Contact
CoCalc Logo Icon
StoreFeaturesDocsShareSupport News AboutSign UpSign In
| Download
Views: 356
Kernel: SageMath 8.1

Locally nonrotating frames and NEC

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

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

Metric

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 Delta1 = 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/Delta1 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 Delta2 = 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/Delta2 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}

Locally nonrotating frames

A1=(r^2+a^2)^2-a^2*Delta1*sin(th)^2 A2=(r^2+a^2)^2-a^2*Delta2*sin(th)^2
b1=XBL1.frame() b1
(M1,(t,r,θ,ϕ))\renewcommand{\Bold}[1]{\mathbf{#1}}\left(M_1, \left(\frac{\partial}{\partial t },\frac{\partial}{\partial r },\frac{\partial}{\partial {\theta} },\frac{\partial}{\partial {\phi} }\right)\right)
change_frame = M1.automorphism_field() change_frame[:] = [[sqrt(A1/(Sigma*Delta1)),0,0,0], [0,sqrt(Delta1/Sigma),0,0],[0,0,1/sqrt(Sigma),0], [2*m1*a*r/sqrt(A1*Sigma*Delta1),0,0,sqrt(Sigma/A1)/sin(th)]]; #old covariant basis in terms of the new covariant basis e1 = b1.new_frame(change_frame, 'e1') ; e1
(M1,(e10,e11,e12,e13))\renewcommand{\Bold}[1]{\mathbf{#1}}\left(M_1, \left(e1_{0},e1_{1},e1_{2},e1_{3}\right)\right)
b2=XBL2.frame(); b2
(M2,(t,r,θ,ϕ))\renewcommand{\Bold}[1]{\mathbf{#1}}\left(M_2, \left(\frac{\partial}{\partial t },\frac{\partial}{\partial r },\frac{\partial}{\partial {\theta} },\frac{\partial}{\partial {\phi} }\right)\right)
change_frame = M2.automorphism_field() change_frame[:] = [[sqrt(A2/(Sigma*Delta2)),0,0,0], [0,sqrt(Delta2/Sigma),0,0],[0,0,1/sqrt(Sigma),0], [2*m2*a*r/sqrt(A2*Sigma*Delta2),0,0,sqrt(Sigma/A2)/sin(th)]]; #old covariant basis in terms of the new covariant basis e2 = b2.new_frame(change_frame, 'e2') ; e2
(M2,(e20,e21,e22,e23))\renewcommand{\Bold}[1]{\mathbf{#1}}\left(M_2, \left(e2_{0},e2_{1},e2_{2},e2_{3}\right)\right)
g.display_comp(e1)
g0000=1g1111=1g2222=1g3333=1\renewcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{lcl} g_{\,0\,0}^{\phantom{\, 0}\phantom{\, 0}} & = & -1 \\ g_{\,1\,1}^{\phantom{\, 1}\phantom{\, 1}} & = & 1 \\ g_{\,2\,2}^{\phantom{\, 2}\phantom{\, 2}} & = & 1 \\ g_{\,3\,3}^{\phantom{\, 3}\phantom{\, 3}} & = & 1 \end{array}
g.display_comp(e2)
g0000=1g1111=1g2222=1g3333=1\renewcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{lcl} g_{\,0\,0}^{\phantom{\, 0}\phantom{\, 0}} & = & -1 \\ g_{\,1\,1}^{\phantom{\, 1}\phantom{\, 1}} & = & 1 \\ g_{\,2\,2}^{\phantom{\, 2}\phantom{\, 2}} & = & 1 \\ g_{\,3\,3}^{\phantom{\, 3}\phantom{\, 3}} & = & 1 \end{array}

Energy momentum tensor

Ricci = g.ricci() Ricci_scal = g.ricci_scalar() G = Ricci-1/2*g*Ricci_scal; print G
Field of symmetric bilinear forms Ric(g)-unnamed metric on the 4-dimensional differentiable manifold M
G1=G.display(e1)
G2=G.display(e2)
G.display()
Ric(g)unnamed metric=(12(3a2b2r7+b2r9+2b4r62b2r8(a4b2r54a4b4r2)cos(θ)4+(a4b2r52a2b2r74a4b4r2+2a2b4r4)cos(θ)2)r15+6b2r12+12b4r9+8b6r6+(a6r9+6a6b2r6+12a6b4r3+8a6b6)cos(θ)6+3(a4r11+6a4b2r8+12a4b4r5+8a4b6r2)cos(θ)4+3(a2r13+6a2b2r10+12a2b4r7+8a2b6r4)cos(θ)2)dtdt+(12((a5b2r5+a3b2r74a5b4r24a3b4r4)sin(θ)4(a5b2r5+4a3b2r7+3ab2r94a5b4r24a3b4r42ab2r8)sin(θ)2)r15+6b2r12+12b4r9+8b6r6+(a6r9+6a6b2r6+12a6b4r3+8a6b6)cos(θ)6+3(a4r11+6a4b2r8+12a4b4r5+8a4b6r2)cos(θ)4+3(a2r13+6a2b2r10+12a2b4r7+8a2b6r4)cos(θ)2)dtdϕ+(12b2r4a2r8+r10+4a2b2r5+4b2r72r9+4a2b4r2+4b4r44b2r6+(a4r6+a2r8+4a4b2r3+4a2b2r52a2r7+4a4b4+4a2b4r24a2b2r4)cos(θ)2)drdr+(12(2b2r72b4r4+(a2b2r54a2b4r2)cos(θ)2)r11+6b2r8+12b4r5+8b6r2+(a2r9+6a2b2r6+12a2b4r3+8a2b6)cos(θ)2)dθdθ+(12((a5b2r5+a3b2r74a5b4r24a3b4r4)sin(θ)4(a5b2r5+4a3b2r7+3ab2r94a5b4r24a3b4r42ab2r8)sin(θ)2)r15+6b2r12+12b4r9+8b6r6+(a6r9+6a6b2r6+12a6b4r3+8a6b6)cos(θ)6+3(a4r11+6a4b2r8+12a4b4r5+8a4b6r2)cos(θ)4+3(a2r13+6a2b2r10+12a2b4r7+8a2b6r4)cos(θ)2)dϕdt+(12((a6b2r5+a4b2r74a6b4r210a4b4r46a2b4r6+2a2b2r8)sin(θ)4(a6b2r5+4a4b2r7+5a2b2r9+2b2r114a6b4r210a4b4r48a2b4r62b4r8)sin(θ)2)r15+6b2r12+12b4r9+8b6r6+(a6r9+6a6b2r6+12a6b4r3+8a6b6)cos(θ)6+3(a4r11+6a4b2r8+12a4b4r5+8a4b6r2)cos(θ)4+3(a2r13+6a2b2r10+12a2b4r7+8a2b6r4)cos(θ)2)dϕdϕ\renewcommand{\Bold}[1]{\mathbf{#1}}\mathrm{Ric}\left(g\right)-\mbox{unnamed metric} = \left( \frac{12 \, {\left(3 \, a^{2} b^{2} r^{7} + b^{2} r^{9} + 2 \, b^{4} r^{6} - 2 \, b^{2} r^{8} - {\left(a^{4} b^{2} r^{5} - 4 \, a^{4} b^{4} r^{2}\right)} \cos\left({\theta}\right)^{4} + {\left(a^{4} b^{2} r^{5} - 2 \, a^{2} b^{2} r^{7} - 4 \, a^{4} b^{4} r^{2} + 2 \, a^{2} b^{4} r^{4}\right)} \cos\left({\theta}\right)^{2}\right)}}{r^{15} + 6 \, b^{2} r^{12} + 12 \, b^{4} r^{9} + 8 \, b^{6} r^{6} + {\left(a^{6} r^{9} + 6 \, a^{6} b^{2} r^{6} + 12 \, a^{6} b^{4} r^{3} + 8 \, a^{6} b^{6}\right)} \cos\left({\theta}\right)^{6} + 3 \, {\left(a^{4} r^{11} + 6 \, a^{4} b^{2} r^{8} + 12 \, a^{4} b^{4} r^{5} + 8 \, a^{4} b^{6} r^{2}\right)} \cos\left({\theta}\right)^{4} + 3 \, {\left(a^{2} r^{13} + 6 \, a^{2} b^{2} r^{10} + 12 \, a^{2} b^{4} r^{7} + 8 \, a^{2} b^{6} r^{4}\right)} \cos\left({\theta}\right)^{2}} \right) \mathrm{d} t\otimes \mathrm{d} t + \left( \frac{12 \, {\left({\left(a^{5} b^{2} r^{5} + a^{3} b^{2} r^{7} - 4 \, a^{5} b^{4} r^{2} - 4 \, a^{3} b^{4} r^{4}\right)} \sin\left({\theta}\right)^{4} - {\left(a^{5} b^{2} r^{5} + 4 \, a^{3} b^{2} r^{7} + 3 \, a b^{2} r^{9} - 4 \, a^{5} b^{4} r^{2} - 4 \, a^{3} b^{4} r^{4} - 2 \, a b^{2} r^{8}\right)} \sin\left({\theta}\right)^{2}\right)}}{r^{15} + 6 \, b^{2} r^{12} + 12 \, b^{4} r^{9} + 8 \, b^{6} r^{6} + {\left(a^{6} r^{9} + 6 \, a^{6} b^{2} r^{6} + 12 \, a^{6} b^{4} r^{3} + 8 \, a^{6} b^{6}\right)} \cos\left({\theta}\right)^{6} + 3 \, {\left(a^{4} r^{11} + 6 \, a^{4} b^{2} r^{8} + 12 \, a^{4} b^{4} r^{5} + 8 \, a^{4} b^{6} r^{2}\right)} \cos\left({\theta}\right)^{4} + 3 \, {\left(a^{2} r^{13} + 6 \, a^{2} b^{2} r^{10} + 12 \, a^{2} b^{4} r^{7} + 8 \, a^{2} b^{6} r^{4}\right)} \cos\left({\theta}\right)^{2}} \right) \mathrm{d} t\otimes \mathrm{d} {\phi} + \left( -\frac{12 \, b^{2} r^{4}}{a^{2} r^{8} + r^{10} + 4 \, a^{2} b^{2} r^{5} + 4 \, b^{2} r^{7} - 2 \, r^{9} + 4 \, a^{2} b^{4} r^{2} + 4 \, b^{4} r^{4} - 4 \, b^{2} r^{6} + {\left(a^{4} r^{6} + a^{2} r^{8} + 4 \, a^{4} b^{2} r^{3} + 4 \, a^{2} b^{2} r^{5} - 2 \, a^{2} r^{7} + 4 \, a^{4} b^{4} + 4 \, a^{2} b^{4} r^{2} - 4 \, a^{2} b^{2} r^{4}\right)} \cos\left({\theta}\right)^{2}} \right) \mathrm{d} r\otimes \mathrm{d} r + \left( \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)}}{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}} \right) \mathrm{d} {\theta}\otimes \mathrm{d} {\theta} + \left( \frac{12 \, {\left({\left(a^{5} b^{2} r^{5} + a^{3} b^{2} r^{7} - 4 \, a^{5} b^{4} r^{2} - 4 \, a^{3} b^{4} r^{4}\right)} \sin\left({\theta}\right)^{4} - {\left(a^{5} b^{2} r^{5} + 4 \, a^{3} b^{2} r^{7} + 3 \, a b^{2} r^{9} - 4 \, a^{5} b^{4} r^{2} - 4 \, a^{3} b^{4} r^{4} - 2 \, a b^{2} r^{8}\right)} \sin\left({\theta}\right)^{2}\right)}}{r^{15} + 6 \, b^{2} r^{12} + 12 \, b^{4} r^{9} + 8 \, b^{6} r^{6} + {\left(a^{6} r^{9} + 6 \, a^{6} b^{2} r^{6} + 12 \, a^{6} b^{4} r^{3} + 8 \, a^{6} b^{6}\right)} \cos\left({\theta}\right)^{6} + 3 \, {\left(a^{4} r^{11} + 6 \, a^{4} b^{2} r^{8} + 12 \, a^{4} b^{4} r^{5} + 8 \, a^{4} b^{6} r^{2}\right)} \cos\left({\theta}\right)^{4} + 3 \, {\left(a^{2} r^{13} + 6 \, a^{2} b^{2} r^{10} + 12 \, a^{2} b^{4} r^{7} + 8 \, a^{2} b^{6} r^{4}\right)} \cos\left({\theta}\right)^{2}} \right) \mathrm{d} {\phi}\otimes \mathrm{d} t + \left( -\frac{12 \, {\left({\left(a^{6} b^{2} r^{5} + a^{4} b^{2} r^{7} - 4 \, a^{6} b^{4} r^{2} - 10 \, a^{4} b^{4} r^{4} - 6 \, a^{2} b^{4} r^{6} + 2 \, a^{2} b^{2} r^{8}\right)} \sin\left({\theta}\right)^{4} - {\left(a^{6} b^{2} r^{5} + 4 \, a^{4} b^{2} r^{7} + 5 \, a^{2} b^{2} r^{9} + 2 \, b^{2} r^{11} - 4 \, a^{6} b^{4} r^{2} - 10 \, a^{4} b^{4} r^{4} - 8 \, a^{2} b^{4} r^{6} - 2 \, b^{4} r^{8}\right)} \sin\left({\theta}\right)^{2}\right)}}{r^{15} + 6 \, b^{2} r^{12} + 12 \, b^{4} r^{9} + 8 \, b^{6} r^{6} + {\left(a^{6} r^{9} + 6 \, a^{6} b^{2} r^{6} + 12 \, a^{6} b^{4} r^{3} + 8 \, a^{6} b^{6}\right)} \cos\left({\theta}\right)^{6} + 3 \, {\left(a^{4} r^{11} + 6 \, a^{4} b^{2} r^{8} + 12 \, a^{4} b^{4} r^{5} + 8 \, a^{4} b^{6} r^{2}\right)} \cos\left({\theta}\right)^{4} + 3 \, {\left(a^{2} r^{13} + 6 \, a^{2} b^{2} r^{10} + 12 \, a^{2} b^{4} r^{7} + 8 \, a^{2} b^{6} r^{4}\right)} \cos\left({\theta}\right)^{2}} \right) \mathrm{d} {\phi}\otimes \mathrm{d} {\phi}

Null Energy Condition

k1 = M1.vector_field('k1') k1[e1,:] = [1,1,0,0] k1.display(e1)
k1=e10+e11\renewcommand{\Bold}[1]{\mathbf{#1}}k1 = e1_{0}+e1_{1}
k1d=k1.down(g) k1k1=k1d['_a']*k1['^a'] k1k1.display()
M1R(t,r,θ,ϕ)0\renewcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{llcl} & M_1 & \longrightarrow & \mathbb{R} \\ & \left(t, r, {\theta}, {\phi}\right) & \longmapsto & 0 \end{array}
k2 = M2.vector_field('k2') k2[e2,:] = [1,1,0,0] k2.display(e2)
k2=e20+e21\renewcommand{\Bold}[1]{\mathbf{#1}}k2 = e2_{0}+e2_{1}
k2d=k2.down(g) k2k2=k2d['_a']*k2['^a'] k2k2.display()
M2R(t,r,θ,ϕ)0\renewcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{llcl} & M_2 & \longrightarrow & \mathbb{R} \\ & \left(t, r, {\theta}, {\phi}\right) & \longmapsto & 0 \end{array}
NN=G['_{ab}']*k1['^a'] NEC1=8*pi*NN['_a']*k1['^a'] NEC1.display()
M1R(t,r,θ,ϕ)96((πa6b2r8+πa4b2r102πa6b4r52πa4b4r72πa4b2r98πa6b6r28πa4b6r4+8πa4b4r6)sin(θ)4(πa6b2r8+4πa4b2r10+3πa2b2r122πa6b4r5+4πa4b4r76πa2b2r118πa6b6r28πa4b6r4+8πa4b4r62(πa4b23πa2b4)r9)sin(θ)2)a2r18+r20+8a2b2r15+24a2b4r12+2(a2+4b2)r17+32a2b6r9+16a2b8r6+12(a2b2+2b4)r14+8(3a2b4+4b6)r11+16(a2b6+b8)r8+(a8r12+a6r14+8a8b2r9+8a6b2r112a6r13+24a8b4r6+24a6b4r812a6b2r10+32a8b6r3+32a6b6r524a6b4r7+16a8b8+16a6b8r216a6b6r4)cos(θ)6+(3a6r14+3a4r16+24a6b2r114a4r15+72a6b4r824a4b2r12+96a6b6r548a4b4r9+48a6b8r232a4b6r6+2(a6+12a4b2)r13+12(a6b2+6a4b4)r10+24(a6b4+4a4b6)r7+16(a6b6+3a4b8)r4)cos(θ)4+(3a4r16+3a2r18+24a4b2r132a2r17+72a4b4r1012a2b2r14+96a4b6r724a2b4r11+48a4b8r416a2b6r8+4(a4+6a2b2)r15+24(a4b2+3a2b4)r12+48(a4b4+2a2b6)r9+16(2a4b6+3a2b8)r6)cos(θ)2\renewcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{llcl} & M_1 & \longrightarrow & \mathbb{R} \\ & \left(t, r, {\theta}, {\phi}\right) & \longmapsto & -\frac{96 \, {\left({\left(\pi a^{6} b^{2} r^{8} + \pi a^{4} b^{2} r^{10} - 2 \, \pi a^{6} b^{4} r^{5} - 2 \, \pi a^{4} b^{4} r^{7} - 2 \, \pi a^{4} b^{2} r^{9} - 8 \, \pi a^{6} b^{6} r^{2} - 8 \, \pi a^{4} b^{6} r^{4} + 8 \, \pi a^{4} b^{4} r^{6}\right)} \sin\left({\theta}\right)^{4} - {\left(\pi a^{6} b^{2} r^{8} + 4 \, \pi a^{4} b^{2} r^{10} + 3 \, \pi a^{2} b^{2} r^{12} - 2 \, \pi a^{6} b^{4} r^{5} + 4 \, \pi a^{4} b^{4} r^{7} - 6 \, \pi a^{2} b^{2} r^{11} - 8 \, \pi a^{6} b^{6} r^{2} - 8 \, \pi a^{4} b^{6} r^{4} + 8 \, \pi a^{4} b^{4} r^{6} - 2 \, {\left(\pi a^{4} b^{2} - 3 \, \pi a^{2} b^{4}\right)} r^{9}\right)} \sin\left({\theta}\right)^{2}\right)}}{a^{2} r^{18} + r^{20} + 8 \, a^{2} b^{2} r^{15} + 24 \, a^{2} b^{4} r^{12} + 2 \, {\left(a^{2} + 4 \, b^{2}\right)} r^{17} + 32 \, a^{2} b^{6} r^{9} + 16 \, a^{2} b^{8} r^{6} + 12 \, {\left(a^{2} b^{2} + 2 \, b^{4}\right)} r^{14} + 8 \, {\left(3 \, a^{2} b^{4} + 4 \, b^{6}\right)} r^{11} + 16 \, {\left(a^{2} b^{6} + b^{8}\right)} r^{8} + {\left(a^{8} r^{12} + a^{6} r^{14} + 8 \, a^{8} b^{2} r^{9} + 8 \, a^{6} b^{2} r^{11} - 2 \, a^{6} r^{13} + 24 \, a^{8} b^{4} r^{6} + 24 \, a^{6} b^{4} r^{8} - 12 \, a^{6} b^{2} r^{10} + 32 \, a^{8} b^{6} r^{3} + 32 \, a^{6} b^{6} r^{5} - 24 \, a^{6} b^{4} r^{7} + 16 \, a^{8} b^{8} + 16 \, a^{6} b^{8} r^{2} - 16 \, a^{6} b^{6} r^{4}\right)} \cos\left({\theta}\right)^{6} + {\left(3 \, a^{6} r^{14} + 3 \, a^{4} r^{16} + 24 \, a^{6} b^{2} r^{11} - 4 \, a^{4} r^{15} + 72 \, a^{6} b^{4} r^{8} - 24 \, a^{4} b^{2} r^{12} + 96 \, a^{6} b^{6} r^{5} - 48 \, a^{4} b^{4} r^{9} + 48 \, a^{6} b^{8} r^{2} - 32 \, a^{4} b^{6} r^{6} + 2 \, {\left(a^{6} + 12 \, a^{4} b^{2}\right)} r^{13} + 12 \, {\left(a^{6} b^{2} + 6 \, a^{4} b^{4}\right)} r^{10} + 24 \, {\left(a^{6} b^{4} + 4 \, a^{4} b^{6}\right)} r^{7} + 16 \, {\left(a^{6} b^{6} + 3 \, a^{4} b^{8}\right)} r^{4}\right)} \cos\left({\theta}\right)^{4} + {\left(3 \, a^{4} r^{16} + 3 \, a^{2} r^{18} + 24 \, a^{4} b^{2} r^{13} - 2 \, a^{2} r^{17} + 72 \, a^{4} b^{4} r^{10} - 12 \, a^{2} b^{2} r^{14} + 96 \, a^{4} b^{6} r^{7} - 24 \, a^{2} b^{4} r^{11} + 48 \, a^{4} b^{8} r^{4} - 16 \, a^{2} b^{6} r^{8} + 4 \, {\left(a^{4} + 6 \, a^{2} b^{2}\right)} r^{15} + 24 \, {\left(a^{4} b^{2} + 3 \, a^{2} b^{4}\right)} r^{12} + 48 \, {\left(a^{4} b^{4} + 2 \, a^{2} b^{6}\right)} r^{9} + 16 \, {\left(2 \, a^{4} b^{6} + 3 \, a^{2} b^{8}\right)} r^{6}\right)} \cos\left({\theta}\right)^{2}} \end{array}
NNN=G['_{ab}']*k2['^a'] NEC2=8*pi*NNN['_a']*k2['^a'] NEC2.display()
M2R(t,r,θ,ϕ)96((πa6b2r8+πa4b2r10+2πa6b4r5+2πa4b4r72πa4b2r98πa6b6r28πa4b6r48πa4b4r6)sin(θ)4(πa6b2r8+4πa4b2r10+3πa2b2r12+2πa6b4r54πa4b4r76πa2b2r118πa6b6r28πa4b6r48πa4b4r62(πa4b2+3πa2b4)r9)sin(θ)2)a2r18+r208a2b2r15+24a2b4r12+2(a24b2)r1732a2b6r9+16a2b8r612(a2b22b4)r14+8(3a2b44b6)r1116(a2b6b8)r8+(a8r12+a6r148a8b2r98a6b2r112a6r13+24a8b4r6+24a6b4r8+12a6b2r1032a8b6r332a6b6r524a6b4r7+16a8b8+16a6b8r2+16a6b6r4)cos(θ)6+(3a6r14+3a4r1624a6b2r114a4r15+72a6b4r8+24a4b2r1296a6b6r548a4b4r9+48a6b8r2+32a4b6r6+2(a612a4b2)r1312(a6b26a4b4)r10+24(a6b44a4b6)r716(a6b63a4b8)r4)cos(θ)4+(3a4r16+3a2r1824a4b2r132a2r17+72a4b4r10+12a2b2r1496a4b6r724a2b4r11+48a4b8r4+16a2b6r8+4(a46a2b2)r1524(a4b23a2b4)r12+48(a4b42a2b6)r916(2a4b63a2b8)r6)cos(θ)2\renewcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{llcl} & M_2 & \longrightarrow & \mathbb{R} \\ & \left(t, r, {\theta}, {\phi}\right) & \longmapsto & \frac{96 \, {\left({\left(\pi a^{6} b^{2} r^{8} + \pi a^{4} b^{2} r^{10} + 2 \, \pi a^{6} b^{4} r^{5} + 2 \, \pi a^{4} b^{4} r^{7} - 2 \, \pi a^{4} b^{2} r^{9} - 8 \, \pi a^{6} b^{6} r^{2} - 8 \, \pi a^{4} b^{6} r^{4} - 8 \, \pi a^{4} b^{4} r^{6}\right)} \sin\left({\theta}\right)^{4} - {\left(\pi a^{6} b^{2} r^{8} + 4 \, \pi a^{4} b^{2} r^{10} + 3 \, \pi a^{2} b^{2} r^{12} + 2 \, \pi a^{6} b^{4} r^{5} - 4 \, \pi a^{4} b^{4} r^{7} - 6 \, \pi a^{2} b^{2} r^{11} - 8 \, \pi a^{6} b^{6} r^{2} - 8 \, \pi a^{4} b^{6} r^{4} - 8 \, \pi a^{4} b^{4} r^{6} - 2 \, {\left(\pi a^{4} b^{2} + 3 \, \pi a^{2} b^{4}\right)} r^{9}\right)} \sin\left({\theta}\right)^{2}\right)}}{a^{2} r^{18} + r^{20} - 8 \, a^{2} b^{2} r^{15} + 24 \, a^{2} b^{4} r^{12} + 2 \, {\left(a^{2} - 4 \, b^{2}\right)} r^{17} - 32 \, a^{2} b^{6} r^{9} + 16 \, a^{2} b^{8} r^{6} - 12 \, {\left(a^{2} b^{2} - 2 \, b^{4}\right)} r^{14} + 8 \, {\left(3 \, a^{2} b^{4} - 4 \, b^{6}\right)} r^{11} - 16 \, {\left(a^{2} b^{6} - b^{8}\right)} r^{8} + {\left(a^{8} r^{12} + a^{6} r^{14} - 8 \, a^{8} b^{2} r^{9} - 8 \, a^{6} b^{2} r^{11} - 2 \, a^{6} r^{13} + 24 \, a^{8} b^{4} r^{6} + 24 \, a^{6} b^{4} r^{8} + 12 \, a^{6} b^{2} r^{10} - 32 \, a^{8} b^{6} r^{3} - 32 \, a^{6} b^{6} r^{5} - 24 \, a^{6} b^{4} r^{7} + 16 \, a^{8} b^{8} + 16 \, a^{6} b^{8} r^{2} + 16 \, a^{6} b^{6} r^{4}\right)} \cos\left({\theta}\right)^{6} + {\left(3 \, a^{6} r^{14} + 3 \, a^{4} r^{16} - 24 \, a^{6} b^{2} r^{11} - 4 \, a^{4} r^{15} + 72 \, a^{6} b^{4} r^{8} + 24 \, a^{4} b^{2} r^{12} - 96 \, a^{6} b^{6} r^{5} - 48 \, a^{4} b^{4} r^{9} + 48 \, a^{6} b^{8} r^{2} + 32 \, a^{4} b^{6} r^{6} + 2 \, {\left(a^{6} - 12 \, a^{4} b^{2}\right)} r^{13} - 12 \, {\left(a^{6} b^{2} - 6 \, a^{4} b^{4}\right)} r^{10} + 24 \, {\left(a^{6} b^{4} - 4 \, a^{4} b^{6}\right)} r^{7} - 16 \, {\left(a^{6} b^{6} - 3 \, a^{4} b^{8}\right)} r^{4}\right)} \cos\left({\theta}\right)^{4} + {\left(3 \, a^{4} r^{16} + 3 \, a^{2} r^{18} - 24 \, a^{4} b^{2} r^{13} - 2 \, a^{2} r^{17} + 72 \, a^{4} b^{4} r^{10} + 12 \, a^{2} b^{2} r^{14} - 96 \, a^{4} b^{6} r^{7} - 24 \, a^{2} b^{4} r^{11} + 48 \, a^{4} b^{8} r^{4} + 16 \, a^{2} b^{6} r^{8} + 4 \, {\left(a^{4} - 6 \, a^{2} b^{2}\right)} r^{15} - 24 \, {\left(a^{4} b^{2} - 3 \, a^{2} b^{4}\right)} r^{12} + 48 \, {\left(a^{4} b^{4} - 2 \, a^{2} b^{6}\right)} r^{9} - 16 \, {\left(2 \, a^{4} b^{6} - 3 \, a^{2} b^{8}\right)} r^{6}\right)} \cos\left({\theta}\right)^{2}} \end{array}
nec1(a,b,r,th)=NEC1.expr() nec2(a,b,r,th)=NEC2.expr()
a=0.9 b=1 plot1=plot((nec1(a,b,r,pi/6), nec1(a,b,r,pi/3), nec1(a,b,r,pi/2)), (r,0,4), ymin=-70,ymax=70, legend_label=[r'$\theta=\frac{\pi}{6}$',r'$\theta=\frac{\pi}{3}$', r'$\theta=\frac{\pi}{2}$'], axes_labels=[r'$r/m$',''], plot_points=400)
a=0.9 b=1 plot2=plot((nec2(a,b,r,pi/6), nec2(a,b,r,pi/3), nec2(a,b,r,pi/2)),(r,-4,0), axes_labels=[r'$r/m$',r'$T_{\hat{\mu}\hat{\nu}}k^{\hat{\mu}}k^{\hat{\nu}}$'], ymin=-70, ymax=70, plot_points=400)
graph = plot1 + plot2 show(graph)
Image in a Jupyter notebook
graph.save('NEC_a=09_b=1.pdf')
ww1(a,b,r,th)=-12*((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)*sin(th)^4 - (a^6*b^2*r^8 + 4*a^4*b^2*r^10 + 3*a^2*b^2*r^12 - 2*a^6*b^4*r^5 + 4*a^4*b^4*r^7 - 6*a^2*b^2*r^11 - 8*a^6*b^6*r^2 - 8*a^4*b^6*r^4 + 8*a^4*b^4*r^6 - 2*(a^4*b^2 - 3*a^2*b^4)*r^9)*sin(th)^2)/(a^2*r^18 + r^20 + 8*a^2*b^2*r^15 + 24*a^2*b^4*r^12 + 2*(a^2 + 4*b^2)*r^17 + 32*a^2*b^6*r^9 + 16*a^2*b^8*r^6 + 12*(a^2*b^2 + 2*b^4)*r^14 + 8*(3*a^2*b^4 + 4*b^6)*r^11 + 16*(a^2*b^6 + b^8)*r^8 + (a^8*r^12 + a^6*r^14 + 8*a^8*b^2*r^9 + 8*a^6*b^2*r^11 - 2*a^6*r^13 + 24*a^8*b^4*r^6 + 24*a^6*b^4*r^8 - 12*a^6*b^2*r^10 + 32*a^8*b^6*r^3 + 32*a^6*b^6*r^5 - 24*a^6*b^4*r^7 + 16*a^8*b^8 + 16*a^6*b^8*r^2 - 16*a^6*b^6*r^4)*cos(th)^6 + (3*a^6*r^14 + 3*a^4*r^16 + 24*a^6*b^2*r^11 - 4*a^4*r^15 + 72*a^6*b^4*r^8 - 24*a^4*b^2*r^12 + 96*a^6*b^6*r^5 - 48*a^4*b^4*r^9 + 48*a^6*b^8*r^2 - 32*a^4*b^6*r^6 + 2*(a^6 + 12*a^4*b^2)*r^13 + 12*(a^6*b^2 + 6*a^4*b^4)*r^10 + 24*(a^6*b^4 + 4*a^4*b^6)*r^7 + 16*(a^6*b^6 + 3*a^4*b^8)*r^4)*cos(th)^4 + (3*a^4*r^16 + 3*a^2*r^18 + 24*a^4*b^2*r^13 - 2*a^2*r^17 + 72*a^4*b^4*r^10 - 12*a^2*b^2*r^14 + 96*a^4*b^6*r^7 - 24*a^2*b^4*r^11 + 48*a^4*b^8*r^4 - 16*a^2*b^6*r^8 + 4*(a^4 + 6*a^2*b^2)*r^15 + 24*(a^4*b^2 + 3*a^2*b^4)*r^12 + 48*(a^4*b^4 + 2*a^2*b^6)*r^9 + 16*(2*a^4*b^6 + 3*a^2*b^8)*r^6)*cos(th)^2)
ww2(a,b,r,th)=12*((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)*sin(th)^4 - (a^6*b^2*r^8 + 4*a^4*b^2*r^10 + 3*a^2*b^2*r^12 + 2*a^6*b^4*r^5 - 4*a^4*b^4*r^7 - 6*a^2*b^2*r^11 - 8*a^6*b^6*r^2 - 8*a^4*b^6*r^4 - 8*a^4*b^4*r^6 - 2*(a^4*b^2 + 3*a^2*b^4)*r^9)*sin(th)^2)/(a^2*r^18 + r^20 - 8*a^2*b^2*r^15 + 24*a^2*b^4*r^12 + 2*(a^2 - 4*b^2)*r^17 - 32*a^2*b^6*r^9 + 16*a^2*b^8*r^6 - 12*(a^2*b^2 - 2*b^4)*r^14 + 8*(3*a^2*b^4 - 4*b^6)*r^11 - 16*(a^2*b^6 - b^8)*r^8 + (a^8*r^12 + a^6*r^14 - 8*a^8*b^2*r^9 - 8*a^6*b^2*r^11 - 2*a^6*r^13 + 24*a^8*b^4*r^6 + 24*a^6*b^4*r^8 + 12*a^6*b^2*r^10 - 32*a^8*b^6*r^3 - 32*a^6*b^6*r^5 - 24*a^6*b^4*r^7 + 16*a^8*b^8 + 16*a^6*b^8*r^2 + 16*a^6*b^6*r^4)*cos(th)^6 + (3*a^6*r^14 + 3*a^4*r^16 - 24*a^6*b^2*r^11 - 4*a^4*r^15 + 72*a^6*b^4*r^8 + 24*a^4*b^2*r^12 - 96*a^6*b^6*r^5 - 48*a^4*b^4*r^9 + 48*a^6*b^8*r^2 + 32*a^4*b^6*r^6 + 2*(a^6 - 12*a^4*b^2)*r^13 - 12*(a^6*b^2 - 6*a^4*b^4)*r^10 + 24*(a^6*b^4 - 4*a^4*b^6)*r^7 - 16*(a^6*b^6 - 3*a^4*b^8)*r^4)*cos(th)^4 + (3*a^4*r^16 + 3*a^2*r^18 - 24*a^4*b^2*r^13 - 2*a^2*r^17 + 72*a^4*b^4*r^10 + 12*a^2*b^2*r^14 - 96*a^4*b^6*r^7 - 24*a^2*b^4*r^11 + 48*a^4*b^8*r^4 + 16*a^2*b^6*r^8 + 4*(a^4 - 6*a^2*b^2)*r^15 - 24*(a^4*b^2 - 3*a^2*b^4)*r^12 + 48*(a^4*b^4 - 2*a^2*b^6)*r^9 - 16*(2*a^4*b^6 - 3*a^2*b^8)*r^6)*cos(th)^2)