M^{4} a^{8} - 2 \, {\Delta} M^{4} a^{6} + {\Delta} M^{4} a^{4}\right)} r^{2}\right)} \sin\left({\theta}\right)^{4} - 2 \, {\left(2 \, M^{2} {\Sigma}^{2} a^{6} - {\Delta} M^{2} {\Sigma}^{2} a^{2}\right)} r^{2} - {\left(16 \, M^{3} {\Sigma} a^{4} r^{5} - 8 \, M^{2} {\Sigma}^{2} a^{4} r^{4} + {\Delta} M^{2} {\Sigma}^{2} a^{6} + 4 \, {\left(4 \, M^{3} {\Sigma} a^{6} + {\Delta} M^{3} {\Sigma} a^{4} - {\Delta} M^{3} {\Sigma} a^{2}\right)} r^{3} - {\left(8 \, M^{2} {\Sigma}^{2} a^{6} + {\Delta} M^{2} {\Sigma}^{2} a^{4}\right)} r^{2} + 4 \, {\left({\Delta} M^{3} {\Sigma} a^{6} - {\Delta} M^{3} {\Sigma} a^{4}\right)} r\right)} \sin\left({\theta}\right)^{2}}{4 \, M^{2} {\Sigma}^{3} r^{6} - 4 \, M {\Sigma}^{4} a^{4} r - 8 \, M {\Sigma}^{4} a^{2} r^{3} - 4 \, M {\Sigma}^{4} r^{5} + {\Sigma}^{5} a^{4} + {\left(8 \, M^{2} {\Sigma}^{3} a^{2} + {\Sigma}^{5}\right)} r^{4} + 4 \, {\left(M^{2} {\Sigma}^{3} a^{4} r^{2} + 4 \, {\left(M^{4} {\Sigma} a^{8} - 2 \, M^{4} {\Sigma} a^{6} + M^{4} {\Sigma} a^{4}\right)} r^{4} + 4 \, {\left(M^{3} {\Sigma}^{2} a^{6} - M^{3} {\Sigma}^{2} a^{4}\right)} r^{3}\right)} \sin\left({\theta}\right)^{4} + 2 \, {\left(2 \, M^{2} {\Sigma}^{3} a^{4} + {\Sigma}^{5} a^{2}\right)} r^{2} + 4 \, {\left(M {\Sigma}^{4} a^{4} r - 4 \, {\left(M^{3} {\Sigma}^{2} a^{4} - M^{3} {\Sigma}^{2} a^{2}\right)} r^{5} + 2 \, {\left(M^{2} {\Sigma}^{3} a^{4} - 2 \, M^{2} {\Sigma}^{3} a^{2}\right)} r^{4} - {\left(4 \, M^{3} {\Sigma}^{2} a^{6} - 4 \, M^{3} {\Sigma}^{2} a^{4} - M {\Sigma}^{4} a^{2}\right)} r^{3} + 2 \, {\left(M^{2} {\Sigma}^{3} a^{6} - 2 \, M^{2} {\Sigma}^{3} a^{4}\right)} r^{2}\right)} \sin\left({\theta}\right)^{2}}
\end{array}\right)\]

```python
Einstein[0,0].simplify()
```

\[\newcommand{\Bold}[1]{\mathbf{#1}}\frac{9 \, {\left(M^{2} a^{4} - M^{2} a^{2}\right)} r^{14} - 54 \, {\left(M^{3} a^{4} - M^{3} a^{2}\right)} r^{13} + 6 \, {\left(M^{2} a^{6} - 18 \, M^{4} a^{2} + {\left(18 \, M^{4} - M^{2}\right)} a^{4}\right)} r^{12} - {\left(M^{2} a^{18} - M^{2} a^{16} - 3 \, {\left(M^{2} a^{14} - M^{2} a^{12}\right)} r^{4} + 8 \, {\left(M^{3} a^{14} - M^{3} a^{12}\right)} r^{3} - 6 \, {\left(M^{2} a^{16} - M^{2} a^{14}\right)} r^{2}\right)} \cos\left({\theta}\right)^{12} - 36 \, {\left(M^{3} a^{6} - 2 \, M^{5} a^{2} + {\left(2 \, M^{5} - M^{3}\right)} a^{4}\right)} r^{11} - {\left({\left(28 \, M^{4} - M^{2}\right)} a^{8} + 72 \, M^{4} a^{4} - {\left(100 \, M^{4} - M^{2}\right)} a^{6}\right)} r^{10} + {\left(M^{2} a^{18} - M^{2} a^{16} + 3 \, {\left(M^{2} a^{12} - M^{2} a^{10}\right)} r^{6} - 42 \, {\left(M^{3} a^{12} - M^{3} a^{10}\right)} r^{5} + {\left({\left(4 \, M^{4} + 15 \, M^{2}\right)} a^{14} - 48 \, M^{4} a^{10} + {\left(44 \, M^{4} - 15 \, M^{2}\right)} a^{12}\right)} r^{4} + 4 \, {\left({\left(2 \, M^{5} - 5 \, M^{3}\right)} a^{14} - {\left(2 \, M^{5} - 5 \, M^{3}\right)} a^{12}\right)} r^{3} - {\left({\left(12 \, M^{4} + 11 \, M^{2}\right)} a^{16} - {\left(12 \, M^{4} + 11 \, M^{2}\right)} a^{14}\right)} r^{2} + 6 \, {\left(M^{3} a^{16} - M^{3} a^{14}\right)} r\right)} \cos\left({\theta}\right)^{10} + 6 \, {\left(8 \, M^{5} a^{4} + {\left(8 \, M^{5} - M^{3}\right)} a^{8} - {\left(16 \, M^{5} - M^{3}\right)} a^{6}\right)} r^{9} - 12 \, {\left(M^{4} a^{10} - 2 \, M^{4} a^{8} + M^{4} a^{6}\right)} r^{8} - {\left(18 \, {\left(M^{2} a^{10} - M^{2} a^{8}\right)} r^{8} + 24 \, {\left(M^{3} a^{10} - M^{3} a^{8}\right)} r^{7} + {\left({\left(4 \, M^{4} - 9 \, M^{2}\right)} a^{12} + 132 \, M^{4} a^{8} - {\left(136 \, M^{4} - 9 \, M^{2}\right)} a^{10}\right)} r^{6} - 2 \, {\left(48 \, M^{5} a^{8} + {\left(16 \, M^{5} - 3 \, M^{3}\right)} a^{12} - {\left(64 \, M^{5} - 3 \, M^{3}\right)} a^{10}\right)} r^{5} + 4 \, {\left(7 \, {\left(M^{4} + M^{2}\right)} a^{14} - 6 \, M^{4} a^{10} - {\left(M^{4} + 7 \, M^{2}\right)} a^{12}\right)} r^{4} + 4 \, {\left({\left(2 \, M^{5} - 13 \, M^{3}\right)} a^{14} - {\left(2 \, M^{5} - 13 \, M^{3}\right)} a^{12}\right)} r^{3} - {\left({\left(24 \, M^{4} + 5 \, M^{2}\right)} a^{16} + 12 \, M^{4} a^{12} - {\left(36 \, M^{4} + 5 \, M^{2}\right)} a^{14}\right)} r^{2} + 6 \, {\left(M^{3} a^{16} - M^{3} a^{14}\right)} r\right)} \cos\left({\theta}\right)^{8} + 8 \, {\left(M^{5} a^{10} - 2 \, M^{5} a^{8} + M^{5} a^{6}\right)} r^{7} - 2 \, {\left(21 \, {\left(M^{2} a^{8} - M^{2} a^{6}\right)} r^{10} - 50 \, {\left(M^{3} a^{8} - M^{3} a^{6}\right)} r^{9} + {\left({\left(10 \, M^{4} - 3 \, M^{2}\right)} a^{10} + 6 \, M^{4} a^{6} - {\left(16 \, M^{4} - 3 \, M^{2}\right)} a^{8}\right)} r^{8} - 4 \, {\left(15 \, M^{5} a^{6} + {\left(5 \, M^{5} + 6 \, M^{3}\right)} a^{10} - 2 \, {\left(10 \, M^{5} + 3 \, M^{3}\right)} a^{8}\right)} r^{7} - {\left({\left(10 \, M^{4} - 11 \, M^{2}\right)} a^{12} - 32 \, M^{6} a^{6} - {\left(32 \, M^{6} + 88 \, M^{4} - 11 \, M^{2}\right)} a^{10} + 2 \, {\left(32 \, M^{6} + 39 \, M^{4}\right)} a^{8}\right)} r^{6} + 2 \, {\left(36 \, M^{5} a^{8} + {\left(20 \, M^{5} - 21 \, M^{3}\right)} a^{12} - 7 \, {\left(8 \, M^{5} - 3 \, M^{3}\right)} a^{10}\right)} r^{5} - 5 \, {\left({\left(6 \, M^{4} + M^{2}\right)} a^{14} + 6 \, M^{4} a^{10} - {\left(12 \, M^{4} + M^{2}\right)} a^{12}\right)} r^{4} + 4 \, {\left(M^{5} a^{10} + {\left(M^{5} + 3 \, M^{3}\right)} a^{14} - {\left(2 \, M^{5} + 3 \, M^{3}\right)} a^{12}\right)} r^{3} + 6 \, {\left(M^{4} a^{16} - 2 \, M^{4} a^{14} + M^{4} a^{12}\right)} r^{2}\right)} \cos\left({\theta}\right)^{6} - {\left(33 \, {\left(M^{2} a^{6} - M^{2} a^{4}\right)} r^{12} - 144 \, {\left(M^{3} a^{6} - M^{3} a^{4}\right)} r^{11} + 12 \, {\left({\left(M^{4} - 2 \, M^{2}\right)} a^{8} - 15 \, M^{4} a^{4} + 2 \, {\left(7 \, M^{4} + M^{2}\right)} a^{6}\right)} r^{10} + 4 \, {\left(12 \, M^{5} a^{4} - {\left(4 \, M^{5} - 5 \, M^{3}\right)} a^{8} - {\left(8 \, M^{5} + 5 \, M^{3}\right)} a^{6}\right)} r^{9} - {\left({\left(76 \, M^{4} + 7 \, M^{2}\right)} a^{10} + 132 \, M^{4} a^{6} - {\left(208 \, M^{4} + 7 \, M^{2}\right)} a^{8}\right)} r^{8} + 8 \, {\left(17 \, M^{5} a^{10} - 44 \, M^{5} a^{8} + 27 \, M^{5} a^{6}\right)} r^{7} - 2 \, {\left(5 \, {\left(2 \, M^{4} + M^{2}\right)} a^{12} + 64 \, M^{6} a^{6} + {\left(64 \, M^{6} - 16 \, M^{4} - 5 \, M^{2}\right)} a^{10} - 2 \, {\left(64 \, M^{6} - 3 \, M^{4}\right)} a^{8}\right)} r^{6} - 4 \, {\left(8 \, M^{5} a^{8} + {\left(8 \, M^{5} - 9 \, M^{3}\right)} a^{12} - {\left(16 \, M^{5} - 9 \, M^{3}\right)} a^{10}\right)} r^{5} + 36 \, {\left(M^{4} a^{14} - 2 \, M^{4} a^{12} + M^{4} a^{10}\right)} r^{4} - 8 \, {\left(M^{5} a^{14} - 2 \, M^{5} a^{12} + M^{5} a^{10}\right)} r^{3}\right)} \cos\left({\theta}\right)^{4} - {\left(9 \, {\left(M^{2} a^{4} - M^{2} a^{2}\right)} r^{14} - 54 \, {\left(M^{3} a^{4} - M^{3} a^{2}\right)} r^{13} - 27 \, {\left(M^{2} a^{6} + 4 \, M^{4} a^{2} - {\left(4 \, M^{4} + M^{2}\right)} a^{4}\right)} r^{12} + 36 \, {\left(3 \, M^{3} a^{6} + 2 \, M^{5} a^{2} - {\left(2 \, M^{5} + 3 \, M^{3}\right)} a^{4}\right)} r^{11} - {\left({\left(40 \, M^{4} + 17 \, M^{2}\right)} a^{8} - 108 \, M^{4} a^{4} + 17 \, {\left(4 \, M^{4} - M^{2}\right)} a^{6}\right)} r^{10} + 2 \, {\left({\left(32 \, M^{5} + 37 \, M^{3}\right)} a^{8} - {\left(32 \, M^{5} + 37 \, M^{3}\right)} a^{6}\right)} r^{9} + {\left({\left(44 \, M^{4} - 5 \, M^{2}\right)} a^{10} + 108 \, M^{4} a^{6} - {\left(152 \, M^{4} - 5 \, M^{2}\right)} a^{8}\right)} r^{8} - 8 \, {\left(11 \, M^{5} a^{6} + {\left(11 \, M^{5} - 3 \, M^{3}\right)} a^{10} - {\left(22 \, M^{5} - 3 \, M^{3}\right)} a^{8}\right)} r^{7} + 4 \, {\left(9 \, M^{4} a^{12} + 16 \, M^{6} a^{6} + 2 \, {\left(8 \, M^{6} - 9 \, M^{4}\right)} a^{10} - {\left(32 \, M^{6} - 9 \, M^{4}\right)} a^{8}\right)} r^{6} - 16 \, {\left(M^{5} a^{12} - 2 \, M^{5} a^{10} + M^{5} a^{8}\right)} r^{5}\right)} \cos\left({\theta}\right)^{2}}{4 \, M a^{2} r^{17} + 4 \, M r^{19} - r^{20} - 2 \, {\left(2 \, M^{2} + a^{2}\right)} r^{18} - {\left({\left(8 \, M^{2} + 1\right)} a^{4} - 8 \, M^{2} a^{2}\right)} r^{16} - {\left(a^{20} - 4 \, M a^{18} r - 4 \, M a^{16} r^{3} + a^{16} r^{4} + 2 \, {\left(2 \, M^{2} a^{16} + a^{18}\right)} r^{2}\right)} \cos\left({\theta}\right)^{16} + 16 \, {\left(M^{3} a^{4} - M^{3} a^{2}\right)} r^{15} - 8 \, {\left(M^{2} a^{6} - M^{2} a^{4}\right)} r^{14} + 8 \, {\left(4 \, M a^{14} r^{5} - a^{14} r^{6} + {\left({\left(M^{2} - 2\right)} a^{16} - 5 \, M^{2} a^{14}\right)} r^{4} + 2 \, {\left(M^{3} a^{14} - {\left(M^{3} - 2 \, M\right)} a^{16}\right)} r^{3} + {\left({\left(M^{2} - 1\right)} a^{18} - M^{2} a^{16}\right)} r^{2}\right)} \cos\left({\theta}\right)^{14} - 16 \, {\left(M^{4} a^{8} - 2 \, M^{4} a^{6} + M^{4} a^{4}\right)} r^{12} + 4 \, {\left(28 \, M a^{12} r^{7} - 7 \, a^{12} r^{8} + 2 \, {\left({\left(6 \, M^{2} - 7\right)} a^{14} - 20 \, M^{2} a^{12}\right)} r^{6} + 4 \, {\left(6 \, M^{3} a^{12} - {\left(6 \, M^{3} - 7 \, M\right)} a^{14}\right)} r^{5} - {\left({\left(4 \, M^{4} - 10 \, M^{2} + 7\right)} a^{16} + 4 \, M^{4} a^{12} - 2 \, {\left(4 \, M^{4} - 5 \, M^{2}\right)} a^{14}\right)} r^{4} + 4 \, {\left(M^{3} a^{16} - M^{3} a^{14}\right)} r^{3} - 2 \, {\left(M^{2} a^{18} - M^{2} a^{16}\right)} r^{2}\right)} \cos\left({\theta}\right)^{12} + 8 \, {\left(28 \, M a^{10} r^{9} - 7 \, a^{10} r^{10} + {\left({\left(15 \, M^{2} - 14\right)} a^{12} - 43 \, M^{2} a^{10}\right)} r^{8} + 2 \, {\left(15 \, M^{3} a^{10} - {\left(15 \, M^{3} - 14 \, M\right)} a^{12}\right)} r^{7} - {\left({\left(8 \, M^{4} - 9 \, M^{2} + 7\right)} a^{14} + 8 \, M^{4} a^{10} - {\left(16 \, M^{4} - 9 \, M^{2}\right)} a^{12}\right)} r^{6} + 12 \, {\left(M^{3} a^{14} - M^{3} a^{12}\right)} r^{5} + 2 \, {\left({\left(2 \, M^{4} - 3 \, M^{2}\right)} a^{16} + 2 \, M^{4} a^{12} - {\left(4 \, M^{4} - 3 \, M^{2}\right)} a^{14}\right)} r^{4}\right)} \cos\left({\theta}\right)^{10} + 2 \, {\left(140 \, M a^{8} r^{11} - 35 \, a^{8} r^{12} + 10 \, {\left({\left(8 \, M^{2} - 7\right)} a^{10} - 22 \, M^{2} a^{8}\right)} r^{10} + 20 \, {\left(8 \, M^{3} a^{8} - {\left(8 \, M^{3} - 7 \, M\right)} a^{10}\right)} r^{9} - {\left({\left(48 \, M^{4} - 20 \, M^{2} + 35\right)} a^{12} + 48 \, M^{4} a^{8} - 4 \, {\left(24 \, M^{4} - 5 \, M^{2}\right)} a^{10}\right)} r^{8} + 120 \, {\left(M^{3} a^{12} - M^{3} a^{10}\right)} r^{7} + 4 \, {\left({\left(16 \, M^{4} - 15 \, M^{2}\right)} a^{14} + 16 \, M^{4} a^{10} - {\left(32 \, M^{4} - 15 \, M^{2}\right)} a^{12}\right)} r^{6} - 8 \, {\left(M^{4} a^{16} - 2 \, M^{4} a^{14} + M^{4} a^{12}\right)} r^{4}\right)} \cos\left({\theta}\right)^{8} + 8 \, {\left(28 \, M a^{6} r^{13} - 7 \, a^{6} r^{14} + {\left({\left(15 \, M^{2} - 14\right)} a^{8} - 43 \, M^{2} a^{6}\right)} r^{12} + 2 \, {\left(15 \, M^{3} a^{6} - {\left(15 \, M^{3} - 14 \, M\right)} a^{8}\right)} r^{11} - {\left({\left(8 \, M^{4} + 5 \, M^{2} + 7\right)} a^{10} + 8 \, M^{4} a^{6} - {\left(16 \, M^{4} + 5 \, M^{2}\right)} a^{8}\right)} r^{10} + 40 \, {\left(M^{3} a^{10} - M^{3} a^{8}\right)} r^{9} + 4 \, {\left({\left(6 \, M^{4} - 5 \, M^{2}\right)} a^{12} + 6 \, M^{4} a^{8} - {\left(12 \, M^{4} - 5 \, M^{2}\right)} a^{10}\right)} r^{8} - 8 \, {\left(M^{4} a^{14} - 2 \, M^{4} a^{12} + M^{4} a^{10}\right)} r^{6}\right)} \cos\left({\theta}\right)^{6} + 4 \, {\left(28 \, M a^{4} r^{15} - 7 \, a^{4} r^{16} + 2 \, {\left({\left(6 \, M^{2} - 7\right)} a^{6} - 20 \, M^{2} a^{4}\right)} r^{14} + 4 \, {\left(6 \, M^{3} a^{4} - {\left(6 \, M^{3} - 7 \, M\right)} a^{6}\right)} r^{13} - {\left({\left(4 \, M^{4} + 18 \, M^{2} + 7\right)} a^{8} + 4 \, M^{4} a^{4} - 2 \, {\left(4 \, M^{4} + 9 \, M^{2}\right)} a^{6}\right)} r^{12} + 60 \, {\left(M^{3} a^{8} - M^{3} a^{6}\right)} r^{11} + 2 \, {\left({\left(16 \, M^{4} - 15 \, M^{2}\right)} a^{10} + 16 \, M^{4} a^{6} - {\left(32 \, M^{4} - 15 \, M^{2}\right)} a^{8}\right)} r^{10} - 24 \, {\left(M^{4} a^{12} - 2 \, M^{4} a^{10} + M^{4} a^{8}\right)} r^{8}\right)} \cos\left({\theta}\right)^{4} + 8 \, {\left(4 \, M a^{2} r^{17} - a^{2} r^{18} + {\left({\left(M^{2} - 2\right)} a^{4} - 5 \, M^{2} a^{2}\right)} r^{16} + 2 \, {\left(M^{3} a^{2} - {\left(M^{3} - 2 \, M\right)} a^{4}\right)} r^{15} - {\left({\left(5 \, M^{2} + 1\right)} a^{6} - 5 \, M^{2} a^{4}\right)} r^{14} + 12 \, {\left(M^{3} a^{6} - M^{3} a^{4}\right)} r^{13} + 2 \, {\left({\left(2 \, M^{4} - 3 \, M^{2}\right)} a^{8} + 2 \, M^{4} a^{4} - {\left(4 \, M^{4} - 3 \, M^{2}\right)} a^{6}\right)} r^{12} - 8 \, {\left(M^{4} a^{10} - 2 \, M^{4} a^{8} + M^{4} a^{6}\right)} r^{10}\right)} \cos\left({\theta}\right)^{2}}\]

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