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Kernel: SageMath (stable)

Extended rotating Hayward metric

This Jupyter/SageMath notebook is related to the article Lamy et al, arXiv:1802.01635.

The metric is that obtained by Bambi & Modesto, Phys. Lett. B 721, 329 (2013) by applying the Newman-Janis transformation to the (non-rotating) Hayward metric for regular black holes (Hayward, PRL 96, 031103 (2006)), extended to cover the region r<0r<0.

version()
'SageMath version 8.1, Release Date: 2017-12-07'
%display latex

To speed up the computation of the Riemann tensor, we ask for parallel computations on 8 threads:

Parallelism().set(nproc=1) # use nproc=1 on CoCalc
M = Manifold(4, 'M') print(M)
4-dimensional differentiable manifold M
M1 = M.open_subset('M_1') XBL1.<t,r,th,ph> = M1.chart(r't r:(0,+oo) th:(0,pi):\theta ph:(0,2*pi):\phi') XBL1
(M1,(t,r,θ,ϕ))\renewcommand{\Bold}[1]{\mathbf{#1}}\left(M_1,(t, r, {\theta}, {\phi})\right)
M2 = M.open_subset('M_2') forget(r>0) XBL2.<t,r,th,ph> = M2.chart(r't r:(-oo,0) th:(0,pi):\theta ph:(0,2*pi):\phi') M._top_charts.append(XBL2) XBL2
(M2,(t,r,θ,ϕ))\renewcommand{\Bold}[1]{\mathbf{#1}}\left(M_2,(t, r, {\theta}, {\phi})\right)
forget(r<0) assumptions()
[txisxreal,rxisxreal,thxisxreal,θ>0,θ[removed]0,ϕ<2π]\renewcommand{\Bold}[1]{\mathbf{#1}}\left[\verb|t|\phantom{\verb!x!}\verb|is|\phantom{\verb!x!}\verb|real|, \verb|r|\phantom{\verb!x!}\verb|is|\phantom{\verb!x!}\verb|real|, \verb|th|\phantom{\verb!x!}\verb|is|\phantom{\verb!x!}\verb|real|, {\theta} > 0, {\theta} [removed] 0, {\phi} < 2 \, \pi\right]
g = M.lorentzian_metric('g')
a, b = var('a b') Sigma = r^2 + a^2*cos(th)^2
m = r^3/(r^3 + 2*b^2) m1 = m Delta = r^2 - 2*m*r + a^2 g1 = g.restrict(M1) g1[0,0] = -(1 - 2*r*m/Sigma) g1[0,3] = -2*a*r*sin(th)^2*m/Sigma g1[1,1] = Sigma/Delta g1[2,2] = Sigma g1[3,3] = (r^2 + a^2 + 2*r*(a*sin(th))^2*m/Sigma)*sin(th)^2 g.display(XBL1.frame())
g=(2r4(a2cos(θ)2+r2)(r3+2b2)1)dtdt2ar4sin(θ)2(a2cos(θ)2+r2)(r3+2b2)dtdϕ+(a2cos(θ)2+r22r4r3+2b2a2r2)drdr+(a2cos(θ)2+r2)dθdθ2ar4sin(θ)2(a2cos(θ)2+r2)(r3+2b2)dϕdt+(2a2r4sin(θ)2(a2cos(θ)2+r2)(r3+2b2)+a2+r2)sin(θ)2dϕdϕ\renewcommand{\Bold}[1]{\mathbf{#1}}g = \left( \frac{2 \, r^{4}}{{\left(a^{2} \cos\left({\theta}\right)^{2} + r^{2}\right)} {\left(r^{3} + 2 \, b^{2}\right)}} - 1 \right) \mathrm{d} t\otimes \mathrm{d} t -\frac{2 \, a r^{4} \sin\left({\theta}\right)^{2}}{{\left(a^{2} \cos\left({\theta}\right)^{2} + r^{2}\right)} {\left(r^{3} + 2 \, b^{2}\right)}} \mathrm{d} t\otimes \mathrm{d} {\phi} + \left( -\frac{a^{2} \cos\left({\theta}\right)^{2} + r^{2}}{\frac{2 \, r^{4}}{r^{3} + 2 \, b^{2}} - a^{2} - 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} -\frac{2 \, a r^{4} \sin\left({\theta}\right)^{2}}{{\left(a^{2} \cos\left({\theta}\right)^{2} + r^{2}\right)} {\left(r^{3} + 2 \, b^{2}\right)}} \mathrm{d} {\phi}\otimes \mathrm{d} t + {\left(\frac{2 \, a^{2} r^{4} \sin\left({\theta}\right)^{2}}{{\left(a^{2} \cos\left({\theta}\right)^{2} + r^{2}\right)} {\left(r^{3} + 2 \, b^{2}\right)}} + a^{2} + r^{2}\right)} \sin\left({\theta}\right)^{2} \mathrm{d} {\phi}\otimes \mathrm{d} {\phi}
g.display_comp(XBL1.frame())
gtttt=2r4(a2cos(θ)2+r2)(r3+2b2)1gtϕtϕ=2ar4sin(θ)2(a2cos(θ)2+r2)(r3+2b2)grrrr=a2cos(θ)2+r22r4r3+2b2a2r2gθθθθ=a2cos(θ)2+r2gϕtϕt=2ar4sin(θ)2(a2cos(θ)2+r2)(r3+2b2)gϕϕϕϕ=(2a2r4sin(θ)2(a2cos(θ)2+r2)(r3+2b2)+a2+r2)sin(θ)2\renewcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{lcl} g_{ \, t \, t }^{ \phantom{\, t}\phantom{\, t} } & = & \frac{2 \, r^{4}}{{\left(a^{2} \cos\left({\theta}\right)^{2} + r^{2}\right)} {\left(r^{3} + 2 \, b^{2}\right)}} - 1 \\ g_{ \, t \, {\phi} }^{ \phantom{\, t}\phantom{\, {\phi}} } & = & -\frac{2 \, a r^{4} \sin\left({\theta}\right)^{2}}{{\left(a^{2} \cos\left({\theta}\right)^{2} + r^{2}\right)} {\left(r^{3} + 2 \, b^{2}\right)}} \\ g_{ \, r \, r }^{ \phantom{\, r}\phantom{\, r} } & = & -\frac{a^{2} \cos\left({\theta}\right)^{2} + r^{2}}{\frac{2 \, r^{4}}{r^{3} + 2 \, b^{2}} - a^{2} - r^{2}} \\ g_{ \, {\theta} \, {\theta} }^{ \phantom{\, {\theta}}\phantom{\, {\theta}} } & = & a^{2} \cos\left({\theta}\right)^{2} + r^{2} \\ g_{ \, {\phi} \, t }^{ \phantom{\, {\phi}}\phantom{\, t} } & = & -\frac{2 \, a r^{4} \sin\left({\theta}\right)^{2}}{{\left(a^{2} \cos\left({\theta}\right)^{2} + r^{2}\right)} {\left(r^{3} + 2 \, b^{2}\right)}} \\ g_{ \, {\phi} \, {\phi} }^{ \phantom{\, {\phi}}\phantom{\, {\phi}} } & = & {\left(\frac{2 \, a^{2} r^{4} \sin\left({\theta}\right)^{2}}{{\left(a^{2} \cos\left({\theta}\right)^{2} + r^{2}\right)} {\left(r^{3} + 2 \, b^{2}\right)}} + a^{2} + r^{2}\right)} \sin\left({\theta}\right)^{2} \end{array}
m = -r^3/(-r^3 + 2*b^2) m2 = m Delta = r^2 - 2*m*r + a^2 g2 = g.restrict(M2) g2[0,0] = -(1 - 2*r*m/Sigma) g2[0,3] = -2*a*r*sin(th)^2*m/Sigma g2[1,1] = Sigma/Delta g2[2,2] = Sigma g2[3,3] = (r^2 + a^2 + 2*r*(a*sin(th))^2*m/Sigma)*sin(th)^2 g.display(XBL2.frame())
g=(2r4(a2cos(θ)2+r2)(r32b2)1)dtdt2ar4sin(θ)2(a2cos(θ)2+r2)(r32b2)dtdϕ+(a2cos(θ)2+r22r4r32b2a2r2)drdr+(a2cos(θ)2+r2)dθdθ2ar4sin(θ)2(a2cos(θ)2+r2)(r32b2)dϕdt+(2a2r4sin(θ)2(a2cos(θ)2+r2)(r32b2)+a2+r2)sin(θ)2dϕdϕ\renewcommand{\Bold}[1]{\mathbf{#1}}g = \left( \frac{2 \, r^{4}}{{\left(a^{2} \cos\left({\theta}\right)^{2} + r^{2}\right)} {\left(r^{3} - 2 \, b^{2}\right)}} - 1 \right) \mathrm{d} t\otimes \mathrm{d} t -\frac{2 \, a r^{4} \sin\left({\theta}\right)^{2}}{{\left(a^{2} \cos\left({\theta}\right)^{2} + r^{2}\right)} {\left(r^{3} - 2 \, b^{2}\right)}} \mathrm{d} t\otimes \mathrm{d} {\phi} + \left( -\frac{a^{2} \cos\left({\theta}\right)^{2} + r^{2}}{\frac{2 \, r^{4}}{r^{3} - 2 \, b^{2}} - a^{2} - 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} -\frac{2 \, a r^{4} \sin\left({\theta}\right)^{2}}{{\left(a^{2} \cos\left({\theta}\right)^{2} + r^{2}\right)} {\left(r^{3} - 2 \, b^{2}\right)}} \mathrm{d} {\phi}\otimes \mathrm{d} t + {\left(\frac{2 \, a^{2} r^{4} \sin\left({\theta}\right)^{2}}{{\left(a^{2} \cos\left({\theta}\right)^{2} + r^{2}\right)} {\left(r^{3} - 2 \, b^{2}\right)}} + a^{2} + r^{2}\right)} \sin\left({\theta}\right)^{2} \mathrm{d} {\phi}\otimes \mathrm{d} {\phi}
g.display_comp(XBL2.frame())
gtttt=2r4(a2cos(θ)2+r2)(r32b2)1gtϕtϕ=2ar4sin(θ)2(a2cos(θ)2+r2)(r32b2)grrrr=a2cos(θ)2+r22r4r32b2a2r2gθθθθ=a2cos(θ)2+r2gϕtϕt=2ar4sin(θ)2(a2cos(θ)2+r2)(r32b2)gϕϕϕϕ=(2a2r4sin(θ)2(a2cos(θ)2+r2)(r32b2)+a2+r2)sin(θ)2\renewcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{lcl} g_{ \, t \, t }^{ \phantom{\, t}\phantom{\, t} } & = & \frac{2 \, r^{4}}{{\left(a^{2} \cos\left({\theta}\right)^{2} + r^{2}\right)} {\left(r^{3} - 2 \, b^{2}\right)}} - 1 \\ g_{ \, t \, {\phi} }^{ \phantom{\, t}\phantom{\, {\phi}} } & = & -\frac{2 \, a r^{4} \sin\left({\theta}\right)^{2}}{{\left(a^{2} \cos\left({\theta}\right)^{2} + r^{2}\right)} {\left(r^{3} - 2 \, b^{2}\right)}} \\ g_{ \, r \, r }^{ \phantom{\, r}\phantom{\, r} } & = & -\frac{a^{2} \cos\left({\theta}\right)^{2} + r^{2}}{\frac{2 \, r^{4}}{r^{3} - 2 \, b^{2}} - a^{2} - r^{2}} \\ g_{ \, {\theta} \, {\theta} }^{ \phantom{\, {\theta}}\phantom{\, {\theta}} } & = & a^{2} \cos\left({\theta}\right)^{2} + r^{2} \\ g_{ \, {\phi} \, t }^{ \phantom{\, {\phi}}\phantom{\, t} } & = & -\frac{2 \, a r^{4} \sin\left({\theta}\right)^{2}}{{\left(a^{2} \cos\left({\theta}\right)^{2} + r^{2}\right)} {\left(r^{3} - 2 \, b^{2}\right)}} \\ g_{ \, {\phi} \, {\phi} }^{ \phantom{\, {\phi}}\phantom{\, {\phi}} } & = & {\left(\frac{2 \, a^{2} r^{4} \sin\left({\theta}\right)^{2}}{{\left(a^{2} \cos\left({\theta}\right)^{2} + r^{2}\right)} {\left(r^{3} - 2 \, b^{2}\right)}} + a^{2} + r^{2}\right)} \sin\left({\theta}\right)^{2} \end{array}
graph = plot(m1.subs(b=1), (r, 0, 8), axes_labels=[r'$r/m$', r'$M(r)/m$'], gridlines=True) graph += plot(m2.subs(b=1), (r, -8, 0)) graph
Image in a Jupyter notebook
gm1 = g.inverse() gm1.display(XBL1.frame())
g1=(2a2r4sin(θ)2+a2r5+r7+2a2b2r2+2b2r4+(a4r3+a2r5+2a4b2+2a2b2r2)cos(θ)2a2r5+r7+2a2b2r2+2b2r42r6+(a4r3+a2r5+2a4b2+2a2b2r22a2r4)cos(θ)2)tt+(2ar4a2r5+r7+2a2b2r2+2b2r42r6+(a4r3+a2r5+2a4b2+2a2b2r22a2r4)cos(θ)2)tϕ+(a2r3+r5+2a2b2+2b2r22r4r5+2b2r2+(a2r3+2a2b2)cos(θ)2)rr+(1a2cos(θ)2+r2)θθ+(2ar4a2r5+r7+2a2b2r2+2b2r42r6+(a4r3+a2r5+2a4b2+2a2b2r22a2r4)cos(θ)2)ϕt+(r5+2b2r22r4+(a2r3+2a2b2)cos(θ)22a2r4sin(θ)4+(a2r5+r7+2a2b2r22r62(a2b2)r4+(a4r3+a2r5+2a4b2+2a2b2r2)cos(θ)2)sin(θ)2)ϕϕ\renewcommand{\Bold}[1]{\mathbf{#1}}g^{-1} = \left( -\frac{2 \, a^{2} r^{4} \sin\left({\theta}\right)^{2} + a^{2} r^{5} + r^{7} + 2 \, a^{2} b^{2} r^{2} + 2 \, b^{2} r^{4} + {\left(a^{4} r^{3} + a^{2} r^{5} + 2 \, a^{4} b^{2} + 2 \, a^{2} b^{2} r^{2}\right)} \cos\left({\theta}\right)^{2}}{a^{2} r^{5} + r^{7} + 2 \, a^{2} b^{2} r^{2} + 2 \, b^{2} r^{4} - 2 \, r^{6} + {\left(a^{4} r^{3} + a^{2} r^{5} + 2 \, a^{4} b^{2} + 2 \, a^{2} b^{2} r^{2} - 2 \, a^{2} r^{4}\right)} \cos\left({\theta}\right)^{2}} \right) \frac{\partial}{\partial t }\otimes \frac{\partial}{\partial t } + \left( -\frac{2 \, a r^{4}}{a^{2} r^{5} + r^{7} + 2 \, a^{2} b^{2} r^{2} + 2 \, b^{2} r^{4} - 2 \, r^{6} + {\left(a^{4} r^{3} + a^{2} r^{5} + 2 \, a^{4} b^{2} + 2 \, a^{2} b^{2} r^{2} - 2 \, a^{2} r^{4}\right)} \cos\left({\theta}\right)^{2}} \right) \frac{\partial}{\partial t }\otimes \frac{\partial}{\partial {\phi} } + \left( \frac{a^{2} r^{3} + r^{5} + 2 \, a^{2} b^{2} + 2 \, b^{2} r^{2} - 2 \, r^{4}}{r^{5} + 2 \, b^{2} r^{2} + {\left(a^{2} r^{3} + 2 \, a^{2} b^{2}\right)} \cos\left({\theta}\right)^{2}} \right) \frac{\partial}{\partial r }\otimes \frac{\partial}{\partial r } + \left( \frac{1}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \right) \frac{\partial}{\partial {\theta} }\otimes \frac{\partial}{\partial {\theta} } + \left( -\frac{2 \, a r^{4}}{a^{2} r^{5} + r^{7} + 2 \, a^{2} b^{2} r^{2} + 2 \, b^{2} r^{4} - 2 \, r^{6} + {\left(a^{4} r^{3} + a^{2} r^{5} + 2 \, a^{4} b^{2} + 2 \, a^{2} b^{2} r^{2} - 2 \, a^{2} r^{4}\right)} \cos\left({\theta}\right)^{2}} \right) \frac{\partial}{\partial {\phi} }\otimes \frac{\partial}{\partial t } + \left( \frac{r^{5} + 2 \, b^{2} r^{2} - 2 \, r^{4} + {\left(a^{2} r^{3} + 2 \, a^{2} b^{2}\right)} \cos\left({\theta}\right)^{2}}{2 \, a^{2} r^{4} \sin\left({\theta}\right)^{4} + {\left(a^{2} r^{5} + r^{7} + 2 \, a^{2} b^{2} r^{2} - 2 \, r^{6} - 2 \, {\left(a^{2} - b^{2}\right)} r^{4} + {\left(a^{4} r^{3} + a^{2} r^{5} + 2 \, a^{4} b^{2} + 2 \, a^{2} b^{2} r^{2}\right)} \cos\left({\theta}\right)^{2}\right)} \sin\left({\theta}\right)^{2}} \right) \frac{\partial}{\partial {\phi} }\otimes \frac{\partial}{\partial {\phi} }
gm1.display(XBL2.frame())
g1=(2a2r4sin(θ)2+a2r5+r72a2b2r22b2r4+(a4r3+a2r52a4b22a2b2r2)cos(θ)2a2r5+r72a2b2r22b2r42r6+(a4r3+a2r52a4b22a2b2r22a2r4)cos(θ)2)tt+(2ar4a2r5+r72a2b2r22b2r42r6+(a4r3+a2r52a4b22a2b2r22a2r4)cos(θ)2)tϕ+(a2r3+r52a2b22b2r22r4r52b2r2+(a2r32a2b2)cos(θ)2)rr+(1a2cos(θ)2+r2)θθ+(2ar4a2r5+r72a2b2r22b2r42r6+(a4r3+a2r52a4b22a2b2r22a2r4)cos(θ)2)ϕt+(r52b2r22r4+(a2r32a2b2)cos(θ)22a2r4sin(θ)4+(a2r5+r72a2b2r22r62(a2+b2)r4+(a4r3+a2r52a4b22a2b2r2)cos(θ)2)sin(θ)2)ϕϕ\renewcommand{\Bold}[1]{\mathbf{#1}}g^{-1} = \left( -\frac{2 \, a^{2} r^{4} \sin\left({\theta}\right)^{2} + a^{2} r^{5} + r^{7} - 2 \, a^{2} b^{2} r^{2} - 2 \, b^{2} r^{4} + {\left(a^{4} r^{3} + a^{2} r^{5} - 2 \, a^{4} b^{2} - 2 \, a^{2} b^{2} r^{2}\right)} \cos\left({\theta}\right)^{2}}{a^{2} r^{5} + r^{7} - 2 \, a^{2} b^{2} r^{2} - 2 \, b^{2} r^{4} - 2 \, r^{6} + {\left(a^{4} r^{3} + a^{2} r^{5} - 2 \, a^{4} b^{2} - 2 \, a^{2} b^{2} r^{2} - 2 \, a^{2} r^{4}\right)} \cos\left({\theta}\right)^{2}} \right) \frac{\partial}{\partial t }\otimes \frac{\partial}{\partial t } + \left( -\frac{2 \, a r^{4}}{a^{2} r^{5} + r^{7} - 2 \, a^{2} b^{2} r^{2} - 2 \, b^{2} r^{4} - 2 \, r^{6} + {\left(a^{4} r^{3} + a^{2} r^{5} - 2 \, a^{4} b^{2} - 2 \, a^{2} b^{2} r^{2} - 2 \, a^{2} r^{4}\right)} \cos\left({\theta}\right)^{2}} \right) \frac{\partial}{\partial t }\otimes \frac{\partial}{\partial {\phi} } + \left( \frac{a^{2} r^{3} + r^{5} - 2 \, a^{2} b^{2} - 2 \, b^{2} r^{2} - 2 \, r^{4}}{r^{5} - 2 \, b^{2} r^{2} + {\left(a^{2} r^{3} - 2 \, a^{2} b^{2}\right)} \cos\left({\theta}\right)^{2}} \right) \frac{\partial}{\partial r }\otimes \frac{\partial}{\partial r } + \left( \frac{1}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \right) \frac{\partial}{\partial {\theta} }\otimes \frac{\partial}{\partial {\theta} } + \left( -\frac{2 \, a r^{4}}{a^{2} r^{5} + r^{7} - 2 \, a^{2} b^{2} r^{2} - 2 \, b^{2} r^{4} - 2 \, r^{6} + {\left(a^{4} r^{3} + a^{2} r^{5} - 2 \, a^{4} b^{2} - 2 \, a^{2} b^{2} r^{2} - 2 \, a^{2} r^{4}\right)} \cos\left({\theta}\right)^{2}} \right) \frac{\partial}{\partial {\phi} }\otimes \frac{\partial}{\partial t } + \left( \frac{r^{5} - 2 \, b^{2} r^{2} - 2 \, r^{4} + {\left(a^{2} r^{3} - 2 \, a^{2} b^{2}\right)} \cos\left({\theta}\right)^{2}}{2 \, a^{2} r^{4} \sin\left({\theta}\right)^{4} + {\left(a^{2} r^{5} + r^{7} - 2 \, a^{2} b^{2} r^{2} - 2 \, r^{6} - 2 \, {\left(a^{2} + b^{2}\right)} r^{4} + {\left(a^{4} r^{3} + a^{2} r^{5} - 2 \, a^{4} b^{2} - 2 \, a^{2} b^{2} r^{2}\right)} \cos\left({\theta}\right)^{2}\right)} \sin\left({\theta}\right)^{2}} \right) \frac{\partial}{\partial {\phi} }\otimes \frac{\partial}{\partial {\phi} }
#g.christoffel_symbols_display(XBL1)
#gam = g.christoffel_symbols(XBL1) #for i in range(4): # for j in range(4): # for k in range(j,4): # print("---------------------") # print("Gamma^{}_{}{} for r >=0:".format(i,j,k)) # if gam[i,j,k] == 0: # print(0) # else: # show(gam[i,j,k].expr().factor()) # print(gam[i,j,k].expr().factor())
#g.christoffel_symbols_display(XBL2)
#gam = g.christoffel_symbols(XBL2) #for i in range(4): # for j in range(4): # for k in range(j,4): # print("--------------------") # print("Gamma^{}_{}{} for r < 0:".format(i,j,k)) # if gam[i,j,k] == 0: # print(0) # else: # show(gam[i,j,k].expr().factor()) # print(gam[i,j,k].expr().factor())

gttg_{tt} component

g.restrict(M1)[0,0]
2r4(a2cos(θ)2+r2)(r3+2b2)1\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{2 \, r^{4}}{{\left(a^{2} \cos\left({\theta}\right)^{2} + r^{2}\right)} {\left(r^{3} + 2 \, b^{2}\right)}} - 1
g.restrict(M2)[0,0]
2r4(a2cos(θ)2+r2)(r32b2)1\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{2 \, r^{4}}{{\left(a^{2} \cos\left({\theta}\right)^{2} + r^{2}\right)} {\left(r^{3} - 2 \, b^{2}\right)}} - 1
def plot_profile(field, param_values, rmin, rmax, y_label, comp=None, ymin=None, ymax=None, gridlines=True): graph = Graphics() rmin0 = max(rmin, 0) rmax0 = min(rmax, 0) if rmax > 0: f1 = field.restrict(M1) if comp is not None: f1 = f1[comp].expr() else: f1 = f1.expr() f1 = f1.subs(param_values) graph += plot(f1.subs(th=0), (r, rmin0, rmax), color='lightblue', legend_label=r'$\theta=0$', axes_labels = [r'$r/m$', y_label], ymin=ymin, ymax=ymax, gridlines=gridlines) graph += plot(f1.subs(th=pi/4), (r, rmin0, rmax), color='magenta', legend_label=r'$\theta=\pi/4$') graph += plot(f1.subs(th=pi/2), (r, rmin0, rmax), color='blue', legend_label=r'$\theta=\pi/2$') if rmin < 0: f1 = field.restrict(M2) if comp is not None: f1 = f1[comp].expr() else: f1 = f1.expr() f1 = f1.subs(param_values) graph += plot(f1.subs(th=0), (r, rmin, rmax0), color='lightblue', axes_labels = [r'$r/m$', y_label], ymin=ymin, ymax=ymax) graph += plot(f1.subs(th=pi/4), (r, rmin, rmax0), color='magenta') graph += plot(f1.subs(th=pi/2), (r, rmin, rmax0), color='blue') return graph
graph = plot_profile(g, {a: 0.9, b: 1}, -8, 8, r'$g_{tt}$', comp=(0,0)) graph.save('g_tt.pdf') graph
Image in a Jupyter notebook

Lapse function

The lapse function NN is deduced from the standard formula gtt=1/N2g^{tt} = - 1/N^2:

M.top_charts()
[(M1,(t,r,θ,ϕ)),(M2,(t,r,θ,ϕ))]\renewcommand{\Bold}[1]{\mathbf{#1}}\left[\left(M_1,(t, r, {\theta}, {\phi})\right), \left(M_2,(t, r, {\theta}, {\phi})\right)\right]
NN = M.scalar_field(coord_expression={XBL1: sqrt(- 1 / g.restrict(M1).inverse()[0,0]), XBL2: sqrt(- 1 / g.restrict(M2).inverse()[0,0])}, name='N') NN.display()
N:MRon M1:(t,r,θ,ϕ)a2r3+r5+2a2b2+2b2r22r4a2cos(θ)2+r22a2r4sin(θ)2+a2r5+r7+2a2b2r2+2b2r4+(a4r3+a2r5+2a4b2+2a2b2r2)cos(θ)2on M2:(t,r,θ,ϕ)a2r3+r52a2b22b2r22r4a2cos(θ)2+r22a2r4sin(θ)2+a2r5+r72a2b2r22b2r4+(a4r3+a2r52a4b22a2b2r2)cos(θ)2\renewcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{llcl} N:& M & \longrightarrow & \mathbb{R} \\ \mbox{on}\ M_1 : & \left(t, r, {\theta}, {\phi}\right) & \longmapsto & \frac{\sqrt{a^{2} r^{3} + r^{5} + 2 \, a^{2} b^{2} + 2 \, b^{2} r^{2} - 2 \, r^{4}} \sqrt{a^{2} \cos\left({\theta}\right)^{2} + r^{2}}}{\sqrt{2 \, a^{2} r^{4} \sin\left({\theta}\right)^{2} + a^{2} r^{5} + r^{7} + 2 \, a^{2} b^{2} r^{2} + 2 \, b^{2} r^{4} + {\left(a^{4} r^{3} + a^{2} r^{5} + 2 \, a^{4} b^{2} + 2 \, a^{2} b^{2} r^{2}\right)} \cos\left({\theta}\right)^{2}}} \\ \mbox{on}\ M_2 : & \left(t, r, {\theta}, {\phi}\right) & \longmapsto & \frac{\sqrt{a^{2} r^{3} + r^{5} - 2 \, a^{2} b^{2} - 2 \, b^{2} r^{2} - 2 \, r^{4}} \sqrt{a^{2} \cos\left({\theta}\right)^{2} + r^{2}}}{\sqrt{2 \, a^{2} r^{4} \sin\left({\theta}\right)^{2} + a^{2} r^{5} + r^{7} - 2 \, a^{2} b^{2} r^{2} - 2 \, b^{2} r^{4} + {\left(a^{4} r^{3} + a^{2} r^{5} - 2 \, a^{4} b^{2} - 2 \, a^{2} b^{2} r^{2}\right)} \cos\left({\theta}\right)^{2}}} \end{array}
graph = plot_profile(NN, {a: 0.9, b: 1}, -8, 8, r'$N$') graph.save('lapse.pdf') graph
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gtϕg_{t\phi} component

graph = plot_profile(g, {a: 0.9, b: 1}, -8, 8, r'$g_{t\phi}$', comp=(0,3)) graph.save('g_tphi.pdf') graph
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Other metric components

graph = plot_profile(g, {a: 2, b: 0.5}, -4, 4, r'$g_{\phi\phi}$', comp=(3,3)) graph.save('g_phiphi.pdf') graph
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g.restrict(M1)[3,3].expr().factor()
(a4r3cos(θ)2+a2r5cos(θ)2+2a4b2cos(θ)2+2a2b2r2cos(θ)2+2a2r4sin(θ)2+a2r5+r7+2a2b2r2+2b2r4)sin(θ)2(a2cos(θ)2+r2)(r3+2b2)\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{{\left(a^{4} r^{3} \cos\left({\theta}\right)^{2} + a^{2} r^{5} \cos\left({\theta}\right)^{2} + 2 \, a^{4} b^{2} \cos\left({\theta}\right)^{2} + 2 \, a^{2} b^{2} r^{2} \cos\left({\theta}\right)^{2} + 2 \, a^{2} r^{4} \sin\left({\theta}\right)^{2} + a^{2} r^{5} + r^{7} + 2 \, a^{2} b^{2} r^{2} + 2 \, b^{2} r^{4}\right)} \sin\left({\theta}\right)^{2}}{{\left(a^{2} \cos\left({\theta}\right)^{2} + r^{2}\right)} {\left(r^{3} + 2 \, b^{2}\right)}}
g.restrict(M2)[3,3].expr().factor()
(a4r3cos(θ)2+a2r5cos(θ)22a4b2cos(θ)22a2b2r2cos(θ)2+2a2r4sin(θ)2+a2r5+r72a2b2r22b2r4)sin(θ)2(a2cos(θ)2+r2)(r32b2)\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{{\left(a^{4} r^{3} \cos\left({\theta}\right)^{2} + a^{2} r^{5} \cos\left({\theta}\right)^{2} - 2 \, a^{4} b^{2} \cos\left({\theta}\right)^{2} - 2 \, a^{2} b^{2} r^{2} \cos\left({\theta}\right)^{2} + 2 \, a^{2} r^{4} \sin\left({\theta}\right)^{2} + a^{2} r^{5} + r^{7} - 2 \, a^{2} b^{2} r^{2} - 2 \, b^{2} r^{4}\right)} \sin\left({\theta}\right)^{2}}{{\left(a^{2} \cos\left({\theta}\right)^{2} + r^{2}\right)} {\left(r^{3} - 2 \, b^{2}\right)}}
graph = plot_profile(g, {a: 0.9, b: 1}, -4, 4, r'$g_{rr}$', comp=(1,1)) graph.save('g_rr.pdf') graph
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graph = plot_profile(g, {a: 0.9, b: 1}, -2, 2, r'$g_{\theta\theta}$', comp=(2,2)) graph.save('g_thth.pdf') graph
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Ricci tensor

Ric = g.ricci() ; print(Ric)
Field of symmetric bilinear forms Ric(g) on the 4-dimensional differentiable manifold M
Ric.display_comp(XBL1.frame())
Ric(g)tttt=12(3a2b2r10+2b2r12+6a2b4r7+2b4r94b2r114b6r6+4b4r8+(a4b2r82a4b4r56a2b4r72a2b2r98a4b6r212a2b6r4+8a2b4r6)cos(θ)2)r18+8b2r15+24b4r12+32b6r9+16b8r6+(a6r12+8a6b2r9+24a6b4r6+32a6b6r3+16a6b8)cos(θ)6+3(a4r14+8a4b2r11+24a4b4r8+32a4b6r5+16a4b8r2)cos(θ)4+3(a2r16+8a2b2r13+24a2b4r10+32a2b6r7+16a2b8r4)cos(θ)2Ric(g)tϕtϕ=12((a5b2r8+a3b2r102a5b4r52a3b4r72a3b2r98a5b6r28a3b6r4+8a3b4r6)sin(θ)4(4a3b2r10+3ab2r122a5b4r5+4a3b4r74ab2r118a5b6r28a3b6r4+8a3b4r62(a3b23ab4)r9+(a5b2+4ab4)r8)sin(θ)2)r18+8b2r15+24b4r12+32b6r9+16b8r6+(a6r12+8a6b2r9+24a6b4r6+32a6b6r3+16a6b8)cos(θ)6+3(a4r14+8a4b2r11+24a4b4r8+32a4b6r5+16a4b8r2)cos(θ)4+3(a2r16+8a2b2r13+24a2b4r10+32a2b6r7+16a2b8r4)cos(θ)2Ric(g)rrrr=12(2b2r72b4r4+(a2b2r54a2b4r2)cos(θ)2)a2r11+r13+6a2b2r8+6b2r102r12+12a2b4r5+12b4r78b2r9+8a2b6r2+8b6r48b4r6+(a4r9+a2r11+6a4b2r6+6a2b2r82a2r10+12a4b4r3+12a2b4r58a2b2r7+8a4b6+8a2b6r28a2b4r4)cos(θ)2Ric(g)θθθθ=12b2r4r8+4b2r5+4b4r2+(a2r6+4a2b2r3+4a2b4)cos(θ)2Ric(g)ϕtϕt=12((a5b2r8+a3b2r102a5b4r52a3b4r72a3b2r98a5b6r28a3b6r4+8a3b4r6)sin(θ)4(4a3b2r10+3ab2r122a5b4r5+4a3b4r74ab2r118a5b6r28a3b6r4+8a3b4r62(a3b23ab4)r9+(a5b2+4ab4)r8)sin(θ)2)r18+8b2r15+24b4r12+32b6r9+16b8r6+(a6r12+8a6b2r9+24a6b4r6+32a6b6r3+16a6b8)cos(θ)6+3(a4r14+8a4b2r11+24a4b4r8+32a4b6r5+16a4b8r2)cos(θ)4+3(a2r16+8a2b2r13+24a2b4r10+32a2b6r7+16a2b8r4)cos(θ)2Ric(g)ϕϕϕϕ=12(3a4b2r10+4a2b2r12+b2r14+6a4b4r7+10a2b4r9+4a2b6r64(a2b2b4)r11+4(a2b4+b6)r8+(a6b2r8+a4b2r102a6b4r52a4b4r72a4b2r98a6b6r28a4b6r4+8a4b4r6)cos(θ)6+2(a2b2r12+2a6b4r5+3a4b4r72a2b2r11+8a6b6r2+6a4b6r4+(2a4b2+a2b4)r9(a6b22a2b4)r82(4a4b4+a2b6)r6)cos(θ)4(4a4b2r10+6a2b2r12+b2r14+2a6b4r5+10a4b4r7+8a6b6r2+4a4b6r48a4b4r64(2a2b2b4)r11+2(a4b2+6a2b4)r9(a6b28a2b44b6)r8)cos(θ)2)r18+8b2r15+24b4r12+32b6r9+16b8r6+(a6r12+8a6b2r9+24a6b4r6+32a6b6r3+16a6b8)cos(θ)6+3(a4r14+8a4b2r11+24a4b4r8+32a4b6r5+16a4b8r2)cos(θ)4+3(a2r16+8a2b2r13+24a2b4r10+32a2b6r7+16a2b8r4)cos(θ)2\renewcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{lcl} \mathrm{Ric}\left(g\right)_{ \, t \, t }^{ \phantom{\, t}\phantom{\, t} } & = & \frac{12 \, {\left(3 \, a^{2} b^{2} r^{10} + 2 \, b^{2} r^{12} + 6 \, a^{2} b^{4} r^{7} + 2 \, b^{4} r^{9} - 4 \, b^{2} r^{11} - 4 \, b^{6} r^{6} + 4 \, b^{4} r^{8} + {\left(a^{4} b^{2} r^{8} - 2 \, a^{4} b^{4} r^{5} - 6 \, a^{2} b^{4} r^{7} - 2 \, a^{2} b^{2} r^{9} - 8 \, a^{4} b^{6} r^{2} - 12 \, a^{2} b^{6} r^{4} + 8 \, a^{2} b^{4} r^{6}\right)} \cos\left({\theta}\right)^{2}\right)}}{r^{18} + 8 \, b^{2} r^{15} + 24 \, b^{4} r^{12} + 32 \, b^{6} r^{9} + 16 \, b^{8} r^{6} + {\left(a^{6} r^{12} + 8 \, a^{6} b^{2} r^{9} + 24 \, a^{6} b^{4} r^{6} + 32 \, a^{6} b^{6} r^{3} + 16 \, a^{6} b^{8}\right)} \cos\left({\theta}\right)^{6} + 3 \, {\left(a^{4} r^{14} + 8 \, a^{4} b^{2} r^{11} + 24 \, a^{4} b^{4} r^{8} + 32 \, a^{4} b^{6} r^{5} + 16 \, a^{4} b^{8} r^{2}\right)} \cos\left({\theta}\right)^{4} + 3 \, {\left(a^{2} r^{16} + 8 \, a^{2} b^{2} r^{13} + 24 \, a^{2} b^{4} r^{10} + 32 \, a^{2} b^{6} r^{7} + 16 \, a^{2} b^{8} r^{4}\right)} \cos\left({\theta}\right)^{2}} \\ \mathrm{Ric}\left(g\right)_{ \, t \, {\phi} }^{ \phantom{\, t}\phantom{\, {\phi}} } & = & \frac{12 \, {\left({\left(a^{5} b^{2} r^{8} + a^{3} b^{2} r^{10} - 2 \, a^{5} b^{4} r^{5} - 2 \, a^{3} b^{4} r^{7} - 2 \, a^{3} b^{2} r^{9} - 8 \, a^{5} b^{6} r^{2} - 8 \, a^{3} b^{6} r^{4} + 8 \, a^{3} b^{4} r^{6}\right)} \sin\left({\theta}\right)^{4} - {\left(4 \, a^{3} b^{2} r^{10} + 3 \, a b^{2} r^{12} - 2 \, a^{5} b^{4} r^{5} + 4 \, a^{3} b^{4} r^{7} - 4 \, a b^{2} r^{11} - 8 \, a^{5} b^{6} r^{2} - 8 \, a^{3} b^{6} r^{4} + 8 \, a^{3} b^{4} r^{6} - 2 \, {\left(a^{3} b^{2} - 3 \, a b^{4}\right)} r^{9} + {\left(a^{5} b^{2} + 4 \, a b^{4}\right)} r^{8}\right)} \sin\left({\theta}\right)^{2}\right)}}{r^{18} + 8 \, b^{2} r^{15} + 24 \, b^{4} r^{12} + 32 \, b^{6} r^{9} + 16 \, b^{8} r^{6} + {\left(a^{6} r^{12} + 8 \, a^{6} b^{2} r^{9} + 24 \, a^{6} b^{4} r^{6} + 32 \, a^{6} b^{6} r^{3} + 16 \, a^{6} b^{8}\right)} \cos\left({\theta}\right)^{6} + 3 \, {\left(a^{4} r^{14} + 8 \, a^{4} b^{2} r^{11} + 24 \, a^{4} b^{4} r^{8} + 32 \, a^{4} b^{6} r^{5} + 16 \, a^{4} b^{8} r^{2}\right)} \cos\left({\theta}\right)^{4} + 3 \, {\left(a^{2} r^{16} + 8 \, a^{2} b^{2} r^{13} + 24 \, a^{2} b^{4} r^{10} + 32 \, a^{2} b^{6} r^{7} + 16 \, a^{2} b^{8} r^{4}\right)} \cos\left({\theta}\right)^{2}} \\ \mathrm{Ric}\left(g\right)_{ \, r \, r }^{ \phantom{\, r}\phantom{\, r} } & = & -\frac{12 \, {\left(2 \, b^{2} r^{7} - 2 \, b^{4} r^{4} + {\left(a^{2} b^{2} r^{5} - 4 \, a^{2} b^{4} r^{2}\right)} \cos\left({\theta}\right)^{2}\right)}}{a^{2} r^{11} + r^{13} + 6 \, a^{2} b^{2} r^{8} + 6 \, b^{2} r^{10} - 2 \, r^{12} + 12 \, a^{2} b^{4} r^{5} + 12 \, b^{4} r^{7} - 8 \, b^{2} r^{9} + 8 \, a^{2} b^{6} r^{2} + 8 \, b^{6} r^{4} - 8 \, b^{4} r^{6} + {\left(a^{4} r^{9} + a^{2} r^{11} + 6 \, a^{4} b^{2} r^{6} + 6 \, a^{2} b^{2} r^{8} - 2 \, a^{2} r^{10} + 12 \, a^{4} b^{4} r^{3} + 12 \, a^{2} b^{4} r^{5} - 8 \, a^{2} b^{2} r^{7} + 8 \, a^{4} b^{6} + 8 \, a^{2} b^{6} r^{2} - 8 \, a^{2} b^{4} r^{4}\right)} \cos\left({\theta}\right)^{2}} \\ \mathrm{Ric}\left(g\right)_{ \, {\theta} \, {\theta} }^{ \phantom{\, {\theta}}\phantom{\, {\theta}} } & = & \frac{12 \, b^{2} r^{4}}{r^{8} + 4 \, b^{2} r^{5} + 4 \, b^{4} r^{2} + {\left(a^{2} r^{6} + 4 \, a^{2} b^{2} r^{3} + 4 \, a^{2} b^{4}\right)} \cos\left({\theta}\right)^{2}} \\ \mathrm{Ric}\left(g\right)_{ \, {\phi} \, t }^{ \phantom{\, {\phi}}\phantom{\, t} } & = & \frac{12 \, {\left({\left(a^{5} b^{2} r^{8} + a^{3} b^{2} r^{10} - 2 \, a^{5} b^{4} r^{5} - 2 \, a^{3} b^{4} r^{7} - 2 \, a^{3} b^{2} r^{9} - 8 \, a^{5} b^{6} r^{2} - 8 \, a^{3} b^{6} r^{4} + 8 \, a^{3} b^{4} r^{6}\right)} \sin\left({\theta}\right)^{4} - {\left(4 \, a^{3} b^{2} r^{10} + 3 \, a b^{2} r^{12} - 2 \, a^{5} b^{4} r^{5} + 4 \, a^{3} b^{4} r^{7} - 4 \, a b^{2} r^{11} - 8 \, a^{5} b^{6} r^{2} - 8 \, a^{3} b^{6} r^{4} + 8 \, a^{3} b^{4} r^{6} - 2 \, {\left(a^{3} b^{2} - 3 \, a b^{4}\right)} r^{9} + {\left(a^{5} b^{2} + 4 \, a b^{4}\right)} r^{8}\right)} \sin\left({\theta}\right)^{2}\right)}}{r^{18} + 8 \, b^{2} r^{15} + 24 \, b^{4} r^{12} + 32 \, b^{6} r^{9} + 16 \, b^{8} r^{6} + {\left(a^{6} r^{12} + 8 \, a^{6} b^{2} r^{9} + 24 \, a^{6} b^{4} r^{6} + 32 \, a^{6} b^{6} r^{3} + 16 \, a^{6} b^{8}\right)} \cos\left({\theta}\right)^{6} + 3 \, {\left(a^{4} r^{14} + 8 \, a^{4} b^{2} r^{11} + 24 \, a^{4} b^{4} r^{8} + 32 \, a^{4} b^{6} r^{5} + 16 \, a^{4} b^{8} r^{2}\right)} \cos\left({\theta}\right)^{4} + 3 \, {\left(a^{2} r^{16} + 8 \, a^{2} b^{2} r^{13} + 24 \, a^{2} b^{4} r^{10} + 32 \, a^{2} b^{6} r^{7} + 16 \, a^{2} b^{8} r^{4}\right)} \cos\left({\theta}\right)^{2}} \\ \mathrm{Ric}\left(g\right)_{ \, {\phi} \, {\phi} }^{ \phantom{\, {\phi}}\phantom{\, {\phi}} } & = & \frac{12 \, {\left(3 \, a^{4} b^{2} r^{10} + 4 \, a^{2} b^{2} r^{12} + b^{2} r^{14} + 6 \, a^{4} b^{4} r^{7} + 10 \, a^{2} b^{4} r^{9} + 4 \, a^{2} b^{6} r^{6} - 4 \, {\left(a^{2} b^{2} - b^{4}\right)} r^{11} + 4 \, {\left(a^{2} b^{4} + b^{6}\right)} r^{8} + {\left(a^{6} b^{2} r^{8} + a^{4} b^{2} r^{10} - 2 \, a^{6} b^{4} r^{5} - 2 \, a^{4} b^{4} r^{7} - 2 \, a^{4} b^{2} r^{9} - 8 \, a^{6} b^{6} r^{2} - 8 \, a^{4} b^{6} r^{4} + 8 \, a^{4} b^{4} r^{6}\right)} \cos\left({\theta}\right)^{6} + 2 \, {\left(a^{2} b^{2} r^{12} + 2 \, a^{6} b^{4} r^{5} + 3 \, a^{4} b^{4} r^{7} - 2 \, a^{2} b^{2} r^{11} + 8 \, a^{6} b^{6} r^{2} + 6 \, a^{4} b^{6} r^{4} + {\left(2 \, a^{4} b^{2} + a^{2} b^{4}\right)} r^{9} - {\left(a^{6} b^{2} - 2 \, a^{2} b^{4}\right)} r^{8} - 2 \, {\left(4 \, a^{4} b^{4} + a^{2} b^{6}\right)} r^{6}\right)} \cos\left({\theta}\right)^{4} - {\left(4 \, a^{4} b^{2} r^{10} + 6 \, a^{2} b^{2} r^{12} + b^{2} r^{14} + 2 \, a^{6} b^{4} r^{5} + 10 \, a^{4} b^{4} r^{7} + 8 \, a^{6} b^{6} r^{2} + 4 \, a^{4} b^{6} r^{4} - 8 \, a^{4} b^{4} r^{6} - 4 \, {\left(2 \, a^{2} b^{2} - b^{4}\right)} r^{11} + 2 \, {\left(a^{4} b^{2} + 6 \, a^{2} b^{4}\right)} r^{9} - {\left(a^{6} b^{2} - 8 \, a^{2} b^{4} - 4 \, b^{6}\right)} r^{8}\right)} \cos\left({\theta}\right)^{2}\right)}}{r^{18} + 8 \, b^{2} r^{15} + 24 \, b^{4} r^{12} + 32 \, b^{6} r^{9} + 16 \, b^{8} r^{6} + {\left(a^{6} r^{12} + 8 \, a^{6} b^{2} r^{9} + 24 \, a^{6} b^{4} r^{6} + 32 \, a^{6} b^{6} r^{3} + 16 \, a^{6} b^{8}\right)} \cos\left({\theta}\right)^{6} + 3 \, {\left(a^{4} r^{14} + 8 \, a^{4} b^{2} r^{11} + 24 \, a^{4} b^{4} r^{8} + 32 \, a^{4} b^{6} r^{5} + 16 \, a^{4} b^{8} r^{2}\right)} \cos\left({\theta}\right)^{4} + 3 \, {\left(a^{2} r^{16} + 8 \, a^{2} b^{2} r^{13} + 24 \, a^{2} b^{4} r^{10} + 32 \, a^{2} b^{6} r^{7} + 16 \, a^{2} b^{8} r^{4}\right)} \cos\left({\theta}\right)^{2}} \end{array}

We check that for b=0b=0, we are dealing with a solution of the vacuum Einstein equation:

all([all([Ric[i,j].expr().subs(b=0) == 0 for i in range(4)]) for j in range(4)])
True\renewcommand{\Bold}[1]{\mathbf{#1}}\mathrm{True}

The Ricci scalar:

Rscal = g.ricci_scalar() Rscal.display()
r(g):MRon M1:(t,r,θ,ϕ)24(b2r54b4r2)r11+6b2r8+12b4r5+8b6r2+(a2r9+6a2b2r6+12a2b4r3+8a2b6)cos(θ)2on M2:(t,r,θ,ϕ)24(b2r5+4b4r2)r116b2r8+12b4r58b6r2+(a2r96a2b2r6+12a2b4r38a2b6)cos(θ)2\renewcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{llcl} \mathrm{r}\left(g\right):& M & \longrightarrow & \mathbb{R} \\ \mbox{on}\ M_1 : & \left(t, r, {\theta}, {\phi}\right) & \longmapsto & -\frac{24 \, {\left(b^{2} r^{5} - 4 \, b^{4} r^{2}\right)}}{r^{11} + 6 \, b^{2} r^{8} + 12 \, b^{4} r^{5} + 8 \, b^{6} r^{2} + {\left(a^{2} r^{9} + 6 \, a^{2} b^{2} r^{6} + 12 \, a^{2} b^{4} r^{3} + 8 \, a^{2} b^{6}\right)} \cos\left({\theta}\right)^{2}} \\ \mbox{on}\ M_2 : & \left(t, r, {\theta}, {\phi}\right) & \longmapsto & \frac{24 \, {\left(b^{2} r^{5} + 4 \, b^{4} r^{2}\right)}}{r^{11} - 6 \, b^{2} r^{8} + 12 \, b^{4} r^{5} - 8 \, b^{6} r^{2} + {\left(a^{2} r^{9} - 6 \, a^{2} b^{2} r^{6} + 12 \, a^{2} b^{4} r^{3} - 8 \, a^{2} b^{6}\right)} \cos\left({\theta}\right)^{2}} \end{array}
graph = plot_profile(Rscal, {a: 0.9, b: 1}, -3, 3, r'$R$') graph.save('ricci_scalar_HT.pdf') graph
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Riemann tensor

R = g.riemann() ; print(R)
Tensor field Riem(g) of type (1,3) on the 4-dimensional differentiable manifold M
R[0,1,2,3]
((a7r9+a5r11+10a7b2r6+10a5b2r82a5r10+16a7b4r3+16a5b4r516a5b2r7)cos(θ)5(3a7r9+8a5r11+5a3r13+30a7b2r6+56a5b2r86a3r12+48a7b4r3+80a5b4r52(a513a3b2)r1016(a5b22a3b4)r7)cos(θ)3+3(3a5r11+5a3r13+2ar15+6a5b2r8+10a3b2r102(a32ab2)r12)cos(θ))sin(θ)a2r15+r17+6a2b2r12+6b2r142r16+12a2b4r9+12b4r118b2r13+8a2b6r6+8b6r88b4r10+(a8r9+a6r11+6a8b2r6+6a6b2r82a6r10+12a8b4r3+12a6b4r58a6b2r7+8a8b6+8a6b6r28a6b4r4)cos(θ)6+3(a6r11+a4r13+6a6b2r8+6a4b2r102a4r12+12a6b4r5+12a4b4r78a4b2r9+8a6b6r2+8a4b6r48a4b4r6)cos(θ)4+3(a4r13+a2r15+6a4b2r10+6a2b2r122a2r14+12a4b4r7+12a2b4r98a2b2r11+8a4b6r4+8a2b6r68a2b4r8)cos(θ)2\renewcommand{\Bold}[1]{\mathbf{#1}}-\frac{{\left({\left(a^{7} r^{9} + a^{5} r^{11} + 10 \, a^{7} b^{2} r^{6} + 10 \, a^{5} b^{2} r^{8} - 2 \, a^{5} r^{10} + 16 \, a^{7} b^{4} r^{3} + 16 \, a^{5} b^{4} r^{5} - 16 \, a^{5} b^{2} r^{7}\right)} \cos\left({\theta}\right)^{5} - {\left(3 \, a^{7} r^{9} + 8 \, a^{5} r^{11} + 5 \, a^{3} r^{13} + 30 \, a^{7} b^{2} r^{6} + 56 \, a^{5} b^{2} r^{8} - 6 \, a^{3} r^{12} + 48 \, a^{7} b^{4} r^{3} + 80 \, a^{5} b^{4} r^{5} - 2 \, {\left(a^{5} - 13 \, a^{3} b^{2}\right)} r^{10} - 16 \, {\left(a^{5} b^{2} - 2 \, a^{3} b^{4}\right)} r^{7}\right)} \cos\left({\theta}\right)^{3} + 3 \, {\left(3 \, a^{5} r^{11} + 5 \, a^{3} r^{13} + 2 \, a r^{15} + 6 \, a^{5} b^{2} r^{8} + 10 \, a^{3} b^{2} r^{10} - 2 \, {\left(a^{3} - 2 \, a b^{2}\right)} r^{12}\right)} \cos\left({\theta}\right)\right)} \sin\left({\theta}\right)}{a^{2} r^{15} + r^{17} + 6 \, a^{2} b^{2} r^{12} + 6 \, b^{2} r^{14} - 2 \, r^{16} + 12 \, a^{2} b^{4} r^{9} + 12 \, b^{4} r^{11} - 8 \, b^{2} r^{13} + 8 \, a^{2} b^{6} r^{6} + 8 \, b^{6} r^{8} - 8 \, b^{4} r^{10} + {\left(a^{8} r^{9} + a^{6} r^{11} + 6 \, a^{8} b^{2} r^{6} + 6 \, a^{6} b^{2} r^{8} - 2 \, a^{6} r^{10} + 12 \, a^{8} b^{4} r^{3} + 12 \, a^{6} b^{4} r^{5} - 8 \, a^{6} b^{2} r^{7} + 8 \, a^{8} b^{6} + 8 \, a^{6} b^{6} r^{2} - 8 \, a^{6} b^{4} r^{4}\right)} \cos\left({\theta}\right)^{6} + 3 \, {\left(a^{6} r^{11} + a^{4} r^{13} + 6 \, a^{6} b^{2} r^{8} + 6 \, a^{4} b^{2} r^{10} - 2 \, a^{4} r^{12} + 12 \, a^{6} b^{4} r^{5} + 12 \, a^{4} b^{4} r^{7} - 8 \, a^{4} b^{2} r^{9} + 8 \, a^{6} b^{6} r^{2} + 8 \, a^{4} b^{6} r^{4} - 8 \, a^{4} b^{4} r^{6}\right)} \cos\left({\theta}\right)^{4} + 3 \, {\left(a^{4} r^{13} + a^{2} r^{15} + 6 \, a^{4} b^{2} r^{10} + 6 \, a^{2} b^{2} r^{12} - 2 \, a^{2} r^{14} + 12 \, a^{4} b^{4} r^{7} + 12 \, a^{2} b^{4} r^{9} - 8 \, a^{2} b^{2} r^{11} + 8 \, a^{4} b^{6} r^{4} + 8 \, a^{2} b^{6} r^{6} - 8 \, a^{2} b^{4} r^{8}\right)} \cos\left({\theta}\right)^{2}}

Kretschmann scalar

The tensor RR^\flat, of components Rabcd=gamR bcdmR_{abcd} = g_{am} R^m_{\ \, bcd}:

dR = R.down(g); print(dR)
Tensor field of type (0,4) on the 4-dimensional differentiable manifold M

The tensor RR^\sharp, of components Rabcd=gbpgcqgdrR pqraR^{abcd} = g^{bp} g^{cq} g^{dr} R^a_{\ \, pqr}:

uR = R.up(g); print(uR)
Tensor field of type (4,0) on the 4-dimensional differentiable manifold M

The Kretschmann scalar K:=RabcdRabcdK := R^{abcd} R_{abcd}:

Kr_scalar = uR['^{abcd}']*dR['_{abcd}'] Kr_scalar.display()
MRon M1:(t,r,θ,ϕ)48(r248b2r21+72b4r1816b6r15+32b8r12+12(a8b4r108a8b6r7+16a8b8r4)cos(θ)8(a6r18+8a6b2r15+36a6b4r12+944a6b6r9+64a6b8r6)cos(θ)6+(15a4r20+200a4b2r17+924a4b4r14+48a4b6r11+352a4b8r8)cos(θ)4(15a2r22+56a2b2r19276a2b4r16+144a2b6r13128a2b8r10)cos(θ)2)r30+12b2r27+60b4r24+160b6r21+240b8r18+192b10r15+64b12r12+(a12r18+12a12b2r15+60a12b4r12+160a12b6r9+240a12b8r6+192a12b10r3+64a12b12)cos(θ)12+6(a10r20+12a10b2r17+60a10b4r14+160a10b6r11+240a10b8r8+192a10b10r5+64a10b12r2)cos(θ)10+15(a8r22+12a8b2r19+60a8b4r16+160a8b6r13+240a8b8r10+192a8b10r7+64a8b12r4)cos(θ)8+20(a6r24+12a6b2r21+60a6b4r18+160a6b6r15+240a6b8r12+192a6b10r9+64a6b12r6)cos(θ)6+15(a4r26+12a4b2r23+60a4b4r20+160a4b6r17+240a4b8r14+192a4b10r11+64a4b12r8)cos(θ)4+6(a2r28+12a2b2r25+60a2b4r22+160a2b6r19+240a2b8r16+192a2b10r13+64a2b12r10)cos(θ)2on M2:(t,r,θ,ϕ)48(r24+8b2r21+72b4r18+16b6r15+32b8r12+12(a8b4r10+8a8b6r7+16a8b8r4)cos(θ)8(a6r188a6b2r15+36a6b4r12944a6b6r9+64a6b8r6)cos(θ)6+(15a4r20200a4b2r17+924a4b4r1448a4b6r11+352a4b8r8)cos(θ)4(15a2r2256a2b2r19276a2b4r16144a2b6r13128a2b8r10)cos(θ)2)r3012b2r27+60b4r24160b6r21+240b8r18192b10r15+64b12r12+(a12r1812a12b2r15+60a12b4r12160a12b6r9+240a12b8r6192a12b10r3+64a12b12)cos(θ)12+6(a10r2012a10b2r17+60a10b4r14160a10b6r11+240a10b8r8192a10b10r5+64a10b12r2)cos(θ)10+15(a8r2212a8b2r19+60a8b4r16160a8b6r13+240a8b8r10192a8b10r7+64a8b12r4)cos(θ)8+20(a6r2412a6b2r21+60a6b4r18160a6b6r15+240a6b8r12192a6b10r9+64a6b12r6)cos(θ)6+15(a4r2612a4b2r23+60a4b4r20160a4b6r17+240a4b8r14192a4b10r11+64a4b12r8)cos(θ)4+6(a2r2812a2b2r25+60a2b4r22160a2b6r19+240a2b8r16192a2b10r13+64a2b12r10)cos(θ)2\renewcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{llcl} & M & \longrightarrow & \mathbb{R} \\ \mbox{on}\ M_1 : & \left(t, r, {\theta}, {\phi}\right) & \longmapsto & \frac{48 \, {\left(r^{24} - 8 \, b^{2} r^{21} + 72 \, b^{4} r^{18} - 16 \, b^{6} r^{15} + 32 \, b^{8} r^{12} + 12 \, {\left(a^{8} b^{4} r^{10} - 8 \, a^{8} b^{6} r^{7} + 16 \, a^{8} b^{8} r^{4}\right)} \cos\left({\theta}\right)^{8} - {\left(a^{6} r^{18} + 8 \, a^{6} b^{2} r^{15} + 36 \, a^{6} b^{4} r^{12} + 944 \, a^{6} b^{6} r^{9} + 64 \, a^{6} b^{8} r^{6}\right)} \cos\left({\theta}\right)^{6} + {\left(15 \, a^{4} r^{20} + 200 \, a^{4} b^{2} r^{17} + 924 \, a^{4} b^{4} r^{14} + 48 \, a^{4} b^{6} r^{11} + 352 \, a^{4} b^{8} r^{8}\right)} \cos\left({\theta}\right)^{4} - {\left(15 \, a^{2} r^{22} + 56 \, a^{2} b^{2} r^{19} - 276 \, a^{2} b^{4} r^{16} + 144 \, a^{2} b^{6} r^{13} - 128 \, a^{2} b^{8} r^{10}\right)} \cos\left({\theta}\right)^{2}\right)}}{r^{30} + 12 \, b^{2} r^{27} + 60 \, b^{4} r^{24} + 160 \, b^{6} r^{21} + 240 \, b^{8} r^{18} + 192 \, b^{10} r^{15} + 64 \, b^{12} r^{12} + {\left(a^{12} r^{18} + 12 \, a^{12} b^{2} r^{15} + 60 \, a^{12} b^{4} r^{12} + 160 \, a^{12} b^{6} r^{9} + 240 \, a^{12} b^{8} r^{6} + 192 \, a^{12} b^{10} r^{3} + 64 \, a^{12} b^{12}\right)} \cos\left({\theta}\right)^{12} + 6 \, {\left(a^{10} r^{20} + 12 \, a^{10} b^{2} r^{17} + 60 \, a^{10} b^{4} r^{14} + 160 \, a^{10} b^{6} r^{11} + 240 \, a^{10} b^{8} r^{8} + 192 \, a^{10} b^{10} r^{5} + 64 \, a^{10} b^{12} r^{2}\right)} \cos\left({\theta}\right)^{10} + 15 \, {\left(a^{8} r^{22} + 12 \, a^{8} b^{2} r^{19} + 60 \, a^{8} b^{4} r^{16} + 160 \, a^{8} b^{6} r^{13} + 240 \, a^{8} b^{8} r^{10} + 192 \, a^{8} b^{10} r^{7} + 64 \, a^{8} b^{12} r^{4}\right)} \cos\left({\theta}\right)^{8} + 20 \, {\left(a^{6} r^{24} + 12 \, a^{6} b^{2} r^{21} + 60 \, a^{6} b^{4} r^{18} + 160 \, a^{6} b^{6} r^{15} + 240 \, a^{6} b^{8} r^{12} + 192 \, a^{6} b^{10} r^{9} + 64 \, a^{6} b^{12} r^{6}\right)} \cos\left({\theta}\right)^{6} + 15 \, {\left(a^{4} r^{26} + 12 \, a^{4} b^{2} r^{23} + 60 \, a^{4} b^{4} r^{20} + 160 \, a^{4} b^{6} r^{17} + 240 \, a^{4} b^{8} r^{14} + 192 \, a^{4} b^{10} r^{11} + 64 \, a^{4} b^{12} r^{8}\right)} \cos\left({\theta}\right)^{4} + 6 \, {\left(a^{2} r^{28} + 12 \, a^{2} b^{2} r^{25} + 60 \, a^{2} b^{4} r^{22} + 160 \, a^{2} b^{6} r^{19} + 240 \, a^{2} b^{8} r^{16} + 192 \, a^{2} b^{10} r^{13} + 64 \, a^{2} b^{12} r^{10}\right)} \cos\left({\theta}\right)^{2}} \\ \mbox{on}\ M_2 : & \left(t, r, {\theta}, {\phi}\right) & \longmapsto & \frac{48 \, {\left(r^{24} + 8 \, b^{2} r^{21} + 72 \, b^{4} r^{18} + 16 \, b^{6} r^{15} + 32 \, b^{8} r^{12} + 12 \, {\left(a^{8} b^{4} r^{10} + 8 \, a^{8} b^{6} r^{7} + 16 \, a^{8} b^{8} r^{4}\right)} \cos\left({\theta}\right)^{8} - {\left(a^{6} r^{18} - 8 \, a^{6} b^{2} r^{15} + 36 \, a^{6} b^{4} r^{12} - 944 \, a^{6} b^{6} r^{9} + 64 \, a^{6} b^{8} r^{6}\right)} \cos\left({\theta}\right)^{6} + {\left(15 \, a^{4} r^{20} - 200 \, a^{4} b^{2} r^{17} + 924 \, a^{4} b^{4} r^{14} - 48 \, a^{4} b^{6} r^{11} + 352 \, a^{4} b^{8} r^{8}\right)} \cos\left({\theta}\right)^{4} - {\left(15 \, a^{2} r^{22} - 56 \, a^{2} b^{2} r^{19} - 276 \, a^{2} b^{4} r^{16} - 144 \, a^{2} b^{6} r^{13} - 128 \, a^{2} b^{8} r^{10}\right)} \cos\left({\theta}\right)^{2}\right)}}{r^{30} - 12 \, b^{2} r^{27} + 60 \, b^{4} r^{24} - 160 \, b^{6} r^{21} + 240 \, b^{8} r^{18} - 192 \, b^{10} r^{15} + 64 \, b^{12} r^{12} + {\left(a^{12} r^{18} - 12 \, a^{12} b^{2} r^{15} + 60 \, a^{12} b^{4} r^{12} - 160 \, a^{12} b^{6} r^{9} + 240 \, a^{12} b^{8} r^{6} - 192 \, a^{12} b^{10} r^{3} + 64 \, a^{12} b^{12}\right)} \cos\left({\theta}\right)^{12} + 6 \, {\left(a^{10} r^{20} - 12 \, a^{10} b^{2} r^{17} + 60 \, a^{10} b^{4} r^{14} - 160 \, a^{10} b^{6} r^{11} + 240 \, a^{10} b^{8} r^{8} - 192 \, a^{10} b^{10} r^{5} + 64 \, a^{10} b^{12} r^{2}\right)} \cos\left({\theta}\right)^{10} + 15 \, {\left(a^{8} r^{22} - 12 \, a^{8} b^{2} r^{19} + 60 \, a^{8} b^{4} r^{16} - 160 \, a^{8} b^{6} r^{13} + 240 \, a^{8} b^{8} r^{10} - 192 \, a^{8} b^{10} r^{7} + 64 \, a^{8} b^{12} r^{4}\right)} \cos\left({\theta}\right)^{8} + 20 \, {\left(a^{6} r^{24} - 12 \, a^{6} b^{2} r^{21} + 60 \, a^{6} b^{4} r^{18} - 160 \, a^{6} b^{6} r^{15} + 240 \, a^{6} b^{8} r^{12} - 192 \, a^{6} b^{10} r^{9} + 64 \, a^{6} b^{12} r^{6}\right)} \cos\left({\theta}\right)^{6} + 15 \, {\left(a^{4} r^{26} - 12 \, a^{4} b^{2} r^{23} + 60 \, a^{4} b^{4} r^{20} - 160 \, a^{4} b^{6} r^{17} + 240 \, a^{4} b^{8} r^{14} - 192 \, a^{4} b^{10} r^{11} + 64 \, a^{4} b^{12} r^{8}\right)} \cos\left({\theta}\right)^{4} + 6 \, {\left(a^{2} r^{28} - 12 \, a^{2} b^{2} r^{25} + 60 \, a^{2} b^{4} r^{22} - 160 \, a^{2} b^{6} r^{19} + 240 \, a^{2} b^{8} r^{16} - 192 \, a^{2} b^{10} r^{13} + 64 \, a^{2} b^{12} r^{10}\right)} \cos\left({\theta}\right)^{2}} \end{array}
K = Kr_scalar.expr() K
48(r248b2r21+72b4r1816b6r15+32b8r12+12(a8b4r108a8b6r7+16a8b8r4)cos(θ)8(a6r18+8a6b2r15+36a6b4r12+944a6b6r9+64a6b8r6)cos(θ)6+(15a4r20+200a4b2r17+924a4b4r14+48a4b6r11+352a4b8r8)cos(θ)4(15a2r22+56a2b2r19276a2b4r16+144a2b6r13128a2b8r10)cos(θ)2)r30+12b2r27+60b4r24+160b6r21+240b8r18+192b10r15+64b12r12+(a12r18+12a12b2r15+60a12b4r12+160a12b6r9+240a12b8r6+192a12b10r3+64a12b12)cos(θ)12+6(a10r20+12a10b2r17+60a10b4r14+160a10b6r11+240a10b8r8+192a10b10r5+64a10b12r2)cos(θ)10+15(a8r22+12a8b2r19+60a8b4r16+160a8b6r13+240a8b8r10+192a8b10r7+64a8b12r4)cos(θ)8+20(a6r24+12a6b2r21+60a6b4r18+160a6b6r15+240a6b8r12+192a6b10r9+64a6b12r6)cos(θ)6+15(a4r26+12a4b2r23+60a4b4r20+160a4b6r17+240a4b8r14+192a4b10r11+64a4b12r8)cos(θ)4+6(a2r28+12a2b2r25+60a2b4r22+160a2b6r19+240a2b8r16+192a2b10r13+64a2b12r10)cos(θ)2\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{48 \, {\left(r^{24} - 8 \, b^{2} r^{21} + 72 \, b^{4} r^{18} - 16 \, b^{6} r^{15} + 32 \, b^{8} r^{12} + 12 \, {\left(a^{8} b^{4} r^{10} - 8 \, a^{8} b^{6} r^{7} + 16 \, a^{8} b^{8} r^{4}\right)} \cos\left({\theta}\right)^{8} - {\left(a^{6} r^{18} + 8 \, a^{6} b^{2} r^{15} + 36 \, a^{6} b^{4} r^{12} + 944 \, a^{6} b^{6} r^{9} + 64 \, a^{6} b^{8} r^{6}\right)} \cos\left({\theta}\right)^{6} + {\left(15 \, a^{4} r^{20} + 200 \, a^{4} b^{2} r^{17} + 924 \, a^{4} b^{4} r^{14} + 48 \, a^{4} b^{6} r^{11} + 352 \, a^{4} b^{8} r^{8}\right)} \cos\left({\theta}\right)^{4} - {\left(15 \, a^{2} r^{22} + 56 \, a^{2} b^{2} r^{19} - 276 \, a^{2} b^{4} r^{16} + 144 \, a^{2} b^{6} r^{13} - 128 \, a^{2} b^{8} r^{10}\right)} \cos\left({\theta}\right)^{2}\right)}}{r^{30} + 12 \, b^{2} r^{27} + 60 \, b^{4} r^{24} + 160 \, b^{6} r^{21} + 240 \, b^{8} r^{18} + 192 \, b^{10} r^{15} + 64 \, b^{12} r^{12} + {\left(a^{12} r^{18} + 12 \, a^{12} b^{2} r^{15} + 60 \, a^{12} b^{4} r^{12} + 160 \, a^{12} b^{6} r^{9} + 240 \, a^{12} b^{8} r^{6} + 192 \, a^{12} b^{10} r^{3} + 64 \, a^{12} b^{12}\right)} \cos\left({\theta}\right)^{12} + 6 \, {\left(a^{10} r^{20} + 12 \, a^{10} b^{2} r^{17} + 60 \, a^{10} b^{4} r^{14} + 160 \, a^{10} b^{6} r^{11} + 240 \, a^{10} b^{8} r^{8} + 192 \, a^{10} b^{10} r^{5} + 64 \, a^{10} b^{12} r^{2}\right)} \cos\left({\theta}\right)^{10} + 15 \, {\left(a^{8} r^{22} + 12 \, a^{8} b^{2} r^{19} + 60 \, a^{8} b^{4} r^{16} + 160 \, a^{8} b^{6} r^{13} + 240 \, a^{8} b^{8} r^{10} + 192 \, a^{8} b^{10} r^{7} + 64 \, a^{8} b^{12} r^{4}\right)} \cos\left({\theta}\right)^{8} + 20 \, {\left(a^{6} r^{24} + 12 \, a^{6} b^{2} r^{21} + 60 \, a^{6} b^{4} r^{18} + 160 \, a^{6} b^{6} r^{15} + 240 \, a^{6} b^{8} r^{12} + 192 \, a^{6} b^{10} r^{9} + 64 \, a^{6} b^{12} r^{6}\right)} \cos\left({\theta}\right)^{6} + 15 \, {\left(a^{4} r^{26} + 12 \, a^{4} b^{2} r^{23} + 60 \, a^{4} b^{4} r^{20} + 160 \, a^{4} b^{6} r^{17} + 240 \, a^{4} b^{8} r^{14} + 192 \, a^{4} b^{10} r^{11} + 64 \, a^{4} b^{12} r^{8}\right)} \cos\left({\theta}\right)^{4} + 6 \, {\left(a^{2} r^{28} + 12 \, a^{2} b^{2} r^{25} + 60 \, a^{2} b^{4} r^{22} + 160 \, a^{2} b^{6} r^{19} + 240 \, a^{2} b^{8} r^{16} + 192 \, a^{2} b^{10} r^{13} + 64 \, a^{2} b^{12} r^{10}\right)} \cos\left({\theta}\right)^{2}}
K_0 = K.subs(r=0) K_0
0\renewcommand{\Bold}[1]{\mathbf{#1}}0

The equatorial value of the Kretschmann scalar is

K_eq = K.subs(th=pi/2).simplify_full() K_eq
48(r128b2r9+72b4r616b6r3+32b8)r18+12b2r15+60b4r12+160b6r9+240b8r6+192b10r3+64b12\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{48 \, {\left(r^{12} - 8 \, b^{2} r^{9} + 72 \, b^{4} r^{6} - 16 \, b^{6} r^{3} + 32 \, b^{8}\right)}}{r^{18} + 12 \, b^{2} r^{15} + 60 \, b^{4} r^{12} + 160 \, b^{6} r^{9} + 240 \, b^{8} r^{6} + 192 \, b^{10} r^{3} + 64 \, b^{12}}
taylor(K_eq, r, 0, 6)
216r6b884r3b6+24b4\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{216 \, r^{6}}{b^{8}} - \frac{84 \, r^{3}}{b^{6}} + \frac{24}{b^{4}}

The limit r0r\rightarrow 0:

K_eq.subs(r=0)
24b4\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{24}{b^{4}}

We recover the same value as that given by Eq. (24) of Bambi & Modesto, Phys. Lett. B 721, 329 (2013) (note that the quantity gg used by Bambi & Modesto is related to our bb by g3=2b2g^3 = 2 b^2).

K1 = K.subs({a: 0.9, b: 1}) plot3d(K1, (r, -1/2, 1/2), (th, 0, pi/2), plot_points=200, aspect_ratio=[2, 1, 0.1], axes_labels=['r', 'theta', 'K'], viewer='threejs', online=True)
MIME type unknown not supported
graph = plot_profile(Kr_scalar, {a: 0.9, b: 1}, -3, 3, r'$K$') graph.save('Kretschmann_scalar_HT.pdf') graph
Image in a Jupyter notebook

Non-rotating limit

Ka0 = K.subs(a=0).simplify_full() Ka0
48(r128b2r9+72b4r616b6r3+32b8)r18+12b2r15+60b4r12+160b6r9+240b8r6+192b10r3+64b12\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{48 \, {\left(r^{12} - 8 \, b^{2} r^{9} + 72 \, b^{4} r^{6} - 16 \, b^{6} r^{3} + 32 \, b^{8}\right)}}{r^{18} + 12 \, b^{2} r^{15} + 60 \, b^{4} r^{12} + 160 \, b^{6} r^{9} + 240 \, b^{8} r^{6} + 192 \, b^{10} r^{3} + 64 \, b^{12}}
taylor(Ka0, r, 0, 6)
216r6b884r3b6+24b4\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{216 \, r^{6}}{b^{8}} - \frac{84 \, r^{3}}{b^{6}} + \frac{24}{b^{4}}

Check: we recover Schwarzschild value when b=0b=0:

Ka0.subs(b=0).simplify_full()
48r6\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{48}{r^{6}}

Chern-Pontryagin scalar

We start by getting the Levi-Civita 4-vector ϵabcd\epsilon^{abcd}:

epsv = g.volume_form(contra=4) print(epsv)
4-vector field on the 4-dimensional differentiable manifold M
epsv.display()
1(a2cos(θ)2+r2)sin(θ)trθϕ\renewcommand{\Bold}[1]{\mathbf{#1}}-\frac{1}{{\left(a^{2} \cos\left({\theta}\right)^{2} + r^{2}\right)} \sin\left({\theta}\right)} \frac{\partial}{\partial t }\wedge \frac{\partial}{\partial r }\wedge \frac{\partial}{\partial {\theta} }\wedge \frac{\partial}{\partial {\phi} }

The dual Riemann tensor is computed as Rabcd=12ϵabmnRmnrsgrcgsd {}^*R^{abcd} = \frac{1}{2} \epsilon^{abmn} R_{mnrs} g^{rc} g^{sd} with 12ϵabmnRmnrs \frac{1}{2} \epsilon^{abmn} R_{mnrs} as a first step:

tmp = 1/2*epsv['^{abmn}']*dR['_{mnrs}'] print(tmp)
Tensor field of type (2,2) on the 4-dimensional differentiable manifold M
dual_Riem = tmp.up(g) print(dual_Riem)
Tensor field of type (4,0) on the 4-dimensional differentiable manifold M
dual_Riem[0,1,2,3]
3a2r9+2r11+4b2r8+3(a4r7+4a4b2r4)cos(θ)4(9a4r7+7a2r9+24a4b2r4+8a2b2r6)cos(θ)2(r16+4b2r13+4b4r10+(a10r6+4a10b2r3+4a10b4)cos(θ)10+5(a8r8+4a8b2r5+4a8b4r2)cos(θ)8+10(a6r10+4a6b2r7+4a6b4r4)cos(θ)6+10(a4r12+4a4b2r9+4a4b4r6)cos(θ)4+5(a2r14+4a2b2r11+4a2b4r8)cos(θ)2)sin(θ)\renewcommand{\Bold}[1]{\mathbf{#1}}-\frac{3 \, a^{2} r^{9} + 2 \, r^{11} + 4 \, b^{2} r^{8} + 3 \, {\left(a^{4} r^{7} + 4 \, a^{4} b^{2} r^{4}\right)} \cos\left({\theta}\right)^{4} - {\left(9 \, a^{4} r^{7} + 7 \, a^{2} r^{9} + 24 \, a^{4} b^{2} r^{4} + 8 \, a^{2} b^{2} r^{6}\right)} \cos\left({\theta}\right)^{2}}{{\left(r^{16} + 4 \, b^{2} r^{13} + 4 \, b^{4} r^{10} + {\left(a^{10} r^{6} + 4 \, a^{10} b^{2} r^{3} + 4 \, a^{10} b^{4}\right)} \cos\left({\theta}\right)^{10} + 5 \, {\left(a^{8} r^{8} + 4 \, a^{8} b^{2} r^{5} + 4 \, a^{8} b^{4} r^{2}\right)} \cos\left({\theta}\right)^{8} + 10 \, {\left(a^{6} r^{10} + 4 \, a^{6} b^{2} r^{7} + 4 \, a^{6} b^{4} r^{4}\right)} \cos\left({\theta}\right)^{6} + 10 \, {\left(a^{4} r^{12} + 4 \, a^{4} b^{2} r^{9} + 4 \, a^{4} b^{4} r^{6}\right)} \cos\left({\theta}\right)^{4} + 5 \, {\left(a^{2} r^{14} + 4 \, a^{2} b^{2} r^{11} + 4 \, a^{2} b^{4} r^{8}\right)} \cos\left({\theta}\right)^{2}\right)} \sin\left({\theta}\right)}

The Chern-Pontryagin scalar is RabcdRabcd{}^*R^{abcd} R_{abcd}:

CP_scalar = dual_Riem['^{abcd}']*dR['_{abcd}'] CP_scalar.display()
MRon M1:(t,r,θ,ϕ)96(2(a7b2r11+4a7b4r832a7b6r5)cos(θ)7+(3a5r16+40a5b2r13+208a5b4r10+64a5b6r7)cos(θ)52(5a3r18+35a3b2r154a3b4r12)cos(θ)3+3(ar204ab2r17)cos(θ))r27+10b2r24+40b4r21+80b6r18+80b8r15+32b10r12+(a12r15+10a12b2r12+40a12b4r9+80a12b6r6+80a12b8r3+32a12b10)cos(θ)12+6(a10r17+10a10b2r14+40a10b4r11+80a10b6r8+80a10b8r5+32a10b10r2)cos(θ)10+15(a8r19+10a8b2r16+40a8b4r13+80a8b6r10+80a8b8r7+32a8b10r4)cos(θ)8+20(a6r21+10a6b2r18+40a6b4r15+80a6b6r12+80a6b8r9+32a6b10r6)cos(θ)6+15(a4r23+10a4b2r20+40a4b4r17+80a4b6r14+80a4b8r11+32a4b10r8)cos(θ)4+6(a2r25+10a2b2r22+40a2b4r19+80a2b6r16+80a2b8r13+32a2b10r10)cos(θ)2on M2:(t,r,θ,ϕ)96(2(a7b2r114a7b4r832a7b6r5)cos(θ)7(3a5r1640a5b2r13+208a5b4r1064a5b6r7)cos(θ)5+2(5a3r1835a3b2r154a3b4r12)cos(θ)33(ar20+4ab2r17)cos(θ))r2710b2r24+40b4r2180b6r18+80b8r1532b10r12+(a12r1510a12b2r12+40a12b4r980a12b6r6+80a12b8r332a12b10)cos(θ)12+6(a10r1710a10b2r14+40a10b4r1180a10b6r8+80a10b8r532a10b10r2)cos(θ)10+15(a8r1910a8b2r16+40a8b4r1380a8b6r10+80a8b8r732a8b10r4)cos(θ)8+20(a6r2110a6b2r18+40a6b4r1580a6b6r12+80a6b8r932a6b10r6)cos(θ)6+15(a4r2310a4b2r20+40a4b4r1780a4b6r14+80a4b8r1132a4b10r8)cos(θ)4+6(a2r2510a2b2r22+40a2b4r1980a2b6r16+80a2b8r1332a2b10r10)cos(θ)2\renewcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{llcl} & M & \longrightarrow & \mathbb{R} \\ \mbox{on}\ M_1 : & \left(t, r, {\theta}, {\phi}\right) & \longmapsto & -\frac{96 \, {\left(2 \, {\left(a^{7} b^{2} r^{11} + 4 \, a^{7} b^{4} r^{8} - 32 \, a^{7} b^{6} r^{5}\right)} \cos\left({\theta}\right)^{7} + {\left(3 \, a^{5} r^{16} + 40 \, a^{5} b^{2} r^{13} + 208 \, a^{5} b^{4} r^{10} + 64 \, a^{5} b^{6} r^{7}\right)} \cos\left({\theta}\right)^{5} - 2 \, {\left(5 \, a^{3} r^{18} + 35 \, a^{3} b^{2} r^{15} - 4 \, a^{3} b^{4} r^{12}\right)} \cos\left({\theta}\right)^{3} + 3 \, {\left(a r^{20} - 4 \, a b^{2} r^{17}\right)} \cos\left({\theta}\right)\right)}}{r^{27} + 10 \, b^{2} r^{24} + 40 \, b^{4} r^{21} + 80 \, b^{6} r^{18} + 80 \, b^{8} r^{15} + 32 \, b^{10} r^{12} + {\left(a^{12} r^{15} + 10 \, a^{12} b^{2} r^{12} + 40 \, a^{12} b^{4} r^{9} + 80 \, a^{12} b^{6} r^{6} + 80 \, a^{12} b^{8} r^{3} + 32 \, a^{12} b^{10}\right)} \cos\left({\theta}\right)^{12} + 6 \, {\left(a^{10} r^{17} + 10 \, a^{10} b^{2} r^{14} + 40 \, a^{10} b^{4} r^{11} + 80 \, a^{10} b^{6} r^{8} + 80 \, a^{10} b^{8} r^{5} + 32 \, a^{10} b^{10} r^{2}\right)} \cos\left({\theta}\right)^{10} + 15 \, {\left(a^{8} r^{19} + 10 \, a^{8} b^{2} r^{16} + 40 \, a^{8} b^{4} r^{13} + 80 \, a^{8} b^{6} r^{10} + 80 \, a^{8} b^{8} r^{7} + 32 \, a^{8} b^{10} r^{4}\right)} \cos\left({\theta}\right)^{8} + 20 \, {\left(a^{6} r^{21} + 10 \, a^{6} b^{2} r^{18} + 40 \, a^{6} b^{4} r^{15} + 80 \, a^{6} b^{6} r^{12} + 80 \, a^{6} b^{8} r^{9} + 32 \, a^{6} b^{10} r^{6}\right)} \cos\left({\theta}\right)^{6} + 15 \, {\left(a^{4} r^{23} + 10 \, a^{4} b^{2} r^{20} + 40 \, a^{4} b^{4} r^{17} + 80 \, a^{4} b^{6} r^{14} + 80 \, a^{4} b^{8} r^{11} + 32 \, a^{4} b^{10} r^{8}\right)} \cos\left({\theta}\right)^{4} + 6 \, {\left(a^{2} r^{25} + 10 \, a^{2} b^{2} r^{22} + 40 \, a^{2} b^{4} r^{19} + 80 \, a^{2} b^{6} r^{16} + 80 \, a^{2} b^{8} r^{13} + 32 \, a^{2} b^{10} r^{10}\right)} \cos\left({\theta}\right)^{2}} \\ \mbox{on}\ M_2 : & \left(t, r, {\theta}, {\phi}\right) & \longmapsto & \frac{96 \, {\left(2 \, {\left(a^{7} b^{2} r^{11} - 4 \, a^{7} b^{4} r^{8} - 32 \, a^{7} b^{6} r^{5}\right)} \cos\left({\theta}\right)^{7} - {\left(3 \, a^{5} r^{16} - 40 \, a^{5} b^{2} r^{13} + 208 \, a^{5} b^{4} r^{10} - 64 \, a^{5} b^{6} r^{7}\right)} \cos\left({\theta}\right)^{5} + 2 \, {\left(5 \, a^{3} r^{18} - 35 \, a^{3} b^{2} r^{15} - 4 \, a^{3} b^{4} r^{12}\right)} \cos\left({\theta}\right)^{3} - 3 \, {\left(a r^{20} + 4 \, a b^{2} r^{17}\right)} \cos\left({\theta}\right)\right)}}{r^{27} - 10 \, b^{2} r^{24} + 40 \, b^{4} r^{21} - 80 \, b^{6} r^{18} + 80 \, b^{8} r^{15} - 32 \, b^{10} r^{12} + {\left(a^{12} r^{15} - 10 \, a^{12} b^{2} r^{12} + 40 \, a^{12} b^{4} r^{9} - 80 \, a^{12} b^{6} r^{6} + 80 \, a^{12} b^{8} r^{3} - 32 \, a^{12} b^{10}\right)} \cos\left({\theta}\right)^{12} + 6 \, {\left(a^{10} r^{17} - 10 \, a^{10} b^{2} r^{14} + 40 \, a^{10} b^{4} r^{11} - 80 \, a^{10} b^{6} r^{8} + 80 \, a^{10} b^{8} r^{5} - 32 \, a^{10} b^{10} r^{2}\right)} \cos\left({\theta}\right)^{10} + 15 \, {\left(a^{8} r^{19} - 10 \, a^{8} b^{2} r^{16} + 40 \, a^{8} b^{4} r^{13} - 80 \, a^{8} b^{6} r^{10} + 80 \, a^{8} b^{8} r^{7} - 32 \, a^{8} b^{10} r^{4}\right)} \cos\left({\theta}\right)^{8} + 20 \, {\left(a^{6} r^{21} - 10 \, a^{6} b^{2} r^{18} + 40 \, a^{6} b^{4} r^{15} - 80 \, a^{6} b^{6} r^{12} + 80 \, a^{6} b^{8} r^{9} - 32 \, a^{6} b^{10} r^{6}\right)} \cos\left({\theta}\right)^{6} + 15 \, {\left(a^{4} r^{23} - 10 \, a^{4} b^{2} r^{20} + 40 \, a^{4} b^{4} r^{17} - 80 \, a^{4} b^{6} r^{14} + 80 \, a^{4} b^{8} r^{11} - 32 \, a^{4} b^{10} r^{8}\right)} \cos\left({\theta}\right)^{4} + 6 \, {\left(a^{2} r^{25} - 10 \, a^{2} b^{2} r^{22} + 40 \, a^{2} b^{4} r^{19} - 80 \, a^{2} b^{6} r^{16} + 80 \, a^{2} b^{8} r^{13} - 32 \, a^{2} b^{10} r^{10}\right)} \cos\left({\theta}\right)^{2}} \end{array}
CP = CP_scalar.expr().factor() CP
96(2a4b2r3cos(θ)4+3a2r8cos(θ)28a4b4cos(θ)4+22a2b2r5cos(θ)2r10+8a2b4r2cos(θ)2+4b2r7)(a2r3cos(θ)2+8a2b2cos(θ)23r5)ar5cos(θ)(a2cos(θ)2+r2)6(r3+2b2)5\renewcommand{\Bold}[1]{\mathbf{#1}}-\frac{96 \, {\left(2 \, a^{4} b^{2} r^{3} \cos\left({\theta}\right)^{4} + 3 \, a^{2} r^{8} \cos\left({\theta}\right)^{2} - 8 \, a^{4} b^{4} \cos\left({\theta}\right)^{4} + 22 \, a^{2} b^{2} r^{5} \cos\left({\theta}\right)^{2} - r^{10} + 8 \, a^{2} b^{4} r^{2} \cos\left({\theta}\right)^{2} + 4 \, b^{2} r^{7}\right)} {\left(a^{2} r^{3} \cos\left({\theta}\right)^{2} + 8 \, a^{2} b^{2} \cos\left({\theta}\right)^{2} - 3 \, r^{5}\right)} a r^{5} \cos\left({\theta}\right)}{{\left(a^{2} \cos\left({\theta}\right)^{2} + r^{2}\right)}^{6} {\left(r^{3} + 2 \, b^{2}\right)}^{5}}

The Kerr value is obtained for b=0b=0:

CP.subs(b=0).factor()
96(3a2cos(θ)2r2)(a2cos(θ)23r2)arcos(θ)(a2cos(θ)2+r2)6\renewcommand{\Bold}[1]{\mathbf{#1}}-\frac{96 \, {\left(3 \, a^{2} \cos\left({\theta}\right)^{2} - r^{2}\right)} {\left(a^{2} \cos\left({\theta}\right)^{2} - 3 \, r^{2}\right)} a r \cos\left({\theta}\right)}{{\left(a^{2} \cos\left({\theta}\right)^{2} + r^{2}\right)}^{6}}
num = _.numerator()/(96*a*cos(th)*r) num
(3a2cos(θ)2r2)(a2cos(θ)23r2)\renewcommand{\Bold}[1]{\mathbf{#1}}-{\left(3 \, a^{2} \cos\left({\theta}\right)^{2} - r^{2}\right)} {\left(a^{2} \cos\left({\theta}\right)^{2} - 3 \, r^{2}\right)}
num.expand()
3a4cos(θ)4+10a2r2cos(θ)23r4\renewcommand{\Bold}[1]{\mathbf{#1}}-3 \, a^{4} \cos\left({\theta}\right)^{4} + 10 \, a^{2} r^{2} \cos\left({\theta}\right)^{2} - 3 \, r^{4}

The above value of the Chern-Pontryagin scalar coincides with K2-K_2 as given by Eq. (32) with Q=0Q=0 of Cherubini et al., IJMPD 11, 827 (2002).

CP1 = CP.subs({a: 0.9, b: 1}) plot3d(CP1, (r, -1/2, 1/2), (th, 0, pi/2), plot_points=200, aspect_ratio=[2, 1, 0.1], axes_labels=['r', 'theta', 'CP'], viewer='threejs', online=True)
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graph = plot_profile(CP_scalar, {a: 0.9, b: 1}, -3, 3, r'$CP$') graph.save('ChernPontryagin_scalar_HT.pdf') graph
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Ricci squared

The Ricci squared is RabRabR_{ab} R^{ab}:

Ric2_scalar = Ric['_ab'] * Ric.up(g)['^ab'] Ric2_scalar.display()
MRon M1:(t,r,θ,ϕ)288(5b4r144b6r11+8b8r8+(a4b4r108a4b6r7+16a4b8r4)cos(θ)4+4(a2b4r125a2b6r9+4a2b8r6)cos(θ)2)r26+12b2r23+60b4r20+160b6r17+240b8r14+192b10r11+64b12r8+(a8r18+12a8b2r15+60a8b4r12+160a8b6r9+240a8b8r6+192a8b10r3+64a8b12)cos(θ)8+4(a6r20+12a6b2r17+60a6b4r14+160a6b6r11+240a6b8r8+192a6b10r5+64a6b12r2)cos(θ)6+6(a4r22+12a4b2r19+60a4b4r16+160a4b6r13+240a4b8r10+192a4b10r7+64a4b12r4)cos(θ)4+4(a2r24+12a2b2r21+60a2b4r18+160a2b6r15+240a2b8r12+192a2b10r9+64a2b12r6)cos(θ)2on M2:(t,r,θ,ϕ)288(5b4r14+4b6r11+8b8r8+(a4b4r10+8a4b6r7+16a4b8r4)cos(θ)4+4(a2b4r12+5a2b6r9+4a2b8r6)cos(θ)2)r2612b2r23+60b4r20160b6r17+240b8r14192b10r11+64b12r8+(a8r1812a8b2r15+60a8b4r12160a8b6r9+240a8b8r6192a8b10r3+64a8b12)cos(θ)8+4(a6r2012a6b2r17+60a6b4r14160a6b6r11+240a6b8r8192a6b10r5+64a6b12r2)cos(θ)6+6(a4r2212a4b2r19+60a4b4r16160a4b6r13+240a4b8r10192a4b10r7+64a4b12r4)cos(θ)4+4(a2r2412a2b2r21+60a2b4r18160a2b6r15+240a2b8r12192a2b10r9+64a2b12r6)cos(θ)2\renewcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{llcl} & M & \longrightarrow & \mathbb{R} \\ \mbox{on}\ M_1 : & \left(t, r, {\theta}, {\phi}\right) & \longmapsto & \frac{288 \, {\left(5 \, b^{4} r^{14} - 4 \, b^{6} r^{11} + 8 \, b^{8} r^{8} + {\left(a^{4} b^{4} r^{10} - 8 \, a^{4} b^{6} r^{7} + 16 \, a^{4} b^{8} r^{4}\right)} \cos\left({\theta}\right)^{4} + 4 \, {\left(a^{2} b^{4} r^{12} - 5 \, a^{2} b^{6} r^{9} + 4 \, a^{2} b^{8} r^{6}\right)} \cos\left({\theta}\right)^{2}\right)}}{r^{26} + 12 \, b^{2} r^{23} + 60 \, b^{4} r^{20} + 160 \, b^{6} r^{17} + 240 \, b^{8} r^{14} + 192 \, b^{10} r^{11} + 64 \, b^{12} r^{8} + {\left(a^{8} r^{18} + 12 \, a^{8} b^{2} r^{15} + 60 \, a^{8} b^{4} r^{12} + 160 \, a^{8} b^{6} r^{9} + 240 \, a^{8} b^{8} r^{6} + 192 \, a^{8} b^{10} r^{3} + 64 \, a^{8} b^{12}\right)} \cos\left({\theta}\right)^{8} + 4 \, {\left(a^{6} r^{20} + 12 \, a^{6} b^{2} r^{17} + 60 \, a^{6} b^{4} r^{14} + 160 \, a^{6} b^{6} r^{11} + 240 \, a^{6} b^{8} r^{8} + 192 \, a^{6} b^{10} r^{5} + 64 \, a^{6} b^{12} r^{2}\right)} \cos\left({\theta}\right)^{6} + 6 \, {\left(a^{4} r^{22} + 12 \, a^{4} b^{2} r^{19} + 60 \, a^{4} b^{4} r^{16} + 160 \, a^{4} b^{6} r^{13} + 240 \, a^{4} b^{8} r^{10} + 192 \, a^{4} b^{10} r^{7} + 64 \, a^{4} b^{12} r^{4}\right)} \cos\left({\theta}\right)^{4} + 4 \, {\left(a^{2} r^{24} + 12 \, a^{2} b^{2} r^{21} + 60 \, a^{2} b^{4} r^{18} + 160 \, a^{2} b^{6} r^{15} + 240 \, a^{2} b^{8} r^{12} + 192 \, a^{2} b^{10} r^{9} + 64 \, a^{2} b^{12} r^{6}\right)} \cos\left({\theta}\right)^{2}} \\ \mbox{on}\ M_2 : & \left(t, r, {\theta}, {\phi}\right) & \longmapsto & \frac{288 \, {\left(5 \, b^{4} r^{14} + 4 \, b^{6} r^{11} + 8 \, b^{8} r^{8} + {\left(a^{4} b^{4} r^{10} + 8 \, a^{4} b^{6} r^{7} + 16 \, a^{4} b^{8} r^{4}\right)} \cos\left({\theta}\right)^{4} + 4 \, {\left(a^{2} b^{4} r^{12} + 5 \, a^{2} b^{6} r^{9} + 4 \, a^{2} b^{8} r^{6}\right)} \cos\left({\theta}\right)^{2}\right)}}{r^{26} - 12 \, b^{2} r^{23} + 60 \, b^{4} r^{20} - 160 \, b^{6} r^{17} + 240 \, b^{8} r^{14} - 192 \, b^{10} r^{11} + 64 \, b^{12} r^{8} + {\left(a^{8} r^{18} - 12 \, a^{8} b^{2} r^{15} + 60 \, a^{8} b^{4} r^{12} - 160 \, a^{8} b^{6} r^{9} + 240 \, a^{8} b^{8} r^{6} - 192 \, a^{8} b^{10} r^{3} + 64 \, a^{8} b^{12}\right)} \cos\left({\theta}\right)^{8} + 4 \, {\left(a^{6} r^{20} - 12 \, a^{6} b^{2} r^{17} + 60 \, a^{6} b^{4} r^{14} - 160 \, a^{6} b^{6} r^{11} + 240 \, a^{6} b^{8} r^{8} - 192 \, a^{6} b^{10} r^{5} + 64 \, a^{6} b^{12} r^{2}\right)} \cos\left({\theta}\right)^{6} + 6 \, {\left(a^{4} r^{22} - 12 \, a^{4} b^{2} r^{19} + 60 \, a^{4} b^{4} r^{16} - 160 \, a^{4} b^{6} r^{13} + 240 \, a^{4} b^{8} r^{10} - 192 \, a^{4} b^{10} r^{7} + 64 \, a^{4} b^{12} r^{4}\right)} \cos\left({\theta}\right)^{4} + 4 \, {\left(a^{2} r^{24} - 12 \, a^{2} b^{2} r^{21} + 60 \, a^{2} b^{4} r^{18} - 160 \, a^{2} b^{6} r^{15} + 240 \, a^{2} b^{8} r^{12} - 192 \, a^{2} b^{10} r^{9} + 64 \, a^{2} b^{12} r^{6}\right)} \cos\left({\theta}\right)^{2}} \end{array}
Ric2 = Ric2_scalar.expr().factor() Ric2
288(a4r6cos(θ)48a4b2r3cos(θ)4+4a2r8cos(θ)2+16a4b4cos(θ)420a2b2r5cos(θ)2+5r10+16a2b4r2cos(θ)24b2r7+8b4r4)b4r4(a2cos(θ)2+r2)4(r3+2b2)6\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{288 \, {\left(a^{4} r^{6} \cos\left({\theta}\right)^{4} - 8 \, a^{4} b^{2} r^{3} \cos\left({\theta}\right)^{4} + 4 \, a^{2} r^{8} \cos\left({\theta}\right)^{2} + 16 \, a^{4} b^{4} \cos\left({\theta}\right)^{4} - 20 \, a^{2} b^{2} r^{5} \cos\left({\theta}\right)^{2} + 5 \, r^{10} + 16 \, a^{2} b^{4} r^{2} \cos\left({\theta}\right)^{2} - 4 \, b^{2} r^{7} + 8 \, b^{4} r^{4}\right)} b^{4} r^{4}}{{\left(a^{2} \cos\left({\theta}\right)^{2} + r^{2}\right)}^{4} {\left(r^{3} + 2 \, b^{2}\right)}^{6}}

The Kerr value is obtained for b=0b=0; we check that it is zero:

Ric2.subs({b: 0})
0\renewcommand{\Bold}[1]{\mathbf{#1}}0
Ric2_plot = Ric2.subs({a: 0.9, b: 1}) plot3d(Ric2_plot, (r, -1/2, 1/2), (th, 0, pi/2), plot_points=200, aspect_ratio=[2, 1, 0.1], axes_labels=['r', 'theta', 'R_{ab} R^{ab}'], viewer='threejs', online=True)
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graph = plot_profile(Ric2_scalar, {a: 0.9, b: 1}, -3, 3, r'$R_{ab} R^{ab}$') graph.save('Ric2_scalar_HT.pdf') graph
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Euler invariant

E_scalar=-Kr_scalar+4*Ric2_scalar-Rscal*Rscal
graph = plot_profile(E_scalar, {a: 0.9, b: 1}, -3, 3, r'$E$') graph.save('Euler_scalar.pdf') graph
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