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5D Kerr-AdS spacetime with a Nambu-Goto string (case a=b)

Project: KerrAdS
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Kernel: SageMath 9.3

5D Kerr-AdS spacetime with a Nambu-Goto string

Case a = b

This SageMath notebook is relative to the article Holographic drag force in 5d Kerr-AdS black hole by Irina Ya. Aref'eva, Anastasia A. Golubtsova and Eric Gourgoulhon, arXiv:2004.12984.

The involved differential geometry computations are based on tools developed through the SageManifolds project.

NB: a version of SageMath at least equal to 8.2 is required to run this notebook:

version()
'SageMath version 9.3, Release Date: 2021-05-09'

First we set up the notebook to display mathematical objects using LaTeX rendering:

%display latex

Since some computations are quite long, we ask for running them in parallel on 8 cores:

Parallelism().set(nproc=1) # only nproc=1 works on CoCalc

Spacetime manifold

We declare the Kerr-AdS spacetime as a 5-dimensional Lorentzian manifold:

M = Manifold(5, 'M', r'\mathcal{M}', structure='Lorentzian', metric_name='G') print(M)
5-dimensional Lorentzian manifold M

Let us define Boyer-Lindquist-type coordinates (rational polynomial version) on M\mathcal{M}, via the method chart(), the argument of which is a string expressing the coordinates names, their ranges (the default is (,+)(-\infty,+\infty)) and their LaTeX symbols:

BL.<t,r,mu,ph,ps> = M.chart(r't r:(0,+oo) mu:(0,1):\mu ph:(0,2*pi):\phi ps:(0,2*pi):\psi') BL
(M,(t,r,μ,ϕ,ψ))\renewcommand{\Bold}[1]{\mathbf{#1}}\left(\mathcal{M},(t, r, {\mu}, {\phi}, {\psi})\right)

The coordinate μ\mu is related to the standard Boyer-Lindquist coordinate θ\theta by μ=cosθ \mu = \cos\theta

The coordinate ranges are

BL.coord_range()
t: (,+);r: (0,+);μ: (0,1);ϕ: (0,2π);ψ: (0,2π)\renewcommand{\Bold}[1]{\mathbf{#1}}t :\ \left( -\infty, +\infty \right) ;\quad r :\ \left( 0 , +\infty \right) ;\quad {\mu} :\ \left( 0 , 1 \right) ;\quad {\phi} :\ \left( 0 , 2 \, \pi \right) ;\quad {\psi} :\ \left( 0 , 2 \, \pi \right)

Note that contrary to the 4-dimensional case, the range of μ\mu is (0,1)(0,1) only (cf. Sec. 1.2 of R.C. Myers, arXiv:1111.1903 or Sec. 2 of G.W. Gibbons, H. Lüb, Don N. Page, C.N. Pope, J. Geom. Phys. 53, 49 (2005)). In other words, the range of θ\theta is (0,π2)\left(0, \frac{\pi}{2}\right) only.

Metric tensor

The 4 parameters mm, aa, bb and \ell of the Kerr-AdS spacetime are declared as symbolic variables, aa and bb being the two angular momentum parameters and \ell being related to the cosmological constant by Λ=62\Lambda = - 6 \ell^2:

var('m a b', domain='real')
(m,a,b)\renewcommand{\Bold}[1]{\mathbf{#1}}\left(m, a, b\right)
var('l', domain='real', latex_name=r'\ell')
\renewcommand{\Bold}[1]{\mathbf{#1}}{\ell}
# Particular cases # m = 0 # a = 0 # b = 0 b = a

Some auxiliary functions:

keep_Delta = False # change to False to provide explicit expression for Delta_r, Xi_a, etc...
sig = (1 + r^2*l^2)/r^2 costh2 = mu^2 sinth2 = 1 - mu^2 rho2 = r^2 + a^2*mu^2 + b^2*sinth2 if keep_Delta: Delta_r = var('Delta_r', latex_name=r'\Delta_r', domain='real') Delta_th = var('Delta_th', latex_name=r'\Delta_\theta', domain='real') if a == b: Xi_a = var('Xi', latex_name=r'\Xi', domain='real') Xi_b = Xi_a else: Xi_a = var('Xi_a', latex_name=r'\Xi_a', domain='real') Xi_b = var('Xi_b', latex_name=r'\Xi_b', domain='real') #Delta_th = 1 - a^2*l^2*mu^2 - b^2*l^2*sinth2 Xi_a = 1 - a^2*l^2 Xi_b = 1 - b^2*l^2 else: Delta_r = (r^2+a^2)*(r^2+b^2)*sig - 2*m Delta_th = 1 - a^2*l^2*mu^2 - b^2*l^2*sinth2 Xi_a = 1 - a^2*l^2 Xi_b = 1 - b^2*l^2

The metric is set by its components in the coordinate frame associated with the Boyer-Lindquist-type coordinates, which is the current manifold's default frame. These components can be deduced from Eq. (5.22) of the article S.W. Hawking, C.J. Hunter & M.M. Taylor-Robinson, Phys. Rev. D 59, 064005 (1999) (the check of agreement with this equation is performed below):

G = M.metric() tmp = 1/rho2*( -Delta_r + Delta_th*(a^2*sinth2 + b^2*mu^2) + a^2*b^2*sig ) G[0,0] = tmp.simplify_full() tmp = a*sinth2/(rho2*Xi_a)*( Delta_r - (r^2+a^2)*(Delta_th + b^2*sig) ) G[0,3] = tmp.simplify_full() tmp = b*mu^2/(rho2*Xi_b)*( Delta_r - (r^2+b^2)*(Delta_th + a^2*sig) ) G[0,4] = tmp.simplify_full() G[1,1] = (rho2/Delta_r).simplify_full() G[2,2] = (rho2/Delta_th/(1-mu^2)).simplify_full() tmp = sinth2/(rho2*Xi_a^2)*( - Delta_r*a^2*sinth2 + (r^2+a^2)^2*(Delta_th + sig*b^2*sinth2) ) G[3,3] = tmp.simplify_full() tmp = a*b*sinth2*mu^2/(rho2*Xi_a*Xi_b)*( - Delta_r + sig*(r^2+a^2)*(r^2+b^2) ) G[3,4] = tmp.simplify_full() tmp = mu^2/(rho2*Xi_b^2)*( - Delta_r*b^2*mu^2 + (r^2+b^2)^2*(Delta_th + sig*a^2*mu^2) ) G[4,4] = tmp.simplify_full() G.display()
G=(a42+2r4+(2a22+1)r2+a22ma2+r2)dtdt+(a52(a2μ2a2)r4(a522am)μ22(a32μ2a32)r22ama42+(a221)r2a2)dtdϕ+(2a32μ2r2+a2μ2r4+(a522am)μ2a42+(a221)r2a2)dtdψ+(a2r2+r42r6+(2a22+1)r4+a4+(a42+2a22m)r2)drdr+(a2+r2a22(a221)μ21)dμdμ+(a52(a2μ2a2)r4(a522am)μ22(a32μ2a32)r22ama42+(a221)r2a2)dϕdt+(a622a2mμ4+(a22(a221)μ21)r4a42a2m(a62a44a2m)μ2+2(a42(a42a2)μ2a2)r2a642a42+(a442a22+1)r2+a2)dϕdϕ+(2(a2mμ4a2mμ2)a642a42+(a442a22+1)r2+a2)dϕdψ+(2a32μ2r2+a2μ2r4+(a522am)μ2a42+(a221)r2a2)dψdt+(2(a2mμ4a2mμ2)a642a42+(a442a22+1)r2+a2)dψdϕ+(2a2mμ4(a221)μ2r42(a42a2)μ2r2(a62a4)μ2a642a42+(a442a22+1)r2+a2)dψdψ\renewcommand{\Bold}[1]{\mathbf{#1}}G = \left( -\frac{a^{4} {\ell}^{2} + {\ell}^{2} r^{4} + {\left(2 \, a^{2} {\ell}^{2} + 1\right)} r^{2} + a^{2} - 2 \, m}{a^{2} + r^{2}} \right) \mathrm{d} t\otimes \mathrm{d} t + \left( -\frac{a^{5} {\ell}^{2} - {\left(a {\ell}^{2} {\mu}^{2} - a {\ell}^{2}\right)} r^{4} - {\left(a^{5} {\ell}^{2} - 2 \, a m\right)} {\mu}^{2} - 2 \, {\left(a^{3} {\ell}^{2} {\mu}^{2} - a^{3} {\ell}^{2}\right)} r^{2} - 2 \, a m}{a^{4} {\ell}^{2} + {\left(a^{2} {\ell}^{2} - 1\right)} r^{2} - a^{2}} \right) \mathrm{d} t\otimes \mathrm{d} {\phi} + \left( -\frac{2 \, a^{3} {\ell}^{2} {\mu}^{2} r^{2} + a {\ell}^{2} {\mu}^{2} r^{4} + {\left(a^{5} {\ell}^{2} - 2 \, a m\right)} {\mu}^{2}}{a^{4} {\ell}^{2} + {\left(a^{2} {\ell}^{2} - 1\right)} r^{2} - a^{2}} \right) \mathrm{d} t\otimes \mathrm{d} {\psi} + \left( \frac{a^{2} r^{2} + r^{4}}{{\ell}^{2} r^{6} + {\left(2 \, a^{2} {\ell}^{2} + 1\right)} r^{4} + a^{4} + {\left(a^{4} {\ell}^{2} + 2 \, a^{2} - 2 \, m\right)} r^{2}} \right) \mathrm{d} r\otimes \mathrm{d} r + \left( -\frac{a^{2} + r^{2}}{a^{2} {\ell}^{2} - {\left(a^{2} {\ell}^{2} - 1\right)} {\mu}^{2} - 1} \right) \mathrm{d} {\mu}\otimes \mathrm{d} {\mu} + \left( -\frac{a^{5} {\ell}^{2} - {\left(a {\ell}^{2} {\mu}^{2} - a {\ell}^{2}\right)} r^{4} - {\left(a^{5} {\ell}^{2} - 2 \, a m\right)} {\mu}^{2} - 2 \, {\left(a^{3} {\ell}^{2} {\mu}^{2} - a^{3} {\ell}^{2}\right)} r^{2} - 2 \, a m}{a^{4} {\ell}^{2} + {\left(a^{2} {\ell}^{2} - 1\right)} r^{2} - a^{2}} \right) \mathrm{d} {\phi}\otimes \mathrm{d} t + \left( -\frac{a^{6} {\ell}^{2} - 2 \, a^{2} m {\mu}^{4} + {\left(a^{2} {\ell}^{2} - {\left(a^{2} {\ell}^{2} - 1\right)} {\mu}^{2} - 1\right)} r^{4} - a^{4} - 2 \, a^{2} m - {\left(a^{6} {\ell}^{2} - a^{4} - 4 \, a^{2} m\right)} {\mu}^{2} + 2 \, {\left(a^{4} {\ell}^{2} - {\left(a^{4} {\ell}^{2} - a^{2}\right)} {\mu}^{2} - a^{2}\right)} r^{2}}{a^{6} {\ell}^{4} - 2 \, a^{4} {\ell}^{2} + {\left(a^{4} {\ell}^{4} - 2 \, a^{2} {\ell}^{2} + 1\right)} r^{2} + a^{2}} \right) \mathrm{d} {\phi}\otimes \mathrm{d} {\phi} + \left( -\frac{2 \, {\left(a^{2} m {\mu}^{4} - a^{2} m {\mu}^{2}\right)}}{a^{6} {\ell}^{4} - 2 \, a^{4} {\ell}^{2} + {\left(a^{4} {\ell}^{4} - 2 \, a^{2} {\ell}^{2} + 1\right)} r^{2} + a^{2}} \right) \mathrm{d} {\phi}\otimes \mathrm{d} {\psi} + \left( -\frac{2 \, a^{3} {\ell}^{2} {\mu}^{2} r^{2} + a {\ell}^{2} {\mu}^{2} r^{4} + {\left(a^{5} {\ell}^{2} - 2 \, a m\right)} {\mu}^{2}}{a^{4} {\ell}^{2} + {\left(a^{2} {\ell}^{2} - 1\right)} r^{2} - a^{2}} \right) \mathrm{d} {\psi}\otimes \mathrm{d} t + \left( -\frac{2 \, {\left(a^{2} m {\mu}^{4} - a^{2} m {\mu}^{2}\right)}}{a^{6} {\ell}^{4} - 2 \, a^{4} {\ell}^{2} + {\left(a^{4} {\ell}^{4} - 2 \, a^{2} {\ell}^{2} + 1\right)} r^{2} + a^{2}} \right) \mathrm{d} {\psi}\otimes \mathrm{d} {\phi} + \left( \frac{2 \, a^{2} m {\mu}^{4} - {\left(a^{2} {\ell}^{2} - 1\right)} {\mu}^{2} r^{4} - 2 \, {\left(a^{4} {\ell}^{2} - a^{2}\right)} {\mu}^{2} r^{2} - {\left(a^{6} {\ell}^{2} - a^{4}\right)} {\mu}^{2}}{a^{6} {\ell}^{4} - 2 \, a^{4} {\ell}^{2} + {\left(a^{4} {\ell}^{4} - 2 \, a^{2} {\ell}^{2} + 1\right)} r^{2} + a^{2}} \right) \mathrm{d} {\psi}\otimes \mathrm{d} {\psi}
G.display_comp(only_nonredundant=True)
Gtttt=a42+2r4+(2a22+1)r2+a22ma2+r2Gtϕtϕ=a52(a2μ2a2)r4(a522am)μ22(a32μ2a32)r22ama42+(a221)r2a2Gtψtψ=2a32μ2r2+a2μ2r4+(a522am)μ2a42+(a221)r2a2Grrrr=a2r2+r42r6+(2a22+1)r4+a4+(a42+2a22m)r2Gμμμμ=a2+r2a22(a221)μ21Gϕϕϕϕ=a622a2mμ4+(a22(a221)μ21)r4a42a2m(a62a44a2m)μ2+2(a42(a42a2)μ2a2)r2a642a42+(a442a22+1)r2+a2Gϕψϕψ=2(a2mμ4a2mμ2)a642a42+(a442a22+1)r2+a2Gψψψψ=2a2mμ4(a221)μ2r42(a42a2)μ2r2(a62a4)μ2a642a42+(a442a22+1)r2+a2\renewcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{lcl} G_{ \, t \, t }^{ \phantom{\, t}\phantom{\, t} } & = & -\frac{a^{4} {\ell}^{2} + {\ell}^{2} r^{4} + {\left(2 \, a^{2} {\ell}^{2} + 1\right)} r^{2} + a^{2} - 2 \, m}{a^{2} + r^{2}} \\ G_{ \, t \, {\phi} }^{ \phantom{\, t}\phantom{\, {\phi}} } & = & -\frac{a^{5} {\ell}^{2} - {\left(a {\ell}^{2} {\mu}^{2} - a {\ell}^{2}\right)} r^{4} - {\left(a^{5} {\ell}^{2} - 2 \, a m\right)} {\mu}^{2} - 2 \, {\left(a^{3} {\ell}^{2} {\mu}^{2} - a^{3} {\ell}^{2}\right)} r^{2} - 2 \, a m}{a^{4} {\ell}^{2} + {\left(a^{2} {\ell}^{2} - 1\right)} r^{2} - a^{2}} \\ G_{ \, t \, {\psi} }^{ \phantom{\, t}\phantom{\, {\psi}} } & = & -\frac{2 \, a^{3} {\ell}^{2} {\mu}^{2} r^{2} + a {\ell}^{2} {\mu}^{2} r^{4} + {\left(a^{5} {\ell}^{2} - 2 \, a m\right)} {\mu}^{2}}{a^{4} {\ell}^{2} + {\left(a^{2} {\ell}^{2} - 1\right)} r^{2} - a^{2}} \\ G_{ \, r \, r }^{ \phantom{\, r}\phantom{\, r} } & = & \frac{a^{2} r^{2} + r^{4}}{{\ell}^{2} r^{6} + {\left(2 \, a^{2} {\ell}^{2} + 1\right)} r^{4} + a^{4} + {\left(a^{4} {\ell}^{2} + 2 \, a^{2} - 2 \, m\right)} r^{2}} \\ G_{ \, {\mu} \, {\mu} }^{ \phantom{\, {\mu}}\phantom{\, {\mu}} } & = & -\frac{a^{2} + r^{2}}{a^{2} {\ell}^{2} - {\left(a^{2} {\ell}^{2} - 1\right)} {\mu}^{2} - 1} \\ G_{ \, {\phi} \, {\phi} }^{ \phantom{\, {\phi}}\phantom{\, {\phi}} } & = & -\frac{a^{6} {\ell}^{2} - 2 \, a^{2} m {\mu}^{4} + {\left(a^{2} {\ell}^{2} - {\left(a^{2} {\ell}^{2} - 1\right)} {\mu}^{2} - 1\right)} r^{4} - a^{4} - 2 \, a^{2} m - {\left(a^{6} {\ell}^{2} - a^{4} - 4 \, a^{2} m\right)} {\mu}^{2} + 2 \, {\left(a^{4} {\ell}^{2} - {\left(a^{4} {\ell}^{2} - a^{2}\right)} {\mu}^{2} - a^{2}\right)} r^{2}}{a^{6} {\ell}^{4} - 2 \, a^{4} {\ell}^{2} + {\left(a^{4} {\ell}^{4} - 2 \, a^{2} {\ell}^{2} + 1\right)} r^{2} + a^{2}} \\ G_{ \, {\phi} \, {\psi} }^{ \phantom{\, {\phi}}\phantom{\, {\psi}} } & = & -\frac{2 \, {\left(a^{2} m {\mu}^{4} - a^{2} m {\mu}^{2}\right)}}{a^{6} {\ell}^{4} - 2 \, a^{4} {\ell}^{2} + {\left(a^{4} {\ell}^{4} - 2 \, a^{2} {\ell}^{2} + 1\right)} r^{2} + a^{2}} \\ G_{ \, {\psi} \, {\psi} }^{ \phantom{\, {\psi}}\phantom{\, {\psi}} } & = & \frac{2 \, a^{2} m {\mu}^{4} - {\left(a^{2} {\ell}^{2} - 1\right)} {\mu}^{2} r^{4} - 2 \, {\left(a^{4} {\ell}^{2} - a^{2}\right)} {\mu}^{2} r^{2} - {\left(a^{6} {\ell}^{2} - a^{4}\right)} {\mu}^{2}}{a^{6} {\ell}^{4} - 2 \, a^{4} {\ell}^{2} + {\left(a^{4} {\ell}^{4} - 2 \, a^{2} {\ell}^{2} + 1\right)} r^{2} + a^{2}} \end{array}

Check of agreement with Eq. (5.22) of Hawking et al or Eq. (2.3) of our paper

We need the 1-forms dt\mathrm{d}t, dr\mathrm{d}r, dμ\mathrm{d}\mu, dϕ\mathrm{d}\phi and dψ\mathrm{d}\psi:

dt, dr, dmu, dph, dps = (BL.coframe()[i] for i in M.irange()) dt, dr, dmu, dph, dps
(dt,dr,dμ,dϕ,dψ)\renewcommand{\Bold}[1]{\mathbf{#1}}\left(\mathrm{d} t, \mathrm{d} r, \mathrm{d} {\mu}, \mathrm{d} {\phi}, \mathrm{d} {\psi}\right)
print(dt)
1-form dt on the 5-dimensional Lorentzian manifold M

In agreement with μ=cosθ\mu = \cos\theta, we introduce the 1-form dθ=dμ/sinθ\mathrm{d}\theta = - \mathrm{d}\mu /\sin\theta , with sinθ=1μ2\sin\theta = \sqrt{1-\mu^2} since θ(0,π2)\theta\in\left(0, \frac{\pi}{2}\right):

dth = - 1/sqrt(1 - mu^2)*dmu
s1 = dt - a*sinth2/Xi_a*dph - b*costh2/Xi_b*dps s1.display()
dt+(aμ2aa221)dϕ+(aμ2a221)dψ\renewcommand{\Bold}[1]{\mathbf{#1}}\mathrm{d} t + \left( -\frac{a {\mu}^{2} - a}{a^{2} {\ell}^{2} - 1} \right) \mathrm{d} {\phi} + \left( \frac{a {\mu}^{2}}{a^{2} {\ell}^{2} - 1} \right) \mathrm{d} {\psi}
s2 = a*dt - (r^2 + a^2)/Xi_a*dph s2.display()
adt+(a2+r2a221)dϕ\renewcommand{\Bold}[1]{\mathbf{#1}}a \mathrm{d} t + \left( \frac{a^{2} + r^{2}}{a^{2} {\ell}^{2} - 1} \right) \mathrm{d} {\phi}
s3 = b*dt - (r^2 + b^2)/Xi_b*dps s3.display()
adt+(a2+r2a221)dψ\renewcommand{\Bold}[1]{\mathbf{#1}}a \mathrm{d} t + \left( \frac{a^{2} + r^{2}}{a^{2} {\ell}^{2} - 1} \right) \mathrm{d} {\psi}
s4 = a*b*dt - b*(r^2 + a^2)*sinth2/Xi_a * dph - a*(r^2 + b^2)*costh2/Xi_b * dps s4.display()
a2dt+(a3μ2a3+(aμ2a)r2a221)dϕ+(a3μ2+aμ2r2a221)dψ\renewcommand{\Bold}[1]{\mathbf{#1}}a^{2} \mathrm{d} t + \left( -\frac{a^{3} {\mu}^{2} - a^{3} + {\left(a {\mu}^{2} - a\right)} r^{2}}{a^{2} {\ell}^{2} - 1} \right) \mathrm{d} {\phi} + \left( \frac{a^{3} {\mu}^{2} + a {\mu}^{2} r^{2}}{a^{2} {\ell}^{2} - 1} \right) \mathrm{d} {\psi}
G0 = - Delta_r/rho2 * s1*s1 + Delta_th*sinth2/rho2 * s2*s2 + Delta_th*costh2/rho2 * s3*s3 \ + rho2/Delta_r * dr*dr + rho2/Delta_th * dth*dth + sig/rho2 * s4*s4 G0.display_comp(only_nonredundant=True)
Xtttt=a42+2r4+(2a22+1)r2+a22ma2+r2Xtϕtϕ=a52(a2μ2a2)r4(a522am)μ22(a32μ2a32)r22ama42+(a221)r2a2Xtψtψ=2a32μ2r2+a2μ2r4+(a522am)μ2a42+(a221)r2a2Xrrrr=a2μ2(μ21)a2+r22m(2r2+1)(a2+r2)2r2Xμμμμ=a2+r2a22(a221)μ21Xϕtϕt=a52(a2μ2a2)r4(a522am)μ22(a32μ2a32)r22ama42+(a221)r2a2Xϕϕϕϕ=a622a2mμ4+(a22(a221)μ21)r4a42a2m(a62a44a2m)μ2+2(a42(a42a2)μ2a2)r2a642a42+(a442a22+1)r2+a2Xϕψϕψ=2(a2mμ4a2mμ2)a642a42+(a442a22+1)r2+a2Xψtψt=2a32μ2r2+a2μ2r4+(a522am)μ2a42+(a221)r2a2Xψϕψϕ=2(a2mμ4a2mμ2)a642a42+(a442a22+1)r2+a2Xψψψψ=2a2mμ4(a221)μ2r42(a42a2)μ2r2(a62a4)μ2a642a42+(a442a22+1)r2+a2\renewcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{lcl} X_{ \, t \, t }^{ \phantom{\, t}\phantom{\, t} } & = & -\frac{a^{4} {\ell}^{2} + {\ell}^{2} r^{4} + {\left(2 \, a^{2} {\ell}^{2} + 1\right)} r^{2} + a^{2} - 2 \, m}{a^{2} + r^{2}} \\ X_{ \, t \, {\phi} }^{ \phantom{\, t}\phantom{\, {\phi}} } & = & -\frac{a^{5} {\ell}^{2} - {\left(a {\ell}^{2} {\mu}^{2} - a {\ell}^{2}\right)} r^{4} - {\left(a^{5} {\ell}^{2} - 2 \, a m\right)} {\mu}^{2} - 2 \, {\left(a^{3} {\ell}^{2} {\mu}^{2} - a^{3} {\ell}^{2}\right)} r^{2} - 2 \, a m}{a^{4} {\ell}^{2} + {\left(a^{2} {\ell}^{2} - 1\right)} r^{2} - a^{2}} \\ X_{ \, t \, {\psi} }^{ \phantom{\, t}\phantom{\, {\psi}} } & = & -\frac{2 \, a^{3} {\ell}^{2} {\mu}^{2} r^{2} + a {\ell}^{2} {\mu}^{2} r^{4} + {\left(a^{5} {\ell}^{2} - 2 \, a m\right)} {\mu}^{2}}{a^{4} {\ell}^{2} + {\left(a^{2} {\ell}^{2} - 1\right)} r^{2} - a^{2}} \\ X_{ \, r \, r }^{ \phantom{\, r}\phantom{\, r} } & = & -\frac{a^{2} {\mu}^{2} - {\left({\mu}^{2} - 1\right)} a^{2} + r^{2}}{2 \, m - \frac{{\left({\ell}^{2} r^{2} + 1\right)} {\left(a^{2} + r^{2}\right)}^{2}}{r^{2}}} \\ X_{ \, {\mu} \, {\mu} }^{ \phantom{\, {\mu}}\phantom{\, {\mu}} } & = & -\frac{a^{2} + r^{2}}{a^{2} {\ell}^{2} - {\left(a^{2} {\ell}^{2} - 1\right)} {\mu}^{2} - 1} \\ X_{ \, {\phi} \, t }^{ \phantom{\, {\phi}}\phantom{\, t} } & = & -\frac{a^{5} {\ell}^{2} - {\left(a {\ell}^{2} {\mu}^{2} - a {\ell}^{2}\right)} r^{4} - {\left(a^{5} {\ell}^{2} - 2 \, a m\right)} {\mu}^{2} - 2 \, {\left(a^{3} {\ell}^{2} {\mu}^{2} - a^{3} {\ell}^{2}\right)} r^{2} - 2 \, a m}{a^{4} {\ell}^{2} + {\left(a^{2} {\ell}^{2} - 1\right)} r^{2} - a^{2}} \\ X_{ \, {\phi} \, {\phi} }^{ \phantom{\, {\phi}}\phantom{\, {\phi}} } & = & -\frac{a^{6} {\ell}^{2} - 2 \, a^{2} m {\mu}^{4} + {\left(a^{2} {\ell}^{2} - {\left(a^{2} {\ell}^{2} - 1\right)} {\mu}^{2} - 1\right)} r^{4} - a^{4} - 2 \, a^{2} m - {\left(a^{6} {\ell}^{2} - a^{4} - 4 \, a^{2} m\right)} {\mu}^{2} + 2 \, {\left(a^{4} {\ell}^{2} - {\left(a^{4} {\ell}^{2} - a^{2}\right)} {\mu}^{2} - a^{2}\right)} r^{2}}{a^{6} {\ell}^{4} - 2 \, a^{4} {\ell}^{2} + {\left(a^{4} {\ell}^{4} - 2 \, a^{2} {\ell}^{2} + 1\right)} r^{2} + a^{2}} \\ X_{ \, {\phi} \, {\psi} }^{ \phantom{\, {\phi}}\phantom{\, {\psi}} } & = & -\frac{2 \, {\left(a^{2} m {\mu}^{4} - a^{2} m {\mu}^{2}\right)}}{a^{6} {\ell}^{4} - 2 \, a^{4} {\ell}^{2} + {\left(a^{4} {\ell}^{4} - 2 \, a^{2} {\ell}^{2} + 1\right)} r^{2} + a^{2}} \\ X_{ \, {\psi} \, t }^{ \phantom{\, {\psi}}\phantom{\, t} } & = & -\frac{2 \, a^{3} {\ell}^{2} {\mu}^{2} r^{2} + a {\ell}^{2} {\mu}^{2} r^{4} + {\left(a^{5} {\ell}^{2} - 2 \, a m\right)} {\mu}^{2}}{a^{4} {\ell}^{2} + {\left(a^{2} {\ell}^{2} - 1\right)} r^{2} - a^{2}} \\ X_{ \, {\psi} \, {\phi} }^{ \phantom{\, {\psi}}\phantom{\, {\phi}} } & = & -\frac{2 \, {\left(a^{2} m {\mu}^{4} - a^{2} m {\mu}^{2}\right)}}{a^{6} {\ell}^{4} - 2 \, a^{4} {\ell}^{2} + {\left(a^{4} {\ell}^{4} - 2 \, a^{2} {\ell}^{2} + 1\right)} r^{2} + a^{2}} \\ X_{ \, {\psi} \, {\psi} }^{ \phantom{\, {\psi}}\phantom{\, {\psi}} } & = & \frac{2 \, a^{2} m {\mu}^{4} - {\left(a^{2} {\ell}^{2} - 1\right)} {\mu}^{2} r^{4} - 2 \, {\left(a^{4} {\ell}^{2} - a^{2}\right)} {\mu}^{2} r^{2} - {\left(a^{6} {\ell}^{2} - a^{4}\right)} {\mu}^{2}}{a^{6} {\ell}^{4} - 2 \, a^{4} {\ell}^{2} + {\left(a^{4} {\ell}^{4} - 2 \, a^{2} {\ell}^{2} + 1\right)} r^{2} + a^{2}} \end{array}
G0 == G
True\renewcommand{\Bold}[1]{\mathbf{#1}}\mathrm{True}

Einstein equation

The Ricci tensor of gg is

if not keep_Delta: # Ric = G.ricci() # print(Ric) pass
if not keep_Delta: # show(Ric.display_comp(only_nonredundant=True)) pass

Let us check that gg is a solution of the vacuum Einstein equation with the cosmological constant Λ=62\Lambda = - 6 \ell^2:

Lambda = -6*l^2 if not keep_Delta: # print(Ric == 2/3*Lambda*G) pass

Check of Eq. (2.10)

One must have a=ba=b and keep_Delta == False for the test to pass:

if a == b and not keep_Delta: G1 = - (1 + rho2*l^2 - 2*m/rho2) * dt*dt + rho2/Delta_r * dr*dr \ + rho2/Delta_th * dth*dth \ + sinth2/Xi_a^2*(rho2*Xi_a + 2*a^2*m/rho2*sinth2) * dph * dph \ + costh2/Xi_a^2*(rho2*Xi_a + 2*a^2*m/rho2*costh2) * dps * dps \ + a*sinth2/Xi_a*(rho2*l^2 - 2*m/rho2) * (dt*dph + dph*dt) \ + a*costh2/Xi_a*(rho2*l^2 - 2*m/rho2) * (dt*dps + dps*dt) \ + 2*m*a^2*sinth2*costh2/Xi_a^2/rho2 * (dph*dps + dps*dph) print(G1 == G)
True

String worldsheet

The string worldsheet as a 2-dimensional Lorentzian manifold:

W = Manifold(2, 'W', structure='Lorentzian') print(W)
2-dimensional Lorentzian manifold W

Let us assume that the string worldsheet is parametrized by (t,r)(t,r):

XW.<t,r> = W.chart(r't r:(0,+oo)') XW
(W,(t,r))\renewcommand{\Bold}[1]{\mathbf{#1}}\left(W,(t, r)\right)

The string embedding in Kerr-AdS spacetime, as an expansion about a straight string solution in AdS (Eq. (4.6)-(4.8) of the paper):

Mu0 = var('Mu0', latex_name=r'\mu_0', domain='real') Phi0 = var('Phi0', latex_name=r'\Phi_0', domain='real') Psi0 = var('Psi0', latex_name=r'\Psi_0', domain='real') beta1 = var('beta1', latex_name=r'\beta_1', domain='real') beta2 = var('beta2', latex_name=r'\beta_2', domain='real') cosTh0 = Mu0 sinTh0 = sqrt(1 - Mu0^2) mu_s = Mu0 + a^2*function('mu_1')(r) ph_s = Phi0 + beta1*a*l^2*t + beta1*a*function('phi_1')(r) ps_s = Psi0 + beta2*a*l^2*t + beta2*a*function('psi_1')(r) F = W.diff_map(M, {(XW, BL): [t, r, mu_s, ph_s, ps_s]}, name='F') F.display()
F:WM(t,r)(t,r,μ,ϕ,ψ)=(t,r,a2μ1(r)+μ0,aβ12t+aβ1ϕ1(r)+Φ0,aβ22t+aβ2ψ1(r)+Ψ0)\renewcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{llcl} F:& W & \longrightarrow & \mathcal{M} \\ & \left(t, r\right) & \longmapsto & \left(t, r, {\mu}, {\phi}, {\psi}\right) = \left(t, r, a^{2} \mu_{1}\left(r\right) + {\mu_0}, a {\beta_1} {\ell}^{2} t + a {\beta_1} \phi_{1}\left(r\right) + {\Phi_0}, a {\beta_2} {\ell}^{2} t + a {\beta_2} \psi_{1}\left(r\right) + {\Psi_0}\right) \end{array}
F.jacobian_matrix()
(10010a2rμ1(r)aβ12aβ1rϕ1(r)aβ22aβ2rψ1(r))\renewcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rr} 1 & 0 \\ 0 & 1 \\ 0 & a^{2} \frac{\partial}{\partial r}\mu_{1}\left(r\right) \\ a {\beta_1} {\ell}^{2} & a {\beta_1} \frac{\partial}{\partial r}\phi_{1}\left(r\right) \\ a {\beta_2} {\ell}^{2} & a {\beta_2} \frac{\partial}{\partial r}\psi_{1}\left(r\right) \end{array}\right)

Induced metric on the string worldsheet

The string worldsheet metric is the metric gg induced by the spacetime metric GG, i.e. the pullback of GG by the embedding FF:

g = W.metric() g.set(F.pullback(G))
# g[0,0].expr().factor()

Nambu-Goto action

detg = g.determinant().expr()

Expanding at second order in aa:

detg_a2 = detg.series(a, 3).truncate().simplify_full() detg_a2
((μ02a2β22(μ021)a2β12+2μ02a2β22(μ021)a2β1)42)r4+((μ021)a2β124r8+2(μ021)a2β122r64(μ021)a2β12mr2+4(μ021)a2β12m2(4(μ021)a2β122m(μ021)a2β12)r4)rϕ1(r)2(μ02a2β224r8+2μ02a2β222r64μ02a2β22mr2+4μ02a2β22m2(4μ02a2β222mμ02a2β22)r4)rψ1(r)2+a22(2(μ02a2β2(μ021)a2β1)21)mr22r4+r22m\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{{\left({\left({\mu_0}^{2} a^{2} {\beta_2}^{2} - {\left({\mu_0}^{2} - 1\right)} a^{2} {\beta_1}^{2} + 2 \, {\mu_0}^{2} a^{2} {\beta_2} - 2 \, {\left({\mu_0}^{2} - 1\right)} a^{2} {\beta_1}\right)} {\ell}^{4} - {\ell}^{2}\right)} r^{4} + {\left({\left({\mu_0}^{2} - 1\right)} a^{2} {\beta_1}^{2} {\ell}^{4} r^{8} + 2 \, {\left({\mu_0}^{2} - 1\right)} a^{2} {\beta_1}^{2} {\ell}^{2} r^{6} - 4 \, {\left({\mu_0}^{2} - 1\right)} a^{2} {\beta_1}^{2} m r^{2} + 4 \, {\left({\mu_0}^{2} - 1\right)} a^{2} {\beta_1}^{2} m^{2} - {\left(4 \, {\left({\mu_0}^{2} - 1\right)} a^{2} {\beta_1}^{2} {\ell}^{2} m - {\left({\mu_0}^{2} - 1\right)} a^{2} {\beta_1}^{2}\right)} r^{4}\right)} \frac{\partial}{\partial r}\phi_{1}\left(r\right)^{2} - {\left({\mu_0}^{2} a^{2} {\beta_2}^{2} {\ell}^{4} r^{8} + 2 \, {\mu_0}^{2} a^{2} {\beta_2}^{2} {\ell}^{2} r^{6} - 4 \, {\mu_0}^{2} a^{2} {\beta_2}^{2} m r^{2} + 4 \, {\mu_0}^{2} a^{2} {\beta_2}^{2} m^{2} - {\left(4 \, {\mu_0}^{2} a^{2} {\beta_2}^{2} {\ell}^{2} m - {\mu_0}^{2} a^{2} {\beta_2}^{2}\right)} r^{4}\right)} \frac{\partial}{\partial r}\psi_{1}\left(r\right)^{2} + a^{2} - 2 \, {\left(2 \, {\left({\mu_0}^{2} a^{2} {\beta_2} - {\left({\mu_0}^{2} - 1\right)} a^{2} {\beta_1}\right)} {\ell}^{2} - 1\right)} m - r^{2}}{{\ell}^{2} r^{4} + r^{2} - 2 \, m}

The Nambu-Goto Lagrangian at second order in aa:

L_a2 = (sqrt(-detg_a2)).series(a, 3).truncate().simplify_full() L_a2
((μ02a2β22(μ021)a2β12+2μ02a2β22(μ021)a2β1)422)r4+((μ021)a2β124r8+2(μ021)a2β122r64(μ021)a2β12mr2+4(μ021)a2β12m2(4(μ021)a2β122m(μ021)a2β12)r4)rϕ1(r)2(μ02a2β224r8+2μ02a2β222r64μ02a2β22mr2+4μ02a2β22m2(4μ02a2β222mμ02a2β22)r4)rψ1(r)2+a24((μ02a2β2(μ021)a2β1)21)m2r22(2r4+r22m)\renewcommand{\Bold}[1]{\mathbf{#1}}-\frac{{\left({\left({\mu_0}^{2} a^{2} {\beta_2}^{2} - {\left({\mu_0}^{2} - 1\right)} a^{2} {\beta_1}^{2} + 2 \, {\mu_0}^{2} a^{2} {\beta_2} - 2 \, {\left({\mu_0}^{2} - 1\right)} a^{2} {\beta_1}\right)} {\ell}^{4} - 2 \, {\ell}^{2}\right)} r^{4} + {\left({\left({\mu_0}^{2} - 1\right)} a^{2} {\beta_1}^{2} {\ell}^{4} r^{8} + 2 \, {\left({\mu_0}^{2} - 1\right)} a^{2} {\beta_1}^{2} {\ell}^{2} r^{6} - 4 \, {\left({\mu_0}^{2} - 1\right)} a^{2} {\beta_1}^{2} m r^{2} + 4 \, {\left({\mu_0}^{2} - 1\right)} a^{2} {\beta_1}^{2} m^{2} - {\left(4 \, {\left({\mu_0}^{2} - 1\right)} a^{2} {\beta_1}^{2} {\ell}^{2} m - {\left({\mu_0}^{2} - 1\right)} a^{2} {\beta_1}^{2}\right)} r^{4}\right)} \frac{\partial}{\partial r}\phi_{1}\left(r\right)^{2} - {\left({\mu_0}^{2} a^{2} {\beta_2}^{2} {\ell}^{4} r^{8} + 2 \, {\mu_0}^{2} a^{2} {\beta_2}^{2} {\ell}^{2} r^{6} - 4 \, {\mu_0}^{2} a^{2} {\beta_2}^{2} m r^{2} + 4 \, {\mu_0}^{2} a^{2} {\beta_2}^{2} m^{2} - {\left(4 \, {\mu_0}^{2} a^{2} {\beta_2}^{2} {\ell}^{2} m - {\mu_0}^{2} a^{2} {\beta_2}^{2}\right)} r^{4}\right)} \frac{\partial}{\partial r}\psi_{1}\left(r\right)^{2} + a^{2} - 4 \, {\left({\left({\mu_0}^{2} a^{2} {\beta_2} - {\left({\mu_0}^{2} - 1\right)} a^{2} {\beta_1}\right)} {\ell}^{2} - 1\right)} m - 2 \, r^{2}}{2 \, {\left({\ell}^{2} r^{4} + r^{2} - 2 \, m\right)}}
L_a2.numerator()
μ02a2β124r8rϕ1(r)2+μ02a2β224r8rψ1(r)2+a2β124r8rϕ1(r)22μ02a2β122r6rϕ1(r)2+2μ02a2β222r6rψ1(r)2+4μ02a2β122mr4rϕ1(r)24μ02a2β222mr4rψ1(r)2+μ02a2β124r4μ02a2β224r4+2a2β122r6rϕ1(r)2+2μ02a2β14r42μ02a2β24r44a2β122mr4rϕ1(r)2a2β124r4μ02a2β12r4rϕ1(r)2+μ02a2β22r4rψ1(r)22a2β14r4+4μ02a2β12mr2rϕ1(r)24μ02a2β22mr2rψ1(r)24μ02a2β12m2rϕ1(r)2+a2β12r4rϕ1(r)2+4μ02a2β22m2rψ1(r)24a2β12mr2rϕ1(r)24μ02a2β12m+4μ02a2β22m+4a2β12m2rϕ1(r)2+4a2β12m+22r4a2+2r24m\renewcommand{\Bold}[1]{\mathbf{#1}}-{\mu_0}^{2} a^{2} {\beta_1}^{2} {\ell}^{4} r^{8} \frac{\partial}{\partial r}\phi_{1}\left(r\right)^{2} + {\mu_0}^{2} a^{2} {\beta_2}^{2} {\ell}^{4} r^{8} \frac{\partial}{\partial r}\psi_{1}\left(r\right)^{2} + a^{2} {\beta_1}^{2} {\ell}^{4} r^{8} \frac{\partial}{\partial r}\phi_{1}\left(r\right)^{2} - 2 \, {\mu_0}^{2} a^{2} {\beta_1}^{2} {\ell}^{2} r^{6} \frac{\partial}{\partial r}\phi_{1}\left(r\right)^{2} + 2 \, {\mu_0}^{2} a^{2} {\beta_2}^{2} {\ell}^{2} r^{6} \frac{\partial}{\partial r}\psi_{1}\left(r\right)^{2} + 4 \, {\mu_0}^{2} a^{2} {\beta_1}^{2} {\ell}^{2} m r^{4} \frac{\partial}{\partial r}\phi_{1}\left(r\right)^{2} - 4 \, {\mu_0}^{2} a^{2} {\beta_2}^{2} {\ell}^{2} m r^{4} \frac{\partial}{\partial r}\psi_{1}\left(r\right)^{2} + {\mu_0}^{2} a^{2} {\beta_1}^{2} {\ell}^{4} r^{4} - {\mu_0}^{2} a^{2} {\beta_2}^{2} {\ell}^{4} r^{4} + 2 \, a^{2} {\beta_1}^{2} {\ell}^{2} r^{6} \frac{\partial}{\partial r}\phi_{1}\left(r\right)^{2} + 2 \, {\mu_0}^{2} a^{2} {\beta_1} {\ell}^{4} r^{4} - 2 \, {\mu_0}^{2} a^{2} {\beta_2} {\ell}^{4} r^{4} - 4 \, a^{2} {\beta_1}^{2} {\ell}^{2} m r^{4} \frac{\partial}{\partial r}\phi_{1}\left(r\right)^{2} - a^{2} {\beta_1}^{2} {\ell}^{4} r^{4} - {\mu_0}^{2} a^{2} {\beta_1}^{2} r^{4} \frac{\partial}{\partial r}\phi_{1}\left(r\right)^{2} + {\mu_0}^{2} a^{2} {\beta_2}^{2} r^{4} \frac{\partial}{\partial r}\psi_{1}\left(r\right)^{2} - 2 \, a^{2} {\beta_1} {\ell}^{4} r^{4} + 4 \, {\mu_0}^{2} a^{2} {\beta_1}^{2} m r^{2} \frac{\partial}{\partial r}\phi_{1}\left(r\right)^{2} - 4 \, {\mu_0}^{2} a^{2} {\beta_2}^{2} m r^{2} \frac{\partial}{\partial r}\psi_{1}\left(r\right)^{2} - 4 \, {\mu_0}^{2} a^{2} {\beta_1}^{2} m^{2} \frac{\partial}{\partial r}\phi_{1}\left(r\right)^{2} + a^{2} {\beta_1}^{2} r^{4} \frac{\partial}{\partial r}\phi_{1}\left(r\right)^{2} + 4 \, {\mu_0}^{2} a^{2} {\beta_2}^{2} m^{2} \frac{\partial}{\partial r}\psi_{1}\left(r\right)^{2} - 4 \, a^{2} {\beta_1}^{2} m r^{2} \frac{\partial}{\partial r}\phi_{1}\left(r\right)^{2} - 4 \, {\mu_0}^{2} a^{2} {\beta_1} {\ell}^{2} m + 4 \, {\mu_0}^{2} a^{2} {\beta_2} {\ell}^{2} m + 4 \, a^{2} {\beta_1}^{2} m^{2} \frac{\partial}{\partial r}\phi_{1}\left(r\right)^{2} + 4 \, a^{2} {\beta_1} {\ell}^{2} m + 2 \, {\ell}^{2} r^{4} - a^{2} + 2 \, r^{2} - 4 \, m
L_a2.denominator()
22r4+2r24m\renewcommand{\Bold}[1]{\mathbf{#1}}2 \, {\ell}^{2} r^{4} + 2 \, r^{2} - 4 \, m

Euler-Lagrange equations

def euler_lagrange(lagr, qs, var): r""" Derive the Euler-Lagrange equations from a given Lagrangian. INPUT: - ``lagr`` -- symbolic expression representing the Lagrangian density - ``qs`` -- either a single symbolic function or a list/tuple of symbolic functions, representing the `q`'s; these functions must appear in ``lagr`` up to at most their first derivatives - ``var`` -- either a single variable, typically `t` (1-dimensional problem) or a list/tuple of symbolic variables OUTPUT: - list of Euler-Lagrange equations; if only one function is involved, the single Euler-Lagrannge equation is returned instead. """ if not isinstance(qs, (list, tuple)): qs = [qs] if not isinstance(var, (list, tuple)): var = [var] n = len(qs) d = len(var) qv = [SR.var('qxxxx{}'.format(q)) for q in qs] dqv = [[SR.var('qxxxx{}_{}'.format(q, v)) for v in var] for q in qs] subs = {qs[i](*var): qv[i] for i in range(n)} subs_inv = {qv[i]: qs[i](*var) for i in range(n)} for i in range(n): subs.update({diff(qs[i](*var), var[j]): dqv[i][j] for j in range(d)}) subs_inv.update({dqv[i][j]: diff(qs[i](*var), var[j]) for j in range(d)}) lg = lagr.substitute(subs) eqs = [] for i in range(n): dLdq = diff(lg, qv[i]).simplify_full() dLdq = dLdq.substitute(subs_inv) ddL = 0 for j in range(d): h = diff(lg, dqv[i][j]).simplify_full() h = h.substitute(subs_inv) ddL += diff(h, var[j]) eqs.append((dLdq - ddL).simplify_full() == 0) if n == 1: return eqs[0] return eqs

We compute the Euler-Lagrange equations at order a2a^2 for ϕ1\phi_1 and ψ1\psi_1:

eqs = euler_lagrange(L_a2, [phi_1, psi_1], r) eqs
[2(2(μ021)a2β122r3+(μ021)a2β12r)rϕ1(r)+((μ021)a2β122r4+(μ021)a2β12r22(μ021)a2β12m)2(r)2ϕ1(r)=0,2(2μ02a2β222r3+μ02a2β22r)rψ1(r)(μ02a2β222r4+μ02a2β22r22μ02a2β22m)2(r)2ψ1(r)=0]\renewcommand{\Bold}[1]{\mathbf{#1}}\left[2 \, {\left(2 \, {\left({\mu_0}^{2} - 1\right)} a^{2} {\beta_1}^{2} {\ell}^{2} r^{3} + {\left({\mu_0}^{2} - 1\right)} a^{2} {\beta_1}^{2} r\right)} \frac{\partial}{\partial r}\phi_{1}\left(r\right) + {\left({\left({\mu_0}^{2} - 1\right)} a^{2} {\beta_1}^{2} {\ell}^{2} r^{4} + {\left({\mu_0}^{2} - 1\right)} a^{2} {\beta_1}^{2} r^{2} - 2 \, {\left({\mu_0}^{2} - 1\right)} a^{2} {\beta_1}^{2} m\right)} \frac{\partial^{2}}{(\partial r)^{2}}\phi_{1}\left(r\right) = 0, -2 \, {\left(2 \, {\mu_0}^{2} a^{2} {\beta_2}^{2} {\ell}^{2} r^{3} + {\mu_0}^{2} a^{2} {\beta_2}^{2} r\right)} \frac{\partial}{\partial r}\psi_{1}\left(r\right) - {\left({\mu_0}^{2} a^{2} {\beta_2}^{2} {\ell}^{2} r^{4} + {\mu_0}^{2} a^{2} {\beta_2}^{2} r^{2} - 2 \, {\mu_0}^{2} a^{2} {\beta_2}^{2} m\right)} \frac{\partial^{2}}{(\partial r)^{2}}\psi_{1}\left(r\right) = 0\right]

Solving the equation for ϕ1\phi_1 (Eq. (4.10))

eq_phi1 = eqs[0] eq_phi1
2(2(μ021)a2β122r3+(μ021)a2β12r)rϕ1(r)+((μ021)a2β122r4+(μ021)a2β12r22(μ021)a2β12m)2(r)2ϕ1(r)=0\renewcommand{\Bold}[1]{\mathbf{#1}}2 \, {\left(2 \, {\left({\mu_0}^{2} - 1\right)} a^{2} {\beta_1}^{2} {\ell}^{2} r^{3} + {\left({\mu_0}^{2} - 1\right)} a^{2} {\beta_1}^{2} r\right)} \frac{\partial}{\partial r}\phi_{1}\left(r\right) + {\left({\left({\mu_0}^{2} - 1\right)} a^{2} {\beta_1}^{2} {\ell}^{2} r^{4} + {\left({\mu_0}^{2} - 1\right)} a^{2} {\beta_1}^{2} r^{2} - 2 \, {\left({\mu_0}^{2} - 1\right)} a^{2} {\beta_1}^{2} m\right)} \frac{\partial^{2}}{(\partial r)^{2}}\phi_{1}\left(r\right) = 0
eq_phi1 = (eq_phi1/(a^2*(Mu0^2-1)*beta1^2)).simplify_full() eq_phi1
2(22r3+r)rϕ1(r)+(2r4+r22m)2(r)2ϕ1(r)=0\renewcommand{\Bold}[1]{\mathbf{#1}}2 \, {\left(2 \, {\ell}^{2} r^{3} + r\right)} \frac{\partial}{\partial r}\phi_{1}\left(r\right) + {\left({\ell}^{2} r^{4} + r^{2} - 2 \, m\right)} \frac{\partial^{2}}{(\partial r)^{2}}\phi_{1}\left(r\right) = 0
phi1_sol(r) = desolve(eq_phi1, phi_1(r), ivar=r) phi1_sol(r)
K112r4+r22mdr+K2\renewcommand{\Bold}[1]{\mathbf{#1}}K_{1} \int \frac{1}{{\ell}^{2} r^{4} + r^{2} - 2 \, m}\,{d r} + K_{2}

We revover Eqs. (4.10) with K1=P/β12K_1 = \mathcal{P}/\beta_1^2 and K2=0K_2=0. In what follows, we introduce P=P/β12\mathcal{P}' = \mathcal{P}/\beta_1^2 instead of P\mathcal{P}.

The symbolic constants K1K_1 and K2K_2 are actually denoted _K1 and _K2 by SageMath, as the print reveals:

print(phi1_sol(r))
_K1*integrate(1/(l^2*r^4 + r^2 - 2*m), r) + _K2

Hence we perform the substitutions with SR.var('_K1') and SR.var('_K2'):

P = var("P", latex_name=r"\mathcal{P}'") phi1_sol(r) = phi1_sol(r).subs({SR.var('_K1'): P, SR.var('_K2'): 0}) print(phi1_sol(r))
P*integrate(1/(l^2*r^4 + r^2 - 2*m), r)

Solving the equation for ψ1\psi_1 (Eq. (4.10))

eq_psi1 = eqs[1] eq_psi1
2(2μ02a2β222r3+μ02a2β22r)rψ1(r)(μ02a2β222r4+μ02a2β22r22μ02a2β22m)2(r)2ψ1(r)=0\renewcommand{\Bold}[1]{\mathbf{#1}}-2 \, {\left(2 \, {\mu_0}^{2} a^{2} {\beta_2}^{2} {\ell}^{2} r^{3} + {\mu_0}^{2} a^{2} {\beta_2}^{2} r\right)} \frac{\partial}{\partial r}\psi_{1}\left(r\right) - {\left({\mu_0}^{2} a^{2} {\beta_2}^{2} {\ell}^{2} r^{4} + {\mu_0}^{2} a^{2} {\beta_2}^{2} r^{2} - 2 \, {\mu_0}^{2} a^{2} {\beta_2}^{2} m\right)} \frac{\partial^{2}}{(\partial r)^{2}}\psi_{1}\left(r\right) = 0
eq_psi1 = (eq_psi1/(a^2*Mu0^2*beta2^2)).simplify_full() eq_psi1
2(22r3+r)rψ1(r)(2r4+r22m)2(r)2ψ1(r)=0\renewcommand{\Bold}[1]{\mathbf{#1}}-2 \, {\left(2 \, {\ell}^{2} r^{3} + r\right)} \frac{\partial}{\partial r}\psi_{1}\left(r\right) - {\left({\ell}^{2} r^{4} + r^{2} - 2 \, m\right)} \frac{\partial^{2}}{(\partial r)^{2}}\psi_{1}\left(r\right) = 0
psi1_sol(r) = desolve(eq_psi1, psi_1(r), ivar=r) psi1_sol(r)
K112r4+r22mdr+K2\renewcommand{\Bold}[1]{\mathbf{#1}}K_{1} \int \frac{1}{{\ell}^{2} r^{4} + r^{2} - 2 \, m}\,{d r} + K_{2}

We revover Eq. (4.10) with K1=Q/β22K_1 = \mathcal{Q}/\beta_2^2 and K2=0K_2=0. In what follows, we introduce Q=Q/β22\mathcal{Q}' = \mathcal{Q}/\beta_2^2 instead of Q\mathcal{Q}:

Q = var('Q', latex_name=r"\mathcal{Q}'") psi1_sol(r) = psi1_sol(r).subs({SR.var('_K1'): Q, SR.var('_K2'): 0}) print(psi1_sol(r))
Q*integrate(1/(l^2*r^4 + r^2 - 2*m), r)

Nambu-Goto Lagrangian at fourth order in aa

detg_a4 = detg.series(a, 5).truncate().simplify_full()
L_a4 = (sqrt(-detg_a4)).series(a, 5).truncate().simplify_full()
eqs = euler_lagrange(L_a4, [phi_1, psi_1, mu_1], r)

The equation for μ1\mu_1

eq_mu1 = eqs[2] eq_mu1
((μ03μ0)a4β12(μ03μ0)a4β22+2(μ03μ0)a4β12(μ03μ0)a4β2)4r44((μ03μ0)a4β1(μ03μ0)a4β2)2m((μ03μ0)a4β124r8+2(μ03μ0)a4β122r64(μ03μ0)a4β12mr2+4(μ03μ0)a4β12m2(4(μ03μ0)a4β122m(μ03μ0)a4β12)r4)rϕ1(r)2+((μ03μ0)a4β224r8+2(μ03μ0)a4β222r64(μ03μ0)a4β22mr2+4(μ03μ0)a4β22m2(4(μ03μ0)a4β222m(μ03μ0)a4β22)r4)rψ1(r)2+2(2a44r7+3a42r52a4mr(4a42ma4)r3)rμ1(r)+(a44r8+2a42r64a4mr2+4a4m2(4a42ma4)r4)2(r)2μ1(r)(μ021)2r4+(μ021)r22(μ021)m=0\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{{\left({\left({\mu_0}^{3} - {\mu_0}\right)} a^{4} {\beta_1}^{2} - {\left({\mu_0}^{3} - {\mu_0}\right)} a^{4} {\beta_2}^{2} + 2 \, {\left({\mu_0}^{3} - {\mu_0}\right)} a^{4} {\beta_1} - 2 \, {\left({\mu_0}^{3} - {\mu_0}\right)} a^{4} {\beta_2}\right)} {\ell}^{4} r^{4} - 4 \, {\left({\left({\mu_0}^{3} - {\mu_0}\right)} a^{4} {\beta_1} - {\left({\mu_0}^{3} - {\mu_0}\right)} a^{4} {\beta_2}\right)} {\ell}^{2} m - {\left({\left({\mu_0}^{3} - {\mu_0}\right)} a^{4} {\beta_1}^{2} {\ell}^{4} r^{8} + 2 \, {\left({\mu_0}^{3} - {\mu_0}\right)} a^{4} {\beta_1}^{2} {\ell}^{2} r^{6} - 4 \, {\left({\mu_0}^{3} - {\mu_0}\right)} a^{4} {\beta_1}^{2} m r^{2} + 4 \, {\left({\mu_0}^{3} - {\mu_0}\right)} a^{4} {\beta_1}^{2} m^{2} - {\left(4 \, {\left({\mu_0}^{3} - {\mu_0}\right)} a^{4} {\beta_1}^{2} {\ell}^{2} m - {\left({\mu_0}^{3} - {\mu_0}\right)} a^{4} {\beta_1}^{2}\right)} r^{4}\right)} \frac{\partial}{\partial r}\phi_{1}\left(r\right)^{2} + {\left({\left({\mu_0}^{3} - {\mu_0}\right)} a^{4} {\beta_2}^{2} {\ell}^{4} r^{8} + 2 \, {\left({\mu_0}^{3} - {\mu_0}\right)} a^{4} {\beta_2}^{2} {\ell}^{2} r^{6} - 4 \, {\left({\mu_0}^{3} - {\mu_0}\right)} a^{4} {\beta_2}^{2} m r^{2} + 4 \, {\left({\mu_0}^{3} - {\mu_0}\right)} a^{4} {\beta_2}^{2} m^{2} - {\left(4 \, {\left({\mu_0}^{3} - {\mu_0}\right)} a^{4} {\beta_2}^{2} {\ell}^{2} m - {\left({\mu_0}^{3} - {\mu_0}\right)} a^{4} {\beta_2}^{2}\right)} r^{4}\right)} \frac{\partial}{\partial r}\psi_{1}\left(r\right)^{2} + 2 \, {\left(2 \, a^{4} {\ell}^{4} r^{7} + 3 \, a^{4} {\ell}^{2} r^{5} - 2 \, a^{4} m r - {\left(4 \, a^{4} {\ell}^{2} m - a^{4}\right)} r^{3}\right)} \frac{\partial}{\partial r}\mu_{1}\left(r\right) + {\left(a^{4} {\ell}^{4} r^{8} + 2 \, a^{4} {\ell}^{2} r^{6} - 4 \, a^{4} m r^{2} + 4 \, a^{4} m^{2} - {\left(4 \, a^{4} {\ell}^{2} m - a^{4}\right)} r^{4}\right)} \frac{\partial^{2}}{(\partial r)^{2}}\mu_{1}\left(r\right)}{{\left({\mu_0}^{2} - 1\right)} {\ell}^{2} r^{4} + {\left({\mu_0}^{2} - 1\right)} r^{2} - 2 \, {\left({\mu_0}^{2} - 1\right)} m} = 0
# eq_mu1.lhs().numerator().simplify_full()
eq_mu1.lhs().denominator().simplify_full()
(μ021)2r4+(μ021)r22(μ021)m\renewcommand{\Bold}[1]{\mathbf{#1}}{\left({\mu_0}^{2} - 1\right)} {\ell}^{2} r^{4} + {\left({\mu_0}^{2} - 1\right)} r^{2} - 2 \, {\left({\mu_0}^{2} - 1\right)} m
eq_mu1 = eq_mu1.lhs().numerator().simplify_full() == 0 #eq_mu1
eq_mu1 = (eq_mu1/a^4).simplify_full() eq_mu1
((μ03μ0)β12(μ03μ0)β22+2(μ03μ0)β12(μ03μ0)β2)4r44((μ03μ0)β1(μ03μ0)β2)2m((μ03μ0)β124r8+2(μ03μ0)β122r64(μ03μ0)β12mr2+4(μ03μ0)β12m2(4(μ03μ0)β122m(μ03μ0)β12)r4)rϕ1(r)2+((μ03μ0)β224r8+2(μ03μ0)β222r64(μ03μ0)β22mr2+4(μ03μ0)β22m2(4(μ03μ0)β222m(μ03μ0)β22)r4)rψ1(r)2+2(24r7+32r5(42m1)r32mr)rμ1(r)+(4r8+22r6(42m1)r44mr2+4m2)2(r)2μ1(r)=0\renewcommand{\Bold}[1]{\mathbf{#1}}{\left({\left({\mu_0}^{3} - {\mu_0}\right)} {\beta_1}^{2} - {\left({\mu_0}^{3} - {\mu_0}\right)} {\beta_2}^{2} + 2 \, {\left({\mu_0}^{3} - {\mu_0}\right)} {\beta_1} - 2 \, {\left({\mu_0}^{3} - {\mu_0}\right)} {\beta_2}\right)} {\ell}^{4} r^{4} - 4 \, {\left({\left({\mu_0}^{3} - {\mu_0}\right)} {\beta_1} - {\left({\mu_0}^{3} - {\mu_0}\right)} {\beta_2}\right)} {\ell}^{2} m - {\left({\left({\mu_0}^{3} - {\mu_0}\right)} {\beta_1}^{2} {\ell}^{4} r^{8} + 2 \, {\left({\mu_0}^{3} - {\mu_0}\right)} {\beta_1}^{2} {\ell}^{2} r^{6} - 4 \, {\left({\mu_0}^{3} - {\mu_0}\right)} {\beta_1}^{2} m r^{2} + 4 \, {\left({\mu_0}^{3} - {\mu_0}\right)} {\beta_1}^{2} m^{2} - {\left(4 \, {\left({\mu_0}^{3} - {\mu_0}\right)} {\beta_1}^{2} {\ell}^{2} m - {\left({\mu_0}^{3} - {\mu_0}\right)} {\beta_1}^{2}\right)} r^{4}\right)} \frac{\partial}{\partial r}\phi_{1}\left(r\right)^{2} + {\left({\left({\mu_0}^{3} - {\mu_0}\right)} {\beta_2}^{2} {\ell}^{4} r^{8} + 2 \, {\left({\mu_0}^{3} - {\mu_0}\right)} {\beta_2}^{2} {\ell}^{2} r^{6} - 4 \, {\left({\mu_0}^{3} - {\mu_0}\right)} {\beta_2}^{2} m r^{2} + 4 \, {\left({\mu_0}^{3} - {\mu_0}\right)} {\beta_2}^{2} m^{2} - {\left(4 \, {\left({\mu_0}^{3} - {\mu_0}\right)} {\beta_2}^{2} {\ell}^{2} m - {\left({\mu_0}^{3} - {\mu_0}\right)} {\beta_2}^{2}\right)} r^{4}\right)} \frac{\partial}{\partial r}\psi_{1}\left(r\right)^{2} + 2 \, {\left(2 \, {\ell}^{4} r^{7} + 3 \, {\ell}^{2} r^{5} - {\left(4 \, {\ell}^{2} m - 1\right)} r^{3} - 2 \, m r\right)} \frac{\partial}{\partial r}\mu_{1}\left(r\right) + {\left({\ell}^{4} r^{8} + 2 \, {\ell}^{2} r^{6} - {\left(4 \, {\ell}^{2} m - 1\right)} r^{4} - 4 \, m r^{2} + 4 \, m^{2}\right)} \frac{\partial^{2}}{(\partial r)^{2}}\mu_{1}\left(r\right) = 0

We plug the solutions obtained previously for ϕ1(r)\phi_1(r) and ψ1(r)\psi_1(r) in this equation:

eq_mu1 = eq_mu1.substitute_function(phi_1, phi1_sol).substitute_function(psi_1, psi1_sol) eq_mu1 = eq_mu1.simplify_full() eq_mu1
((μ03μ0)β12(μ03μ0)β22+2(μ03μ0)β12(μ03μ0)β2)4r4(μ03μ0)P2β12+(μ03μ0)Q2β224((μ03μ0)β1(μ03μ0)β2)2m+2(24r7+32r5(42m1)r32mr)rμ1(r)+(4r8+22r6(42m1)r44mr2+4m2)2(r)2μ1(r)=0\renewcommand{\Bold}[1]{\mathbf{#1}}{\left({\left({\mu_0}^{3} - {\mu_0}\right)} {\beta_1}^{2} - {\left({\mu_0}^{3} - {\mu_0}\right)} {\beta_2}^{2} + 2 \, {\left({\mu_0}^{3} - {\mu_0}\right)} {\beta_1} - 2 \, {\left({\mu_0}^{3} - {\mu_0}\right)} {\beta_2}\right)} {\ell}^{4} r^{4} - {\left({\mu_0}^{3} - {\mu_0}\right)} {\mathcal{P}'}^{2} {\beta_1}^{2} + {\left({\mu_0}^{3} - {\mu_0}\right)} {\mathcal{Q}'}^{2} {\beta_2}^{2} - 4 \, {\left({\left({\mu_0}^{3} - {\mu_0}\right)} {\beta_1} - {\left({\mu_0}^{3} - {\mu_0}\right)} {\beta_2}\right)} {\ell}^{2} m + 2 \, {\left(2 \, {\ell}^{4} r^{7} + 3 \, {\ell}^{2} r^{5} - {\left(4 \, {\ell}^{2} m - 1\right)} r^{3} - 2 \, m r\right)} \frac{\partial}{\partial r}\mu_{1}\left(r\right) + {\left({\ell}^{4} r^{8} + 2 \, {\ell}^{2} r^{6} - {\left(4 \, {\ell}^{2} m - 1\right)} r^{4} - 4 \, m r^{2} + 4 \, m^{2}\right)} \frac{\partial^{2}}{(\partial r)^{2}}\mu_{1}\left(r\right) = 0

Check of Eq. (4.11)

lhs = eq_mu1.lhs() lhs = lhs.simplify_full() lhs
((μ03μ0)β12(μ03μ0)β22+2(μ03μ0)β12(μ03μ0)β2)4r4(μ03μ0)P2β12+(μ03μ0)Q2β224((μ03μ0)β1(μ03μ0)β2)2m+2(24r7+32r5(42m1)r32mr)rμ1(r)+(4r8+22r6(42m1)r44mr2+4m2)2(r)2μ1(r)\renewcommand{\Bold}[1]{\mathbf{#1}}{\left({\left({\mu_0}^{3} - {\mu_0}\right)} {\beta_1}^{2} - {\left({\mu_0}^{3} - {\mu_0}\right)} {\beta_2}^{2} + 2 \, {\left({\mu_0}^{3} - {\mu_0}\right)} {\beta_1} - 2 \, {\left({\mu_0}^{3} - {\mu_0}\right)} {\beta_2}\right)} {\ell}^{4} r^{4} - {\left({\mu_0}^{3} - {\mu_0}\right)} {\mathcal{P}'}^{2} {\beta_1}^{2} + {\left({\mu_0}^{3} - {\mu_0}\right)} {\mathcal{Q}'}^{2} {\beta_2}^{2} - 4 \, {\left({\left({\mu_0}^{3} - {\mu_0}\right)} {\beta_1} - {\left({\mu_0}^{3} - {\mu_0}\right)} {\beta_2}\right)} {\ell}^{2} m + 2 \, {\left(2 \, {\ell}^{4} r^{7} + 3 \, {\ell}^{2} r^{5} - {\left(4 \, {\ell}^{2} m - 1\right)} r^{3} - 2 \, m r\right)} \frac{\partial}{\partial r}\mu_{1}\left(r\right) + {\left({\ell}^{4} r^{8} + 2 \, {\ell}^{2} r^{6} - {\left(4 \, {\ell}^{2} m - 1\right)} r^{4} - 4 \, m r^{2} + 4 \, m^{2}\right)} \frac{\partial^{2}}{(\partial r)^{2}}\mu_{1}\left(r\right)
s = lhs.coefficient(diff(mu_1(r), r, 2)) # coefficient of mu_1'' s.factor()
(2r4+r22m)2\renewcommand{\Bold}[1]{\mathbf{#1}}{\left({\ell}^{2} r^{4} + r^{2} - 2 \, m\right)}^{2}
s1 = (lhs/s - diff(mu_1(r), r, 2)).simplify_full() s1
((μ03μ0)β12(μ03μ0)β22+2(μ03μ0)β12(μ03μ0)β2)4r4(μ03μ0)P2β12+(μ03μ0)Q2β224((μ03μ0)β1(μ03μ0)β2)2m+2(24r7+32r5(42m1)r32mr)rμ1(r)4r8+22r6(42m1)r44mr2+4m2\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{{\left({\left({\mu_0}^{3} - {\mu_0}\right)} {\beta_1}^{2} - {\left({\mu_0}^{3} - {\mu_0}\right)} {\beta_2}^{2} + 2 \, {\left({\mu_0}^{3} - {\mu_0}\right)} {\beta_1} - 2 \, {\left({\mu_0}^{3} - {\mu_0}\right)} {\beta_2}\right)} {\ell}^{4} r^{4} - {\left({\mu_0}^{3} - {\mu_0}\right)} {\mathcal{P}'}^{2} {\beta_1}^{2} + {\left({\mu_0}^{3} - {\mu_0}\right)} {\mathcal{Q}'}^{2} {\beta_2}^{2} - 4 \, {\left({\left({\mu_0}^{3} - {\mu_0}\right)} {\beta_1} - {\left({\mu_0}^{3} - {\mu_0}\right)} {\beta_2}\right)} {\ell}^{2} m + 2 \, {\left(2 \, {\ell}^{4} r^{7} + 3 \, {\ell}^{2} r^{5} - {\left(4 \, {\ell}^{2} m - 1\right)} r^{3} - 2 \, m r\right)} \frac{\partial}{\partial r}\mu_{1}\left(r\right)}{{\ell}^{4} r^{8} + 2 \, {\ell}^{2} r^{6} - {\left(4 \, {\ell}^{2} m - 1\right)} r^{4} - 4 \, m r^{2} + 4 \, m^{2}}
b1 = s1.coefficient(diff(mu_1(r), r)).factor() b1
2(22r2+1)r2r4+r22m\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{2 \, {\left(2 \, {\ell}^{2} r^{2} + 1\right)} r}{{\ell}^{2} r^{4} + r^{2} - 2 \, m}
s2 = (s1 - b1*diff(mu_1(r), r)).simplify_full() s2
((μ03μ0)β12(μ03μ0)β22+2(μ03μ0)β12(μ03μ0)β2)4r4(μ03μ0)P2β12+(μ03μ0)Q2β224((μ03μ0)β1(μ03μ0)β2)2m4r8+22r6(42m1)r44mr2+4m2\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{{\left({\left({\mu_0}^{3} - {\mu_0}\right)} {\beta_1}^{2} - {\left({\mu_0}^{3} - {\mu_0}\right)} {\beta_2}^{2} + 2 \, {\left({\mu_0}^{3} - {\mu_0}\right)} {\beta_1} - 2 \, {\left({\mu_0}^{3} - {\mu_0}\right)} {\beta_2}\right)} {\ell}^{4} r^{4} - {\left({\mu_0}^{3} - {\mu_0}\right)} {\mathcal{P}'}^{2} {\beta_1}^{2} + {\left({\mu_0}^{3} - {\mu_0}\right)} {\mathcal{Q}'}^{2} {\beta_2}^{2} - 4 \, {\left({\left({\mu_0}^{3} - {\mu_0}\right)} {\beta_1} - {\left({\mu_0}^{3} - {\mu_0}\right)} {\beta_2}\right)} {\ell}^{2} m}{{\ell}^{4} r^{8} + 2 \, {\ell}^{2} r^{6} - {\left(4 \, {\ell}^{2} m - 1\right)} r^{4} - 4 \, m r^{2} + 4 \, m^{2}}
s2.factor()
(β124r4β224r4+2β14r42β24r4P2β12+Q2β224β12m+4β22m)(μ0+1)(μ01)μ0(2r4+r22m)2\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{{\left({\beta_1}^{2} {\ell}^{4} r^{4} - {\beta_2}^{2} {\ell}^{4} r^{4} + 2 \, {\beta_1} {\ell}^{4} r^{4} - 2 \, {\beta_2} {\ell}^{4} r^{4} - {\mathcal{P}'}^{2} {\beta_1}^{2} + {\mathcal{Q}'}^{2} {\beta_2}^{2} - 4 \, {\beta_1} {\ell}^{2} m + 4 \, {\beta_2} {\ell}^{2} m\right)} {\left({\mu_0} + 1\right)} {\left({\mu_0} - 1\right)} {\mu_0}}{{\left({\ell}^{2} r^{4} + r^{2} - 2 \, m\right)}^{2}}

The equation for μ1\mu_1 is thus:

eq_mu1 = diff(mu_1(r), r, 2) + b1*diff(mu_1(r), r) + s2.factor() == 0 eq_mu1
(β124r4β224r4+2β14r42β24r4P2β12+Q2β224β12m+4β22m)(μ0+1)(μ01)μ0(2r4+r22m)2+2(22r2+1)rrμ1(r)2r4+r22m+2(r)2μ1(r)=0\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{{\left({\beta_1}^{2} {\ell}^{4} r^{4} - {\beta_2}^{2} {\ell}^{4} r^{4} + 2 \, {\beta_1} {\ell}^{4} r^{4} - 2 \, {\beta_2} {\ell}^{4} r^{4} - {\mathcal{P}'}^{2} {\beta_1}^{2} + {\mathcal{Q}'}^{2} {\beta_2}^{2} - 4 \, {\beta_1} {\ell}^{2} m + 4 \, {\beta_2} {\ell}^{2} m\right)} {\left({\mu_0} + 1\right)} {\left({\mu_0} - 1\right)} {\mu_0}}{{\left({\ell}^{2} r^{4} + r^{2} - 2 \, m\right)}^{2}} + \frac{2 \, {\left(2 \, {\ell}^{2} r^{2} + 1\right)} r \frac{\partial}{\partial r}\mu_{1}\left(r\right)}{{\ell}^{2} r^{4} + r^{2} - 2 \, m} + \frac{\partial^{2}}{(\partial r)^{2}}\mu_{1}\left(r\right) = 0
h(r) = l^2 + 1/r^2 - 2*m/r^4 h(r)
2+1r22mr4\renewcommand{\Bold}[1]{\mathbf{#1}}{\ell}^{2} + \frac{1}{r^{2}} - \frac{2 \, m}{r^{4}}
s3 = (s2 / (Mu0*(1-Mu0^2))* r^8*h(r)^2).simplify_full() s3
(β12β22+2β12β2)4r4+P2β12Q2β22+4(β1β2)2m\renewcommand{\Bold}[1]{\mathbf{#1}}-{\left({\beta_1}^{2} - {\beta_2}^{2} + 2 \, {\beta_1} - 2 \, {\beta_2}\right)} {\ell}^{4} r^{4} + {\mathcal{P}'}^{2} {\beta_1}^{2} - {\mathcal{Q}'}^{2} {\beta_2}^{2} + 4 \, {\left({\beta_1} - {\beta_2}\right)} {\ell}^{2} m

Given that μ1(r)=sinΘ0  θ1(r)=1μ02  θ1(r),sin2Θ0=2μ01μ02 \mu_1(r) = - \sin\Theta_0 \; \theta_1(r) = - \sqrt{1-\mu_0^2} \; \theta_1(r), \qquad \sin2\Theta_0 = 2\mu_0\sqrt{1-\mu_0^2} and P=P/β12andQ=Q/β12,\mathcal{P}' = \mathcal{P}/\beta_1^2 \qquad\mbox{and}\qquad \mathcal{Q}' = \mathcal{Q}/\beta_1^2, we get for the equation for θ1\theta_1: θ1+2r(22r2+1)r4hθ1+β22Q2β12P24(β1β2)2m+(β12β22+2(β1β2))4r42r8h2sin(2Θ0)=0 \theta_1'' + \frac{2r(2\ell^2 r^2 + 1)}{r^4 h} \, \theta_1' + \frac{\beta_2^{-2}\mathcal{Q}^2 - \beta_1^{-2}\mathcal{P}^2 - 4 (\beta_1 - \beta_2) \ell^2 m + (\beta_1^2 - \beta_2^2 + 2 (\beta_1 - \beta_2)) \ell^4 r^4}{2 r^8 h^2}\sin(2\Theta_0) = 0

This agrees with Eq. (4.11) of the paper.

Solving the equation for μ1\mu_1

mu1_sol(r) = desolve(eq_mu1, mu_1(r), ivar=r) mu1_sol(r)
K2((μ03μ0)β12(μ03μ0)β22+2(μ03μ0)β12(μ03μ0)β2)2r(μ03μ0)(β12β22+2β12β2)2r2+P2β12Q2β222(β12β22)2m2r4+r22mdrK12r4+r22mdr\renewcommand{\Bold}[1]{\mathbf{#1}}K_{2} - \int \frac{{\left({\left({\mu_0}^{3} - {\mu_0}\right)} {\beta_1}^{2} - {\left({\mu_0}^{3} - {\mu_0}\right)} {\beta_2}^{2} + 2 \, {\left({\mu_0}^{3} - {\mu_0}\right)} {\beta_1} - 2 \, {\left({\mu_0}^{3} - {\mu_0}\right)} {\beta_2}\right)} {\ell}^{2} r - {\left({\mu_0}^{3} - {\mu_0}\right)} \int \frac{{\left({\beta_1}^{2} - {\beta_2}^{2} + 2 \, {\beta_1} - 2 \, {\beta_2}\right)} {\ell}^{2} r^{2} + {\mathcal{P}'}^{2} {\beta_1}^{2} - {\mathcal{Q}'}^{2} {\beta_2}^{2} - 2 \, {\left({\beta_1}^{2} - {\beta_2}^{2}\right)} {\ell}^{2} m}{{\ell}^{2} r^{4} + r^{2} - 2 \, m}\,{d r} - K_{1}}{{\ell}^{2} r^{4} + r^{2} - 2 \, m}\,{d r}
print(mu1_sol(r))
_K2 - integrate((((Mu0^3 - Mu0)*beta1^2 - (Mu0^3 - Mu0)*beta2^2 + 2*(Mu0^3 - Mu0)*beta1 - 2*(Mu0^3 - Mu0)*beta2)*l^2*r + (Mu0^3 - Mu0)*integrate(-((beta1^2 - beta2^2 + 2*beta1 - 2*beta2)*l^2*r^2 + P^2*beta1^2 - Q^2*beta2^2 - 2*(beta1^2 - beta2^2)*l^2*m)/(l^2*r^4 + r^2 - 2*m), r) - _K1)/(l^2*r^4 + r^2 - 2*m), r)

Let us check that mu1_sol is indeed a solution of the equation for μ1\mu_1:

eq_mu1.substitute_function(mu_1, mu1_sol).simplify_full()
0=0\renewcommand{\Bold}[1]{\mathbf{#1}}0 = 0
mu1_sol(r)
K2((μ03μ0)β12(μ03μ0)β22+2(μ03μ0)β12(μ03μ0)β2)2r(μ03μ0)(β12β22+2β12β2)2r2+P2β12Q2β222(β12β22)2m2r4+r22mdrK12r4+r22mdr\renewcommand{\Bold}[1]{\mathbf{#1}}K_{2} - \int \frac{{\left({\left({\mu_0}^{3} - {\mu_0}\right)} {\beta_1}^{2} - {\left({\mu_0}^{3} - {\mu_0}\right)} {\beta_2}^{2} + 2 \, {\left({\mu_0}^{3} - {\mu_0}\right)} {\beta_1} - 2 \, {\left({\mu_0}^{3} - {\mu_0}\right)} {\beta_2}\right)} {\ell}^{2} r - {\left({\mu_0}^{3} - {\mu_0}\right)} \int \frac{{\left({\beta_1}^{2} - {\beta_2}^{2} + 2 \, {\beta_1} - 2 \, {\beta_2}\right)} {\ell}^{2} r^{2} + {\mathcal{P}'}^{2} {\beta_1}^{2} - {\mathcal{Q}'}^{2} {\beta_2}^{2} - 2 \, {\left({\beta_1}^{2} - {\beta_2}^{2}\right)} {\ell}^{2} m}{{\ell}^{2} r^{4} + r^{2} - 2 \, m}\,{d r} - K_{1}}{{\ell}^{2} r^{4} + r^{2} - 2 \, m}\,{d r}

The innermost integral can be written (β12β22+2(β1β2))2  s1(r)+(P2β12Qβ222(β12β22)2m)  s2(r) (\beta_1^2 - \beta_2^2 + 2 (\beta_1-\beta_2)) \ell^2 \; s_1(r) + \left({\mathcal{P}'}^2 \beta_1^2 - {\mathcal{Q}'}\beta_2^2 - 2 (\beta_1^2-\beta_2^2)\ell^2 m \right) \; s_2(r) with s1(r):=rrˉ22rˉ4+rˉ22mdrˉands2(r):=rdrˉ2rˉ4+rˉ22m. s_1(r) := \int^r \frac{\bar{r}^2}{\ell^2 \bar{r}^4 + \bar{r}^2 - 2m} \, \mathrm{d}\bar{r} \qquad \mbox{and}\qquad s_2(r) := \int^r \frac{\mathrm{d}\bar{r}}{\ell^2 \bar{r}^4 + \bar{r}^2 - 2m} .

Let us evaluate s1s_1 by means of FriCAS:

s1 = integrate(r^2/(l^2*r^4 + r^2 - 2*m), r, algorithm='fricas') s1
121284m+286m+4+184m+2log(12(84m+2)84m+286m+4+184m+286m+4+r)121284m+286m+4+184m+2log(12(84m+2)84m+286m+4+184m+286m+4+r)121284m+286m+4184m+2log(12(84m+2)84m+286m+4184m+286m+4+r)+121284m+286m+4184m+2log(12(84m+2)84m+286m+4184m+286m+4+r)\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{2} \, \sqrt{\frac{1}{2}} \sqrt{-\frac{\frac{8 \, {\ell}^{4} m + {\ell}^{2}}{\sqrt{8 \, {\ell}^{6} m + {\ell}^{4}}} + 1}{8 \, {\ell}^{4} m + {\ell}^{2}}} \log\left(\frac{\sqrt{\frac{1}{2}} {\left(8 \, {\ell}^{4} m + {\ell}^{2}\right)} \sqrt{-\frac{\frac{8 \, {\ell}^{4} m + {\ell}^{2}}{\sqrt{8 \, {\ell}^{6} m + {\ell}^{4}}} + 1}{8 \, {\ell}^{4} m + {\ell}^{2}}}}{\sqrt{8 \, {\ell}^{6} m + {\ell}^{4}}} + r\right) - \frac{1}{2} \, \sqrt{\frac{1}{2}} \sqrt{-\frac{\frac{8 \, {\ell}^{4} m + {\ell}^{2}}{\sqrt{8 \, {\ell}^{6} m + {\ell}^{4}}} + 1}{8 \, {\ell}^{4} m + {\ell}^{2}}} \log\left(-\frac{\sqrt{\frac{1}{2}} {\left(8 \, {\ell}^{4} m + {\ell}^{2}\right)} \sqrt{-\frac{\frac{8 \, {\ell}^{4} m + {\ell}^{2}}{\sqrt{8 \, {\ell}^{6} m + {\ell}^{4}}} + 1}{8 \, {\ell}^{4} m + {\ell}^{2}}}}{\sqrt{8 \, {\ell}^{6} m + {\ell}^{4}}} + r\right) - \frac{1}{2} \, \sqrt{\frac{1}{2}} \sqrt{\frac{\frac{8 \, {\ell}^{4} m + {\ell}^{2}}{\sqrt{8 \, {\ell}^{6} m + {\ell}^{4}}} - 1}{8 \, {\ell}^{4} m + {\ell}^{2}}} \log\left(\frac{\sqrt{\frac{1}{2}} {\left(8 \, {\ell}^{4} m + {\ell}^{2}\right)} \sqrt{\frac{\frac{8 \, {\ell}^{4} m + {\ell}^{2}}{\sqrt{8 \, {\ell}^{6} m + {\ell}^{4}}} - 1}{8 \, {\ell}^{4} m + {\ell}^{2}}}}{\sqrt{8 \, {\ell}^{6} m + {\ell}^{4}}} + r\right) + \frac{1}{2} \, \sqrt{\frac{1}{2}} \sqrt{\frac{\frac{8 \, {\ell}^{4} m + {\ell}^{2}}{\sqrt{8 \, {\ell}^{6} m + {\ell}^{4}}} - 1}{8 \, {\ell}^{4} m + {\ell}^{2}}} \log\left(-\frac{\sqrt{\frac{1}{2}} {\left(8 \, {\ell}^{4} m + {\ell}^{2}\right)} \sqrt{\frac{\frac{8 \, {\ell}^{4} m + {\ell}^{2}}{\sqrt{8 \, {\ell}^{6} m + {\ell}^{4}}} - 1}{8 \, {\ell}^{4} m + {\ell}^{2}}}}{\sqrt{8 \, {\ell}^{6} m + {\ell}^{4}}} + r\right)
s1 = s1.canonicalize_radical().simplify_log() s1
282m82m+1+1log(2(82m+1)14r82m82m+1+12(82m+1)14r+82m82m+1+1)+282m82m+11log(2(82m+1)14r+82m82m+112(82m+1)14r82m82m+11)4(82m+1)34\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{\sqrt{2} \sqrt{8 \, {\ell}^{2} m - \sqrt{8 \, {\ell}^{2} m + 1} + 1} \log\left(\frac{\sqrt{2} {\left(8 \, {\ell}^{2} m + 1\right)}^{\frac{1}{4}} {\ell} r - \sqrt{8 \, {\ell}^{2} m - \sqrt{8 \, {\ell}^{2} m + 1} + 1}}{\sqrt{2} {\left(8 \, {\ell}^{2} m + 1\right)}^{\frac{1}{4}} {\ell} r + \sqrt{8 \, {\ell}^{2} m - \sqrt{8 \, {\ell}^{2} m + 1} + 1}}\right) + \sqrt{2} \sqrt{-8 \, {\ell}^{2} m - \sqrt{8 \, {\ell}^{2} m + 1} - 1} \log\left(\frac{\sqrt{2} {\left(8 \, {\ell}^{2} m + 1\right)}^{\frac{1}{4}} {\ell} r + \sqrt{-8 \, {\ell}^{2} m - \sqrt{8 \, {\ell}^{2} m + 1} - 1}}{\sqrt{2} {\left(8 \, {\ell}^{2} m + 1\right)}^{\frac{1}{4}} {\ell} r - \sqrt{-8 \, {\ell}^{2} m - \sqrt{8 \, {\ell}^{2} m + 1} - 1}}\right)}{4 \, {\left(8 \, {\ell}^{2} m + 1\right)}^{\frac{3}{4}} {\ell}}

Check:

diff(s1, r).simplify_full()
r22r4+r22m\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{r^{2}}{{\ell}^{2} r^{4} + r^{2} - 2 \, m}

Similarly, we evaluate s2s_2 by means of FriCAS:

s2 = integrate(1/(l^2*r^4 + r^2 - 2*m), r, algorithm='fricas') s2
1482m2+m82m3+m2+182m2+mlog(22r+12(82m82m2+m82m3+m2+1)82m2+m82m3+m2+182m2+m)+1482m2+m82m3+m2+182m2+mlog(22r12(82m82m2+m82m3+m2+1)82m2+m82m3+m2+182m2+m)1482m2+m82m3+m2182m2+mlog(22r+12(82m+82m2+m82m3+m2+1)82m2+m82m3+m2182m2+m)+1482m2+m82m3+m2182m2+mlog(22r12(82m+82m2+m82m3+m2+1)82m2+m82m3+m2182m2+m)\renewcommand{\Bold}[1]{\mathbf{#1}}-\frac{1}{4} \, \sqrt{\frac{\frac{8 \, {\ell}^{2} m^{2} + m}{\sqrt{8 \, {\ell}^{2} m^{3} + m^{2}}} + 1}{8 \, {\ell}^{2} m^{2} + m}} \log\left(2 \, {\ell}^{2} r + \frac{1}{2} \, {\left(8 \, {\ell}^{2} m - \frac{8 \, {\ell}^{2} m^{2} + m}{\sqrt{8 \, {\ell}^{2} m^{3} + m^{2}}} + 1\right)} \sqrt{\frac{\frac{8 \, {\ell}^{2} m^{2} + m}{\sqrt{8 \, {\ell}^{2} m^{3} + m^{2}}} + 1}{8 \, {\ell}^{2} m^{2} + m}}\right) + \frac{1}{4} \, \sqrt{\frac{\frac{8 \, {\ell}^{2} m^{2} + m}{\sqrt{8 \, {\ell}^{2} m^{3} + m^{2}}} + 1}{8 \, {\ell}^{2} m^{2} + m}} \log\left(2 \, {\ell}^{2} r - \frac{1}{2} \, {\left(8 \, {\ell}^{2} m - \frac{8 \, {\ell}^{2} m^{2} + m}{\sqrt{8 \, {\ell}^{2} m^{3} + m^{2}}} + 1\right)} \sqrt{\frac{\frac{8 \, {\ell}^{2} m^{2} + m}{\sqrt{8 \, {\ell}^{2} m^{3} + m^{2}}} + 1}{8 \, {\ell}^{2} m^{2} + m}}\right) - \frac{1}{4} \, \sqrt{-\frac{\frac{8 \, {\ell}^{2} m^{2} + m}{\sqrt{8 \, {\ell}^{2} m^{3} + m^{2}}} - 1}{8 \, {\ell}^{2} m^{2} + m}} \log\left(2 \, {\ell}^{2} r + \frac{1}{2} \, {\left(8 \, {\ell}^{2} m + \frac{8 \, {\ell}^{2} m^{2} + m}{\sqrt{8 \, {\ell}^{2} m^{3} + m^{2}}} + 1\right)} \sqrt{-\frac{\frac{8 \, {\ell}^{2} m^{2} + m}{\sqrt{8 \, {\ell}^{2} m^{3} + m^{2}}} - 1}{8 \, {\ell}^{2} m^{2} + m}}\right) + \frac{1}{4} \, \sqrt{-\frac{\frac{8 \, {\ell}^{2} m^{2} + m}{\sqrt{8 \, {\ell}^{2} m^{3} + m^{2}}} - 1}{8 \, {\ell}^{2} m^{2} + m}} \log\left(2 \, {\ell}^{2} r - \frac{1}{2} \, {\left(8 \, {\ell}^{2} m + \frac{8 \, {\ell}^{2} m^{2} + m}{\sqrt{8 \, {\ell}^{2} m^{3} + m^{2}}} + 1\right)} \sqrt{-\frac{\frac{8 \, {\ell}^{2} m^{2} + m}{\sqrt{8 \, {\ell}^{2} m^{3} + m^{2}}} - 1}{8 \, {\ell}^{2} m^{2} + m}}\right)
s2 = s2.canonicalize_radical().simplify_log() s2
82m+82m+11log(4(82m+1)142mr82m+82m+11(82m+1+1)4(82m+1)142mr+82m+82m+11(82m+1+1))+82m+82m+1+1log(4(82m+1)142mr82m+82m+1+1(82m+11)4(82m+1)142mr+82m+82m+1+1(82m+11))4(82m+1)34m\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{\sqrt{-8 \, {\ell}^{2} m + \sqrt{8 \, {\ell}^{2} m + 1} - 1} \log\left(\frac{4 \, {\left(8 \, {\ell}^{2} m + 1\right)}^{\frac{1}{4}} {\ell}^{2} \sqrt{m} r - \sqrt{-8 \, {\ell}^{2} m + \sqrt{8 \, {\ell}^{2} m + 1} - 1} {\left(\sqrt{8 \, {\ell}^{2} m + 1} + 1\right)}}{4 \, {\left(8 \, {\ell}^{2} m + 1\right)}^{\frac{1}{4}} {\ell}^{2} \sqrt{m} r + \sqrt{-8 \, {\ell}^{2} m + \sqrt{8 \, {\ell}^{2} m + 1} - 1} {\left(\sqrt{8 \, {\ell}^{2} m + 1} + 1\right)}}\right) + \sqrt{8 \, {\ell}^{2} m + \sqrt{8 \, {\ell}^{2} m + 1} + 1} \log\left(\frac{4 \, {\left(8 \, {\ell}^{2} m + 1\right)}^{\frac{1}{4}} {\ell}^{2} \sqrt{m} r - \sqrt{8 \, {\ell}^{2} m + \sqrt{8 \, {\ell}^{2} m + 1} + 1} {\left(\sqrt{8 \, {\ell}^{2} m + 1} - 1\right)}}{4 \, {\left(8 \, {\ell}^{2} m + 1\right)}^{\frac{1}{4}} {\ell}^{2} \sqrt{m} r + \sqrt{8 \, {\ell}^{2} m + \sqrt{8 \, {\ell}^{2} m + 1} + 1} {\left(\sqrt{8 \, {\ell}^{2} m + 1} - 1\right)}}\right)}{4 \, {\left(8 \, {\ell}^{2} m + 1\right)}^{\frac{3}{4}} \sqrt{m}}

Check:

diff(s2, r).simplify_full()
12r4+r22m\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{{\ell}^{2} r^{4} + r^{2} - 2 \, m}

In the above expressions for s1(r)s_1(r) and s2(r)s_2(r) there appears the factor P=1+82m,\mathfrak{P} = \sqrt{1 + 8\ell^2 m}, which we represent by the symbolic variable B

B = var('B') assume(B > 1)

Let us make BB appear in s1s_1:

s1 = s1.subs({l^2: (B^2 - 1)/(8*m)}).simplify_full() s1
2B2Blog(2BrB2B2Br+B2B)+2B2Blog(2Br+B2B2BrB2B)4B32\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{\sqrt{2} \sqrt{B^{2} - B} \log\left(\frac{\sqrt{2} \sqrt{B} {\ell} r - \sqrt{B^{2} - B}}{\sqrt{2} \sqrt{B} {\ell} r + \sqrt{B^{2} - B}}\right) + \sqrt{2} \sqrt{-B^{2} - B} \log\left(\frac{\sqrt{2} \sqrt{B} {\ell} r + \sqrt{-B^{2} - B}}{\sqrt{2} \sqrt{B} {\ell} r - \sqrt{-B^{2} - B}}\right)}{4 \, B^{\frac{3}{2}} {\ell}}

In this expression, there appears the term B2B\sqrt{-B^2-B} which is imaginary since B>1B>1. We there rewrite it as iBB+1i\sqrt{B}\sqrt{B+1}:

s1 = s1.subs({sqrt(-B^2 - B): I*sqrt(B)*sqrt(B + 1), sqrt(B^2 - B): sqrt(B)*sqrt(B - 1)}) s1
i2B+1Blog(2Br+iB+1B2BriB+1B)+2B1Blog(2BrB1B2Br+B1B)4B32\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{i \, \sqrt{2} \sqrt{B + 1} \sqrt{B} \log\left(\frac{\sqrt{2} \sqrt{B} {\ell} r + i \, \sqrt{B + 1} \sqrt{B}}{\sqrt{2} \sqrt{B} {\ell} r - i \, \sqrt{B + 1} \sqrt{B}}\right) + \sqrt{2} \sqrt{B - 1} \sqrt{B} \log\left(\frac{\sqrt{2} \sqrt{B} {\ell} r - \sqrt{B - 1} \sqrt{B}}{\sqrt{2} \sqrt{B} {\ell} r + \sqrt{B - 1} \sqrt{B}}\right)}{4 \, B^{\frac{3}{2}} {\ell}}
s1 = s1.simplify_log() s1
i2B+1log(2r+iB+12riB+1)+2B1log(2rB12r+B1)4B\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{i \, \sqrt{2} \sqrt{B + 1} \log\left(\frac{\sqrt{2} {\ell} r + i \, \sqrt{B + 1}}{\sqrt{2} {\ell} r - i \, \sqrt{B + 1}}\right) + \sqrt{2} \sqrt{B - 1} \log\left(\frac{\sqrt{2} {\ell} r - \sqrt{B - 1}}{\sqrt{2} {\ell} r + \sqrt{B - 1}}\right)}{4 \, B {\ell}}

In the first log\log, we recognize the arctan\mathrm{arctan} function, via the identity arctanx=i2ln(i+xix), \mathrm{arctan}\, x = \frac{i}{2} \ln\left( \frac{i + x}{i - x} \right), which we use in the form iln(x+ixi)=2arctan(x)π i \ln\left( \frac{x + i}{x - i} \right) = 2 \mathrm{arctan}(x) - \pi as we can check:

taylor(I*ln((x+I)/(x-I)) - 2*atan(x) + pi, x, 0, 10)
0\renewcommand{\Bold}[1]{\mathbf{#1}}0

Thus, we set, disregarding the additive constant π-\pi,

s1 = sqrt(2)/(4*B*l)*(2*sqrt(B+1)*atan(sqrt(2)*l/sqrt(B+1)*r) + sqrt(B-1)*ln((sqrt(2)*l/sqrt(B-1)*r - 1)/(sqrt(2)*l/sqrt(B-1)*r + 1))) s1
2(2B+1arctan(2rB+1)+B1log(2rB112rB1+1))4B\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{\sqrt{2} {\left(2 \, \sqrt{B + 1} \arctan\left(\frac{\sqrt{2} {\ell} r}{\sqrt{B + 1}}\right) + \sqrt{B - 1} \log\left(\frac{\frac{\sqrt{2} {\ell} r}{\sqrt{B - 1}} - 1}{\frac{\sqrt{2} {\ell} r}{\sqrt{B - 1}} + 1}\right)\right)}}{4 \, B {\ell}}

Let us check that we have indeed a primitive of rr22r4+r22mr\mapsto \frac{r^2}{\ell^2 r^4 + r^2 - 2m}:

Ds1 = diff(s1, r).simplify_full() Ds1
42r244r4+42r2B2+1\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{4 \, {\ell}^{2} r^{2}}{4 \, {\ell}^{4} r^{4} + 4 \, {\ell}^{2} r^{2} - B^{2} + 1}
Ds1.subs({B: sqrt(1 + 8*l^2*m)}).simplify_full()
r22r4+r22m\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{r^{2}}{{\ell}^{2} r^{4} + r^{2} - 2 \, m}

Similarly, we can express s2s_2 in terms of BB:

s2 = s2.subs({l^2: (B^2 - 1)/(8*m)}).simplify_full() s2
B2+Blog((B+1)Bmr2B2+Bm(B+1)Bmr+2B2+Bm)+B2+Blog((B1)Bmr2B2+Bm(B1)Bmr+2B2+Bm)4B32m\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{\sqrt{B^{2} + B} \log\left(\frac{{\left(B + 1\right)} \sqrt{B} \sqrt{m} r - 2 \, \sqrt{B^{2} + B} m}{{\left(B + 1\right)} \sqrt{B} \sqrt{m} r + 2 \, \sqrt{B^{2} + B} m}\right) + \sqrt{-B^{2} + B} \log\left(\frac{{\left(B - 1\right)} \sqrt{B} \sqrt{m} r - 2 \, \sqrt{-B^{2} + B} m}{{\left(B - 1\right)} \sqrt{B} \sqrt{m} r + 2 \, \sqrt{-B^{2} + B} m}\right)}{4 \, B^{\frac{3}{2}} \sqrt{m}}

Since B>1B>1, we replace B2+B\sqrt{-B^2 + B} by iBB1i\sqrt{B}\sqrt{B-1}:

s2 = s2.subs({sqrt(-B^2 + B): I*sqrt(B)*sqrt(B - 1), sqrt(B^2 + B): sqrt(B)*sqrt(B + 1)}) s2
B+1Blog((B+1)Bmr2B+1Bm(B+1)Bmr+2B+1Bm)+iB1Blog((B1)Bmr2iB1Bm(B1)Bmr+2iB1Bm)4B32m\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{\sqrt{B + 1} \sqrt{B} \log\left(\frac{{\left(B + 1\right)} \sqrt{B} \sqrt{m} r - 2 \, \sqrt{B + 1} \sqrt{B} m}{{\left(B + 1\right)} \sqrt{B} \sqrt{m} r + 2 \, \sqrt{B + 1} \sqrt{B} m}\right) + i \, \sqrt{B - 1} \sqrt{B} \log\left(\frac{{\left(B - 1\right)} \sqrt{B} \sqrt{m} r - 2 i \, \sqrt{B - 1} \sqrt{B} m}{{\left(B - 1\right)} \sqrt{B} \sqrt{m} r + 2 i \, \sqrt{B - 1} \sqrt{B} m}\right)}{4 \, B^{\frac{3}{2}} \sqrt{m}}
s2 = s2.simplify_log() s2
B+1log((B+1)mr2B+1m(B+1)mr+2B+1m)+iB1log((B1)mr2iB1m(B1)mr+2iB1m)4Bm\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{\sqrt{B + 1} \log\left(\frac{{\left(B + 1\right)} \sqrt{m} r - 2 \, \sqrt{B + 1} m}{{\left(B + 1\right)} \sqrt{m} r + 2 \, \sqrt{B + 1} m}\right) + i \, \sqrt{B - 1} \log\left(\frac{{\left(B - 1\right)} \sqrt{m} r - 2 i \, \sqrt{B - 1} m}{{\left(B - 1\right)} \sqrt{m} r + 2 i \, \sqrt{B - 1} m}\right)}{4 \, B \sqrt{m}}

Again, we use the identity iln(x+ixi)=2arctan(x)π i \ln\left( \frac{x + i}{x - i} \right) = 2 \mathrm{arctan}(x) - \pi to rewrite s2s_2 as

s2 = 1/(4*B*sqrt(m))*(sqrt(B+1)*ln( (sqrt(B+1)/(2*sqrt(m))*r - 1) /(sqrt(B+1)/(2*sqrt(m))*r + 1) ) - 2*sqrt(B-1)*atan(sqrt(B-1)/(2*sqrt(m))*r)) s2
2B1arctan(B1r2m)B+1log(B+1rm2B+1rm+2)4Bm\renewcommand{\Bold}[1]{\mathbf{#1}}-\frac{2 \, \sqrt{B - 1} \arctan\left(\frac{\sqrt{B - 1} r}{2 \, \sqrt{m}}\right) - \sqrt{B + 1} \log\left(\frac{\frac{\sqrt{B + 1} r}{\sqrt{m}} - 2}{\frac{\sqrt{B + 1} r}{\sqrt{m}} + 2}\right)}{4 \, B \sqrt{m}}

Let us check that we have indeed a primitive of r12r4+r22mr\mapsto \frac{1}{\ell^2 r^4 + r^2 - 2m}:

Ds2 = diff(s2, r).simplify_full() Ds2
8m(B21)r4+8mr216m2\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{8 \, m}{{\left(B^{2} - 1\right)} r^{4} + 8 \, m r^{2} - 16 \, m^{2}}
Ds2.subs({B: sqrt(1 + 8*l^2*m)}).simplify_full()
12r4+r22m\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{{\ell}^{2} r^{4} + r^{2} - 2 \, m}

Given the above expressions for s1(r)s_1(r) and s2(r)s_2(r) we rewrite the solution

mu1_sol(r)
K2((μ03μ0)β12(μ03μ0)β22+2(μ03μ0)β12(μ03μ0)β2)2r(μ03μ0)(β12β22+2β12β2)2r2+P2β12Q2β222(β12β22)2m2r4+r22mdrK12r4+r22mdr\renewcommand{\Bold}[1]{\mathbf{#1}}K_{2} - \int \frac{{\left({\left({\mu_0}^{3} - {\mu_0}\right)} {\beta_1}^{2} - {\left({\mu_0}^{3} - {\mu_0}\right)} {\beta_2}^{2} + 2 \, {\left({\mu_0}^{3} - {\mu_0}\right)} {\beta_1} - 2 \, {\left({\mu_0}^{3} - {\mu_0}\right)} {\beta_2}\right)} {\ell}^{2} r - {\left({\mu_0}^{3} - {\mu_0}\right)} \int \frac{{\left({\beta_1}^{2} - {\beta_2}^{2} + 2 \, {\beta_1} - 2 \, {\beta_2}\right)} {\ell}^{2} r^{2} + {\mathcal{P}'}^{2} {\beta_1}^{2} - {\mathcal{Q}'}^{2} {\beta_2}^{2} - 2 \, {\left({\beta_1}^{2} - {\beta_2}^{2}\right)} {\ell}^{2} m}{{\ell}^{2} r^{4} + r^{2} - 2 \, m}\,{d r} - K_{1}}{{\ell}^{2} r^{4} + r^{2} - 2 \, m}\,{d r}

as

Instead of K1K_1 and K2K_2, let us introduce the constants C1C_1 and C2C_2:

C1, C2 = var('C_1', 'C_2')
# mu1 / mu0(1-mu0^2) : mu1s0 = -C2/(Mu0*sqrt(1-Mu0^2)) - C1/(Mu0*sqrt(1-Mu0^2))*s2 \ + integrate(((beta1^2 - beta2^2 + 2*(beta1 - beta2))*l^2*r - (beta1^2 - beta2^2 +2*(beta1-beta2))*l^2 * s1 - (P^2*beta1^2 - Q^2*beta2^2 - 2*(beta1^2 - beta2^2)*l^2*m) * s2) / (l^2*r^4 + r^2 - 2*m), r, hold=True) mu1s0
C2μ02+1μ0+(2B1arctan(B1r2m)B+1log(B+1rm2B+1rm+2))C14μ02+1Bμ0m+4(β12β22+2β12β2)2r2(β12β22+2β12β2)(2B+1arctan(2rB+1)+B1log(2rB112rB1+1))B+(P2β12Q2β222(β12β22)2m)(2B1arctan(B1r2m)B+1log(B+1rm2B+1rm+2))Bm4(2r4+r22m)dr\renewcommand{\Bold}[1]{\mathbf{#1}}-\frac{C_{2}}{\sqrt{-{\mu_0}^{2} + 1} {\mu_0}} + \frac{{\left(2 \, \sqrt{B - 1} \arctan\left(\frac{\sqrt{B - 1} r}{2 \, \sqrt{m}}\right) - \sqrt{B + 1} \log\left(\frac{\frac{\sqrt{B + 1} r}{\sqrt{m}} - 2}{\frac{\sqrt{B + 1} r}{\sqrt{m}} + 2}\right)\right)} C_{1}}{4 \, \sqrt{-{\mu_0}^{2} + 1} B {\mu_0} \sqrt{m}} + \int \frac{4 \, {\left({\beta_1}^{2} - {\beta_2}^{2} + 2 \, {\beta_1} - 2 \, {\beta_2}\right)} {\ell}^{2} r - \frac{\sqrt{2} {\left({\beta_1}^{2} - {\beta_2}^{2} + 2 \, {\beta_1} - 2 \, {\beta_2}\right)} {\left(2 \, \sqrt{B + 1} \arctan\left(\frac{\sqrt{2} {\ell} r}{\sqrt{B + 1}}\right) + \sqrt{B - 1} \log\left(\frac{\frac{\sqrt{2} {\ell} r}{\sqrt{B - 1}} - 1}{\frac{\sqrt{2} {\ell} r}{\sqrt{B - 1}} + 1}\right)\right)} {\ell}}{B} + \frac{{\left({\mathcal{P}'}^{2} {\beta_1}^{2} - {\mathcal{Q}'}^{2} {\beta_2}^{2} - 2 \, {\left({\beta_1}^{2} - {\beta_2}^{2}\right)} {\ell}^{2} m\right)} {\left(2 \, \sqrt{B - 1} \arctan\left(\frac{\sqrt{B - 1} r}{2 \, \sqrt{m}}\right) - \sqrt{B + 1} \log\left(\frac{\frac{\sqrt{B + 1} r}{\sqrt{m}} - 2}{\frac{\sqrt{B + 1} r}{\sqrt{m}} + 2}\right)\right)}}{B \sqrt{m}}}{4 \, {\left({\ell}^{2} r^{4} + r^{2} - 2 \, m\right)}}\,{d r}
mu1_sol(r) = mu1s0 * Mu0*(1-Mu0^2) mu1_sol(r)
14(μ021)μ0(4C2μ02+1μ0(2B1arctan(B1r2m)B+1log(B+1rm2B+1rm+2))C1μ02+1Bμ0m44(β12β22+2β12β2)2r2(β12β22+2β12β2)(2B+1arctan(2rB+1)+B1log(2rB112rB1+1))B+(P2β12Q2β222(β12β22)2m)(2B1arctan(B1r2m)B+1log(B+1rm2B+1rm+2))Bm4(2r4+r22m)dr)\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{4} \, {\left({\mu_0}^{2} - 1\right)} {\mu_0} {\left(\frac{4 \, C_{2}}{\sqrt{-{\mu_0}^{2} + 1} {\mu_0}} - \frac{{\left(2 \, \sqrt{B - 1} \arctan\left(\frac{\sqrt{B - 1} r}{2 \, \sqrt{m}}\right) - \sqrt{B + 1} \log\left(\frac{\frac{\sqrt{B + 1} r}{\sqrt{m}} - 2}{\frac{\sqrt{B + 1} r}{\sqrt{m}} + 2}\right)\right)} C_{1}}{\sqrt{-{\mu_0}^{2} + 1} B {\mu_0} \sqrt{m}} - 4 \, \int \frac{4 \, {\left({\beta_1}^{2} - {\beta_2}^{2} + 2 \, {\beta_1} - 2 \, {\beta_2}\right)} {\ell}^{2} r - \frac{\sqrt{2} {\left({\beta_1}^{2} - {\beta_2}^{2} + 2 \, {\beta_1} - 2 \, {\beta_2}\right)} {\left(2 \, \sqrt{B + 1} \arctan\left(\frac{\sqrt{2} {\ell} r}{\sqrt{B + 1}}\right) + \sqrt{B - 1} \log\left(\frac{\frac{\sqrt{2} {\ell} r}{\sqrt{B - 1}} - 1}{\frac{\sqrt{2} {\ell} r}{\sqrt{B - 1}} + 1}\right)\right)} {\ell}}{B} + \frac{{\left({\mathcal{P}'}^{2} {\beta_1}^{2} - {\mathcal{Q}'}^{2} {\beta_2}^{2} - 2 \, {\left({\beta_1}^{2} - {\beta_2}^{2}\right)} {\ell}^{2} m\right)} {\left(2 \, \sqrt{B - 1} \arctan\left(\frac{\sqrt{B - 1} r}{2 \, \sqrt{m}}\right) - \sqrt{B + 1} \log\left(\frac{\frac{\sqrt{B + 1} r}{\sqrt{m}} - 2}{\frac{\sqrt{B + 1} r}{\sqrt{m}} + 2}\right)\right)}}{B \sqrt{m}}}{4 \, {\left({\ell}^{2} r^{4} + r^{2} - 2 \, m\right)}}\,{d r}\right)}

Let us check that we do have a solution of the equation for μ1\mu_1:

eq_mu1.substitute_function(mu_1, mu1_sol).simplify_full().subs({B: sqrt(1 + 8*l^2*m)}).simplify_full()
0=0\renewcommand{\Bold}[1]{\mathbf{#1}}0 = 0

Conjugate momenta

def conjugate_momenta(lagr, qs, var): r""" Compute the conjugate momenta from a given Lagrangian. INPUT: - ``lagr`` -- symbolic expression representing the Lagrangian density - ``qs`` -- either a single symbolic function or a list/tuple of symbolic functions, representing the `q`'s; these functions must appear in ``lagr`` up to at most their first derivatives - ``var`` -- either a single variable, typically `t` (1-dimensional problem) or a list/tuple of symbolic variables; in the latter case the time coordinate must the first one OUTPUT: - list of conjugate momenta; if only one function is involved, the single conjugate momentum is returned instead. """ if not isinstance(qs, (list, tuple)): qs = [qs] if not isinstance(var, (list, tuple)): var = [var] n = len(qs) d = len(var) dqvt = [SR.var('qxxxx{}_t'.format(q)) for q in qs] subs = {diff(qs[i](*var), var[0]): dqvt[i] for i in range(n)} subs_inv = {dqvt[i]: diff(qs[i](*var), var[0]) for i in range(n)} lg = lagr.substitute(subs) ps = [diff(lg, dotq).simplify_full().substitute(subs_inv) for dotq in dqvt] if n == 1: return ps[0] return ps
pis = conjugate_momenta(L_a2, [phi_1, psi_1], r) pis
[(μ021)a2β122r4rϕ1(r)(μ021)a2β12r2rϕ1(r)+2(μ021)a2β12mrϕ1(r),μ02a2β222r4rψ1(r)+μ02a2β22r2rψ1(r)2μ02a2β22mrψ1(r)]\renewcommand{\Bold}[1]{\mathbf{#1}}\left[-{\left({\mu_0}^{2} - 1\right)} a^{2} {\beta_1}^{2} {\ell}^{2} r^{4} \frac{\partial}{\partial r}\phi_{1}\left(r\right) - {\left({\mu_0}^{2} - 1\right)} a^{2} {\beta_1}^{2} r^{2} \frac{\partial}{\partial r}\phi_{1}\left(r\right) + 2 \, {\left({\mu_0}^{2} - 1\right)} a^{2} {\beta_1}^{2} m \frac{\partial}{\partial r}\phi_{1}\left(r\right), {\mu_0}^{2} a^{2} {\beta_2}^{2} {\ell}^{2} r^{4} \frac{\partial}{\partial r}\psi_{1}\left(r\right) + {\mu_0}^{2} a^{2} {\beta_2}^{2} r^{2} \frac{\partial}{\partial r}\psi_{1}\left(r\right) - 2 \, {\mu_0}^{2} a^{2} {\beta_2}^{2} m \frac{\partial}{\partial r}\psi_{1}\left(r\right)\right]

Check of Eq. (4.14):

pi_phi_r = (pis[0]/(a*beta1)).substitute_function(phi_1, phi1_sol).simplify_full() pi_phi_r
(μ021)Paβ1\renewcommand{\Bold}[1]{\mathbf{#1}}-{\left({\mu_0}^{2} - 1\right)} {\mathcal{P}'} a {\beta_1}

Check of Eq. (4.15):

pi_psi_r = (pis[1]/(a*beta2)).substitute_function(psi_1, psi1_sol).simplify_full() pi_psi_r
μ02Qaβ2\renewcommand{\Bold}[1]{\mathbf{#1}}{\mu_0}^{2} {\mathcal{Q}'} a {\beta_2}

Check of Eq. (4.13):

pis4 = conjugate_momenta(L_a4, [phi_1, psi_1, mu_1], r)
pis4[2]
a42r4rμ1(r)+a4r2rμ1(r)2a4mrμ1(r)μ021\renewcommand{\Bold}[1]{\mathbf{#1}}-\frac{a^{4} {\ell}^{2} r^{4} \frac{\partial}{\partial r}\mu_{1}\left(r\right) + a^{4} r^{2} \frac{\partial}{\partial r}\mu_{1}\left(r\right) - 2 \, a^{4} m \frac{\partial}{\partial r}\mu_{1}\left(r\right)}{{\mu_0}^{2} - 1}

The quantity πθr/(a2sinΘ0cosΘ0)\pi_\theta^r / (a^2 \sin\Theta_0\cos\Theta_0):

pi_theta_r_a2sT0 = (- pis4[2] / (a^4*Mu0)).substitute_function(mu_1, mu1_sol).simplify_full() pi_theta_r_a2sT0 = pi_theta_r_a2sT0.subs({B: sqrt(1 + 8*l^2*m)}).simplify_full() pi_theta_r_a2sT0
(2(2μ0β122μ0β22+22μ0β122μ0β2)marctan(2r82m+1+1)+(μ0P2β12μ0Q2β222(μ0β12μ0β22)2m)log(mr82m+1+12mmr82m+1+1+2m))μ02+182m+1+1+((2μ0β122μ0β22+22μ0β122μ0β2)mlog(2r82m+1182m+1+12r82m+11+82m+11)2(μ0P2β12μ0Q2β222(μ0β12μ0β22)2m)arctan(r82m+112m))μ02+182m+114((μ0β12μ0β22+2μ0β12μ0β2)μ02+12mrC1m)82m+1482m+1μ02+1μ0m\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{{\left(2 \, {\left(\sqrt{2} {\mu_0} {\beta_1}^{2} - \sqrt{2} {\mu_0} {\beta_2}^{2} + 2 \, \sqrt{2} {\mu_0} {\beta_1} - 2 \, \sqrt{2} {\mu_0} {\beta_2}\right)} {\ell} \sqrt{m} \arctan\left(\frac{\sqrt{2} {\ell} r}{\sqrt{\sqrt{8 \, {\ell}^{2} m + 1} + 1}}\right) + {\left({\mu_0} {\mathcal{P}'}^{2} {\beta_1}^{2} - {\mu_0} {\mathcal{Q}'}^{2} {\beta_2}^{2} - 2 \, {\left({\mu_0} {\beta_1}^{2} - {\mu_0} {\beta_2}^{2}\right)} {\ell}^{2} m\right)} \log\left(\frac{\sqrt{m} r \sqrt{\sqrt{8 \, {\ell}^{2} m + 1} + 1} - 2 \, m}{\sqrt{m} r \sqrt{\sqrt{8 \, {\ell}^{2} m + 1} + 1} + 2 \, m}\right)\right)} \sqrt{-{\mu_0}^{2} + 1} \sqrt{\sqrt{8 \, {\ell}^{2} m + 1} + 1} + {\left({\left(\sqrt{2} {\mu_0} {\beta_1}^{2} - \sqrt{2} {\mu_0} {\beta_2}^{2} + 2 \, \sqrt{2} {\mu_0} {\beta_1} - 2 \, \sqrt{2} {\mu_0} {\beta_2}\right)} {\ell} \sqrt{m} \log\left(\frac{\sqrt{2} {\ell} r \sqrt{\sqrt{8 \, {\ell}^{2} m + 1} - 1} - \sqrt{8 \, {\ell}^{2} m + 1} + 1}{\sqrt{2} {\ell} r \sqrt{\sqrt{8 \, {\ell}^{2} m + 1} - 1} + \sqrt{8 \, {\ell}^{2} m + 1} - 1}\right) - 2 \, {\left({\mu_0} {\mathcal{P}'}^{2} {\beta_1}^{2} - {\mu_0} {\mathcal{Q}'}^{2} {\beta_2}^{2} - 2 \, {\left({\mu_0} {\beta_1}^{2} - {\mu_0} {\beta_2}^{2}\right)} {\ell}^{2} m\right)} \arctan\left(\frac{r \sqrt{\sqrt{8 \, {\ell}^{2} m + 1} - 1}}{2 \, \sqrt{m}}\right)\right)} \sqrt{-{\mu_0}^{2} + 1} \sqrt{\sqrt{8 \, {\ell}^{2} m + 1} - 1} - 4 \, {\left({\left({\mu_0} {\beta_1}^{2} - {\mu_0} {\beta_2}^{2} + 2 \, {\mu_0} {\beta_1} - 2 \, {\mu_0} {\beta_2}\right)} \sqrt{-{\mu_0}^{2} + 1} {\ell}^{2} \sqrt{m} r - C_{1} \sqrt{m}\right)} \sqrt{8 \, {\ell}^{2} m + 1}}{4 \, \sqrt{8 \, {\ell}^{2} m + 1} \sqrt{-{\mu_0}^{2} + 1} {\mu_0} \sqrt{m}}

The quantity πθr(a2/2)sin2Θ0+(β12β22+2(β1β2))2rC1sinΘ0cosΘ0\frac{\pi_\theta^r}{(a^2/2) \sin 2\Theta_0} + (\beta_1^2 - \beta_2^2 + 2 (\beta_1-\beta_2))\ell^2 r - \frac{C_1}{\sin\Theta_0\cos\Theta_0}

part1 = - (beta1^2 - beta2^2 + 2*(beta1-beta2))*l^2*r + C1/(Mu0*sqrt(1-Mu0^2)) s = (pi_theta_r_a2sT0 - part1).simplify_full() s
(2(2β122β22+22β122β2)marctan(2r82m+1+1)+(P2β12Q2β222(β12β22)2m)log(mr82m+1+12mmr82m+1+1+2m))82m+1+1+((2β122β22+22β122β2)mlog(2r82m+1182m+1+12r82m+11+82m+11)2(P2β12Q2β222(β12β22)2m)arctan(r82m+112m))82m+11482m+1m\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{{\left(2 \, {\left(\sqrt{2} {\beta_1}^{2} - \sqrt{2} {\beta_2}^{2} + 2 \, \sqrt{2} {\beta_1} - 2 \, \sqrt{2} {\beta_2}\right)} {\ell} \sqrt{m} \arctan\left(\frac{\sqrt{2} {\ell} r}{\sqrt{\sqrt{8 \, {\ell}^{2} m + 1} + 1}}\right) + {\left({\mathcal{P}'}^{2} {\beta_1}^{2} - {\mathcal{Q}'}^{2} {\beta_2}^{2} - 2 \, {\left({\beta_1}^{2} - {\beta_2}^{2}\right)} {\ell}^{2} m\right)} \log\left(\frac{\sqrt{m} r \sqrt{\sqrt{8 \, {\ell}^{2} m + 1} + 1} - 2 \, m}{\sqrt{m} r \sqrt{\sqrt{8 \, {\ell}^{2} m + 1} + 1} + 2 \, m}\right)\right)} \sqrt{\sqrt{8 \, {\ell}^{2} m + 1} + 1} + {\left({\left(\sqrt{2} {\beta_1}^{2} - \sqrt{2} {\beta_2}^{2} + 2 \, \sqrt{2} {\beta_1} - 2 \, \sqrt{2} {\beta_2}\right)} {\ell} \sqrt{m} \log\left(\frac{\sqrt{2} {\ell} r \sqrt{\sqrt{8 \, {\ell}^{2} m + 1} - 1} - \sqrt{8 \, {\ell}^{2} m + 1} + 1}{\sqrt{2} {\ell} r \sqrt{\sqrt{8 \, {\ell}^{2} m + 1} - 1} + \sqrt{8 \, {\ell}^{2} m + 1} - 1}\right) - 2 \, {\left({\mathcal{P}'}^{2} {\beta_1}^{2} - {\mathcal{Q}'}^{2} {\beta_2}^{2} - 2 \, {\left({\beta_1}^{2} - {\beta_2}^{2}\right)} {\ell}^{2} m\right)} \arctan\left(\frac{r \sqrt{\sqrt{8 \, {\ell}^{2} m + 1} - 1}}{2 \, \sqrt{m}}\right)\right)} \sqrt{\sqrt{8 \, {\ell}^{2} m + 1} - 1}}{4 \, \sqrt{8 \, {\ell}^{2} m + 1} \sqrt{m}}

Let us perform an expansion in 1/r1/r for r+r\rightarrow +\infty:

u = var('u') assume(u > 0) s = s.subs({r: 1/u}).simplify_log() assume(l>0) s = s.taylor(u, 0, 2) s = s.subs({u: 1/r}) s
(2πβ122πβ22+22πβ122πβ2)m82m+1+1(πP2β12πQ2β222(πβ12πβ22)2m)82m+11482m+1mβ12β22+2β12β2r\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{{\left(\sqrt{2} \pi {\beta_1}^{2} - \sqrt{2} \pi {\beta_2}^{2} + 2 \, \sqrt{2} \pi {\beta_1} - 2 \, \sqrt{2} \pi {\beta_2}\right)} {\ell} \sqrt{m} \sqrt{\sqrt{8 \, {\ell}^{2} m + 1} + 1} - {\left(\pi {\mathcal{P}'}^{2} {\beta_1}^{2} - \pi {\mathcal{Q}'}^{2} {\beta_2}^{2} - 2 \, {\left(\pi {\beta_1}^{2} - \pi {\beta_2}^{2}\right)} {\ell}^{2} m\right)} \sqrt{\sqrt{8 \, {\ell}^{2} m + 1} - 1}}{4 \, \sqrt{8 \, {\ell}^{2} m + 1} \sqrt{m}} - \frac{{\beta_1}^{2} - {\beta_2}^{2} + 2 \, {\beta_1} - 2 \, {\beta_2}}{r}
s
(2πβ122πβ22+22πβ122πβ2)m82m+1+1(πP2β12πQ2β222(πβ12πβ22)2m)82m+11482m+1mβ12β22+2β12β2r\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{{\left(\sqrt{2} \pi {\beta_1}^{2} - \sqrt{2} \pi {\beta_2}^{2} + 2 \, \sqrt{2} \pi {\beta_1} - 2 \, \sqrt{2} \pi {\beta_2}\right)} {\ell} \sqrt{m} \sqrt{\sqrt{8 \, {\ell}^{2} m + 1} + 1} - {\left(\pi {\mathcal{P}'}^{2} {\beta_1}^{2} - \pi {\mathcal{Q}'}^{2} {\beta_2}^{2} - 2 \, {\left(\pi {\beta_1}^{2} - \pi {\beta_2}^{2}\right)} {\ell}^{2} m\right)} \sqrt{\sqrt{8 \, {\ell}^{2} m + 1} - 1}}{4 \, \sqrt{8 \, {\ell}^{2} m + 1} \sqrt{m}} - \frac{{\beta_1}^{2} - {\beta_2}^{2} + 2 \, {\beta_1} - 2 \, {\beta_2}}{r}

Final result for πθr(a2/2)sin2Θ0\frac{\pi_\theta^r}{(a^2/2) \sin 2\Theta_0}:

part1 + s
(β12β22+2β12β2)2r+(2πβ122πβ22+22πβ122πβ2)m82m+1+1(πP2β12πQ2β222(πβ12πβ22)2m)82m+11482m+1mβ12β22+2β12β2r+C1μ02+1μ0\renewcommand{\Bold}[1]{\mathbf{#1}}-{\left({\beta_1}^{2} - {\beta_2}^{2} + 2 \, {\beta_1} - 2 \, {\beta_2}\right)} {\ell}^{2} r + \frac{{\left(\sqrt{2} \pi {\beta_1}^{2} - \sqrt{2} \pi {\beta_2}^{2} + 2 \, \sqrt{2} \pi {\beta_1} - 2 \, \sqrt{2} \pi {\beta_2}\right)} {\ell} \sqrt{m} \sqrt{\sqrt{8 \, {\ell}^{2} m + 1} + 1} - {\left(\pi {\mathcal{P}'}^{2} {\beta_1}^{2} - \pi {\mathcal{Q}'}^{2} {\beta_2}^{2} - 2 \, {\left(\pi {\beta_1}^{2} - \pi {\beta_2}^{2}\right)} {\ell}^{2} m\right)} \sqrt{\sqrt{8 \, {\ell}^{2} m + 1} - 1}}{4 \, \sqrt{8 \, {\ell}^{2} m + 1} \sqrt{m}} - \frac{{\beta_1}^{2} - {\beta_2}^{2} + 2 \, {\beta_1} - 2 \, {\beta_2}}{r} + \frac{C_{1}}{\sqrt{-{\mu_0}^{2} + 1} {\mu_0}}

The terms in rr, 1/r1/r and C1C_1 agree with Eq. (4.13).