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5D Kerr-AdS spacetime with a Nambu-Goto string (case b=0)

Project: KerrAdS
Views: 478
Kernel: SageMath 9.3

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

Case b = 0

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=(a42μ42r4(a42+a2)μ2(a22+1)r2+2ma2μ2+r2)dtdt+(a52μ4+(a2μ2a2)r4(a52+2am)μ2+(a32μ4a32)r2+2am(a42a2)μ2+(a221)r2)dtdϕ+(a2μ2+r22r4+(a22+1)r2+a22m)drdr+(a2μ2+r2a22μ4(a22+1)μ2+1)dμdμ+(a52μ4+(a2μ2a2)r4(a52+2am)μ2+(a32μ4a32)r2+2am(a42a2)μ2+(a221)r2)dϕdt+((a62a4+2a2m)μ4(a22(a221)μ21)r4+2a2m(a62a4+4a2m)μ2(a42(a42a2)μ4a2)r2(a642a42+a2)μ2+(a442a22+1)r2)dϕdϕ+μ2r2dψdψ\renewcommand{\Bold}[1]{\mathbf{#1}}G = \left( \frac{a^{4} {\ell}^{2} {\mu}^{4} - {\ell}^{2} r^{4} - {\left(a^{4} {\ell}^{2} + a^{2}\right)} {\mu}^{2} - {\left(a^{2} {\ell}^{2} + 1\right)} r^{2} + 2 \, m}{a^{2} {\mu}^{2} + r^{2}} \right) \mathrm{d} t\otimes \mathrm{d} t + \left( \frac{a^{5} {\ell}^{2} {\mu}^{4} + {\left(a {\ell}^{2} {\mu}^{2} - a {\ell}^{2}\right)} r^{4} - {\left(a^{5} {\ell}^{2} + 2 \, a m\right)} {\mu}^{2} + {\left(a^{3} {\ell}^{2} {\mu}^{4} - a^{3} {\ell}^{2}\right)} r^{2} + 2 \, a m}{{\left(a^{4} {\ell}^{2} - a^{2}\right)} {\mu}^{2} + {\left(a^{2} {\ell}^{2} - 1\right)} r^{2}} \right) \mathrm{d} t\otimes \mathrm{d} {\phi} + \left( \frac{a^{2} {\mu}^{2} + r^{2}}{{\ell}^{2} r^{4} + {\left(a^{2} {\ell}^{2} + 1\right)} r^{2} + a^{2} - 2 \, m} \right) \mathrm{d} r\otimes \mathrm{d} r + \left( \frac{a^{2} {\mu}^{2} + r^{2}}{a^{2} {\ell}^{2} {\mu}^{4} - {\left(a^{2} {\ell}^{2} + 1\right)} {\mu}^{2} + 1} \right) \mathrm{d} {\mu}\otimes \mathrm{d} {\mu} + \left( \frac{a^{5} {\ell}^{2} {\mu}^{4} + {\left(a {\ell}^{2} {\mu}^{2} - a {\ell}^{2}\right)} r^{4} - {\left(a^{5} {\ell}^{2} + 2 \, a m\right)} {\mu}^{2} + {\left(a^{3} {\ell}^{2} {\mu}^{4} - a^{3} {\ell}^{2}\right)} r^{2} + 2 \, a m}{{\left(a^{4} {\ell}^{2} - a^{2}\right)} {\mu}^{2} + {\left(a^{2} {\ell}^{2} - 1\right)} r^{2}} \right) \mathrm{d} {\phi}\otimes \mathrm{d} t + \left( \frac{{\left(a^{6} {\ell}^{2} - a^{4} + 2 \, a^{2} m\right)} {\mu}^{4} - {\left(a^{2} {\ell}^{2} - {\left(a^{2} {\ell}^{2} - 1\right)} {\mu}^{2} - 1\right)} r^{4} + 2 \, a^{2} m - {\left(a^{6} {\ell}^{2} - a^{4} + 4 \, a^{2} m\right)} {\mu}^{2} - {\left(a^{4} {\ell}^{2} - {\left(a^{4} {\ell}^{2} - a^{2}\right)} {\mu}^{4} - a^{2}\right)} r^{2}}{{\left(a^{6} {\ell}^{4} - 2 \, a^{4} {\ell}^{2} + a^{2}\right)} {\mu}^{2} + {\left(a^{4} {\ell}^{4} - 2 \, a^{2} {\ell}^{2} + 1\right)} r^{2}} \right) \mathrm{d} {\phi}\otimes \mathrm{d} {\phi} + {\mu}^{2} r^{2} \mathrm{d} {\psi}\otimes \mathrm{d} {\psi}
G.display_comp(only_nonredundant=True)
Gtttt=a42μ42r4(a42+a2)μ2(a22+1)r2+2ma2μ2+r2Gtϕtϕ=a52μ4+(a2μ2a2)r4(a52+2am)μ2+(a32μ4a32)r2+2am(a42a2)μ2+(a221)r2Grrrr=a2μ2+r22r4+(a22+1)r2+a22mGμμμμ=a2μ2+r2a22μ4(a22+1)μ2+1Gϕϕϕϕ=(a62a4+2a2m)μ4(a22(a221)μ21)r4+2a2m(a62a4+4a2m)μ2(a42(a42a2)μ4a2)r2(a642a42+a2)μ2+(a442a22+1)r2Gψψψψ=μ2r2\renewcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{lcl} G_{ \, t \, t }^{ \phantom{\, t}\phantom{\, t} } & = & \frac{a^{4} {\ell}^{2} {\mu}^{4} - {\ell}^{2} r^{4} - {\left(a^{4} {\ell}^{2} + a^{2}\right)} {\mu}^{2} - {\left(a^{2} {\ell}^{2} + 1\right)} r^{2} + 2 \, m}{a^{2} {\mu}^{2} + r^{2}} \\ G_{ \, t \, {\phi} }^{ \phantom{\, t}\phantom{\, {\phi}} } & = & \frac{a^{5} {\ell}^{2} {\mu}^{4} + {\left(a {\ell}^{2} {\mu}^{2} - a {\ell}^{2}\right)} r^{4} - {\left(a^{5} {\ell}^{2} + 2 \, a m\right)} {\mu}^{2} + {\left(a^{3} {\ell}^{2} {\mu}^{4} - a^{3} {\ell}^{2}\right)} r^{2} + 2 \, a m}{{\left(a^{4} {\ell}^{2} - a^{2}\right)} {\mu}^{2} + {\left(a^{2} {\ell}^{2} - 1\right)} r^{2}} \\ G_{ \, r \, r }^{ \phantom{\, r}\phantom{\, r} } & = & \frac{a^{2} {\mu}^{2} + r^{2}}{{\ell}^{2} r^{4} + {\left(a^{2} {\ell}^{2} + 1\right)} r^{2} + a^{2} - 2 \, m} \\ G_{ \, {\mu} \, {\mu} }^{ \phantom{\, {\mu}}\phantom{\, {\mu}} } & = & \frac{a^{2} {\mu}^{2} + r^{2}}{a^{2} {\ell}^{2} {\mu}^{4} - {\left(a^{2} {\ell}^{2} + 1\right)} {\mu}^{2} + 1} \\ G_{ \, {\phi} \, {\phi} }^{ \phantom{\, {\phi}}\phantom{\, {\phi}} } & = & \frac{{\left(a^{6} {\ell}^{2} - a^{4} + 2 \, a^{2} m\right)} {\mu}^{4} - {\left(a^{2} {\ell}^{2} - {\left(a^{2} {\ell}^{2} - 1\right)} {\mu}^{2} - 1\right)} r^{4} + 2 \, a^{2} m - {\left(a^{6} {\ell}^{2} - a^{4} + 4 \, a^{2} m\right)} {\mu}^{2} - {\left(a^{4} {\ell}^{2} - {\left(a^{4} {\ell}^{2} - a^{2}\right)} {\mu}^{4} - a^{2}\right)} r^{2}}{{\left(a^{6} {\ell}^{4} - 2 \, a^{4} {\ell}^{2} + a^{2}\right)} {\mu}^{2} + {\left(a^{4} {\ell}^{4} - 2 \, a^{2} {\ell}^{2} + 1\right)} r^{2}} \\ G_{ \, {\psi} \, {\psi} }^{ \phantom{\, {\psi}}\phantom{\, {\psi}} } & = & {\mu}^{2} r^{2} \end{array}

Check of agreement with Eq. (5.22) of Hawking et al or Eq. (2.3) of 5dKerr-AdS-AGG-5-12

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\renewcommand{\Bold}[1]{\mathbf{#1}}\mathrm{d} t + \left( -\frac{a {\mu}^{2} - a}{a^{2} {\ell}^{2} - 1} \right) \mathrm{d} {\phi}
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()
r2dψ\renewcommand{\Bold}[1]{\mathbf{#1}}-r^{2} \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()
aμ2r2dψ\renewcommand{\Bold}[1]{\mathbf{#1}}-a {\mu}^{2} r^{2} \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=a42μ42r4(a42+a2)μ2(a22+1)r2+2ma2μ2+r2Xtϕtϕ=a52μ4+(a2μ2a2)r4(a52+2am)μ2+(a32μ4a32)r2+2am(a42a2)μ2+(a221)r2Xrrrr=a2μ2+r2(2r2+1)(a2+r2)2mXμμμμ=a2μ2+r2a22μ4(a22+1)μ2+1Xϕtϕt=a52μ4+(a2μ2a2)r4(a52+2am)μ2+(a32μ4a32)r2+2am(a42a2)μ2+(a221)r2Xϕϕϕϕ=(a62a4+2a2m)μ4(a22(a221)μ21)r4+2a2m(a62a4+4a2m)μ2(a42(a42a2)μ4a2)r2(a642a42+a2)μ2+(a442a22+1)r2Xψψψψ=μ2r2\renewcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{lcl} X_{ \, t \, t }^{ \phantom{\, t}\phantom{\, t} } & = & \frac{a^{4} {\ell}^{2} {\mu}^{4} - {\ell}^{2} r^{4} - {\left(a^{4} {\ell}^{2} + a^{2}\right)} {\mu}^{2} - {\left(a^{2} {\ell}^{2} + 1\right)} r^{2} + 2 \, m}{a^{2} {\mu}^{2} + r^{2}} \\ X_{ \, t \, {\phi} }^{ \phantom{\, t}\phantom{\, {\phi}} } & = & \frac{a^{5} {\ell}^{2} {\mu}^{4} + {\left(a {\ell}^{2} {\mu}^{2} - a {\ell}^{2}\right)} r^{4} - {\left(a^{5} {\ell}^{2} + 2 \, a m\right)} {\mu}^{2} + {\left(a^{3} {\ell}^{2} {\mu}^{4} - a^{3} {\ell}^{2}\right)} r^{2} + 2 \, a m}{{\left(a^{4} {\ell}^{2} - a^{2}\right)} {\mu}^{2} + {\left(a^{2} {\ell}^{2} - 1\right)} r^{2}} \\ X_{ \, r \, r }^{ \phantom{\, r}\phantom{\, r} } & = & \frac{a^{2} {\mu}^{2} + r^{2}}{{\left({\ell}^{2} r^{2} + 1\right)} {\left(a^{2} + r^{2}\right)} - 2 \, m} \\ X_{ \, {\mu} \, {\mu} }^{ \phantom{\, {\mu}}\phantom{\, {\mu}} } & = & \frac{a^{2} {\mu}^{2} + r^{2}}{a^{2} {\ell}^{2} {\mu}^{4} - {\left(a^{2} {\ell}^{2} + 1\right)} {\mu}^{2} + 1} \\ X_{ \, {\phi} \, t }^{ \phantom{\, {\phi}}\phantom{\, t} } & = & \frac{a^{5} {\ell}^{2} {\mu}^{4} + {\left(a {\ell}^{2} {\mu}^{2} - a {\ell}^{2}\right)} r^{4} - {\left(a^{5} {\ell}^{2} + 2 \, a m\right)} {\mu}^{2} + {\left(a^{3} {\ell}^{2} {\mu}^{4} - a^{3} {\ell}^{2}\right)} r^{2} + 2 \, a m}{{\left(a^{4} {\ell}^{2} - a^{2}\right)} {\mu}^{2} + {\left(a^{2} {\ell}^{2} - 1\right)} r^{2}} \\ X_{ \, {\phi} \, {\phi} }^{ \phantom{\, {\phi}}\phantom{\, {\phi}} } & = & \frac{{\left(a^{6} {\ell}^{2} - a^{4} + 2 \, a^{2} m\right)} {\mu}^{4} - {\left(a^{2} {\ell}^{2} - {\left(a^{2} {\ell}^{2} - 1\right)} {\mu}^{2} - 1\right)} r^{4} + 2 \, a^{2} m - {\left(a^{6} {\ell}^{2} - a^{4} + 4 \, a^{2} m\right)} {\mu}^{2} - {\left(a^{4} {\ell}^{2} - {\left(a^{4} {\ell}^{2} - a^{2}\right)} {\mu}^{4} - a^{2}\right)} r^{2}}{{\left(a^{6} {\ell}^{4} - 2 \, a^{4} {\ell}^{2} + a^{2}\right)} {\mu}^{2} + {\left(a^{4} {\ell}^{4} - 2 \, a^{2} {\ell}^{2} + 1\right)} r^{2}} \\ X_{ \, {\psi} \, {\psi} }^{ \phantom{\, {\psi}}\phantom{\, {\psi}} } & = & {\mu}^{2} r^{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)

Check of Eq. (2.14)

One must have b=0b=0 and keep_Delta == False for the test to pass:

if b == 0 and not keep_Delta: s1 = dt - a*sinth2/Xi_a*dph s2 = a*dt - (r^2 + a^2)/Xi_a*dph G2 = - Delta_r/rho2* s1*s1 + rho2/Delta_r * dr*dr + rho2/Delta_th * dth*dth \ + Delta_th*sinth2/rho2* s2*s2 + r^2*costh2 * dps*dps print(G2 == 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:

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]
(μ04+μ02)a42+((μ04μ02)a8β12+2(μ04μ02)a8β1+(μ04μ02)a8)6((μ04μ02)a6β12+2(μ04μ02)a6β1+(2μ04μ02)a6)4+(μ02a6β228(2μ02a4β22(μ021)a4β122(μ021)a4β1+a4)6+(μ02a2β22(μ021)a2β122(μ021)a2β1+2a2)42)r4+(2a12β124m+a122+(a16β12+2a16β1+a16)6(a14β12+2a14β1+2a14)4+(a16β228+(a14β122a14β22+2a14β1)6(a12β12a12β22+2a12β1)4)r2)μ1(r)4μ02a2+4(2μ0a10β124m+μ0a102+(μ0a14β12+2μ0a14β1+μ0a14)6(μ0a12β12+2μ0a12β1+2μ0a12)4+(μ0a14β228+(μ0a12β122μ0a12β22+2μ0a12β1)6(μ0a10β12μ0a10β22+2μ0a10β1)4)r2)μ1(r)3+(μ04a8β228(2μ04a6β22(μ041)a6β122(μ041)a6β1+a6)6+(μ04a4β22(μ041)a4β122(μ041)a4β1+a4)4+a221)r2+((6μ02+1)a82+((6μ021)a12β12+2(6μ021)a12β1+(6μ021)a12)6a6((6μ021)a10β12+2(6μ021)a10β1+(12μ021)a10)4+(a10β228+(a8β122a8β22+2a8β1)6(a6β12a6β22+2a6β1)4)r4+6(μ02a12β228+(μ02a10β122μ02a10β22+2μ02a10β1)6(μ02a8β12μ02a8β22+2μ02a8β1)4)r2+4(a6β12+((3μ021)a8β12a8β1)4)m)μ1(r)2+2(((μ042μ02+1)a4β122(μ021)a4β1+a4)4+2((μ021)a2β1a2)2+1)m+2((2μ03+μ0)a62+((2μ03μ0)a10β12+2(2μ03μ0)a10β1+(2μ03μ0)a10)6μ0a4((2μ03μ0)a8β12+2(2μ03μ0)a8β1+(4μ03μ0)a8)4+(μ0a8β228+(μ0a6β122μ0a6β22+2μ0a6β1)6(μ0a4β12μ0a4β22+2μ0a4β1)4)r4+2(μ03a10β228+(μ03a8β122μ03a8β22+2μ03a8β1)6(μ03a6β12μ03a6β22+2μ03a6β1)4)r2+4(μ0a4β12+((μ03μ0)a6β12μ0a6β1)4)m)μ1(r)μ02a642μ02a42+μ02a2+(a442a22+1)r2+(a1042a82+a6)μ1(r)2+2(μ0a842μ0a62+μ0a4)μ1(r)\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{{\left({\mu_0}^{4} + {\mu_0}^{2}\right)} a^{4} {\ell}^{2} + {\left({\left({\mu_0}^{4} - {\mu_0}^{2}\right)} a^{8} {\beta_1}^{2} + 2 \, {\left({\mu_0}^{4} - {\mu_0}^{2}\right)} a^{8} {\beta_1} + {\left({\mu_0}^{4} - {\mu_0}^{2}\right)} a^{8}\right)} {\ell}^{6} - {\left({\left({\mu_0}^{4} - {\mu_0}^{2}\right)} a^{6} {\beta_1}^{2} + 2 \, {\left({\mu_0}^{4} - {\mu_0}^{2}\right)} a^{6} {\beta_1} + {\left(2 \, {\mu_0}^{4} - {\mu_0}^{2}\right)} a^{6}\right)} {\ell}^{4} + {\left({\mu_0}^{2} a^{6} {\beta_2}^{2} {\ell}^{8} - {\left(2 \, {\mu_0}^{2} a^{4} {\beta_2}^{2} - {\left({\mu_0}^{2} - 1\right)} a^{4} {\beta_1}^{2} - 2 \, {\left({\mu_0}^{2} - 1\right)} a^{4} {\beta_1} + a^{4}\right)} {\ell}^{6} + {\left({\mu_0}^{2} a^{2} {\beta_2}^{2} - {\left({\mu_0}^{2} - 1\right)} a^{2} {\beta_1}^{2} - 2 \, {\left({\mu_0}^{2} - 1\right)} a^{2} {\beta_1} + 2 \, a^{2}\right)} {\ell}^{4} - {\ell}^{2}\right)} r^{4} + {\left(2 \, a^{12} {\beta_1}^{2} {\ell}^{4} m + a^{12} {\ell}^{2} + {\left(a^{16} {\beta_1}^{2} + 2 \, a^{16} {\beta_1} + a^{16}\right)} {\ell}^{6} - {\left(a^{14} {\beta_1}^{2} + 2 \, a^{14} {\beta_1} + 2 \, a^{14}\right)} {\ell}^{4} + {\left(a^{16} {\beta_2}^{2} {\ell}^{8} + {\left(a^{14} {\beta_1}^{2} - 2 \, a^{14} {\beta_2}^{2} + 2 \, a^{14} {\beta_1}\right)} {\ell}^{6} - {\left(a^{12} {\beta_1}^{2} - a^{12} {\beta_2}^{2} + 2 \, a^{12} {\beta_1}\right)} {\ell}^{4}\right)} r^{2}\right)} \mu_{1}\left(r\right)^{4} - {\mu_0}^{2} a^{2} + 4 \, {\left(2 \, {\mu_0} a^{10} {\beta_1}^{2} {\ell}^{4} m + {\mu_0} a^{10} {\ell}^{2} + {\left({\mu_0} a^{14} {\beta_1}^{2} + 2 \, {\mu_0} a^{14} {\beta_1} + {\mu_0} a^{14}\right)} {\ell}^{6} - {\left({\mu_0} a^{12} {\beta_1}^{2} + 2 \, {\mu_0} a^{12} {\beta_1} + 2 \, {\mu_0} a^{12}\right)} {\ell}^{4} + {\left({\mu_0} a^{14} {\beta_2}^{2} {\ell}^{8} + {\left({\mu_0} a^{12} {\beta_1}^{2} - 2 \, {\mu_0} a^{12} {\beta_2}^{2} + 2 \, {\mu_0} a^{12} {\beta_1}\right)} {\ell}^{6} - {\left({\mu_0} a^{10} {\beta_1}^{2} - {\mu_0} a^{10} {\beta_2}^{2} + 2 \, {\mu_0} a^{10} {\beta_1}\right)} {\ell}^{4}\right)} r^{2}\right)} \mu_{1}\left(r\right)^{3} + {\left({\mu_0}^{4} a^{8} {\beta_2}^{2} {\ell}^{8} - {\left(2 \, {\mu_0}^{4} a^{6} {\beta_2}^{2} - {\left({\mu_0}^{4} - 1\right)} a^{6} {\beta_1}^{2} - 2 \, {\left({\mu_0}^{4} - 1\right)} a^{6} {\beta_1} + a^{6}\right)} {\ell}^{6} + {\left({\mu_0}^{4} a^{4} {\beta_2}^{2} - {\left({\mu_0}^{4} - 1\right)} a^{4} {\beta_1}^{2} - 2 \, {\left({\mu_0}^{4} - 1\right)} a^{4} {\beta_1} + a^{4}\right)} {\ell}^{4} + a^{2} {\ell}^{2} - 1\right)} r^{2} + {\left({\left(6 \, {\mu_0}^{2} + 1\right)} a^{8} {\ell}^{2} + {\left({\left(6 \, {\mu_0}^{2} - 1\right)} a^{12} {\beta_1}^{2} + 2 \, {\left(6 \, {\mu_0}^{2} - 1\right)} a^{12} {\beta_1} + {\left(6 \, {\mu_0}^{2} - 1\right)} a^{12}\right)} {\ell}^{6} - a^{6} - {\left({\left(6 \, {\mu_0}^{2} - 1\right)} a^{10} {\beta_1}^{2} + 2 \, {\left(6 \, {\mu_0}^{2} - 1\right)} a^{10} {\beta_1} + {\left(12 \, {\mu_0}^{2} - 1\right)} a^{10}\right)} {\ell}^{4} + {\left(a^{10} {\beta_2}^{2} {\ell}^{8} + {\left(a^{8} {\beta_1}^{2} - 2 \, a^{8} {\beta_2}^{2} + 2 \, a^{8} {\beta_1}\right)} {\ell}^{6} - {\left(a^{6} {\beta_1}^{2} - a^{6} {\beta_2}^{2} + 2 \, a^{6} {\beta_1}\right)} {\ell}^{4}\right)} r^{4} + 6 \, {\left({\mu_0}^{2} a^{12} {\beta_2}^{2} {\ell}^{8} + {\left({\mu_0}^{2} a^{10} {\beta_1}^{2} - 2 \, {\mu_0}^{2} a^{10} {\beta_2}^{2} + 2 \, {\mu_0}^{2} a^{10} {\beta_1}\right)} {\ell}^{6} - {\left({\mu_0}^{2} a^{8} {\beta_1}^{2} - {\mu_0}^{2} a^{8} {\beta_2}^{2} + 2 \, {\mu_0}^{2} a^{8} {\beta_1}\right)} {\ell}^{4}\right)} r^{2} + 4 \, {\left(a^{6} {\beta_1} {\ell}^{2} + {\left({\left(3 \, {\mu_0}^{2} - 1\right)} a^{8} {\beta_1}^{2} - a^{8} {\beta_1}\right)} {\ell}^{4}\right)} m\right)} \mu_{1}\left(r\right)^{2} + 2 \, {\left({\left({\left({\mu_0}^{4} - 2 \, {\mu_0}^{2} + 1\right)} a^{4} {\beta_1}^{2} - 2 \, {\left({\mu_0}^{2} - 1\right)} a^{4} {\beta_1} + a^{4}\right)} {\ell}^{4} + 2 \, {\left({\left({\mu_0}^{2} - 1\right)} a^{2} {\beta_1} - a^{2}\right)} {\ell}^{2} + 1\right)} m + 2 \, {\left({\left(2 \, {\mu_0}^{3} + {\mu_0}\right)} a^{6} {\ell}^{2} + {\left({\left(2 \, {\mu_0}^{3} - {\mu_0}\right)} a^{10} {\beta_1}^{2} + 2 \, {\left(2 \, {\mu_0}^{3} - {\mu_0}\right)} a^{10} {\beta_1} + {\left(2 \, {\mu_0}^{3} - {\mu_0}\right)} a^{10}\right)} {\ell}^{6} - {\mu_0} a^{4} - {\left({\left(2 \, {\mu_0}^{3} - {\mu_0}\right)} a^{8} {\beta_1}^{2} + 2 \, {\left(2 \, {\mu_0}^{3} - {\mu_0}\right)} a^{8} {\beta_1} + {\left(4 \, {\mu_0}^{3} - {\mu_0}\right)} a^{8}\right)} {\ell}^{4} + {\left({\mu_0} a^{8} {\beta_2}^{2} {\ell}^{8} + {\left({\mu_0} a^{6} {\beta_1}^{2} - 2 \, {\mu_0} a^{6} {\beta_2}^{2} + 2 \, {\mu_0} a^{6} {\beta_1}\right)} {\ell}^{6} - {\left({\mu_0} a^{4} {\beta_1}^{2} - {\mu_0} a^{4} {\beta_2}^{2} + 2 \, {\mu_0} a^{4} {\beta_1}\right)} {\ell}^{4}\right)} r^{4} + 2 \, {\left({\mu_0}^{3} a^{10} {\beta_2}^{2} {\ell}^{8} + {\left({\mu_0}^{3} a^{8} {\beta_1}^{2} - 2 \, {\mu_0}^{3} a^{8} {\beta_2}^{2} + 2 \, {\mu_0}^{3} a^{8} {\beta_1}\right)} {\ell}^{6} - {\left({\mu_0}^{3} a^{6} {\beta_1}^{2} - {\mu_0}^{3} a^{6} {\beta_2}^{2} + 2 \, {\mu_0}^{3} a^{6} {\beta_1}\right)} {\ell}^{4}\right)} r^{2} + 4 \, {\left({\mu_0} a^{4} {\beta_1} {\ell}^{2} + {\left({\left({\mu_0}^{3} - {\mu_0}\right)} a^{6} {\beta_1}^{2} - {\mu_0} a^{6} {\beta_1}\right)} {\ell}^{4}\right)} m\right)} \mu_{1}\left(r\right)}{{\mu_0}^{2} a^{6} {\ell}^{4} - 2 \, {\mu_0}^{2} a^{4} {\ell}^{2} + {\mu_0}^{2} a^{2} + {\left(a^{4} {\ell}^{4} - 2 \, a^{2} {\ell}^{2} + 1\right)} r^{2} + {\left(a^{10} {\ell}^{4} - 2 \, a^{8} {\ell}^{2} + a^{6}\right)} \mu_{1}\left(r\right)^{2} + 2 \, {\left({\mu_0} a^{8} {\ell}^{4} - 2 \, {\mu_0} a^{6} {\ell}^{2} + {\mu_0} a^{4}\right)} \mu_{1}\left(r\right)}
# g[0,0].expr().factor()

Reduced form of gtrg_{tr}:

if b == 0 and not keep_Delta: gtr = a^2*sinth2/(rho2*Xi_a^2)*(beta1*Xi_a*(Delta_r - Delta_th*(r^2 + a^2)) \ + beta1^2*l^2*(Delta_th*(r^2 + a^2)^2 - a^2*Delta_r*sinth2)) \ * diff(phi_1(r), r) \ + a^2*l^2*beta2^2*r^2*costh2 * diff(psi_1(r), r) gtr = gtr.subs({mu: mu_s, ph: ph_s, ps: ps_s}) print(g[0,1] == gtr)
True

Reduced form of gttg_{tt}:

if b == 0 and not keep_Delta: gtt = (a^2*Delta_th*sinth2 - Delta_r)/rho2 \ + (2*beta1*a^2*l^2*sinth2*(Delta_r - Delta_th*(r^2 + a^2)))/(rho2*Xi_a) \ + beta1^2*a^2*l^4*sinth2/(rho2*Xi_a^2)*(Delta_th*(r^2 + a^2)^2 - a^2*Delta_r*sinth2) \ + beta2^2*a^2*l^4*r^2*costh2 gtt = gtt.subs({mu: mu_s, ph: ph_s, ps: ps_s}) print(g[0,0] == gtt)
True

Reduced form of grrg_{rr}:

if b == 0 and not keep_Delta: grr = rho2/Delta_r + a^2*beta1^2*sinth2/(Xi_a^2*rho2)*(Delta_th*(r^2 + a^2)^2 - Delta_r*a^2*sinth2)*diff(phi_1(r), r)^2 \ + a^4*rho2/Delta_th/sinth2*diff(mu_1(r), r)^2 \ + beta2^2*a^2*r^2*costh2*diff(psi_1(r), r)^2 grr = grr.subs({mu: mu_s, ph: ph_s, ps: ps_s}) print(g[1,1] == grr)
True

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)a2β122(μ021)a2β1)42)r4(μ021)a2+((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+2(2(μ021)a2β12+1)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 \, {\left({\mu_0}^{2} - 1\right)} a^{2} {\beta_1}\right)} {\ell}^{4} - {\ell}^{2}\right)} r^{4} - {\left({\mu_0}^{2} - 1\right)} a^{2} + {\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} + 2 \, {\left(2 \, {\left({\mu_0}^{2} - 1\right)} a^{2} {\beta_1} {\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)a2β122(μ021)a2β1)422)r4(μ021)a2+((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+4((μ021)a2β12+1)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 \, {\left({\mu_0}^{2} - 1\right)} a^{2} {\beta_1}\right)} {\ell}^{4} - 2 \, {\ell}^{2}\right)} r^{4} - {\left({\mu_0}^{2} - 1\right)} a^{2} + {\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} + 4 \, {\left({\left({\mu_0}^{2} - 1\right)} a^{2} {\beta_1} {\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+2μ02a2β14r44a2β122mr4rϕ1(r)2a2β124r4μ02a2β12r4rϕ1(r)2+μ02a2β22r4rψ1(r)22a2β14r4+4μ02a2β12mr2rϕ1(r)24μ02a2β22mr2rψ1(r)24μ02a2β12m2rϕ1(r)2+a2β12r4rϕ1(r)2+4μ02a2β22m2rψ1(r)24a2β12mr2rϕ1(r)24μ02a2β12m+4a2β12m2rϕ1(r)2+4a2β12m+22r4+μ02a2a2+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} - 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 \, 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} + {\mu_0}^{2} a^{2} - 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_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. (3.35)-(3.36) of the paper 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_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. (3.35)-(3.36) of the paper 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
4(μ03μ0)a4β12m((μ03μ0)a4β12(μ03μ0)a4β22+2(μ03μ0)a4β1)4r4(μ03μ0)a4+((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)22(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{4 \, {\left({\mu_0}^{3} - {\mu_0}\right)} a^{4} {\beta_1} {\ell}^{2} m - {\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}\right)} {\ell}^{4} r^{4} - {\left({\mu_0}^{3} - {\mu_0}\right)} a^{4} + {\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/a^4).simplify_full() eq_mu1
((μ03μ0)β12(μ03μ0)β22+2(μ03μ0)β1)4r44(μ03μ0)β12m+μ03((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)μ0=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}\right)} {\ell}^{4} r^{4} - 4 \, {\left({\mu_0}^{3} - {\mu_0}\right)} {\beta_1} {\ell}^{2} m + {\mu_0}^{3} - {\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) - {\mu_0} = 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)β1)4r4(μ03μ0)P2β12+(μ03μ0)Q2β224(μ03μ0)β12m+μ03+2(24r7+32r5(42m1)r32mr)rμ1(r)+(4r8+22r6(42m1)r44mr2+4m2)2(r)2μ1(r)μ0=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}\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({\mu_0}^{3} - {\mu_0}\right)} {\beta_1} {\ell}^{2} m + {\mu_0}^{3} + 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) - {\mu_0} = 0

Check of Eq. (3.37)

lhs = eq_mu1.lhs() lhs = lhs.simplify_full() lhs
((μ03μ0)β12(μ03μ0)β22+2(μ03μ0)β1)4r4(μ03μ0)P2β12+(μ03μ0)Q2β224(μ03μ0)β12m+μ03+2(24r7+32r5(42m1)r32mr)rμ1(r)+(4r8+22r6(42m1)r44mr2+4m2)2(r)2μ1(r)μ0\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}\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({\mu_0}^{3} - {\mu_0}\right)} {\beta_1} {\ell}^{2} m + {\mu_0}^{3} + 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) - {\mu_0}
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)β1)4r4(μ03μ0)P2β12+(μ03μ0)Q2β224(μ03μ0)β12m+μ03+2(24r7+32r5(42m1)r32mr)rμ1(r)μ04r8+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}\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({\mu_0}^{3} - {\mu_0}\right)} {\beta_1} {\ell}^{2} m + {\mu_0}^{3} + 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) - {\mu_0}}{{\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)β1)4r4(μ03μ0)P2β12+(μ03μ0)Q2β224(μ03μ0)β12m+μ03μ04r8+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}\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({\mu_0}^{3} - {\mu_0}\right)} {\beta_1} {\ell}^{2} m + {\mu_0}^{3} - {\mu_0}}{{\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+2β14r4P2β12+Q2β224β12m+1)(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} - {\mathcal{P}'}^{2} {\beta_1}^{2} + {\mathcal{Q}'}^{2} {\beta_2}^{2} - 4 \, {\beta_1} {\ell}^{2} m + 1\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+2β14r4P2β12+Q2β224β12m+1)(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} - {\mathcal{P}'}^{2} {\beta_1}^{2} + {\mathcal{Q}'}^{2} {\beta_2}^{2} - 4 \, {\beta_1} {\ell}^{2} m + 1\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+2β1)4r4+P2β12Q2β22+4β12m1\renewcommand{\Bold}[1]{\mathbf{#1}}-{\left({\beta_1}^{2} - {\beta_2}^{2} + 2 \, {\beta_1}\right)} {\ell}^{4} r^{4} + {\mathcal{P}'}^{2} {\beta_1}^{2} - {\mathcal{Q}'}^{2} {\beta_2}^{2} + 4 \, {\beta_1} {\ell}^{2} m - 1

Given that μ1(r)=sinΘ0  θ1(r)=1μ02  θ1(r)andsin2Θ0=2μ01μ02, \mu_1(r) = - \sin\Theta_0 \; \theta_1(r) = - \sqrt{1-\mu_0^2} \; \theta_1(r) \qquad\mbox{and} \qquad \sin2\Theta_0 = 2\mu_0\sqrt{1-\mu_0^2}, we get for the equation for θ1\theta_1: θ1+2r(1+2r2)r4hθ1+1+β22Q2β12P24β12m+(β12+2β1β22)4r42r8h2sin(2Θ0)=0 \theta_1'' + \frac{2r(1 + \ell^2 r^2)}{r^4 h} \, \theta_1' + \frac{1 + \beta_2^{-2}\mathcal{Q}^2 - \beta_1^{-2}\mathcal{P}^2 - 4 \beta_1 \ell^2 m + (\beta_1^2 + 2 \beta_1 - \beta_2^2) \ell^4 r^4}{2 r^8 h^2}\sin(2\Theta_0) = 0

This agrees with Eq. (3.37) 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)β1)2r(μ03μ0)(β12β22+2β1)2r2+P2β12Q2β222(β12β22)2m12r4+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}\right)} {\ell}^{2} r - {\left({\mu_0}^{3} - {\mu_0}\right)} \int \frac{{\left({\beta_1}^{2} - {\beta_2}^{2} + 2 \, {\beta_1}\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 - 1}{{\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)*l^2*r + (Mu0^3 - Mu0)*integrate(-((beta1^2 - beta2^2 + 2*beta1)*l^2*r^2 + P^2*beta1^2 - Q^2*beta2^2 - 2*(beta1^2 - beta2^2)*l^2*m - 1)/(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)β1)2r(μ03μ0)(β12β22+2β1)2r2+P2β12Q2β222(β12β22)2m12r4+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}\right)} {\ell}^{2} r - {\left({\mu_0}^{3} - {\mu_0}\right)} \int \frac{{\left({\beta_1}^{2} - {\beta_2}^{2} + 2 \, {\beta_1}\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 - 1}{{\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 (β12β2+2β1)2  s1(r)+(P2β12Qβ222(β12β22)2m1)  s2(r) (\beta_1^2 - \beta^2 + 2 \beta_1) \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 - 1\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)β1)2r(μ03μ0)(β12β22+2β1)2r2+P2β12Q2β222(β12β22)2m12r4+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}\right)} {\ell}^{2} r - {\left({\mu_0}^{3} - {\mu_0}\right)} \int \frac{{\left({\beta_1}^{2} - {\beta_2}^{2} + 2 \, {\beta_1}\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 - 1}{{\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 same constants C1C_1 and C2C_2 as in Eq. (3.75) of 5dKerr-AdS-AGG-5-12:

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 + 2*beta1 - beta2^2)*l^2*r - (beta1^2 + 2*beta1 - beta2^2)*l^2 * s1 - (P^2*beta1^2 - Q^2*beta2^2 - 2*(beta1^2 - beta2^2)*l^2*m - 1) * 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+2β1)2r2(β12β22+2β1)(2B+1arctan(2rB+1)+B1log(2rB112rB1+1))B+(P2β12Q2β222(β12β22)2m1)(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}\right)} {\ell}^{2} r - \frac{\sqrt{2} {\left({\beta_1}^{2} - {\beta_2}^{2} + 2 \, {\beta_1}\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 - 1\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+2β1)2r2(β12β22+2β1)(2B+1arctan(2rB+1)+B1log(2rB112rB1+1))B+(P2β12Q2β222(β12β22)2m1)(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}\right)} {\ell}^{2} r - \frac{\sqrt{2} {\left({\beta_1}^{2} - {\beta_2}^{2} + 2 \, {\beta_1}\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 - 1\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

Check of the solution (3.38)

Blm = sqrt(1 + 8*m*l^2) Blm
82m+1\renewcommand{\Bold}[1]{\mathbf{#1}}\sqrt{8 \, {\ell}^{2} m + 1}
A = beta1^2 - beta2^2 + 2*beta1 A
β12β22+2β1\renewcommand{\Bold}[1]{\mathbf{#1}}{\beta_1}^{2} - {\beta_2}^{2} + 2 \, {\beta_1}
J = 4*l^2*m*(beta1^2 - beta2^2) - 2*beta1^2*P^2 + 2*beta2^2*Q^2 + 2 + A J
2P2β12+2Q2β22+4(β12β22)2m+β12β22+2β1+2\renewcommand{\Bold}[1]{\mathbf{#1}}-2 \, {\mathcal{P}'}^{2} {\beta_1}^{2} + 2 \, {\mathcal{Q}'}^{2} {\beta_2}^{2} + 4 \, {\left({\beta_1}^{2} - {\beta_2}^{2}\right)} {\ell}^{2} m + {\beta_1}^{2} - {\beta_2}^{2} + 2 \, {\beta_1} + 2

From Eq. (3.38):

theta1 = C2 + C1*integrate(1/(l^2*r^4 + r^2 - 2*m), r, hold=True) - Mu0*sqrt(1-Mu0^2)\ *integrate((A*l^2*r + l/(sqrt(2)*Blm)*( (J - A*Blm)/sqrt(1-Blm)*atan(sqrt(2)*l*r/sqrt(1-Blm)) - (J + A*Blm)/sqrt(1+Blm)*atan(sqrt(2)*l*r/sqrt(1+Blm)) )) /(l^2*r^4 + r^2 - 2*m), r, hold=True) theta1
μ02+1μ02(β12β22+2β1)2r+2((2P2β122Q2β224(β12β22)2mβ12+β2282m+1(β12β22+2β1)2β12)arctan(2r82m+1+1)82m+1+1(2P2β122Q2β224(β12β22)2mβ12+β22+82m+1(β12β22+2β1)2β12)arctan(2r82m+1+1)82m+1+1)82m+12(2r4+r22m)dr+C112r4+r22mdr+C2\renewcommand{\Bold}[1]{\mathbf{#1}}-\sqrt{-{\mu_0}^{2} + 1} {\mu_0} \int \frac{2 \, {\left({\beta_1}^{2} - {\beta_2}^{2} + 2 \, {\beta_1}\right)} {\ell}^{2} r + \frac{\sqrt{2} {\left(\frac{{\left(2 \, {\mathcal{P}'}^{2} {\beta_1}^{2} - 2 \, {\mathcal{Q}'}^{2} {\beta_2}^{2} - 4 \, {\left({\beta_1}^{2} - {\beta_2}^{2}\right)} {\ell}^{2} m - {\beta_1}^{2} + {\beta_2}^{2} - \sqrt{8 \, {\ell}^{2} m + 1} {\left({\beta_1}^{2} - {\beta_2}^{2} + 2 \, {\beta_1}\right)} - 2 \, {\beta_1} - 2\right)} \arctan\left(\frac{\sqrt{2} {\ell} r}{\sqrt{\sqrt{8 \, {\ell}^{2} m + 1} + 1}}\right)}{\sqrt{\sqrt{8 \, {\ell}^{2} m + 1} + 1}} - \frac{{\left(2 \, {\mathcal{P}'}^{2} {\beta_1}^{2} - 2 \, {\mathcal{Q}'}^{2} {\beta_2}^{2} - 4 \, {\left({\beta_1}^{2} - {\beta_2}^{2}\right)} {\ell}^{2} m - {\beta_1}^{2} + {\beta_2}^{2} + \sqrt{8 \, {\ell}^{2} m + 1} {\left({\beta_1}^{2} - {\beta_2}^{2} + 2 \, {\beta_1}\right)} - 2 \, {\beta_1} - 2\right)} \arctan\left(\frac{\sqrt{2} {\ell} r}{\sqrt{-\sqrt{8 \, {\ell}^{2} m + 1} + 1}}\right)}{\sqrt{-\sqrt{8 \, {\ell}^{2} m + 1} + 1}}\right)} {\ell}}{\sqrt{8 \, {\ell}^{2} m + 1}}}{2 \, {\left({\ell}^{2} r^{4} + r^{2} - 2 \, m\right)}}\,{d r} + C_{1} \int \frac{1}{{\ell}^{2} r^{4} + r^{2} - 2 \, m}\,{d r} + C_{2}
mu1_sol2(r) = - sinTh0 * theta1 eq_mu1.substitute_function(mu_1, mu1_sol2).simplify_full()
0=0\renewcommand{\Bold}[1]{\mathbf{#1}}0 = 0

Hence Eq. (3.38) provides a solution of the equation for μ1\mu_1.

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. (3.45)

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. (3.46)

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. (3.44)

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+22μ0β1)marctan(2r82m+1+1)+(μ0P2β12μ0Q2β222(μ0β12μ0β22)2mμ0)log(mr82m+1+12mmr82m+1+1+2m))μ02+182m+1+1+((2μ0β122μ0β22+22μ0β1)mlog(2r82m+1182m+1+12r82m+11+82m+11)2(μ0P2β12μ0Q2β222(μ0β12μ0β22)2mμ0)arctan(r82m+112m))μ02+182m+114((μ0β12μ0β22+2μ0β1)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}\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 - {\mu_0}\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}\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 - {\mu_0}\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}\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+2β1)2rC1sinΘ0cosΘ0\frac{\pi_\theta^r}{(a^2/2) \sin 2\Theta_0} + (\beta_1^2 - \beta_2^2 + 2 \beta_1)\ell^2 r - \frac{C_1}{\sin\Theta_0\cos\Theta_0}

part1 = - (beta1^2 - beta2^2 + 2*beta1)*l^2*r + C1/(Mu0*sqrt(1-Mu0^2)) s = (pi_theta_r_a2sT0 - part1).simplify_full() s
(2(2β122β22+22β1)marctan(2r82m+1+1)+(P2β12Q2β222(β12β22)2m1)log(mr82m+1+12mmr82m+1+1+2m))82m+1+1+((2β122β22+22β1)mlog(2r82m+1182m+1+12r82m+11+82m+11)2(P2β12Q2β222(β12β22)2m1)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}\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 - 1\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}\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 - 1\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+22πβ1)m82m+1+1+(ππP2β12+πQ2β22+2(πβ12πβ22)2m)82m+11482m+1mβ12β22+2β1r\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}\right)} {\ell} \sqrt{m} \sqrt{\sqrt{8 \, {\ell}^{2} m + 1} + 1} + {\left(\pi - \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}}{r}
s
(2πβ122πβ22+22πβ1)m82m+1+1+(ππP2β12+πQ2β22+2(πβ12πβ22)2m)82m+11482m+1mβ12β22+2β1r\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}\right)} {\ell} \sqrt{m} \sqrt{\sqrt{8 \, {\ell}^{2} m + 1} + 1} + {\left(\pi - \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}}{r}

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

part1 + s
(β12β22+2β1)2r+(2πβ122πβ22+22πβ1)m82m+1+1+(ππP2β12+πQ2β22+2(πβ12πβ22)2m)82m+11482m+1mβ12β22+2β1r+C1μ02+1μ0\renewcommand{\Bold}[1]{\mathbf{#1}}-{\left({\beta_1}^{2} - {\beta_2}^{2} + 2 \, {\beta_1}\right)} {\ell}^{2} r + \frac{{\left(\sqrt{2} \pi {\beta_1}^{2} - \sqrt{2} \pi {\beta_2}^{2} + 2 \, \sqrt{2} \pi {\beta_1}\right)} {\ell} \sqrt{m} \sqrt{\sqrt{8 \, {\ell}^{2} m + 1} + 1} + {\left(\pi - \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}}{r} + \frac{C_{1}}{\sqrt{-{\mu_0}^{2} + 1} {\mu_0}}

The terms in rr, 1/r1/r and C1C_1 agree with Eq. (3.44) of the paper.