CoCalc Public FilesKerr Spacetime.ipynb
Author: Chan Park
Views : 253
Description: Kerr Spacetime
In [1]:
%display latex

In [2]:
M = Manifold(4, 'M', latex_name=r'\mathcal{M}', structure='Lorentzian'); M

$\mathcal{M}$
In [3]:
var('a', domain='real')
assume(a > 0)
assume(a < 1)

In [4]:
BL.<t,r,th,ph> = M.chart(r't r:(1+sqrt(1-a^2),+oo) th:(0,pi):\theta ph:(0,2*pi):\phi'); BL

$\left(\mathcal{M},(t, r, {\theta}, {\phi})\right)$
In [5]:
g = M.metric()

In [6]:
Sigma = r^2 + (a*cos(th))^2
Delta = r^2 - 2*r + a^2
g[0,0] = -(1-2*r/Sigma)
g[0,3] = -2*a*r*sin(th)^2/Sigma
g[1,1] = Sigma/Delta
g[2,2] = Sigma
g[3,3] = (r^2 + a^2 + 2*r*(a*sin(th))^2/Sigma)*sin(th)^2
g.display()

$g = \left( \frac{2 \, r}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} - 1 \right) \mathrm{d} t\otimes \mathrm{d} t + \left( -\frac{2 \, a r \sin\left({\theta}\right)^{2}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \right) \mathrm{d} t\otimes \mathrm{d} {\phi} + \left( \frac{a^{2} \cos\left({\theta}\right)^{2} + r^{2}}{a^{2} + r^{2} - 2 \, r} \right) \mathrm{d} r\otimes \mathrm{d} r + \left( a^{2} \cos\left({\theta}\right)^{2} + r^{2} \right) \mathrm{d} {\theta}\otimes \mathrm{d} {\theta} + \left( -\frac{2 \, a r \sin\left({\theta}\right)^{2}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \right) \mathrm{d} {\phi}\otimes \mathrm{d} t + {\left(\frac{2 \, a^{2} r \sin\left({\theta}\right)^{2}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} + a^{2} + r^{2}\right)} \sin\left({\theta}\right)^{2} \mathrm{d} {\phi}\otimes \mathrm{d} {\phi}$
In [7]:
g.christoffel_symbols_display()

$\begin{array}{lcl} \Gamma_{ \phantom{\, t} \, t \, r }^{ \, t \phantom{\, t} \phantom{\, r} } & = & -\frac{a^{4} - r^{4} - {\left(a^{4} + a^{2} r^{2}\right)} \sin\left({\theta}\right)^{2}}{a^{2} r^{4} + r^{6} - 2 \, r^{5} + {\left(a^{6} + a^{4} r^{2} - 2 \, a^{4} r\right)} \cos\left({\theta}\right)^{4} + 2 \, {\left(a^{4} r^{2} + a^{2} r^{4} - 2 \, a^{2} r^{3}\right)} \cos\left({\theta}\right)^{2}} \\ \Gamma_{ \phantom{\, t} \, t \, {\theta} }^{ \, t \phantom{\, t} \phantom{\, {\theta}} } & = & -\frac{2 \, a^{2} r \cos\left({\theta}\right) \sin\left({\theta}\right)}{a^{4} \cos\left({\theta}\right)^{4} + 2 \, a^{2} r^{2} \cos\left({\theta}\right)^{2} + r^{4}} \\ \Gamma_{ \phantom{\, t} \, r \, {\phi} }^{ \, t \phantom{\, r} \phantom{\, {\phi}} } & = & -\frac{{\left(a^{3} r^{2} + 3 \, a r^{4} - {\left(a^{5} - a^{3} r^{2}\right)} \cos\left({\theta}\right)^{2}\right)} \sin\left({\theta}\right)^{2}}{a^{2} r^{4} + r^{6} - 2 \, r^{5} + {\left(a^{6} + a^{4} r^{2} - 2 \, a^{4} r\right)} \cos\left({\theta}\right)^{4} + 2 \, {\left(a^{4} r^{2} + a^{2} r^{4} - 2 \, a^{2} r^{3}\right)} \cos\left({\theta}\right)^{2}} \\ \Gamma_{ \phantom{\, t} \, {\theta} \, {\phi} }^{ \, t \phantom{\, {\theta}} \phantom{\, {\phi}} } & = & -\frac{2 \, {\left(a^{5} r \cos\left({\theta}\right) \sin\left({\theta}\right)^{5} - {\left(a^{5} r + a^{3} r^{3}\right)} \cos\left({\theta}\right) \sin\left({\theta}\right)^{3}\right)}}{a^{6} \cos\left({\theta}\right)^{6} + 3 \, a^{4} r^{2} \cos\left({\theta}\right)^{4} + 3 \, a^{2} r^{4} \cos\left({\theta}\right)^{2} + r^{6}} \\ \Gamma_{ \phantom{\, r} \, t \, t }^{ \, r \phantom{\, t} \phantom{\, t} } & = & \frac{a^{2} r^{2} + r^{4} - 2 \, r^{3} - {\left(a^{4} + a^{2} r^{2} - 2 \, a^{2} r\right)} \cos\left({\theta}\right)^{2}}{a^{6} \cos\left({\theta}\right)^{6} + 3 \, a^{4} r^{2} \cos\left({\theta}\right)^{4} + 3 \, a^{2} r^{4} \cos\left({\theta}\right)^{2} + r^{6}} \\ \Gamma_{ \phantom{\, r} \, t \, {\phi} }^{ \, r \phantom{\, t} \phantom{\, {\phi}} } & = & -\frac{{\left(a^{3} r^{2} + a r^{4} - 2 \, a r^{3} - {\left(a^{5} + a^{3} r^{2} - 2 \, a^{3} r\right)} \cos\left({\theta}\right)^{2}\right)} \sin\left({\theta}\right)^{2}}{a^{6} \cos\left({\theta}\right)^{6} + 3 \, a^{4} r^{2} \cos\left({\theta}\right)^{4} + 3 \, a^{2} r^{4} \cos\left({\theta}\right)^{2} + r^{6}} \\ \Gamma_{ \phantom{\, r} \, r \, r }^{ \, r \phantom{\, r} \phantom{\, r} } & = & \frac{{\left(a^{2} r - a^{2}\right)} \sin\left({\theta}\right)^{2} + a^{2} - r^{2}}{a^{2} r^{2} + r^{4} - 2 \, r^{3} + {\left(a^{4} + a^{2} r^{2} - 2 \, a^{2} r\right)} \cos\left({\theta}\right)^{2}} \\ \Gamma_{ \phantom{\, r} \, r \, {\theta} }^{ \, r \phantom{\, r} \phantom{\, {\theta}} } & = & -\frac{a^{2} \cos\left({\theta}\right) \sin\left({\theta}\right)}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \\ \Gamma_{ \phantom{\, r} \, {\theta} \, {\theta} }^{ \, r \phantom{\, {\theta}} \phantom{\, {\theta}} } & = & -\frac{a^{2} r + r^{3} - 2 \, r^{2}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \\ \Gamma_{ \phantom{\, r} \, {\phi} \, {\phi} }^{ \, r \phantom{\, {\phi}} \phantom{\, {\phi}} } & = & \frac{{\left(a^{4} r^{2} + a^{2} r^{4} - 2 \, a^{2} r^{3} - {\left(a^{6} + a^{4} r^{2} - 2 \, a^{4} r\right)} \cos\left({\theta}\right)^{2}\right)} \sin\left({\theta}\right)^{4} - {\left(a^{2} r^{5} + r^{7} - 2 \, r^{6} + {\left(a^{6} r + a^{4} r^{3} - 2 \, a^{4} r^{2}\right)} \cos\left({\theta}\right)^{4} + 2 \, {\left(a^{4} r^{3} + a^{2} r^{5} - 2 \, a^{2} r^{4}\right)} \cos\left({\theta}\right)^{2}\right)} \sin\left({\theta}\right)^{2}}{a^{6} \cos\left({\theta}\right)^{6} + 3 \, a^{4} r^{2} \cos\left({\theta}\right)^{4} + 3 \, a^{2} r^{4} \cos\left({\theta}\right)^{2} + r^{6}} \\ \Gamma_{ \phantom{\, {\theta}} \, t \, t }^{ \, {\theta} \phantom{\, t} \phantom{\, t} } & = & -\frac{2 \, a^{2} r \cos\left({\theta}\right) \sin\left({\theta}\right)}{a^{6} \cos\left({\theta}\right)^{6} + 3 \, a^{4} r^{2} \cos\left({\theta}\right)^{4} + 3 \, a^{2} r^{4} \cos\left({\theta}\right)^{2} + r^{6}} \\ \Gamma_{ \phantom{\, {\theta}} \, t \, {\phi} }^{ \, {\theta} \phantom{\, t} \phantom{\, {\phi}} } & = & \frac{2 \, {\left(a^{3} r + a r^{3}\right)} \cos\left({\theta}\right) \sin\left({\theta}\right)}{a^{6} \cos\left({\theta}\right)^{6} + 3 \, a^{4} r^{2} \cos\left({\theta}\right)^{4} + 3 \, a^{2} r^{4} \cos\left({\theta}\right)^{2} + r^{6}} \\ \Gamma_{ \phantom{\, {\theta}} \, r \, r }^{ \, {\theta} \phantom{\, r} \phantom{\, r} } & = & \frac{a^{2} \cos\left({\theta}\right) \sin\left({\theta}\right)}{a^{2} r^{2} + r^{4} - 2 \, r^{3} + {\left(a^{4} + a^{2} r^{2} - 2 \, a^{2} r\right)} \cos\left({\theta}\right)^{2}} \\ \Gamma_{ \phantom{\, {\theta}} \, r \, {\theta} }^{ \, {\theta} \phantom{\, r} \phantom{\, {\theta}} } & = & \frac{r}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \\ \Gamma_{ \phantom{\, {\theta}} \, {\theta} \, {\theta} }^{ \, {\theta} \phantom{\, {\theta}} \phantom{\, {\theta}} } & = & -\frac{a^{2} \cos\left({\theta}\right) \sin\left({\theta}\right)}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \\ \Gamma_{ \phantom{\, {\theta}} \, {\phi} \, {\phi} }^{ \, {\theta} \phantom{\, {\phi}} \phantom{\, {\phi}} } & = & -\frac{{\left({\left(a^{6} + a^{4} r^{2} - 2 \, a^{4} r\right)} \cos\left({\theta}\right)^{5} + 2 \, {\left(a^{4} r^{2} + a^{2} r^{4} - 2 \, a^{2} r^{3}\right)} \cos\left({\theta}\right)^{3} + {\left(a^{2} r^{4} + r^{6} + 2 \, a^{4} r + 4 \, a^{2} r^{3}\right)} \cos\left({\theta}\right)\right)} \sin\left({\theta}\right)}{a^{6} \cos\left({\theta}\right)^{6} + 3 \, a^{4} r^{2} \cos\left({\theta}\right)^{4} + 3 \, a^{2} r^{4} \cos\left({\theta}\right)^{2} + r^{6}} \\ \Gamma_{ \phantom{\, {\phi}} \, t \, r }^{ \, {\phi} \phantom{\, t} \phantom{\, r} } & = & -\frac{a^{3} \cos\left({\theta}\right)^{2} - a r^{2}}{a^{2} r^{4} + r^{6} - 2 \, r^{5} + {\left(a^{6} + a^{4} r^{2} - 2 \, a^{4} r\right)} \cos\left({\theta}\right)^{4} + 2 \, {\left(a^{4} r^{2} + a^{2} r^{4} - 2 \, a^{2} r^{3}\right)} \cos\left({\theta}\right)^{2}} \\ \Gamma_{ \phantom{\, {\phi}} \, t \, {\theta} }^{ \, {\phi} \phantom{\, t} \phantom{\, {\theta}} } & = & -\frac{2 \, a r \cos\left({\theta}\right)}{{\left(a^{4} \cos\left({\theta}\right)^{4} + 2 \, a^{2} r^{2} \cos\left({\theta}\right)^{2} + r^{4}\right)} \sin\left({\theta}\right)} \\ \Gamma_{ \phantom{\, {\phi}} \, r \, {\phi} }^{ \, {\phi} \phantom{\, r} \phantom{\, {\phi}} } & = & \frac{r^{5} + {\left(a^{4} r - a^{4}\right)} \cos\left({\theta}\right)^{4} - a^{2} r^{2} - 2 \, r^{4} + {\left(2 \, a^{2} r^{3} + a^{4} - a^{2} r^{2}\right)} \cos\left({\theta}\right)^{2}}{a^{2} r^{4} + r^{6} - 2 \, r^{5} + {\left(a^{6} + a^{4} r^{2} - 2 \, a^{4} r\right)} \cos\left({\theta}\right)^{4} + 2 \, {\left(a^{4} r^{2} + a^{2} r^{4} - 2 \, a^{2} r^{3}\right)} \cos\left({\theta}\right)^{2}} \\ \Gamma_{ \phantom{\, {\phi}} \, {\theta} \, {\phi} }^{ \, {\phi} \phantom{\, {\theta}} \phantom{\, {\phi}} } & = & \frac{a^{4} \cos\left({\theta}\right)^{5} + 2 \, {\left(a^{2} r^{2} - a^{2} r\right)} \cos\left({\theta}\right)^{3} + {\left(r^{4} + 2 \, a^{2} r\right)} \cos\left({\theta}\right)}{{\left(a^{4} \cos\left({\theta}\right)^{4} + 2 \, a^{2} r^{2} \cos\left({\theta}\right)^{2} + r^{4}\right)} \sin\left({\theta}\right)} \end{array}$
In [ ]: