Oppenheimer_Snyder_curvature.ipynb

Kernel: SageMath 9.5.beta7

Curvature tensor in the Oppenheimer-Snyder interior

This Jupyter/SageMath notebook is relative to the lectures Geometry and physics of black holes.

The computations make use of tools developed through the SageManifolds project.

version()

'SageMath version 9.5.beta7, Release Date: 2021-11-18'

%display latex

Interior spacetime for Oppenheimer-Snyder collapse from rest

M = Manifold(4, 'M', structure='Lorentzian')

Chart of conformal coordinates:

CC.<et,ch,th,ph> = M.chart(r"et:(0,pi):\eta ch:(0,pi/2):\chi th:(0,pi):\theta ph:(0,2*pi):periodic:\varphi")
CC

$$\renewcommand{\Bold}[1]{\mathbf{#1}}\left(M,({\eta}, {\chi}, {\theta}, {\varphi})\right)$$

CC.coord_range()

$$\renewcommand{\Bold}[1]{\mathbf{#1}}{\eta} :\ \left( 0 , \pi \right) ;\quad {\chi} :\ \left( 0 , \frac{1}{2} \, \pi \right) ;\quad {\theta} :\ \left( 0 , \pi \right) ;\quad {\varphi} :\ \left[ 0 , 2 \, \pi \right] \mbox{(periodic)}$$

g = M.metric()
a0 = var('a0', domain='real')
cf = a0^2/4*(1 + cos(et))^2
g[0,0] = - cf
g[1,1] = cf
g[2,2] = cf*sin(ch)^2
g[3,3] = cf*sin(ch)^2*sin(th)^2
g.display()

$$\renewcommand{\Bold}[1]{\mathbf{#1}}g = -\frac{1}{4} \, a_{0}^{2} {\left(\cos\left({\eta}\right) + 1\right)}^{2} \mathrm{d} {\eta}\otimes \mathrm{d} {\eta} + \frac{1}{4} \, a_{0}^{2} {\left(\cos\left({\eta}\right) + 1\right)}^{2} \mathrm{d} {\chi}\otimes \mathrm{d} {\chi} + \frac{1}{4} \, a_{0}^{2} {\left(\cos\left({\eta}\right) + 1\right)}^{2} \sin\left({\chi}\right)^{2} \mathrm{d} {\theta}\otimes \mathrm{d} {\theta} + \frac{1}{4} \, a_{0}^{2} {\left(\cos\left({\eta}\right) + 1\right)}^{2} \sin\left({\chi}\right)^{2} \sin\left({\theta}\right)^{2} \mathrm{d} {\varphi}\otimes \mathrm{d} {\varphi}$$

Ricci tensor

Ric = g.ricci()
Ric.display_comp()

$$\renewcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{lcl} \mathrm{Ric}\left(g\right)_{ \, {\eta} \, {\eta} }^{ \phantom{\, {\eta}}\phantom{\, {\eta}} } & = & \frac{3}{\cos\left({\eta}\right) + 1} \\ \mathrm{Ric}\left(g\right)_{ \, {\chi} \, {\chi} }^{ \phantom{\, {\chi}}\phantom{\, {\chi}} } & = & \frac{3}{\cos\left({\eta}\right) + 1} \\ \mathrm{Ric}\left(g\right)_{ \, {\theta} \, {\theta} }^{ \phantom{\, {\theta}}\phantom{\, {\theta}} } & = & -\frac{3 \, {\left({\left(\cos\left({\eta}\right) \sin\left({\chi}\right)^{2} + 5 \, \sin\left({\chi}\right)^{2}\right)} \sin\left({\eta}\right)^{4} + 16 \, \cos\left({\eta}\right) \sin\left({\chi}\right)^{2} - 4 \, {\left(3 \, \cos\left({\eta}\right) \sin\left({\chi}\right)^{2} + 5 \, \sin\left({\chi}\right)^{2}\right)} \sin\left({\eta}\right)^{2} + 16 \, \sin\left({\chi}\right)^{2}\right)}}{\sin\left({\eta}\right)^{6} - 6 \, {\left(\cos\left({\eta}\right) + 3\right)} \sin\left({\eta}\right)^{4} + 16 \, {\left(2 \, \cos\left({\eta}\right) + 3\right)} \sin\left({\eta}\right)^{2} - 32 \, \cos\left({\eta}\right) - 32} \\ \mathrm{Ric}\left(g\right)_{ \, {\varphi} \, {\varphi} }^{ \phantom{\, {\varphi}}\phantom{\, {\varphi}} } & = & -\frac{3 \, {\left({\left(\cos\left({\eta}\right) \sin\left({\chi}\right)^{2} + 5 \, \sin\left({\chi}\right)^{2}\right)} \sin\left({\eta}\right)^{4} + 16 \, \cos\left({\eta}\right) \sin\left({\chi}\right)^{2} - 4 \, {\left(3 \, \cos\left({\eta}\right) \sin\left({\chi}\right)^{2} + 5 \, \sin\left({\chi}\right)^{2}\right)} \sin\left({\eta}\right)^{2} + 16 \, \sin\left({\chi}\right)^{2}\right)} \sin\left({\theta}\right)^{2}}{\sin\left({\eta}\right)^{6} - 6 \, {\left(\cos\left({\eta}\right) + 3\right)} \sin\left({\eta}\right)^{4} + 16 \, {\left(2 \, \cos\left({\eta}\right) + 3\right)} \sin\left({\eta}\right)^{2} - 32 \, \cos\left({\eta}\right) - 32} \end{array}$$

Some trigonometric simplifications are in order:

Ric.apply_map(lambda x: x.subs({sin(et): sqrt(1 - cos(et)^2)}).factor())
Ric.display_comp()

$$\renewcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{lcl} \mathrm{Ric}\left(g\right)_{ \, {\eta} \, {\eta} }^{ \phantom{\, {\eta}}\phantom{\, {\eta}} } & = & \frac{3}{\cos\left({\eta}\right) + 1} \\ \mathrm{Ric}\left(g\right)_{ \, {\chi} \, {\chi} }^{ \phantom{\, {\chi}}\phantom{\, {\chi}} } & = & \frac{3}{\cos\left({\eta}\right) + 1} \\ \mathrm{Ric}\left(g\right)_{ \, {\theta} \, {\theta} }^{ \phantom{\, {\theta}}\phantom{\, {\theta}} } & = & \frac{3 \, \sin\left({\chi}\right)^{2}}{\cos\left({\eta}\right) + 1} \\ \mathrm{Ric}\left(g\right)_{ \, {\varphi} \, {\varphi} }^{ \phantom{\, {\varphi}}\phantom{\, {\varphi}} } & = & \frac{3 \, \sin\left({\chi}\right)^{2} \sin\left({\theta}\right)^{2}}{\cos\left({\eta}\right) + 1} \end{array}$$

g.ricci_scalar().display()

$$\renewcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{llcl} \mathrm{r}\left(g\right):& M & \longrightarrow & \mathbb{R} \\ & \left({\eta}, {\chi}, {\theta}, {\varphi}\right) & \longmapsto & \frac{24}{4 \, a_{0}^{2} \cos\left({\eta}\right) - {\left(a_{0}^{2} \cos\left({\eta}\right) + 3 \, a_{0}^{2}\right)} \sin\left({\eta}\right)^{2} + 4 \, a_{0}^{2}} \end{array}$$

R = g.ricci_scalar().expr().factor()
R

$$\renewcommand{\Bold}[1]{\mathbf{#1}}-\frac{24}{{\left(\cos\left({\eta}\right) \sin\left({\eta}\right)^{2} + 3 \, \sin\left({\eta}\right)^{2} - 4 \, \cos\left({\eta}\right) - 4\right)} a_{0}^{2}}$$

R = R.subs({sin(et): sqrt(1 - cos(et)^2)})
R.factor()

$$\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{24}{a_{0}^{2} {\left(\cos\left({\eta}\right) + 1\right)}^{3}}$$

Einstein tensor

G = Ric - R/2*g
G.set_name('G')
G.display()

$$\renewcommand{\Bold}[1]{\mathbf{#1}}G = \left( \frac{6}{\cos\left({\eta}\right) + 1} \right) \mathrm{d} {\eta}\otimes \mathrm{d} {\eta}$$

Energy momentum tensor

The fluid 4-velocity:

u = M.vector_field(2/(a0*(1+cos(et))), 0, 0, 0, name='u')
u.display()

$$\renewcommand{\Bold}[1]{\mathbf{#1}}u = \frac{2}{a_{0} {\left(\cos\left({\eta}\right) + 1\right)}} \frac{\partial}{\partial {\eta} }$$

Check that $u$ is a unit vector:

g(u,u).expr()

$$\renewcommand{\Bold}[1]{\mathbf{#1}}-1$$

uf = u.down(g)
uf.display()

$$\renewcommand{\Bold}[1]{\mathbf{#1}}\left( -\frac{1}{2} \, a_{0} \cos\left({\eta}\right) - \frac{1}{2} \, a_{0} \right) \mathrm{d} {\eta}$$

rho = 3/(pi*a0^2*(1 + cos(et))^3)
T = rho*uf*uf
T.set_name('T')
T.display()

$$\renewcommand{\Bold}[1]{\mathbf{#1}}T = \frac{3}{4 \, {\left(\pi + \pi \cos\left({\eta}\right)\right)}} \mathrm{d} {\eta}\otimes \mathrm{d} {\eta}$$

Check of Einstein equation

G == 8*pi*T

$$\renewcommand{\Bold}[1]{\mathbf{#1}}\mathrm{True}$$

Riemann tensor

Riem = g.riemann()
Riem.display_comp(only_nonredundant=True)

$$\renewcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{lcl} \mathrm{Riem}\left(g\right)_{ \phantom{\, {\eta}} \, {\chi} \, {\eta} \, {\chi} }^{ \, {\eta} \phantom{\, {\chi}} \phantom{\, {\eta}} \phantom{\, {\chi}} } & = & -\frac{1}{\cos\left({\eta}\right) + 1} \\ \mathrm{Riem}\left(g\right)_{ \phantom{\, {\eta}} \, {\theta} \, {\eta} \, {\theta} }^{ \, {\eta} \phantom{\, {\theta}} \phantom{\, {\eta}} \phantom{\, {\theta}} } & = & \frac{\cos\left({\eta}\right)^{2} \sin\left({\chi}\right)^{2} + 2 \, \cos\left({\eta}\right) \sin\left({\chi}\right)^{2} + \sin\left({\chi}\right)^{2}}{{\left(\cos\left({\eta}\right) + 3\right)} \sin\left({\eta}\right)^{2} - 4 \, \cos\left({\eta}\right) - 4} \\ \mathrm{Riem}\left(g\right)_{ \phantom{\, {\eta}} \, {\varphi} \, {\eta} \, {\varphi} }^{ \, {\eta} \phantom{\, {\varphi}} \phantom{\, {\eta}} \phantom{\, {\varphi}} } & = & \frac{{\left(\cos\left({\eta}\right)^{2} \sin\left({\chi}\right)^{2} + 2 \, \cos\left({\eta}\right) \sin\left({\chi}\right)^{2} + \sin\left({\chi}\right)^{2}\right)} \sin\left({\theta}\right)^{2}}{{\left(\cos\left({\eta}\right) + 3\right)} \sin\left({\eta}\right)^{2} - 4 \, \cos\left({\eta}\right) - 4} \\ \mathrm{Riem}\left(g\right)_{ \phantom{\, {\chi}} \, {\eta} \, {\eta} \, {\chi} }^{ \, {\chi} \phantom{\, {\eta}} \phantom{\, {\eta}} \phantom{\, {\chi}} } & = & -\frac{1}{\cos\left({\eta}\right) + 1} \\ \mathrm{Riem}\left(g\right)_{ \phantom{\, {\chi}} \, {\theta} \, {\chi} \, {\theta} }^{ \, {\chi} \phantom{\, {\theta}} \phantom{\, {\chi}} \phantom{\, {\theta}} } & = & \frac{2 \, \sin\left({\chi}\right)^{2}}{\cos\left({\eta}\right) + 1} \\ \mathrm{Riem}\left(g\right)_{ \phantom{\, {\chi}} \, {\varphi} \, {\chi} \, {\varphi} }^{ \, {\chi} \phantom{\, {\varphi}} \phantom{\, {\chi}} \phantom{\, {\varphi}} } & = & \frac{2 \, \sin\left({\chi}\right)^{2} \sin\left({\theta}\right)^{2}}{\cos\left({\eta}\right) + 1} \\ \mathrm{Riem}\left(g\right)_{ \phantom{\, {\theta}} \, {\eta} \, {\eta} \, {\theta} }^{ \, {\theta} \phantom{\, {\eta}} \phantom{\, {\eta}} \phantom{\, {\theta}} } & = & \frac{\cos\left({\eta}\right)^{5} + 5 \, \cos\left({\eta}\right)^{4} + 10 \, \cos\left({\eta}\right)^{3} + 10 \, \cos\left({\eta}\right)^{2} + 5 \, \cos\left({\eta}\right) + 1}{\sin\left({\eta}\right)^{6} - 6 \, {\left(\cos\left({\eta}\right) + 3\right)} \sin\left({\eta}\right)^{4} + 16 \, {\left(2 \, \cos\left({\eta}\right) + 3\right)} \sin\left({\eta}\right)^{2} - 32 \, \cos\left({\eta}\right) - 32} \\ \mathrm{Riem}\left(g\right)_{ \phantom{\, {\theta}} \, {\chi} \, {\chi} \, {\theta} }^{ \, {\theta} \phantom{\, {\chi}} \phantom{\, {\chi}} \phantom{\, {\theta}} } & = & \frac{2 \, {\left(\cos\left({\eta}\right) + 1\right)}}{\sin\left({\eta}\right)^{2} - 2 \, \cos\left({\eta}\right) - 2} \\ \mathrm{Riem}\left(g\right)_{ \phantom{\, {\theta}} \, {\varphi} \, {\theta} \, {\varphi} }^{ \, {\theta} \phantom{\, {\varphi}} \phantom{\, {\theta}} \phantom{\, {\varphi}} } & = & -\frac{2 \, {\left(\cos\left({\eta}\right) \sin\left({\chi}\right)^{2} + \sin\left({\chi}\right)^{2}\right)} \sin\left({\theta}\right)^{2}}{\sin\left({\eta}\right)^{2} - 2 \, \cos\left({\eta}\right) - 2} \\ \mathrm{Riem}\left(g\right)_{ \phantom{\, {\varphi}} \, {\eta} \, {\eta} \, {\varphi} }^{ \, {\varphi} \phantom{\, {\eta}} \phantom{\, {\eta}} \phantom{\, {\varphi}} } & = & \frac{\cos\left({\eta}\right)^{5} + 5 \, \cos\left({\eta}\right)^{4} + 10 \, \cos\left({\eta}\right)^{3} + 10 \, \cos\left({\eta}\right)^{2} + 5 \, \cos\left({\eta}\right) + 1}{\sin\left({\eta}\right)^{6} - 6 \, {\left(\cos\left({\eta}\right) + 3\right)} \sin\left({\eta}\right)^{4} + 16 \, {\left(2 \, \cos\left({\eta}\right) + 3\right)} \sin\left({\eta}\right)^{2} - 32 \, \cos\left({\eta}\right) - 32} \\ \mathrm{Riem}\left(g\right)_{ \phantom{\, {\varphi}} \, {\chi} \, {\chi} \, {\varphi} }^{ \, {\varphi} \phantom{\, {\chi}} \phantom{\, {\chi}} \phantom{\, {\varphi}} } & = & \frac{2 \, {\left(\cos\left({\eta}\right) + 1\right)}}{\sin\left({\eta}\right)^{2} - 2 \, \cos\left({\eta}\right) - 2} \\ \mathrm{Riem}\left(g\right)_{ \phantom{\, {\varphi}} \, {\theta} \, {\theta} \, {\varphi} }^{ \, {\varphi} \phantom{\, {\theta}} \phantom{\, {\theta}} \phantom{\, {\varphi}} } & = & \frac{2 \, {\left(\cos\left({\eta}\right) \sin\left({\chi}\right)^{2} + \sin\left({\chi}\right)^{2}\right)}}{\sin\left({\eta}\right)^{2} - 2 \, \cos\left({\eta}\right) - 2} \end{array}$$

Riem.apply_map(lambda x: x.subs({sin(et): sqrt(1 - cos(et)^2)}).factor())
Riem.display_comp(only_nonredundant=True)

$$\renewcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{lcl} \mathrm{Riem}\left(g\right)_{ \phantom{\, {\eta}} \, {\chi} \, {\eta} \, {\chi} }^{ \, {\eta} \phantom{\, {\chi}} \phantom{\, {\eta}} \phantom{\, {\chi}} } & = & -\frac{1}{\cos\left({\eta}\right) + 1} \\ \mathrm{Riem}\left(g\right)_{ \phantom{\, {\eta}} \, {\theta} \, {\eta} \, {\theta} }^{ \, {\eta} \phantom{\, {\theta}} \phantom{\, {\eta}} \phantom{\, {\theta}} } & = & -\frac{\sin\left({\chi}\right)^{2}}{\cos\left({\eta}\right) + 1} \\ \mathrm{Riem}\left(g\right)_{ \phantom{\, {\eta}} \, {\varphi} \, {\eta} \, {\varphi} }^{ \, {\eta} \phantom{\, {\varphi}} \phantom{\, {\eta}} \phantom{\, {\varphi}} } & = & -\frac{\sin\left({\chi}\right)^{2} \sin\left({\theta}\right)^{2}}{\cos\left({\eta}\right) + 1} \\ \mathrm{Riem}\left(g\right)_{ \phantom{\, {\chi}} \, {\eta} \, {\eta} \, {\chi} }^{ \, {\chi} \phantom{\, {\eta}} \phantom{\, {\eta}} \phantom{\, {\chi}} } & = & -\frac{1}{\cos\left({\eta}\right) + 1} \\ \mathrm{Riem}\left(g\right)_{ \phantom{\, {\chi}} \, {\theta} \, {\chi} \, {\theta} }^{ \, {\chi} \phantom{\, {\theta}} \phantom{\, {\chi}} \phantom{\, {\theta}} } & = & \frac{2 \, \sin\left({\chi}\right)^{2}}{\cos\left({\eta}\right) + 1} \\ \mathrm{Riem}\left(g\right)_{ \phantom{\, {\chi}} \, {\varphi} \, {\chi} \, {\varphi} }^{ \, {\chi} \phantom{\, {\varphi}} \phantom{\, {\chi}} \phantom{\, {\varphi}} } & = & \frac{2 \, \sin\left({\chi}\right)^{2} \sin\left({\theta}\right)^{2}}{\cos\left({\eta}\right) + 1} \\ \mathrm{Riem}\left(g\right)_{ \phantom{\, {\theta}} \, {\eta} \, {\eta} \, {\theta} }^{ \, {\theta} \phantom{\, {\eta}} \phantom{\, {\eta}} \phantom{\, {\theta}} } & = & -\frac{1}{\cos\left({\eta}\right) + 1} \\ \mathrm{Riem}\left(g\right)_{ \phantom{\, {\theta}} \, {\chi} \, {\chi} \, {\theta} }^{ \, {\theta} \phantom{\, {\chi}} \phantom{\, {\chi}} \phantom{\, {\theta}} } & = & -\frac{2}{\cos\left({\eta}\right) + 1} \\ \mathrm{Riem}\left(g\right)_{ \phantom{\, {\theta}} \, {\varphi} \, {\theta} \, {\varphi} }^{ \, {\theta} \phantom{\, {\varphi}} \phantom{\, {\theta}} \phantom{\, {\varphi}} } & = & \frac{2 \, \sin\left({\chi}\right)^{2} \sin\left({\theta}\right)^{2}}{\cos\left({\eta}\right) + 1} \\ \mathrm{Riem}\left(g\right)_{ \phantom{\, {\varphi}} \, {\eta} \, {\eta} \, {\varphi} }^{ \, {\varphi} \phantom{\, {\eta}} \phantom{\, {\eta}} \phantom{\, {\varphi}} } & = & -\frac{1}{\cos\left({\eta}\right) + 1} \\ \mathrm{Riem}\left(g\right)_{ \phantom{\, {\varphi}} \, {\chi} \, {\chi} \, {\varphi} }^{ \, {\varphi} \phantom{\, {\chi}} \phantom{\, {\chi}} \phantom{\, {\varphi}} } & = & -\frac{2}{\cos\left({\eta}\right) + 1} \\ \mathrm{Riem}\left(g\right)_{ \phantom{\, {\varphi}} \, {\theta} \, {\theta} \, {\varphi} }^{ \, {\varphi} \phantom{\, {\theta}} \phantom{\, {\theta}} \phantom{\, {\varphi}} } & = & -\frac{2 \, \sin\left({\chi}\right)^{2}}{\cos\left({\eta}\right) + 1} \end{array}$$

Kretschmann scalar

K = Riem.down(g)['_{abcd}'] * Riem.up(g)['^{abcd}']
K.display()

$$\renewcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{llcl} & M & \longrightarrow & \mathbb{R} \\ & \left({\eta}, {\chi}, {\theta}, {\varphi}\right) & \longmapsto & -\frac{960 \, {\left(\cos\left({\eta}\right) + 1\right)}}{{\left(a_{0}^{4} \cos\left({\eta}\right) + 7 \, a_{0}^{4}\right)} \sin\left({\eta}\right)^{6} - 64 \, a_{0}^{4} \cos\left({\eta}\right) - 8 \, {\left(3 \, a_{0}^{4} \cos\left({\eta}\right) + 7 \, a_{0}^{4}\right)} \sin\left({\eta}\right)^{4} - 64 \, a_{0}^{4} + 16 \, {\left(5 \, a_{0}^{4} \cos\left({\eta}\right) + 7 \, a_{0}^{4}\right)} \sin\left({\eta}\right)^{2}} \end{array}$$

K = K.expr().factor()
K

$$\renewcommand{\Bold}[1]{\mathbf{#1}}-\frac{960 \, {\left(\cos\left({\eta}\right) + 1\right)}}{{\left(\cos\left({\eta}\right) \sin\left({\eta}\right)^{6} + 7 \, \sin\left({\eta}\right)^{6} - 24 \, \cos\left({\eta}\right) \sin\left({\eta}\right)^{4} - 56 \, \sin\left({\eta}\right)^{4} + 80 \, \cos\left({\eta}\right) \sin\left({\eta}\right)^{2} + 112 \, \sin\left({\eta}\right)^{2} - 64 \, \cos\left({\eta}\right) - 64\right)} a_{0}^{4}}$$

K = K.subs({sin(et): sqrt(1 - cos(et)^2)}).factor()
K

$$\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{960}{a_{0}^{4} {\left(\cos\left({\eta}\right) + 1\right)}^{6}}$$

Ricci squared

Ric2 = Ric['_{ab}'] * Ric.up(g)['^{ab}']
Ric2.display()

$$\renewcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{llcl} & M & \longrightarrow & \mathbb{R} \\ & \left({\eta}, {\chi}, {\theta}, {\varphi}\right) & \longmapsto & -\frac{576}{a_{0}^{4} \sin\left({\eta}\right)^{6} - 32 \, a_{0}^{4} \cos\left({\eta}\right) - 6 \, {\left(a_{0}^{4} \cos\left({\eta}\right) + 3 \, a_{0}^{4}\right)} \sin\left({\eta}\right)^{4} - 32 \, a_{0}^{4} + 16 \, {\left(2 \, a_{0}^{4} \cos\left({\eta}\right) + 3 \, a_{0}^{4}\right)} \sin\left({\eta}\right)^{2}} \end{array}$$

S = Ric2.expr().factor()
S

$$\renewcommand{\Bold}[1]{\mathbf{#1}}-\frac{576}{{\left(\sin\left({\eta}\right)^{6} - 6 \, \cos\left({\eta}\right) \sin\left({\eta}\right)^{4} - 18 \, \sin\left({\eta}\right)^{4} + 32 \, \cos\left({\eta}\right) \sin\left({\eta}\right)^{2} + 48 \, \sin\left({\eta}\right)^{2} - 32 \, \cos\left({\eta}\right) - 32\right)} a_{0}^{4}}$$

S = S.subs({sin(et): sqrt(1 - cos(et)^2)})
S.factor()

$$\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{576}{a_{0}^{4} {\left(\cos\left({\eta}\right) + 1\right)}^{6}}$$