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

Project: KerrAdS

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Kernel: SageMath 9.3

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

Case b = n a

This SageMath notebook is relative to the article Heavy quarks in rotating plasma via holography by Anastasia A. Golubtsova, Eric Gourgoulhon and Marina K. Usova, arXiv:2107.11672.

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

NB: a version of SageMath at least equal to 9.1 is required to run this notebook:

In [1]:

version()

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

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

In [2]:

%display latex

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

In [3]:

Parallelism().set(nproc=8)

Spacetime manifold

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

In [4]:

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 $\mathcal{M}$ , 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:

In [5]:

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

\renewcommand{\Bold}[1]{\mathbf{#1}}\left(\mathcal{M},(t, r, {\mu}, {\phi}, {\psi})\right)

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

The coordinate ranges are

In [6]:

BL.coord_range()

\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 $\mu$ is $(0,1)$ 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 $\theta$ is $\left(0, \frac{\pi}{2}\right)$ only.

Metric tensor

The 4 parameters $m$ , $a$ , $b$ and $\ell$ of the Kerr-AdS spacetime are declared as symbolic variables, $a$ and $b$ being the two angular momentum parameters and $\ell$ being related to the cosmological constant by $\Lambda = - 6 \ell^2$ :

In [7]:

var('m a b', domain='real')

\renewcommand{\Bold}[1]{\mathbf{#1}}\left(m, a, b\right)

In [8]:

var('l', domain='real', latex_name=r'\ell')

\renewcommand{\Bold}[1]{\mathbf{#1}}{\ell}

We assume that $b = n a$ :

In [9]:

n = var('n', domain='real')
b = n*a

Some auxiliary functions:

In [10]:

keep_Delta = True  # change to False to provide explicit expression for Delta_r, Xi_a, etc...

In [11]:

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
# Explicit expressions:
Delta_r_expr = (r^2+a^2)*(r^2+b^2)*sig - 2*m
Delta_th_expr = 1 - a^2*l^2*costh2 - b^2*l^2*sinth2
Xi_a_expr = 1 - a^2*l^2
Xi_b_expr = 1 - b^2*l^2
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')
else:
    Delta_r = Delta_r_expr
    Delta_th = Delta_th_expr
    Xi_a = Xi_a_expr
    Xi_b = Xi_b_expr

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):

In [12]:

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()

In [13]:

G.display_comp(only_nonredundant=True)

\renewcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{lcl} G_{ \, t \, t }^{ \phantom{\, t}\phantom{\, t} } & = & \frac{a^{4} n^{2} - {\left({\Delta_\theta} a^{2} {\mu}^{2} - {\Delta_\theta} a^{2} - {\left(a^{4} {\ell}^{2} + {\Delta_\theta} a^{2} {\mu}^{2}\right)} n^{2} + {\Delta_r}\right)} r^{2}}{r^{4} + {\left(a^{2} {\mu}^{2} - {\left(a^{2} {\mu}^{2} - a^{2}\right)} n^{2}\right)} r^{2}} \\ G_{ \, t \, {\phi} }^{ \phantom{\, t}\phantom{\, {\phi}} } & = & \frac{{\left({\Delta_\theta} a {\mu}^{2} + {\left(a^{3} {\ell}^{2} {\mu}^{2} - a^{3} {\ell}^{2}\right)} n^{2} - {\Delta_\theta} a\right)} r^{4} + {\left(a^{5} {\mu}^{2} - a^{5}\right)} n^{2} - {\left({\Delta_\theta} a^{3} - {\left({\Delta_\theta} a^{3} - {\Delta_r} a\right)} {\mu}^{2} + {\left(a^{5} {\ell}^{2} + a^{3} - {\left(a^{5} {\ell}^{2} + a^{3}\right)} {\mu}^{2}\right)} n^{2} - {\Delta_r} a\right)} r^{2}}{{\Xi_a} r^{4} + {\left({\Xi_a} a^{2} {\mu}^{2} - {\left({\Xi_a} a^{2} {\mu}^{2} - {\Xi_a} a^{2}\right)} n^{2}\right)} r^{2}} \\ G_{ \, t \, {\psi} }^{ \phantom{\, t}\phantom{\, {\psi}} } & = & -\frac{a^{5} {\mu}^{2} n^{3} + {\left(a^{3} {\ell}^{2} + {\Delta_\theta} a\right)} {\mu}^{2} n r^{4} + {\left({\left(a^{5} {\ell}^{2} + {\Delta_\theta} a^{3}\right)} {\mu}^{2} n^{3} + {\left(a^{3} - {\Delta_r} a\right)} {\mu}^{2} n\right)} r^{2}}{{\Xi_b} r^{4} + {\left({\Xi_b} a^{2} {\mu}^{2} - {\left({\Xi_b} a^{2} {\mu}^{2} - {\Xi_b} a^{2}\right)} n^{2}\right)} r^{2}} \\ G_{ \, r \, r }^{ \phantom{\, r}\phantom{\, r} } & = & \frac{a^{2} {\mu}^{2} - {\left(a^{2} {\mu}^{2} - a^{2}\right)} n^{2} + r^{2}}{{\Delta_r}} \\ G_{ \, {\mu} \, {\mu} }^{ \phantom{\, {\mu}}\phantom{\, {\mu}} } & = & -\frac{a^{2} {\mu}^{2} - {\left(a^{2} {\mu}^{2} - a^{2}\right)} n^{2} + r^{2}}{{\Delta_\theta} {\mu}^{2} - {\Delta_\theta}} \\ G_{ \, {\phi} \, {\phi} }^{ \phantom{\, {\phi}}\phantom{\, {\phi}} } & = & -\frac{{\left({\Delta_\theta} {\mu}^{2} - {\left(a^{2} {\ell}^{2} {\mu}^{4} - 2 \, a^{2} {\ell}^{2} {\mu}^{2} + a^{2} {\ell}^{2}\right)} n^{2} - {\Delta_\theta}\right)} r^{6} + {\left(2 \, {\Delta_\theta} a^{2} {\mu}^{2} - 2 \, {\Delta_\theta} a^{2} - {\left(2 \, a^{4} {\ell}^{2} + {\left(2 \, a^{4} {\ell}^{2} + a^{2}\right)} {\mu}^{4} - 2 \, {\left(2 \, a^{4} {\ell}^{2} + a^{2}\right)} {\mu}^{2} + a^{2}\right)} n^{2}\right)} r^{4} - {\left(a^{6} {\mu}^{4} - 2 \, a^{6} {\mu}^{2} + a^{6}\right)} n^{2} + {\left({\Delta_r} a^{2} {\mu}^{4} - {\Delta_\theta} a^{4} + {\Delta_r} a^{2} + {\left({\Delta_\theta} a^{4} - 2 \, {\Delta_r} a^{2}\right)} {\mu}^{2} - {\left(a^{6} {\ell}^{2} + {\left(a^{6} {\ell}^{2} + 2 \, a^{4}\right)} {\mu}^{4} + 2 \, a^{4} - 2 \, {\left(a^{6} {\ell}^{2} + 2 \, a^{4}\right)} {\mu}^{2}\right)} n^{2}\right)} r^{2}}{{\Xi_a}^{2} r^{4} + {\left({\Xi_a}^{2} a^{2} {\mu}^{2} - {\left({\Xi_a}^{2} a^{2} {\mu}^{2} - {\Xi_a}^{2} a^{2}\right)} n^{2}\right)} r^{2}} \\ G_{ \, {\phi} \, {\psi} }^{ \phantom{\, {\phi}}\phantom{\, {\psi}} } & = & -\frac{{\left(a^{2} {\ell}^{2} {\mu}^{4} - a^{2} {\ell}^{2} {\mu}^{2}\right)} n r^{6} + {\left({\left(a^{4} {\ell}^{2} {\mu}^{4} - a^{4} {\ell}^{2} {\mu}^{2}\right)} n^{3} + {\left({\left(a^{4} {\ell}^{2} + a^{2}\right)} {\mu}^{4} - {\left(a^{4} {\ell}^{2} + a^{2}\right)} {\mu}^{2}\right)} n\right)} r^{4} + {\left(a^{6} {\mu}^{4} - a^{6} {\mu}^{2}\right)} n^{3} + {\left({\left({\left(a^{6} {\ell}^{2} + a^{4}\right)} {\mu}^{4} - {\left(a^{6} {\ell}^{2} + a^{4}\right)} {\mu}^{2}\right)} n^{3} + {\left({\left(a^{4} - {\Delta_r} a^{2}\right)} {\mu}^{4} - {\left(a^{4} - {\Delta_r} a^{2}\right)} {\mu}^{2}\right)} n\right)} r^{2}}{{\Xi_a} {\Xi_b} r^{4} + {\left({\Xi_a} {\Xi_b} a^{2} {\mu}^{2} - {\left({\Xi_a} {\Xi_b} a^{2} {\mu}^{2} - {\Xi_a} {\Xi_b} a^{2}\right)} n^{2}\right)} r^{2}} \\ G_{ \, {\psi} \, {\psi} }^{ \phantom{\, {\psi}}\phantom{\, {\psi}} } & = & \frac{a^{6} {\mu}^{4} n^{4} + {\left(a^{2} {\ell}^{2} {\mu}^{4} + {\Delta_\theta} {\mu}^{2}\right)} r^{6} + {\left(a^{2} {\mu}^{4} + 2 \, {\left(a^{4} {\ell}^{2} {\mu}^{4} + {\Delta_\theta} a^{2} {\mu}^{2}\right)} n^{2}\right)} r^{4} + {\left({\left(2 \, a^{4} - {\Delta_r} a^{2}\right)} {\mu}^{4} n^{2} + {\left(a^{6} {\ell}^{2} {\mu}^{4} + {\Delta_\theta} a^{4} {\mu}^{2}\right)} n^{4}\right)} r^{2}}{{\Xi_b}^{2} r^{4} + {\left({\Xi_b}^{2} a^{2} {\mu}^{2} - {\left({\Xi_b}^{2} a^{2} {\mu}^{2} - {\Xi_b}^{2} a^{2}\right)} n^{2}\right)} r^{2}} \end{array}

Check of Eq. (2.9)

We need the 1-forms $\mathrm{d}t$ , $\mathrm{d}r$ , $\mathrm{d}\mu$ , $\mathrm{d}\phi$ and $\mathrm{d}\psi$ :

In [14]:

dt, dr, dmu, dph, dps = (BL.coframe()[i] for i in M.irange())
dt, dr, dmu, dph, dps

\renewcommand{\Bold}[1]{\mathbf{#1}}\left(\mathrm{d} t, \mathrm{d} r, \mathrm{d} {\mu}, \mathrm{d} {\phi}, \mathrm{d} {\psi}\right)

In [15]:

print(dt)

1-form dt on the 5-dimensional Lorentzian manifold M

In agreement with $\mu = \cos\theta$ , we introduce the 1-form $\mathrm{d}\theta = - \mathrm{d}\mu /\sin\theta ,$ with $\sin\theta = \sqrt{1-\mu^2}$ since $\theta\in\left(0, \frac{\pi}{2}\right)$ :

In [16]:

dth = - 1/sqrt(1 - mu^2)*dmu

In [17]:

s1 = dt - a*sinth2/Xi_a*dph - b*costh2/Xi_b*dps
s1.display()

\renewcommand{\Bold}[1]{\mathbf{#1}}\mathrm{d} t + \left( \frac{a {\mu}^{2} - a}{{\Xi_a}} \right) \mathrm{d} {\phi} -\frac{a {\mu}^{2} n}{{\Xi_b}} \mathrm{d} {\psi}

In [18]:

s2 = a*dt - (r^2 + a^2)/Xi_a*dph
s2.display()

\renewcommand{\Bold}[1]{\mathbf{#1}}a \mathrm{d} t + \left( -\frac{a^{2} + r^{2}}{{\Xi_a}} \right) \mathrm{d} {\phi}

In [19]:

s3 = b*dt - (r^2 + b^2)/Xi_b*dps
s3.display()

\renewcommand{\Bold}[1]{\mathbf{#1}}a n \mathrm{d} t + \left( -\frac{a^{2} n^{2} + r^{2}}{{\Xi_b}} \right) \mathrm{d} {\psi}

In [20]:

s4 = a*b*dt - b*(r^2 + a^2)*sinth2/Xi_a * dph - a*(r^2 + b^2)*costh2/Xi_b * dps
s4.display()

\renewcommand{\Bold}[1]{\mathbf{#1}}a^{2} n \mathrm{d} t + \left( \frac{{\left(a {\mu}^{2} - a\right)} n r^{2} + {\left(a^{3} {\mu}^{2} - a^{3}\right)} n}{{\Xi_a}} \right) \mathrm{d} {\phi} + \left( -\frac{a^{3} {\mu}^{2} n^{2} + a {\mu}^{2} r^{2}}{{\Xi_b}} \right) \mathrm{d} {\psi}

In [21]:

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

Check of Eq. (2.9):

In [22]:

G0 == G

\renewcommand{\Bold}[1]{\mathbf{#1}}\mathrm{True}

Einstein equation

The Ricci tensor of $g$ is

In [23]:

if not keep_Delta:
    # Ric = G.ricci()
    # print(Ric)
    pass

In [24]:

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

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

In [25]:

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

String worldsheet

The string worldsheet as a 2-dimensional Lorentzian submanifold of $\mathcal{M}$ :

In [26]:

W = Manifold(2, 'W', ambient=M, structure='Lorentzian', 
             latex_name=r'\mathcal{W}')
print(W)

2-dimensional Lorentzian submanifold W immersed in the 5-dimensional Lorentzian manifold M

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

In [27]:

XW.<t,r> = W.chart(r't r:(0,+oo)')
XW

\renewcommand{\Bold}[1]{\mathbf{#1}}\left(\mathcal{W},(t, r)\right)

The string embedding in Kerr-AdS spacetime, as an expansion about a straight string solution in AdS (Eq. (4.5) of the paper):

In [28]:

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')

cosTh0 = Mu0
sinTh0 = sqrt(1 - Mu0^2)

mu_s = Mu0 + (a+b)^2*function('mu_1')(r)
ph_s = Phi0 + a*l^2*t + a*function('phi_1')(r)
ps_s = Psi0 + b*l^2*t + b*function('psi_1')(r)

F = W.diff_map(M, {(XW, BL): [t, r, mu_s, ph_s, ps_s]}, name='F') 

W.set_embedding(F)

F.display()

\renewcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{llcl} F:& \mathcal{W} & \longrightarrow & \mathcal{M} \\ & \left(t, r\right) & \longmapsto & \left(t, r, {\mu}, {\phi}, {\psi}\right) = \left(t, r, {\left(a n + a\right)}^{2} \mu_{1}\left(r\right) + {\mu_0}, a {\ell}^{2} t + a \phi_{1}\left(r\right) + {\Phi_0}, a {\ell}^{2} n t + a n \psi_{1}\left(r\right) + {\Psi_0}\right) \end{array}

In [29]:

F.jacobian_matrix()

\renewcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rr} 1 & 0 \\ 0 & 1 \\ 0 & {\left(a^{2} n^{2} + 2 \, a^{2} n + a^{2}\right)} \frac{\partial}{\partial r}\mu_{1}\left(r\right) \\ a {\ell}^{2} & a \frac{\partial}{\partial r}\phi_{1}\left(r\right) \\ a {\ell}^{2} n & a n \frac{\partial}{\partial r}\psi_{1}\left(r\right) \end{array}\right)

Induced metric on the string worldsheet

Because of the bug #27492, which impedes parallel computations involving symbolic functions, such as $\mu_1$ , we switch back to serial computations:

In [30]:

Parallelism().set(nproc=1)

The metric on the string worldsheet $\mathcal{W}$ is the metric $g$ induced by the spacetime metric $G$ , i.e. the pullback of $G$ by the embedding $F$ :

In [31]:

g = W.induced_metric()

Nambu-Goto action

The determinant of $g$ is

In [32]:

detg = g.determinant().expr()

Expanding at fourth order in $a$ (will be required latter):

In [33]:

detg_a4 = detg.series(a, 5).truncate().simplify_full()

In [34]:

detg_a40 = detg_a4

In [35]:

detg_a4 = detg_a40

In [36]:

detg_a4 = detg_a4.subs({Xi_a: Xi_a_expr, Xi_b: Xi_b_expr, 
                        Delta_r: Delta_r_expr, Delta_th: Delta_th_expr})
detg_a4 = detg_a4.subs({mu: mu_s})

In [37]:

detg_a4 = detg_a4.series(a, 5).truncate().simplify_full()

In [38]:

detg_a4.variables()

\renewcommand{\Bold}[1]{\mathbf{#1}}\left({\mu_0}, a, {\ell}, m, n, r\right)

For the time being, only the expansion at second order in $a$ is required:

In [39]:

detg_a2 = detg_a4.series(a, 3).truncate().simplify_full()

The Nambu-Goto Lagrangian at second order in $a$ :

In [40]:

L_a2 = (sqrt(-detg_a2)).series(a, 3).truncate().simplify_full()
L_a2

\renewcommand{\Bold}[1]{\mathbf{#1}}-\frac{{\left(3 \, {\mu_0}^{2} a^{2} {\ell}^{4} n^{2} - 3 \, {\left({\mu_0}^{2} - 1\right)} a^{2} {\ell}^{4} - 2 \, {\ell}^{2}\right)} r^{4} - {\left({\mu_0}^{2} - 1\right)} a^{2} - {\left(4 \, {\mu_0}^{2} a^{2} {\ell}^{2} m - {\mu_0}^{2} a^{2}\right)} n^{2} + {\left({\left({\mu_0}^{2} - 1\right)} a^{2} {\ell}^{4} r^{8} + 2 \, {\left({\mu_0}^{2} - 1\right)} a^{2} {\ell}^{2} r^{6} - 4 \, {\left({\mu_0}^{2} - 1\right)} a^{2} m r^{2} + 4 \, {\left({\mu_0}^{2} - 1\right)} a^{2} m^{2} - {\left(4 \, {\left({\mu_0}^{2} - 1\right)} a^{2} {\ell}^{2} m - {\left({\mu_0}^{2} - 1\right)} a^{2}\right)} r^{4}\right)} \frac{\partial}{\partial r}\phi_{1}\left(r\right)^{2} - {\left({\mu_0}^{2} a^{2} {\ell}^{4} n^{2} r^{8} + 2 \, {\mu_0}^{2} a^{2} {\ell}^{2} n^{2} r^{6} - 4 \, {\mu_0}^{2} a^{2} m n^{2} r^{2} + 4 \, {\mu_0}^{2} a^{2} m^{2} n^{2} - {\left(4 \, {\mu_0}^{2} a^{2} {\ell}^{2} m - {\mu_0}^{2} a^{2}\right)} n^{2} r^{4}\right)} \frac{\partial}{\partial r}\psi_{1}\left(r\right)^{2} + 4 \, {\left({\left({\mu_0}^{2} - 1\right)} a^{2} {\ell}^{2} + 1\right)} m - 2 \, r^{2}}{2 \, {\left({\ell}^{2} r^{4} + r^{2} - 2 \, m\right)}}

In [41]:

L_a2.numerator()

\renewcommand{\Bold}[1]{\mathbf{#1}}{\mu_0}^{2} a^{2} {\ell}^{4} n^{2} r^{8} \frac{\partial}{\partial r}\psi_{1}\left(r\right)^{2} - {\mu_0}^{2} a^{2} {\ell}^{4} r^{8} \frac{\partial}{\partial r}\phi_{1}\left(r\right)^{2} + a^{2} {\ell}^{4} r^{8} \frac{\partial}{\partial r}\phi_{1}\left(r\right)^{2} + 2 \, {\mu_0}^{2} a^{2} {\ell}^{2} n^{2} r^{6} \frac{\partial}{\partial r}\psi_{1}\left(r\right)^{2} - 4 \, {\mu_0}^{2} a^{2} {\ell}^{2} m n^{2} r^{4} \frac{\partial}{\partial r}\psi_{1}\left(r\right)^{2} - 3 \, {\mu_0}^{2} a^{2} {\ell}^{4} n^{2} r^{4} - 2 \, {\mu_0}^{2} a^{2} {\ell}^{2} r^{6} \frac{\partial}{\partial r}\phi_{1}\left(r\right)^{2} + 4 \, {\mu_0}^{2} a^{2} {\ell}^{2} m r^{4} \frac{\partial}{\partial r}\phi_{1}\left(r\right)^{2} + 3 \, {\mu_0}^{2} a^{2} {\ell}^{4} r^{4} + 2 \, a^{2} {\ell}^{2} r^{6} \frac{\partial}{\partial r}\phi_{1}\left(r\right)^{2} + {\mu_0}^{2} a^{2} n^{2} r^{4} \frac{\partial}{\partial r}\psi_{1}\left(r\right)^{2} - 4 \, a^{2} {\ell}^{2} m r^{4} \frac{\partial}{\partial r}\phi_{1}\left(r\right)^{2} - 4 \, {\mu_0}^{2} a^{2} m n^{2} r^{2} \frac{\partial}{\partial r}\psi_{1}\left(r\right)^{2} - 3 \, a^{2} {\ell}^{4} r^{4} - {\mu_0}^{2} a^{2} r^{4} \frac{\partial}{\partial r}\phi_{1}\left(r\right)^{2} + 4 \, {\mu_0}^{2} a^{2} m^{2} n^{2} \frac{\partial}{\partial r}\psi_{1}\left(r\right)^{2} + 4 \, {\mu_0}^{2} a^{2} {\ell}^{2} m n^{2} + 4 \, {\mu_0}^{2} a^{2} m r^{2} \frac{\partial}{\partial r}\phi_{1}\left(r\right)^{2} - 4 \, {\mu_0}^{2} a^{2} m^{2} \frac{\partial}{\partial r}\phi_{1}\left(r\right)^{2} + a^{2} r^{4} \frac{\partial}{\partial r}\phi_{1}\left(r\right)^{2} - 4 \, {\mu_0}^{2} a^{2} {\ell}^{2} m - 4 \, a^{2} m r^{2} \frac{\partial}{\partial r}\phi_{1}\left(r\right)^{2} - {\mu_0}^{2} a^{2} n^{2} + 2 \, {\ell}^{2} r^{4} + 4 \, a^{2} m^{2} \frac{\partial}{\partial r}\phi_{1}\left(r\right)^{2} + 4 \, a^{2} {\ell}^{2} m + {\mu_0}^{2} a^{2} - a^{2} + 2 \, r^{2} - 4 \, m

In [42]:

L_a2.denominator()

\renewcommand{\Bold}[1]{\mathbf{#1}}2 \, {\ell}^{2} r^{4} + 2 \, r^{2} - 4 \, m

Euler-Lagrange equations

In [43]:

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 $a^2$ for $\phi_1$ and $\psi_1$ :

In [44]:

eqs = euler_lagrange(L_a2, [phi_1, psi_1], r)
eqs

\renewcommand{\Bold}[1]{\mathbf{#1}}\left[2 \, {\left(2 \, {\left({\mu_0}^{2} - 1\right)} a^{2} {\ell}^{2} r^{3} + {\left({\mu_0}^{2} - 1\right)} a^{2} r\right)} \frac{\partial}{\partial r}\phi_{1}\left(r\right) + {\left({\left({\mu_0}^{2} - 1\right)} a^{2} {\ell}^{2} r^{4} + {\left({\mu_0}^{2} - 1\right)} a^{2} r^{2} - 2 \, {\left({\mu_0}^{2} - 1\right)} a^{2} m\right)} \frac{\partial^{2}}{(\partial r)^{2}}\phi_{1}\left(r\right) = 0, -2 \, {\left(2 \, {\mu_0}^{2} a^{2} {\ell}^{2} n^{2} r^{3} + {\mu_0}^{2} a^{2} n^{2} r\right)} \frac{\partial}{\partial r}\psi_{1}\left(r\right) - {\left({\mu_0}^{2} a^{2} {\ell}^{2} n^{2} r^{4} + {\mu_0}^{2} a^{2} n^{2} r^{2} - 2 \, {\mu_0}^{2} a^{2} m n^{2}\right)} \frac{\partial^{2}}{(\partial r)^{2}}\psi_{1}\left(r\right) = 0\right]

Solving the equation for $\phi_1$ (Eq. (4.8))

In [45]:

eq_phi1 = eqs[0]
eq_phi1

\renewcommand{\Bold}[1]{\mathbf{#1}}2 \, {\left(2 \, {\left({\mu_0}^{2} - 1\right)} a^{2} {\ell}^{2} r^{3} + {\left({\mu_0}^{2} - 1\right)} a^{2} r\right)} \frac{\partial}{\partial r}\phi_{1}\left(r\right) + {\left({\left({\mu_0}^{2} - 1\right)} a^{2} {\ell}^{2} r^{4} + {\left({\mu_0}^{2} - 1\right)} a^{2} r^{2} - 2 \, {\left({\mu_0}^{2} - 1\right)} a^{2} m\right)} \frac{\partial^{2}}{(\partial r)^{2}}\phi_{1}\left(r\right) = 0

In [46]:

eq_phi1 = (eq_phi1/(a^2*(Mu0^2-1))).simplify_full()
eq_phi1

\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

In [47]:

phi1_sol(r) = desolve(eq_phi1, phi_1(r), ivar=r)
phi1_sol(r)

\renewcommand{\Bold}[1]{\mathbf{#1}}K_{1} \int \frac{1}{{\ell}^{2} r^{4} + r^{2} - 2 \, m}\,{d r} + K_{2}

We recover Eqs. (4.8) with $K_1 = \mathfrak{p}$ and $K_2=0$ .

The symbolic constants $K_1$ and $K_2$ are actually denoted _K1 and _K2 by SageMath, as the print reveals:

In [48]:

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'):

In [49]:

pf = var("pf", latex_name=r"\mathfrak{p}")
phi1_sol(r) = phi1_sol(r).subs({SR.var('_K1'): pf, SR.var('_K2'): 0})
phi1_sol(r)

\renewcommand{\Bold}[1]{\mathbf{#1}}{\mathfrak{p}} \int \frac{1}{{\ell}^{2} r^{4} + r^{2} - 2 \, m}\,{d r}

Solving the equation for $\psi_1$ (Eq. (4.8))

In [50]:

eq_psi1 = eqs[1]
eq_psi1

\renewcommand{\Bold}[1]{\mathbf{#1}}-2 \, {\left(2 \, {\mu_0}^{2} a^{2} {\ell}^{2} n^{2} r^{3} + {\mu_0}^{2} a^{2} n^{2} r\right)} \frac{\partial}{\partial r}\psi_{1}\left(r\right) - {\left({\mu_0}^{2} a^{2} {\ell}^{2} n^{2} r^{4} + {\mu_0}^{2} a^{2} n^{2} r^{2} - 2 \, {\mu_0}^{2} a^{2} m n^{2}\right)} \frac{\partial^{2}}{(\partial r)^{2}}\psi_{1}\left(r\right) = 0

In [51]:

eq_psi1 = (eq_psi1/(a^2*Mu0^2)).simplify_full()
eq_psi1

\renewcommand{\Bold}[1]{\mathbf{#1}}-2 \, {\left(2 \, {\ell}^{2} n^{2} r^{3} + n^{2} r\right)} \frac{\partial}{\partial r}\psi_{1}\left(r\right) - {\left({\ell}^{2} n^{2} r^{4} + n^{2} r^{2} - 2 \, m n^{2}\right)} \frac{\partial^{2}}{(\partial r)^{2}}\psi_{1}\left(r\right) = 0

In [52]:

psi1_sol(r) = desolve(eq_psi1, psi_1(r), ivar=r)
psi1_sol(r)

\renewcommand{\Bold}[1]{\mathbf{#1}}K_{1} \int \frac{1}{{\ell}^{2} r^{4} + r^{2} - 2 \, m}\,{d r} + K_{2}

We recover Eq. (4.8) with $K_1 = \mathfrak{q}$ and $K_2=0$ .

In [53]:

qf = var('qf', latex_name=r"\mathfrak{q}")
psi1_sol(r) = psi1_sol(r).subs({SR.var('_K1'): qf, SR.var('_K2'): 0})
psi1_sol(r)

\renewcommand{\Bold}[1]{\mathbf{#1}}{\mathfrak{q}} \int \frac{1}{{\ell}^{2} r^{4} + r^{2} - 2 \, m}\,{d r}

Nambu-Goto Lagrangian at fourth order in $a$

In [54]:

L_a4 = (sqrt(-detg_a4)).series(a, 5).truncate().simplify_full()

In [55]:

eqs = euler_lagrange(L_a4, [phi_1, psi_1, mu_1], r)

The equation for $\mu_1$

In [56]:

eq_mu1 = eqs[2]
eq_mu1

\renewcommand{\Bold}[1]{\mathbf{#1}}-\frac{4 \, {\left({\mu_0}^{3} - {\mu_0}\right)} a^{4} {\ell}^{2} m - {\left({\mu_0}^{3} - {\mu_0}\right)} a^{4} - {\left(4 \, {\left({\mu_0}^{3} - {\mu_0}\right)} a^{4} {\ell}^{2} m - {\left({\mu_0}^{3} - {\mu_0}\right)} a^{4}\right)} n^{4} + 3 \, {\left({\left({\mu_0}^{3} - {\mu_0}\right)} a^{4} {\ell}^{4} n^{4} + 2 \, {\left({\mu_0}^{3} - {\mu_0}\right)} a^{4} {\ell}^{4} n^{3} - 2 \, {\left({\mu_0}^{3} - {\mu_0}\right)} a^{4} {\ell}^{4} n - {\left({\mu_0}^{3} - {\mu_0}\right)} a^{4} {\ell}^{4}\right)} r^{4} - 2 \, {\left(4 \, {\left({\mu_0}^{3} - {\mu_0}\right)} a^{4} {\ell}^{2} m - {\left({\mu_0}^{3} - {\mu_0}\right)} a^{4}\right)} n^{3} + {\left(4 \, {\left({\mu_0}^{3} - {\mu_0}\right)} a^{4} m^{2} n^{2} + {\left({\left({\mu_0}^{3} - {\mu_0}\right)} a^{4} {\ell}^{4} n^{2} + 2 \, {\left({\mu_0}^{3} - {\mu_0}\right)} a^{4} {\ell}^{4} n + {\left({\mu_0}^{3} - {\mu_0}\right)} a^{4} {\ell}^{4}\right)} r^{8} + 8 \, {\left({\mu_0}^{3} - {\mu_0}\right)} a^{4} m^{2} n + 4 \, {\left({\mu_0}^{3} - {\mu_0}\right)} a^{4} m^{2} + 2 \, {\left({\left({\mu_0}^{3} - {\mu_0}\right)} a^{4} {\ell}^{2} n^{2} + 2 \, {\left({\mu_0}^{3} - {\mu_0}\right)} a^{4} {\ell}^{2} n + {\left({\mu_0}^{3} - {\mu_0}\right)} a^{4} {\ell}^{2}\right)} r^{6} - {\left(4 \, {\left({\mu_0}^{3} - {\mu_0}\right)} a^{4} {\ell}^{2} m - {\left({\mu_0}^{3} - {\mu_0}\right)} a^{4} + {\left(4 \, {\left({\mu_0}^{3} - {\mu_0}\right)} a^{4} {\ell}^{2} m - {\left({\mu_0}^{3} - {\mu_0}\right)} a^{4}\right)} n^{2} + 2 \, {\left(4 \, {\left({\mu_0}^{3} - {\mu_0}\right)} a^{4} {\ell}^{2} m - {\left({\mu_0}^{3} - {\mu_0}\right)} a^{4}\right)} n\right)} r^{4} - 4 \, {\left({\left({\mu_0}^{3} - {\mu_0}\right)} a^{4} m n^{2} + 2 \, {\left({\mu_0}^{3} - {\mu_0}\right)} a^{4} m n + {\left({\mu_0}^{3} - {\mu_0}\right)} a^{4} m\right)} r^{2}\right)} \frac{\partial}{\partial r}\phi_{1}\left(r\right)^{2} - {\left(4 \, {\left({\mu_0}^{3} - {\mu_0}\right)} a^{4} m^{2} n^{4} + 8 \, {\left({\mu_0}^{3} - {\mu_0}\right)} a^{4} m^{2} n^{3} + 4 \, {\left({\mu_0}^{3} - {\mu_0}\right)} a^{4} m^{2} n^{2} + {\left({\left({\mu_0}^{3} - {\mu_0}\right)} a^{4} {\ell}^{4} n^{4} + 2 \, {\left({\mu_0}^{3} - {\mu_0}\right)} a^{4} {\ell}^{4} n^{3} + {\left({\mu_0}^{3} - {\mu_0}\right)} a^{4} {\ell}^{4} n^{2}\right)} r^{8} + 2 \, {\left({\left({\mu_0}^{3} - {\mu_0}\right)} a^{4} {\ell}^{2} n^{4} + 2 \, {\left({\mu_0}^{3} - {\mu_0}\right)} a^{4} {\ell}^{2} n^{3} + {\left({\mu_0}^{3} - {\mu_0}\right)} a^{4} {\ell}^{2} n^{2}\right)} r^{6} - {\left({\left(4 \, {\left({\mu_0}^{3} - {\mu_0}\right)} a^{4} {\ell}^{2} m - {\left({\mu_0}^{3} - {\mu_0}\right)} a^{4}\right)} n^{4} + 2 \, {\left(4 \, {\left({\mu_0}^{3} - {\mu_0}\right)} a^{4} {\ell}^{2} m - {\left({\mu_0}^{3} - {\mu_0}\right)} a^{4}\right)} n^{3} + {\left(4 \, {\left({\mu_0}^{3} - {\mu_0}\right)} a^{4} {\ell}^{2} m - {\left({\mu_0}^{3} - {\mu_0}\right)} a^{4}\right)} n^{2}\right)} r^{4} - 4 \, {\left({\left({\mu_0}^{3} - {\mu_0}\right)} a^{4} m n^{4} + 2 \, {\left({\mu_0}^{3} - {\mu_0}\right)} a^{4} m n^{3} + {\left({\mu_0}^{3} - {\mu_0}\right)} a^{4} m n^{2}\right)} r^{2}\right)} \frac{\partial}{\partial r}\psi_{1}\left(r\right)^{2} + 2 \, {\left(4 \, {\left({\mu_0}^{3} - {\mu_0}\right)} a^{4} {\ell}^{2} m - {\left({\mu_0}^{3} - {\mu_0}\right)} a^{4}\right)} n - 2 \, {\left(2 \, {\left(a^{4} {\ell}^{4} n^{4} + 4 \, a^{4} {\ell}^{4} n^{3} + 6 \, a^{4} {\ell}^{4} n^{2} + 4 \, a^{4} {\ell}^{4} n + a^{4} {\ell}^{4}\right)} r^{7} + 3 \, {\left(a^{4} {\ell}^{2} n^{4} + 4 \, a^{4} {\ell}^{2} n^{3} + 6 \, a^{4} {\ell}^{2} n^{2} + 4 \, a^{4} {\ell}^{2} n + a^{4} {\ell}^{2}\right)} r^{5} - {\left(4 \, a^{4} {\ell}^{2} m + {\left(4 \, a^{4} {\ell}^{2} m - a^{4}\right)} n^{4} - a^{4} + 4 \, {\left(4 \, a^{4} {\ell}^{2} m - a^{4}\right)} n^{3} + 6 \, {\left(4 \, a^{4} {\ell}^{2} m - a^{4}\right)} n^{2} + 4 \, {\left(4 \, a^{4} {\ell}^{2} m - a^{4}\right)} n\right)} r^{3} - 2 \, {\left(a^{4} m n^{4} + 4 \, a^{4} m n^{3} + 6 \, a^{4} m n^{2} + 4 \, a^{4} m n + a^{4} m\right)} r\right)} \frac{\partial}{\partial r}\mu_{1}\left(r\right) - {\left(4 \, a^{4} m^{2} n^{4} + 16 \, a^{4} m^{2} n^{3} + {\left(a^{4} {\ell}^{4} n^{4} + 4 \, a^{4} {\ell}^{4} n^{3} + 6 \, a^{4} {\ell}^{4} n^{2} + 4 \, a^{4} {\ell}^{4} n + a^{4} {\ell}^{4}\right)} r^{8} + 24 \, a^{4} m^{2} n^{2} + 16 \, a^{4} m^{2} n + 2 \, {\left(a^{4} {\ell}^{2} n^{4} + 4 \, a^{4} {\ell}^{2} n^{3} + 6 \, a^{4} {\ell}^{2} n^{2} + 4 \, a^{4} {\ell}^{2} n + a^{4} {\ell}^{2}\right)} r^{6} + 4 \, a^{4} m^{2} - {\left(4 \, a^{4} {\ell}^{2} m + {\left(4 \, a^{4} {\ell}^{2} m - a^{4}\right)} n^{4} - a^{4} + 4 \, {\left(4 \, a^{4} {\ell}^{2} m - a^{4}\right)} n^{3} + 6 \, {\left(4 \, a^{4} {\ell}^{2} m - a^{4}\right)} n^{2} + 4 \, {\left(4 \, a^{4} {\ell}^{2} m - a^{4}\right)} n\right)} r^{4} - 4 \, {\left(a^{4} m n^{4} + 4 \, a^{4} m n^{3} + 6 \, a^{4} m n^{2} + 4 \, a^{4} m n + a^{4} m\right)} r^{2}\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

In [57]:

eq_mu1.lhs().denominator().simplify_full()

\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

In [58]:

eq_mu1 = eq_mu1.lhs().numerator().simplify_full() == 0
#eq_mu1

In [59]:

eq_mu1 = (eq_mu1/a^4).simplify_full()
eq_mu1

\renewcommand{\Bold}[1]{\mathbf{#1}}{\left(4 \, {\left({\mu_0}^{3} - {\mu_0}\right)} {\ell}^{2} m - {\mu_0}^{3} + {\mu_0}\right)} n^{4} - 3 \, {\left({\left({\mu_0}^{3} - {\mu_0}\right)} {\ell}^{4} n^{4} + 2 \, {\left({\mu_0}^{3} - {\mu_0}\right)} {\ell}^{4} n^{3} - 2 \, {\left({\mu_0}^{3} - {\mu_0}\right)} {\ell}^{4} n - {\left({\mu_0}^{3} - {\mu_0}\right)} {\ell}^{4}\right)} r^{4} - 4 \, {\left({\mu_0}^{3} - {\mu_0}\right)} {\ell}^{2} m + 2 \, {\left(4 \, {\left({\mu_0}^{3} - {\mu_0}\right)} {\ell}^{2} m - {\mu_0}^{3} + {\mu_0}\right)} n^{3} + {\mu_0}^{3} - {\left({\left({\left({\mu_0}^{3} - {\mu_0}\right)} {\ell}^{4} n^{2} + 2 \, {\left({\mu_0}^{3} - {\mu_0}\right)} {\ell}^{4} n + {\left({\mu_0}^{3} - {\mu_0}\right)} {\ell}^{4}\right)} r^{8} + 2 \, {\left({\left({\mu_0}^{3} - {\mu_0}\right)} {\ell}^{2} n^{2} + 2 \, {\left({\mu_0}^{3} - {\mu_0}\right)} {\ell}^{2} n + {\left({\mu_0}^{3} - {\mu_0}\right)} {\ell}^{2}\right)} r^{6} + 4 \, {\left({\mu_0}^{3} - {\mu_0}\right)} m^{2} n^{2} - {\left(4 \, {\left({\mu_0}^{3} - {\mu_0}\right)} {\ell}^{2} m - {\mu_0}^{3} + {\left(4 \, {\left({\mu_0}^{3} - {\mu_0}\right)} {\ell}^{2} m - {\mu_0}^{3} + {\mu_0}\right)} n^{2} + 2 \, {\left(4 \, {\left({\mu_0}^{3} - {\mu_0}\right)} {\ell}^{2} m - {\mu_0}^{3} + {\mu_0}\right)} n + {\mu_0}\right)} r^{4} + 8 \, {\left({\mu_0}^{3} - {\mu_0}\right)} m^{2} n + 4 \, {\left({\mu_0}^{3} - {\mu_0}\right)} m^{2} - 4 \, {\left({\left({\mu_0}^{3} - {\mu_0}\right)} m n^{2} + 2 \, {\left({\mu_0}^{3} - {\mu_0}\right)} m n + {\left({\mu_0}^{3} - {\mu_0}\right)} m\right)} r^{2}\right)} \frac{\partial}{\partial r}\phi_{1}\left(r\right)^{2} + {\left({\left({\left({\mu_0}^{3} - {\mu_0}\right)} {\ell}^{4} n^{4} + 2 \, {\left({\mu_0}^{3} - {\mu_0}\right)} {\ell}^{4} n^{3} + {\left({\mu_0}^{3} - {\mu_0}\right)} {\ell}^{4} n^{2}\right)} r^{8} + 4 \, {\left({\mu_0}^{3} - {\mu_0}\right)} m^{2} n^{4} + 2 \, {\left({\left({\mu_0}^{3} - {\mu_0}\right)} {\ell}^{2} n^{4} + 2 \, {\left({\mu_0}^{3} - {\mu_0}\right)} {\ell}^{2} n^{3} + {\left({\mu_0}^{3} - {\mu_0}\right)} {\ell}^{2} n^{2}\right)} r^{6} + 8 \, {\left({\mu_0}^{3} - {\mu_0}\right)} m^{2} n^{3} + 4 \, {\left({\mu_0}^{3} - {\mu_0}\right)} m^{2} n^{2} - {\left({\left(4 \, {\left({\mu_0}^{3} - {\mu_0}\right)} {\ell}^{2} m - {\mu_0}^{3} + {\mu_0}\right)} n^{4} + 2 \, {\left(4 \, {\left({\mu_0}^{3} - {\mu_0}\right)} {\ell}^{2} m - {\mu_0}^{3} + {\mu_0}\right)} n^{3} + {\left(4 \, {\left({\mu_0}^{3} - {\mu_0}\right)} {\ell}^{2} m - {\mu_0}^{3} + {\mu_0}\right)} n^{2}\right)} r^{4} - 4 \, {\left({\left({\mu_0}^{3} - {\mu_0}\right)} m n^{4} + 2 \, {\left({\mu_0}^{3} - {\mu_0}\right)} m n^{3} + {\left({\mu_0}^{3} - {\mu_0}\right)} m n^{2}\right)} r^{2}\right)} \frac{\partial}{\partial r}\psi_{1}\left(r\right)^{2} - 2 \, {\left(4 \, {\left({\mu_0}^{3} - {\mu_0}\right)} {\ell}^{2} m - {\mu_0}^{3} + {\mu_0}\right)} n + 2 \, {\left(2 \, {\left({\ell}^{4} n^{4} + 4 \, {\ell}^{4} n^{3} + 6 \, {\ell}^{4} n^{2} + 4 \, {\ell}^{4} n + {\ell}^{4}\right)} r^{7} + 3 \, {\left({\ell}^{2} n^{4} + 4 \, {\ell}^{2} n^{3} + 6 \, {\ell}^{2} n^{2} + 4 \, {\ell}^{2} n + {\ell}^{2}\right)} r^{5} - {\left({\left(4 \, {\ell}^{2} m - 1\right)} n^{4} + 4 \, {\left(4 \, {\ell}^{2} m - 1\right)} n^{3} + 4 \, {\ell}^{2} m + 6 \, {\left(4 \, {\ell}^{2} m - 1\right)} n^{2} + 4 \, {\left(4 \, {\ell}^{2} m - 1\right)} n - 1\right)} r^{3} - 2 \, {\left(m n^{4} + 4 \, m n^{3} + 6 \, m n^{2} + 4 \, m n + m\right)} r\right)} \frac{\partial}{\partial r}\mu_{1}\left(r\right) + {\left({\left({\ell}^{4} n^{4} + 4 \, {\ell}^{4} n^{3} + 6 \, {\ell}^{4} n^{2} + 4 \, {\ell}^{4} n + {\ell}^{4}\right)} r^{8} + 2 \, {\left({\ell}^{2} n^{4} + 4 \, {\ell}^{2} n^{3} + 6 \, {\ell}^{2} n^{2} + 4 \, {\ell}^{2} n + {\ell}^{2}\right)} r^{6} + 4 \, m^{2} n^{4} + 16 \, m^{2} n^{3} - {\left({\left(4 \, {\ell}^{2} m - 1\right)} n^{4} + 4 \, {\left(4 \, {\ell}^{2} m - 1\right)} n^{3} + 4 \, {\ell}^{2} m + 6 \, {\left(4 \, {\ell}^{2} m - 1\right)} n^{2} + 4 \, {\left(4 \, {\ell}^{2} m - 1\right)} n - 1\right)} r^{4} + 24 \, m^{2} n^{2} + 16 \, m^{2} n - 4 \, {\left(m n^{4} + 4 \, m n^{3} + 6 \, m n^{2} + 4 \, m n + m\right)} 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 $\phi_1(r)$ and $\psi_1(r)$ in this equation:

In [60]:

eq_mu1 = eq_mu1.substitute_function(phi_1, phi1_sol).substitute_function(psi_1, psi1_sol)
eq_mu1 = eq_mu1.simplify_full()
eq_mu1

\renewcommand{\Bold}[1]{\mathbf{#1}}{\left(4 \, {\left({\mu_0}^{3} - {\mu_0}\right)} {\ell}^{2} m - {\mu_0}^{3} + {\mu_0}\right)} n^{4} - 3 \, {\left({\left({\mu_0}^{3} - {\mu_0}\right)} {\ell}^{4} n^{4} + 2 \, {\left({\mu_0}^{3} - {\mu_0}\right)} {\ell}^{4} n^{3} - 2 \, {\left({\mu_0}^{3} - {\mu_0}\right)} {\ell}^{4} n - {\left({\mu_0}^{3} - {\mu_0}\right)} {\ell}^{4}\right)} r^{4} - 4 \, {\left({\mu_0}^{3} - {\mu_0}\right)} {\ell}^{2} m + 2 \, {\left(4 \, {\left({\mu_0}^{3} - {\mu_0}\right)} {\ell}^{2} m - {\mu_0}^{3} + {\mu_0}\right)} n^{3} + {\mu_0}^{3} - {\left({\mu_0}^{3} + {\left({\mu_0}^{3} - {\mu_0}\right)} n^{2} + 2 \, {\left({\mu_0}^{3} - {\mu_0}\right)} n - {\mu_0}\right)} {\mathfrak{p}}^{2} + {\left({\left({\mu_0}^{3} - {\mu_0}\right)} n^{4} + 2 \, {\left({\mu_0}^{3} - {\mu_0}\right)} n^{3} + {\left({\mu_0}^{3} - {\mu_0}\right)} n^{2}\right)} {\mathfrak{q}}^{2} - 2 \, {\left(4 \, {\left({\mu_0}^{3} - {\mu_0}\right)} {\ell}^{2} m - {\mu_0}^{3} + {\mu_0}\right)} n + 2 \, {\left(2 \, {\left({\ell}^{4} n^{4} + 4 \, {\ell}^{4} n^{3} + 6 \, {\ell}^{4} n^{2} + 4 \, {\ell}^{4} n + {\ell}^{4}\right)} r^{7} + 3 \, {\left({\ell}^{2} n^{4} + 4 \, {\ell}^{2} n^{3} + 6 \, {\ell}^{2} n^{2} + 4 \, {\ell}^{2} n + {\ell}^{2}\right)} r^{5} - {\left({\left(4 \, {\ell}^{2} m - 1\right)} n^{4} + 4 \, {\left(4 \, {\ell}^{2} m - 1\right)} n^{3} + 4 \, {\ell}^{2} m + 6 \, {\left(4 \, {\ell}^{2} m - 1\right)} n^{2} + 4 \, {\left(4 \, {\ell}^{2} m - 1\right)} n - 1\right)} r^{3} - 2 \, {\left(m n^{4} + 4 \, m n^{3} + 6 \, m n^{2} + 4 \, m n + m\right)} r\right)} \frac{\partial}{\partial r}\mu_{1}\left(r\right) + {\left({\left({\ell}^{4} n^{4} + 4 \, {\ell}^{4} n^{3} + 6 \, {\ell}^{4} n^{2} + 4 \, {\ell}^{4} n + {\ell}^{4}\right)} r^{8} + 2 \, {\left({\ell}^{2} n^{4} + 4 \, {\ell}^{2} n^{3} + 6 \, {\ell}^{2} n^{2} + 4 \, {\ell}^{2} n + {\ell}^{2}\right)} r^{6} + 4 \, m^{2} n^{4} + 16 \, m^{2} n^{3} - {\left({\left(4 \, {\ell}^{2} m - 1\right)} n^{4} + 4 \, {\left(4 \, {\ell}^{2} m - 1\right)} n^{3} + 4 \, {\ell}^{2} m + 6 \, {\left(4 \, {\ell}^{2} m - 1\right)} n^{2} + 4 \, {\left(4 \, {\ell}^{2} m - 1\right)} n - 1\right)} r^{4} + 24 \, m^{2} n^{2} + 16 \, m^{2} n - 4 \, {\left(m n^{4} + 4 \, m n^{3} + 6 \, m n^{2} + 4 \, m n + m\right)} r^{2} + 4 \, m^{2}\right)} \frac{\partial^{2}}{(\partial r)^{2}}\mu_{1}\left(r\right) - {\mu_0} = 0

Check of Eq. (4.9)

In [61]:

lhs = eq_mu1.lhs()
lhs = lhs.simplify_full()
lhs

\renewcommand{\Bold}[1]{\mathbf{#1}}{\left(4 \, {\left({\mu_0}^{3} - {\mu_0}\right)} {\ell}^{2} m - {\mu_0}^{3} + {\mu_0}\right)} n^{4} - 3 \, {\left({\left({\mu_0}^{3} - {\mu_0}\right)} {\ell}^{4} n^{4} + 2 \, {\left({\mu_0}^{3} - {\mu_0}\right)} {\ell}^{4} n^{3} - 2 \, {\left({\mu_0}^{3} - {\mu_0}\right)} {\ell}^{4} n - {\left({\mu_0}^{3} - {\mu_0}\right)} {\ell}^{4}\right)} r^{4} - 4 \, {\left({\mu_0}^{3} - {\mu_0}\right)} {\ell}^{2} m + 2 \, {\left(4 \, {\left({\mu_0}^{3} - {\mu_0}\right)} {\ell}^{2} m - {\mu_0}^{3} + {\mu_0}\right)} n^{3} + {\mu_0}^{3} - {\left({\mu_0}^{3} + {\left({\mu_0}^{3} - {\mu_0}\right)} n^{2} + 2 \, {\left({\mu_0}^{3} - {\mu_0}\right)} n - {\mu_0}\right)} {\mathfrak{p}}^{2} + {\left({\left({\mu_0}^{3} - {\mu_0}\right)} n^{4} + 2 \, {\left({\mu_0}^{3} - {\mu_0}\right)} n^{3} + {\left({\mu_0}^{3} - {\mu_0}\right)} n^{2}\right)} {\mathfrak{q}}^{2} - 2 \, {\left(4 \, {\left({\mu_0}^{3} - {\mu_0}\right)} {\ell}^{2} m - {\mu_0}^{3} + {\mu_0}\right)} n + 2 \, {\left(2 \, {\left({\ell}^{4} n^{4} + 4 \, {\ell}^{4} n^{3} + 6 \, {\ell}^{4} n^{2} + 4 \, {\ell}^{4} n + {\ell}^{4}\right)} r^{7} + 3 \, {\left({\ell}^{2} n^{4} + 4 \, {\ell}^{2} n^{3} + 6 \, {\ell}^{2} n^{2} + 4 \, {\ell}^{2} n + {\ell}^{2}\right)} r^{5} - {\left({\left(4 \, {\ell}^{2} m - 1\right)} n^{4} + 4 \, {\left(4 \, {\ell}^{2} m - 1\right)} n^{3} + 4 \, {\ell}^{2} m + 6 \, {\left(4 \, {\ell}^{2} m - 1\right)} n^{2} + 4 \, {\left(4 \, {\ell}^{2} m - 1\right)} n - 1\right)} r^{3} - 2 \, {\left(m n^{4} + 4 \, m n^{3} + 6 \, m n^{2} + 4 \, m n + m\right)} r\right)} \frac{\partial}{\partial r}\mu_{1}\left(r\right) + {\left({\left({\ell}^{4} n^{4} + 4 \, {\ell}^{4} n^{3} + 6 \, {\ell}^{4} n^{2} + 4 \, {\ell}^{4} n + {\ell}^{4}\right)} r^{8} + 2 \, {\left({\ell}^{2} n^{4} + 4 \, {\ell}^{2} n^{3} + 6 \, {\ell}^{2} n^{2} + 4 \, {\ell}^{2} n + {\ell}^{2}\right)} r^{6} + 4 \, m^{2} n^{4} + 16 \, m^{2} n^{3} - {\left({\left(4 \, {\ell}^{2} m - 1\right)} n^{4} + 4 \, {\left(4 \, {\ell}^{2} m - 1\right)} n^{3} + 4 \, {\ell}^{2} m + 6 \, {\left(4 \, {\ell}^{2} m - 1\right)} n^{2} + 4 \, {\left(4 \, {\ell}^{2} m - 1\right)} n - 1\right)} r^{4} + 24 \, m^{2} n^{2} + 16 \, m^{2} n - 4 \, {\left(m n^{4} + 4 \, m n^{3} + 6 \, m n^{2} + 4 \, m n + m\right)} r^{2} + 4 \, m^{2}\right)} \frac{\partial^{2}}{(\partial r)^{2}}\mu_{1}\left(r\right) - {\mu_0}

In [62]:

s = lhs.coefficient(diff(mu_1(r), r, 2))  # coefficient of mu_1''
s.factor()

\renewcommand{\Bold}[1]{\mathbf{#1}}{\left({\ell}^{2} r^{4} + r^{2} - 2 \, m\right)}^{2} {\left(n + 1\right)}^{4}

In [63]:

s1 = (lhs/s - diff(mu_1(r), r, 2)).simplify_full()
s1

\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{{\left({\mu_0}^{3} - {\mu_0}\right)} n^{2} {\mathfrak{q}}^{2} - 3 \, {\left({\left({\mu_0}^{3} - {\mu_0}\right)} {\ell}^{4} n^{2} - {\left({\mu_0}^{3} - {\mu_0}\right)} {\ell}^{4}\right)} r^{4} - 4 \, {\left({\mu_0}^{3} - {\mu_0}\right)} {\ell}^{2} m + {\mu_0}^{3} + {\left(4 \, {\left({\mu_0}^{3} - {\mu_0}\right)} {\ell}^{2} m - {\mu_0}^{3} + {\mu_0}\right)} n^{2} - {\left({\mu_0}^{3} - {\mu_0}\right)} {\mathfrak{p}}^{2} + 2 \, {\left(2 \, {\left({\ell}^{4} n^{2} + 2 \, {\ell}^{4} n + {\ell}^{4}\right)} r^{7} + 3 \, {\left({\ell}^{2} n^{2} + 2 \, {\ell}^{2} n + {\ell}^{2}\right)} r^{5} - {\left(4 \, {\ell}^{2} m + {\left(4 \, {\ell}^{2} m - 1\right)} n^{2} + 2 \, {\left(4 \, {\ell}^{2} m - 1\right)} n - 1\right)} r^{3} - 2 \, {\left(m n^{2} + 2 \, m n + m\right)} r\right)} \frac{\partial}{\partial r}\mu_{1}\left(r\right) - {\mu_0}}{{\left({\ell}^{4} n^{2} + 2 \, {\ell}^{4} n + {\ell}^{4}\right)} r^{8} + 2 \, {\left({\ell}^{2} n^{2} + 2 \, {\ell}^{2} n + {\ell}^{2}\right)} r^{6} - {\left(4 \, {\ell}^{2} m + {\left(4 \, {\ell}^{2} m - 1\right)} n^{2} + 2 \, {\left(4 \, {\ell}^{2} m - 1\right)} n - 1\right)} r^{4} + 4 \, m^{2} n^{2} + 8 \, m^{2} n - 4 \, {\left(m n^{2} + 2 \, m n + m\right)} r^{2} + 4 \, m^{2}}

In [64]:

b1 = s1.coefficient(diff(mu_1(r), r)).factor()
b1

\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{2 \, {\left(2 \, {\ell}^{2} r^{2} + 1\right)} r}{{\ell}^{2} r^{4} + r^{2} - 2 \, m}

In [65]:

b2 = (s1 - b1*diff(mu_1(r), r)).simplify_full().factor()
b2

\renewcommand{\Bold}[1]{\mathbf{#1}}-\frac{{\left(3 \, {\ell}^{4} n^{2} r^{4} - 3 \, {\ell}^{4} r^{4} - 4 \, {\ell}^{2} m n^{2} - n^{2} {\mathfrak{q}}^{2} + 4 \, {\ell}^{2} m + n^{2} + {\mathfrak{p}}^{2} - 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} {\left(n + 1\right)}^{2}}

The equation for $\mu_1$ is thus:

In [66]:

eq_mu1 = diff(mu_1(r), r, 2) + b1*diff(mu_1(r), r) + b2 == 0
eq_mu1

\renewcommand{\Bold}[1]{\mathbf{#1}}\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{{\left(3 \, {\ell}^{4} n^{2} r^{4} - 3 \, {\ell}^{4} r^{4} - 4 \, {\ell}^{2} m n^{2} - n^{2} {\mathfrak{q}}^{2} + 4 \, {\ell}^{2} m + n^{2} + {\mathfrak{p}}^{2} - 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} {\left(n + 1\right)}^{2}} + \frac{\partial^{2}}{(\partial r)^{2}}\mu_{1}\left(r\right) = 0

Let us define $\Theta_2 := 2 \Theta_0$

In [67]:

Th2 = var('Th2', latex_name=r'\Theta_2', 
          domain='real')

Given that $\mu_1(r) = - \sin\Theta_0 \; \theta_1(r) = - \sqrt{1-\mu_0^2} \; \theta_1(r)$ and $\sin2\Theta_0 = 2\mu_0\sqrt{1-\mu_0^2},$ we get the equation for $\Upsilon := \theta_1' = - \frac{\mu_1'}{\sqrt{1 - \mu_0}}$ :

In [68]:

Y = function('Y', latex_name=r'\Upsilon')
eq_Y = diff(Y(r), r) + b1*Y(r) \
       - (b2/(2*(1-Mu0^2)*Mu0)*sin(Th2)).factor() == 0
eq_Y

\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{2 \, {\left(2 \, {\ell}^{2} r^{2} + 1\right)} r \Upsilon\left(r\right)}{{\ell}^{2} r^{4} + r^{2} - 2 \, m} - \frac{{\left(3 \, {\ell}^{4} n^{2} r^{4} - 3 \, {\ell}^{4} r^{4} - 4 \, {\ell}^{2} m n^{2} - n^{2} {\mathfrak{q}}^{2} + 4 \, {\ell}^{2} m + n^{2} + {\mathfrak{p}}^{2} - 1\right)} \sin\left({\Theta_2}\right)}{2 \, {\left({\ell}^{2} r^{4} + r^{2} - 2 \, m\right)}^{2} {\left(n + 1\right)}^{2}} + \frac{\partial}{\partial r}\Upsilon\left(r\right) = 0

This agrees with Eq. (4.9) of the paper.

Solving the equation for $\Upsilon := \theta_1'$

We use the function desolve to solve the differential equation for $\Upsilon$ :

In [69]:

Y_sol(r) = desolve(eq_Y, Y(r), ivar=r)

The solution involves an integral that SageMath is not capable to evaluate with the default integrator. Trying to display Y_sol would make SageMath hang. Instead, we print Y_sol to get the unvaluated form of the integral, in order to compute it by means of FriCAS:

In [70]:

print(Y_sol(r))

1/2*(2*_C + 3*(l^2*n*sin(Th2) - l^2*sin(Th2))*r/(n + 1) - integrate((n^2*qf^2 + 2*l^2*m - (2*l^2*m + 1)*n^2 + 3*(l^2*n^2 - l^2)*r^2 - pf^2 + 1)/(l^2*r^4 + r^2 - 2*m), r)*sin(Th2)/(n^2 + 2*n + 1))/(l^2*r^4 + r^2 - 2*m)

The solution involves some constant, denoted _C by SageMath. We rename it C_1 and rewrite the above solution as

In [71]:

C_1 = var('C_1')
Integ(r) = function('Integ')(r)
Y_sol0(r) = 1/2*(2*C_1 + 3*(l^2*n*sin(Th2) - l^2*sin(Th2))*r/(n + 1) \
                 - Integ(r)*sin(Th2)/(n^2 + 2*n + 1))/(l^2*r^4 + r^2 - 2*m)
Y_sol0(r)

\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{2 \, C_{1} + \frac{3 \, {\left({\ell}^{2} n \sin\left({\Theta_2}\right) - {\ell}^{2} \sin\left({\Theta_2}\right)\right)} r}{n + 1} - \frac{{\rm Integ}\left(r\right) \sin\left({\Theta_2}\right)}{n^{2} + 2 \, n + 1}}{2 \, {\left({\ell}^{2} r^{4} + r^{2} - 2 \, m\right)}}

Integ(r) represents the integral $I(r)$ , whose integrand, $F(r)$ say, is read from the output of print(Y_sol(r)):

In [72]:

F(r) = (n^2*qf^2 + 2*l^2*m - (2*l^2*m + 1)*n^2 + 3*(l^2*n^2 - l^2)*r^2 - pf^2 + 1) \
       /(l^2*r^4 + r^2 - 2*m)
F(r)

\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{n^{2} {\mathfrak{q}}^{2} + 2 \, {\ell}^{2} m - {\left(2 \, {\ell}^{2} m + 1\right)} n^{2} + 3 \, {\left({\ell}^{2} n^{2} - {\ell}^{2}\right)} r^{2} - {\mathfrak{p}}^{2} + 1}{{\ell}^{2} r^{4} + r^{2} - 2 \, m}

We split the integral in two parts: $I(r) = F_1 \; s_1(r) + F_2 \; s_2(r)$ with $s_1(r) := \int^r \frac{\bar{r}^2}{\ell^2 \bar{r}^4 + \bar{r}^2 - 2m} \, \mathrm{d}\bar{r}, \qquad s_2(r) := \int^r \frac{\mathrm{d}\bar{r}}{\ell^2 \bar{r}^4 + \bar{r}^2 - 2m}$ and

In [73]:

F1 = 3*(l^2*n^2 - l^2)
F1

\renewcommand{\Bold}[1]{\mathbf{#1}}3 \, {\ell}^{2} n^{2} - 3 \, {\ell}^{2}

In [74]:

F2 = n^2*qf^2 + 2*l^2*m - (2*l^2*m + 1)*n^2 - pf^2 + 1
F2

\renewcommand{\Bold}[1]{\mathbf{#1}}n^{2} {\mathfrak{q}}^{2} + 2 \, {\ell}^{2} m - {\left(2 \, {\ell}^{2} m + 1\right)} n^{2} - {\mathfrak{p}}^{2} + 1

Check:

In [75]:

bool(F(r) == F1*r^2/(l^2*r^4 + r^2 - 2*m) + F2/(l^2*r^4 + r^2 - 2*m))

\renewcommand{\Bold}[1]{\mathbf{#1}}\mathrm{True}

Let us evaluate $s_1(r)$ by means of FriCAS:

In [76]:

s1 = integrate(r^2/(l^2*r^4 + r^2 - 2*m), r, algorithm='fricas')
s1

\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)

In [77]:

s1 = s1.canonicalize_radical().simplify_log()
s1

\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:

In [78]:

diff(s1, r).simplify_full()

\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{r^{2}}{{\ell}^{2} r^{4} + r^{2} - 2 \, m}

Similarly, we evaluate $s_2(r)$ by means of FriCAS:

In [79]:

s2 = integrate(1/(l^2*r^4 + r^2 - 2*m), r, algorithm='fricas')
s2

\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)

In [80]:

s2 = s2.canonicalize_radical().simplify_log()
s2

\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:

In [81]:

diff(s2, r).simplify_full()

\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{{\ell}^{2} r^{4} + r^{2} - 2 \, m}

In the above expressions for $s_1(r)$ and $s_2(r)$ there appears $\sqrt{1 + 8 \ell^2 m}$ , which can be rewritten $\sqrt{1 + 8 \ell^2 m} = 2 \ell^2 r_H^2 + 1$ where $r_H$ is the positive root of $\ell^2 r_H^4 + r_H^2 - 2m = 0$ . More precisely, we perform the following substitution: $m = \frac{1}{2} r_H^2 (\ell^2 r_H^2 + 1)$

In [82]:

rH = var('rH', latex_name=r'r_H', domain='real')
assume(rH > 0)
m_rH = rH^2*(l^2*rH^2 + 1)/2
s1 = s1.subs({m: m_rH}).canonicalize_radical().simplify_log()
s1

\renewcommand{\Bold}[1]{\mathbf{#1}}-\frac{{\ell} {r_H} \log\left(\frac{r + {r_H}}{r - {r_H}}\right) + i \, \sqrt{{\ell}^{2} {r_H}^{2} + 1} \log\left(\frac{{\ell} r - i \, \sqrt{{\ell}^{2} {r_H}^{2} + 1}}{{\ell} r + i \, \sqrt{{\ell}^{2} {r_H}^{2} + 1}}\right)}{2 \, {\left(2 \, {\ell}^{3} {r_H}^{2} + {\ell}\right)}}

In the second $\log$ , we recognize the $\mathrm{arccot}$ function, via the identity $\mathrm{arccot}\, x = \frac{i}{2} \ln\left( \frac{x - i}{x + i} \right) .$ Given that $\mathrm{arccot}\, x = \pi/2 - \mathrm{arctan}\, x$ , we use this identity as $i \ln\left( \frac{x - i}{x + i} \right) = - 2 \, \mathrm{arctan}(x) + \pi$

Thus, we perform the following substitution, disregarding the additive constant $\pi$ :

In [83]:

s1 = s1.subs({I*sqrt(l^2*rH^2 + 1)*log((l*r - I*sqrt(l^2*rH^2 + 1))/(l*r + I*sqrt(l^2*rH^2 + 1))):
              -2*sqrt(l^2*rH^2 + 1)*atan(l*r/sqrt(l^2*rH^2 + 1))})
s1

\renewcommand{\Bold}[1]{\mathbf{#1}}-\frac{{\ell} {r_H} \log\left(\frac{r + {r_H}}{r - {r_H}}\right) - 2 \, \sqrt{{\ell}^{2} {r_H}^{2} + 1} \arctan\left(\frac{{\ell} r}{\sqrt{{\ell}^{2} {r_H}^{2} + 1}}\right)}{2 \, {\left(2 \, {\ell}^{3} {r_H}^{2} + {\ell}\right)}}

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

In [84]:

Ds1 = diff(s1, r).simplify_full()
Ds1

\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{r^{2}}{{\ell}^{2} r^{4} - {\ell}^{2} {r_H}^{4} + r^{2} - {r_H}^{2}}

In [85]:

rH_m = sqrt(sqrt(1 + 8*l^2*m) - 1)/(sqrt(2)*l)
Ds1.subs({rH: rH_m}).simplify_full()

\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{r^{2}}{{\ell}^{2} r^{4} + r^{2} - 2 \, m}

Similarly, let us express $s_2$ in terms of $r_H$ :

In [86]:

s2 = s2.subs({m: m_rH}).canonicalize_radical().simplify_log()
s2

\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{i \, {\ell} {r_H} \log\left(\frac{-i \, {\ell}^{2} {r_H}^{2} + \sqrt{{\ell}^{2} {r_H}^{2} + 1} {\ell} r - i}{i \, {\ell}^{2} {r_H}^{2} + \sqrt{{\ell}^{2} {r_H}^{2} + 1} {\ell} r + i}\right) + \sqrt{{\ell}^{2} {r_H}^{2} + 1} \log\left(\frac{r - {r_H}}{r + {r_H}}\right)}{2 \, {\left(2 \, {\ell}^{2} {r_H}^{3} + {r_H}\right)} \sqrt{{\ell}^{2} {r_H}^{2} + 1}}

Again, we use the identity $i \ln\left( \frac{x - i}{x + i} \right) = - 2 \, \mathrm{arctan}(x) + \pi$ to rewrite $s_2$ as

In [87]:

s2 = s2.subs({I*l*rH*log((-I*l^2*rH^2 + sqrt(l^2*rH^2 + 1)*l*r - I)/(I*l^2*rH^2 + sqrt(l^2*rH^2 + 1)*l*r + I)):
             -2*l*rH*atan(l*r/sqrt(l^2*rH^2 + 1))})
s2

\renewcommand{\Bold}[1]{\mathbf{#1}}-\frac{2 \, {\ell} {r_H} \arctan\left(\frac{{\ell} r}{\sqrt{{\ell}^{2} {r_H}^{2} + 1}}\right) - \sqrt{{\ell}^{2} {r_H}^{2} + 1} \log\left(\frac{r - {r_H}}{r + {r_H}}\right)}{2 \, {\left(2 \, {\ell}^{2} {r_H}^{3} + {r_H}\right)} \sqrt{{\ell}^{2} {r_H}^{2} + 1}}

Let us also replace $\ln\left(\frac{r - r_H}{r + r_H}\right)$ by $-\ln\left(\frac{r + r_H}{r - r_H}\right)$ in order to have the same log term as in $s_1(r)$ :

In [88]:

s2 = s2.subs({log((r - rH)/(r + rH)): - log((r + rH)/(r - rH))})
s2

\renewcommand{\Bold}[1]{\mathbf{#1}}-\frac{2 \, {\ell} {r_H} \arctan\left(\frac{{\ell} r}{\sqrt{{\ell}^{2} {r_H}^{2} + 1}}\right) + \sqrt{{\ell}^{2} {r_H}^{2} + 1} \log\left(\frac{r + {r_H}}{r - {r_H}}\right)}{2 \, {\left(2 \, {\ell}^{2} {r_H}^{3} + {r_H}\right)} \sqrt{{\ell}^{2} {r_H}^{2} + 1}}

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

In [89]:

Ds2 = diff(s2, r).simplify_full()
Ds2

\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{{\ell}^{2} r^{4} - {\ell}^{2} {r_H}^{4} + r^{2} - {r_H}^{2}}

In [90]:

Ds2.subs({rH: rH_m}).simplify_full()

\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{{\ell}^{2} r^{4} + r^{2} - 2 \, m}

The full integral is thus

In [91]:

Integ0 = (F1*s1 + F2*s2).simplify_full()
Integ0

\renewcommand{\Bold}[1]{\mathbf{#1}}-\frac{{\left(n^{2} {\mathfrak{q}}^{2} + 2 \, {\ell}^{2} m - {\left(2 \, {\ell}^{2} m + 1\right)} n^{2} + 3 \, {\left({\ell}^{2} n^{2} - {\ell}^{2}\right)} {r_H}^{2} - {\mathfrak{p}}^{2} + 1\right)} \sqrt{{\ell}^{2} {r_H}^{2} + 1} \log\left(\frac{r + {r_H}}{r - {r_H}}\right) - 2 \, {\left(3 \, {\left({\ell}^{3} n^{2} - {\ell}^{3}\right)} {r_H}^{3} - {\left({\ell} n^{2} {\mathfrak{q}}^{2} + 2 \, {\ell}^{3} m - 2 \, {\left({\ell}^{3} m + 2 \, {\ell}\right)} n^{2} - {\ell} {\mathfrak{p}}^{2} + 4 \, {\ell}\right)} {r_H}\right)} \arctan\left(\frac{{\ell} r}{\sqrt{{\ell}^{2} {r_H}^{2} + 1}}\right)}{2 \, {\left(2 \, {\ell}^{2} {r_H}^{3} + {r_H}\right)} \sqrt{{\ell}^{2} {r_H}^{2} + 1}}

so that the solution is

In [92]:

Y_sol(r) = Y_sol0(r).subs({Integ(r): Integ0}).simplify_full()
Y_sol(r)

\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{2 \, {\left(3 \, {\left({\ell}^{3} n^{2} \sin\left({\Theta_2}\right) - {\ell}^{3} \sin\left({\Theta_2}\right)\right)} {r_H}^{3} - {\left({\ell} n^{2} {\mathfrak{q}}^{2} \sin\left({\Theta_2}\right) + 2 \, {\ell}^{3} m \sin\left({\Theta_2}\right) - {\ell} {\mathfrak{p}}^{2} \sin\left({\Theta_2}\right) - 2 \, {\left({\ell}^{3} m \sin\left({\Theta_2}\right) + 2 \, {\ell} \sin\left({\Theta_2}\right)\right)} n^{2} + 4 \, {\ell} \sin\left({\Theta_2}\right)\right)} {r_H}\right)} \arctan\left(\frac{{\ell} r}{\sqrt{{\ell}^{2} {r_H}^{2} + 1}}\right) - \sqrt{{\ell}^{2} {r_H}^{2} + 1} {\left(4 \, {\left(2 \, C_{1} {\ell}^{2} n^{2} + 4 \, C_{1} {\ell}^{2} n + 2 \, C_{1} {\ell}^{2} + 3 \, {\left({\ell}^{4} n^{2} \sin\left({\Theta_2}\right) - {\ell}^{4} \sin\left({\Theta_2}\right)\right)} r\right)} {r_H}^{3} + 2 \, {\left(2 \, C_{1} n^{2} + 4 \, C_{1} n + 3 \, {\left({\ell}^{2} n^{2} \sin\left({\Theta_2}\right) - {\ell}^{2} \sin\left({\Theta_2}\right)\right)} r + 2 \, C_{1}\right)} {r_H} + {\left(n^{2} {\mathfrak{q}}^{2} \sin\left({\Theta_2}\right) + 2 \, {\ell}^{2} m \sin\left({\Theta_2}\right) - {\left(2 \, {\ell}^{2} m \sin\left({\Theta_2}\right) + \sin\left({\Theta_2}\right)\right)} n^{2} + 3 \, {\left({\ell}^{2} n^{2} \sin\left({\Theta_2}\right) - {\ell}^{2} \sin\left({\Theta_2}\right)\right)} {r_H}^{2} - {\mathfrak{p}}^{2} \sin\left({\Theta_2}\right) + \sin\left({\Theta_2}\right)\right)} \log\left(\frac{r + {r_H}}{r - {r_H}}\right)\right)}}{4 \, \sqrt{{\ell}^{2} {r_H}^{2} + 1} {\left(2 \, {\left(2 \, {\ell}^{2} m n^{2} - {\left({\ell}^{4} n^{2} + 2 \, {\ell}^{4} n + {\ell}^{4}\right)} r^{4} + 4 \, {\ell}^{2} m n + 2 \, {\ell}^{2} m - {\left({\ell}^{2} n^{2} + 2 \, {\ell}^{2} n + {\ell}^{2}\right)} r^{2}\right)} {r_H}^{3} - {\left({\left({\ell}^{2} n^{2} + 2 \, {\ell}^{2} n + {\ell}^{2}\right)} r^{4} - 2 \, m n^{2} + {\left(n^{2} + 2 \, n + 1\right)} r^{2} - 4 \, m n - 2 \, m\right)} {r_H}\right)}}

In [93]:

Y_sol(r).numerator().simplify_full()

\renewcommand{\Bold}[1]{\mathbf{#1}}-2 \, {\left(3 \, {\left({\ell}^{3} n^{2} \sin\left({\Theta_2}\right) - {\ell}^{3} \sin\left({\Theta_2}\right)\right)} {r_H}^{3} - {\left({\ell} n^{2} {\mathfrak{q}}^{2} \sin\left({\Theta_2}\right) + 2 \, {\ell}^{3} m \sin\left({\Theta_2}\right) - {\ell} {\mathfrak{p}}^{2} \sin\left({\Theta_2}\right) - 2 \, {\left({\ell}^{3} m \sin\left({\Theta_2}\right) + 2 \, {\ell} \sin\left({\Theta_2}\right)\right)} n^{2} + 4 \, {\ell} \sin\left({\Theta_2}\right)\right)} {r_H}\right)} \arctan\left(\frac{{\ell} r}{\sqrt{{\ell}^{2} {r_H}^{2} + 1}}\right) + \sqrt{{\ell}^{2} {r_H}^{2} + 1} {\left(4 \, {\left(2 \, C_{1} {\ell}^{2} n^{2} + 4 \, C_{1} {\ell}^{2} n + 2 \, C_{1} {\ell}^{2} + 3 \, {\left({\ell}^{4} n^{2} \sin\left({\Theta_2}\right) - {\ell}^{4} \sin\left({\Theta_2}\right)\right)} r\right)} {r_H}^{3} + 2 \, {\left(2 \, C_{1} n^{2} + 4 \, C_{1} n + 3 \, {\left({\ell}^{2} n^{2} \sin\left({\Theta_2}\right) - {\ell}^{2} \sin\left({\Theta_2}\right)\right)} r + 2 \, C_{1}\right)} {r_H} + {\left(n^{2} {\mathfrak{q}}^{2} \sin\left({\Theta_2}\right) + 2 \, {\ell}^{2} m \sin\left({\Theta_2}\right) - {\left(2 \, {\ell}^{2} m \sin\left({\Theta_2}\right) + \sin\left({\Theta_2}\right)\right)} n^{2} + 3 \, {\left({\ell}^{2} n^{2} \sin\left({\Theta_2}\right) - {\ell}^{2} \sin\left({\Theta_2}\right)\right)} {r_H}^{2} - {\mathfrak{p}}^{2} \sin\left({\Theta_2}\right) + \sin\left({\Theta_2}\right)\right)} \log\left(\frac{r + {r_H}}{r - {r_H}}\right)\right)}

In [94]:

Y_sol(r).denominator().factor()

\renewcommand{\Bold}[1]{\mathbf{#1}}4 \, {\left({\ell}^{2} r^{4} + r^{2} - 2 \, m\right)} {\left(2 \, {\ell}^{2} {r_H}^{2} + 1\right)} \sqrt{{\ell}^{2} {r_H}^{2} + 1} {\left(n + 1\right)}^{2} {r_H}

Let us check that Y_sol is indeed a solution of the differential equation for $\Upsilon$ :

In [95]:

eq_Y.substitute_function(Y, Y_sol).subs({rH: rH_m}).simplify_full()

\renewcommand{\Bold}[1]{\mathbf{#1}}0 = 0

In [96]:

print(Y_sol(r))

1/4*(2*(3*(l^3*n^2*sin(Th2) - l^3*sin(Th2))*rH^3 - (l*n^2*qf^2*sin(Th2) + 2*l^3*m*sin(Th2) - l*pf^2*sin(Th2) - 2*(l^3*m*sin(Th2) + 2*l*sin(Th2))*n^2 + 4*l*sin(Th2))*rH)*arctan(l*r/sqrt(l^2*rH^2 + 1)) - sqrt(l^2*rH^2 + 1)*(4*(2*C_1*l^2*n^2 + 4*C_1*l^2*n + 2*C_1*l^2 + 3*(l^4*n^2*sin(Th2) - l^4*sin(Th2))*r)*rH^3 + 2*(2*C_1*n^2 + 4*C_1*n + 3*(l^2*n^2*sin(Th2) - l^2*sin(Th2))*r + 2*C_1)*rH + (n^2*qf^2*sin(Th2) + 2*l^2*m*sin(Th2) - (2*l^2*m*sin(Th2) + sin(Th2))*n^2 + 3*(l^2*n^2*sin(Th2) - l^2*sin(Th2))*rH^2 - pf^2*sin(Th2) + sin(Th2))*log((r + rH)/(r - rH))))/(sqrt(l^2*rH^2 + 1)*(2*(2*l^2*m*n^2 - (l^4*n^2 + 2*l^4*n + l^4)*r^4 + 4*l^2*m*n + 2*l^2*m - (l^2*n^2 + 2*l^2*n + l^2)*r^2)*rH^3 - ((l^2*n^2 + 2*l^2*n + l^2)*r^4 - 2*m*n^2 + (n^2 + 2*n + 1)*r^2 - 4*m*n - 2*m)*rH))

Check of Eq. (4.10) (expression of $\theta'_1 = \Upsilon$ )

The term involving the constant $C_1$ agrees with that of Eq. (4.10):

In [97]:

s = Y_sol(r).coefficient(C_1).simplify_full()
s

\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{{\ell}^{2} r^{4} + r^{2} - 2 \, m}

In [98]:

fr4 = l^2*r^4 + r^2 - 2*m
bool(s == 1/fr4)

\renewcommand{\Bold}[1]{\mathbf{#1}}\mathrm{True}

Let us remove it from $\Upsilon$ and divide the result by $\sin(2\Theta_0)$ :

In [99]:

Y1 = ((Y_sol(r) - s*C_1)/sin(Th2)).simplify_full()
Y1

\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{2 \, {\left(3 \, {\left({\ell}^{3} n^{2} - {\ell}^{3}\right)} {r_H}^{3} - {\left({\ell} n^{2} {\mathfrak{q}}^{2} + 2 \, {\ell}^{3} m - 2 \, {\left({\ell}^{3} m + 2 \, {\ell}\right)} n^{2} - {\ell} {\mathfrak{p}}^{2} + 4 \, {\ell}\right)} {r_H}\right)} \arctan\left(\frac{{\ell} r}{\sqrt{{\ell}^{2} {r_H}^{2} + 1}}\right) - {\left(12 \, {\left({\ell}^{4} n^{2} - {\ell}^{4}\right)} r {r_H}^{3} + 6 \, {\left({\ell}^{2} n^{2} - {\ell}^{2}\right)} r {r_H} + {\left(n^{2} {\mathfrak{q}}^{2} + 2 \, {\ell}^{2} m - {\left(2 \, {\ell}^{2} m + 1\right)} n^{2} + 3 \, {\left({\ell}^{2} n^{2} - {\ell}^{2}\right)} {r_H}^{2} - {\mathfrak{p}}^{2} + 1\right)} \log\left(\frac{r + {r_H}}{r - {r_H}}\right)\right)} \sqrt{{\ell}^{2} {r_H}^{2} + 1}}{4 \, \sqrt{{\ell}^{2} {r_H}^{2} + 1} {\left(2 \, {\left(2 \, {\ell}^{2} m n^{2} - {\left({\ell}^{4} n^{2} + 2 \, {\ell}^{4} n + {\ell}^{4}\right)} r^{4} + 4 \, {\ell}^{2} m n + 2 \, {\ell}^{2} m - {\left({\ell}^{2} n^{2} + 2 \, {\ell}^{2} n + {\ell}^{2}\right)} r^{2}\right)} {r_H}^{3} - {\left({\left({\ell}^{2} n^{2} + 2 \, {\ell}^{2} n + {\ell}^{2}\right)} r^{4} - 2 \, m n^{2} + {\left(n^{2} + 2 \, n + 1\right)} r^{2} - 4 \, m n - 2 \, m\right)} {r_H}\right)}}

The coefficient of the arctan term is

In [100]:

s = Y1.coefficient(arctan(l*r/sqrt(l^2*rH^2 + 1))).simplify_full().factor()
s

\renewcommand{\Bold}[1]{\mathbf{#1}}-\frac{{\left(3 \, {\ell}^{2} n^{2} {r_H}^{2} + 2 \, {\ell}^{2} m n^{2} - n^{2} {\mathfrak{q}}^{2} - 3 \, {\ell}^{2} {r_H}^{2} - 2 \, {\ell}^{2} m + 4 \, n^{2} + {\mathfrak{p}}^{2} - 4\right)} {\ell}}{2 \, {\left({\ell}^{2} r^{4} + r^{2} - 2 \, m\right)} {\left(2 \, {\ell}^{2} {r_H}^{2} + 1\right)} \sqrt{{\ell}^{2} {r_H}^{2} + 1} {\left(n + 1\right)}^{2}}

The numerator of this term agrees with Eq. (4.10):

In [101]:

s.numerator().subs({m: m_rH}).factor()

\renewcommand{\Bold}[1]{\mathbf{#1}}-{\left({\ell}^{4} n^{2} {r_H}^{4} - {\ell}^{4} {r_H}^{4} + 4 \, {\ell}^{2} n^{2} {r_H}^{2} - n^{2} {\mathfrak{q}}^{2} - 4 \, {\ell}^{2} {r_H}^{2} + 4 \, n^{2} + {\mathfrak{p}}^{2} - 4\right)} {\ell}

In [102]:

bool(s.numerator().subs({m: m_rH}) 
     == l*((1 - n^2)*(l^2*rH^2 + 2)^2 - pf^2 + n^2*qf^2))

\renewcommand{\Bold}[1]{\mathbf{#1}}\mathrm{True}

The denominator agrees with Eq. (4.10) as well:

In [103]:

s.denominator()

\renewcommand{\Bold}[1]{\mathbf{#1}}2 \, {\left({\ell}^{2} r^{4} + r^{2} - 2 \, m\right)} {\left(2 \, {\ell}^{2} {r_H}^{2} + 1\right)} \sqrt{{\ell}^{2} {r_H}^{2} + 1} {\left(n + 1\right)}^{2}

Let us remove the arctan term from $\Upsilon$ :

In [104]:

Y2 = (Y1 - s*arctan(l*r/sqrt(l^2*rH^2 + 1))).simplify_full()
Y2

\renewcommand{\Bold}[1]{\mathbf{#1}}-\frac{12 \, {\left({\ell}^{4} n^{2} - {\ell}^{4}\right)} r {r_H}^{3} + 6 \, {\left({\ell}^{2} n^{2} - {\ell}^{2}\right)} r {r_H} + {\left(n^{2} {\mathfrak{q}}^{2} + 2 \, {\ell}^{2} m - {\left(2 \, {\ell}^{2} m + 1\right)} n^{2} + 3 \, {\left({\ell}^{2} n^{2} - {\ell}^{2}\right)} {r_H}^{2} - {\mathfrak{p}}^{2} + 1\right)} \log\left(\frac{r + {r_H}}{r - {r_H}}\right)}{4 \, {\left(2 \, {\left(2 \, {\ell}^{2} m n^{2} - {\left({\ell}^{4} n^{2} + 2 \, {\ell}^{4} n + {\ell}^{4}\right)} r^{4} + 4 \, {\ell}^{2} m n + 2 \, {\ell}^{2} m - {\left({\ell}^{2} n^{2} + 2 \, {\ell}^{2} n + {\ell}^{2}\right)} r^{2}\right)} {r_H}^{3} - {\left({\left({\ell}^{2} n^{2} + 2 \, {\ell}^{2} n + {\ell}^{2}\right)} r^{4} - 2 \, m n^{2} + {\left(n^{2} + 2 \, n + 1\right)} r^{2} - 4 \, m n - 2 \, m\right)} {r_H}\right)}}

In [105]:

Y2.numerator().simplify_full()

\renewcommand{\Bold}[1]{\mathbf{#1}}12 \, {\left({\ell}^{4} n^{2} - {\ell}^{4}\right)} r {r_H}^{3} + 6 \, {\left({\ell}^{2} n^{2} - {\ell}^{2}\right)} r {r_H} + {\left(n^{2} {\mathfrak{q}}^{2} + 2 \, {\ell}^{2} m - {\left(2 \, {\ell}^{2} m + 1\right)} n^{2} + 3 \, {\left({\ell}^{2} n^{2} - {\ell}^{2}\right)} {r_H}^{2} - {\mathfrak{p}}^{2} + 1\right)} \log\left(\frac{r + {r_H}}{r - {r_H}}\right)

In [106]:

Y2.denominator().factor()

\renewcommand{\Bold}[1]{\mathbf{#1}}4 \, {\left({\ell}^{2} r^{4} + r^{2} - 2 \, m\right)} {\left(2 \, {\ell}^{2} {r_H}^{2} + 1\right)} {\left(n + 1\right)}^{2} {r_H}

The coefficient of the log term is

In [107]:

s = Y2.coefficient(log((r + rH)/(r - rH))).simplify_full().factor()
s

\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{3 \, {\ell}^{2} n^{2} {r_H}^{2} - 2 \, {\ell}^{2} m n^{2} + n^{2} {\mathfrak{q}}^{2} - 3 \, {\ell}^{2} {r_H}^{2} + 2 \, {\ell}^{2} m - n^{2} - {\mathfrak{p}}^{2} + 1}{4 \, {\left({\ell}^{2} r^{4} + r^{2} - 2 \, m\right)} {\left(2 \, {\ell}^{2} {r_H}^{2} + 1\right)} {\left(n + 1\right)}^{2} {r_H}}

In [108]:

s.numerator().subs({m: m_rH}).factor()

\renewcommand{\Bold}[1]{\mathbf{#1}}-{\ell}^{4} n^{2} {r_H}^{4} + {\ell}^{4} {r_H}^{4} + 2 \, {\ell}^{2} n^{2} {r_H}^{2} + n^{2} {\mathfrak{q}}^{2} - 2 \, {\ell}^{2} {r_H}^{2} - n^{2} - {\mathfrak{p}}^{2} + 1

Check against Eq. (4.10):

In [109]:

bool(s.numerator().subs({m: m_rH}) == (1 - n^2)*(l^2*rH^2 - 1)^2 - pf^2 + n^2*qf^2)

\renewcommand{\Bold}[1]{\mathbf{#1}}\mathrm{True}

In [110]:

s.denominator()

\renewcommand{\Bold}[1]{\mathbf{#1}}4 \, {\left({\ell}^{2} r^{4} + r^{2} - 2 \, m\right)} {\left(2 \, {\ell}^{2} {r_H}^{2} + 1\right)} {\left(n + 1\right)}^{2} {r_H}

Given that $\mathrm{artanh}\, x = \frac{1}{2} \ln\left( \frac{1 + x}{1 - x} \right)$ we have $\ln \left( \frac{x + 1}{x - 1} \right) = 2\, \mathrm{artanh}\left(\frac{1}{x}\right)$

Hence the term in $\ln\left(\frac{r + r_H}{r - r_H}\right)$ agrees with the corresponding term in Eq. (4.10).

Finally, the last term in $\Upsilon$ is

In [111]:

Y3 = (Y2 - s*log((r + rH)/(r - rH))).simplify_full()
Y3.factor()

\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{3 \, {\ell}^{2} {\left(n - 1\right)} r}{2 \, {\left({\ell}^{2} r^{4} + r^{2} - 2 \, m\right)} {\left(n + 1\right)}}

This term agrees with Eq. (4.10), given the simplification $\frac{1 - n^2}{(1 + n)^2} = -\frac{n - 1}{n + 1}$ .

Conclusion: we have full agreement with Eq. (4.10).

Conjugate momenta

In [112]:

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

In [113]:

pis = conjugate_momenta(L_a2, [phi_1, psi_1], r)
pis

\renewcommand{\Bold}[1]{\mathbf{#1}}\left[-{\left({\mu_0}^{2} - 1\right)} a^{2} {\ell}^{2} r^{4} \frac{\partial}{\partial r}\phi_{1}\left(r\right) - {\left({\mu_0}^{2} - 1\right)} a^{2} r^{2} \frac{\partial}{\partial r}\phi_{1}\left(r\right) + 2 \, {\left({\mu_0}^{2} - 1\right)} a^{2} m \frac{\partial}{\partial r}\phi_{1}\left(r\right), {\mu_0}^{2} a^{2} {\ell}^{2} n^{2} r^{4} \frac{\partial}{\partial r}\psi_{1}\left(r\right) + {\mu_0}^{2} a^{2} n^{2} r^{2} \frac{\partial}{\partial r}\psi_{1}\left(r\right) - 2 \, {\mu_0}^{2} a^{2} m n^{2} \frac{\partial}{\partial r}\psi_{1}\left(r\right)\right]

Check of Eq. (4.15):

In [114]:

pi_phi_r = (pis[0]/a).substitute_function(phi_1, phi1_sol).simplify_full()
pi_phi_r

\renewcommand{\Bold}[1]{\mathbf{#1}}-{\left({\mu_0}^{2} - 1\right)} a {\mathfrak{p}}

In [115]:

pi_psi_r = (pis[1]/a).substitute_function(psi_1, psi1_sol).simplify_full()
pi_psi_r

\renewcommand{\Bold}[1]{\mathbf{#1}}{\mu_0}^{2} a n^{2} {\mathfrak{q}}

Check of Eq. (4.14):

We start from $\pi^r_\theta$ as given by Eq. (4.13):

In [116]:

pi_theta = (fr4*(a + b)^2*Y_sol(r)).simplify_full()
pi_theta

\renewcommand{\Bold}[1]{\mathbf{#1}}-\frac{2 \, {\left(3 \, {\left(a^{2} {\ell}^{3} n^{2} \sin\left({\Theta_2}\right) - a^{2} {\ell}^{3} \sin\left({\Theta_2}\right)\right)} {r_H}^{3} - {\left(a^{2} {\ell} n^{2} {\mathfrak{q}}^{2} \sin\left({\Theta_2}\right) + 2 \, a^{2} {\ell}^{3} m \sin\left({\Theta_2}\right) - a^{2} {\ell} {\mathfrak{p}}^{2} \sin\left({\Theta_2}\right) + 4 \, a^{2} {\ell} \sin\left({\Theta_2}\right) - 2 \, {\left(a^{2} {\ell}^{3} m \sin\left({\Theta_2}\right) + 2 \, a^{2} {\ell} \sin\left({\Theta_2}\right)\right)} n^{2}\right)} {r_H}\right)} \arctan\left(\frac{{\ell} r}{\sqrt{{\ell}^{2} {r_H}^{2} + 1}}\right) - \sqrt{{\ell}^{2} {r_H}^{2} + 1} {\left(4 \, {\left(2 \, C_{1} a^{2} {\ell}^{2} n^{2} + 4 \, C_{1} a^{2} {\ell}^{2} n + 2 \, C_{1} a^{2} {\ell}^{2} + 3 \, {\left(a^{2} {\ell}^{4} n^{2} \sin\left({\Theta_2}\right) - a^{2} {\ell}^{4} \sin\left({\Theta_2}\right)\right)} r\right)} {r_H}^{3} + 2 \, {\left(2 \, C_{1} a^{2} n^{2} + 4 \, C_{1} a^{2} n + 2 \, C_{1} a^{2} + 3 \, {\left(a^{2} {\ell}^{2} n^{2} \sin\left({\Theta_2}\right) - a^{2} {\ell}^{2} \sin\left({\Theta_2}\right)\right)} r\right)} {r_H} + {\left(a^{2} n^{2} {\mathfrak{q}}^{2} \sin\left({\Theta_2}\right) + 2 \, a^{2} {\ell}^{2} m \sin\left({\Theta_2}\right) - a^{2} {\mathfrak{p}}^{2} \sin\left({\Theta_2}\right) - {\left(2 \, a^{2} {\ell}^{2} m \sin\left({\Theta_2}\right) + a^{2} \sin\left({\Theta_2}\right)\right)} n^{2} + 3 \, {\left(a^{2} {\ell}^{2} n^{2} \sin\left({\Theta_2}\right) - a^{2} {\ell}^{2} \sin\left({\Theta_2}\right)\right)} {r_H}^{2} + a^{2} \sin\left({\Theta_2}\right)\right)} \log\left(\frac{r + {r_H}}{r - {r_H}}\right)\right)}}{4 \, {\left(2 \, {\ell}^{2} {r_H}^{3} + {r_H}\right)} \sqrt{{\ell}^{2} {r_H}^{2} + 1}}

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

In [117]:

assume(l > 0)

u = var('u')
assume(u > 0)
s = pi_theta.subs({r: 1/u}).simplify_log()
s = s.taylor(u, 0, 2)
s = s.subs({u: 1/r})
s

\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{3}{2} \, {\left(a^{2} {\ell}^{2} n^{2} \sin\left({\Theta_2}\right) - a^{2} {\ell}^{2} \sin\left({\Theta_2}\right)\right)} r + \frac{3 \, {\left(a^{2} n^{2} \sin\left({\Theta_2}\right) - a^{2} \sin\left({\Theta_2}\right)\right)}}{2 \, r} + \frac{\pi a^{2} {\ell} n^{2} {\mathfrak{q}}^{2} \sin\left({\Theta_2}\right) + 2 \, \pi a^{2} {\ell}^{3} m \sin\left({\Theta_2}\right) - \pi a^{2} {\ell} {\mathfrak{p}}^{2} \sin\left({\Theta_2}\right) + 4 \, \pi a^{2} {\ell} \sin\left({\Theta_2}\right) - 2 \, {\left(\pi a^{2} {\ell}^{3} m \sin\left({\Theta_2}\right) + 2 \, \pi a^{2} {\ell} \sin\left({\Theta_2}\right)\right)} n^{2} - 3 \, {\left(\pi a^{2} {\ell}^{3} n^{2} \sin\left({\Theta_2}\right) - \pi a^{2} {\ell}^{3} \sin\left({\Theta_2}\right)\right)} {r_H}^{2} + 4 \, {\left(C_{1} a^{2} n^{2} + 2 \, C_{1} a^{2} n + C_{1} a^{2} + 2 \, {\left(C_{1} a^{2} {\ell}^{2} n^{2} + 2 \, C_{1} a^{2} {\ell}^{2} n + C_{1} a^{2} {\ell}^{2}\right)} {r_H}^{2}\right)} \sqrt{{\ell}^{2} {r_H}^{2} + 1}}{4 \, {\left(2 \, {\ell}^{2} {r_H}^{2} + 1\right)} \sqrt{{\ell}^{2} {r_H}^{2} + 1}}

We consider $\frac{\pi^r_\theta}{(a^2/2)\sin(2\Theta_0)}$ :

In [118]:

s1 = (s/(a^2/2*sin(Th2))).expand()
s1

\renewcommand{\Bold}[1]{\mathbf{#1}}-\frac{3 \, \pi {\ell}^{3} n^{2} {r_H}^{2}}{2 \, {\left(2 \, {\ell}^{2} {r_H}^{2} + 1\right)} \sqrt{{\ell}^{2} {r_H}^{2} + 1}} - \frac{\pi {\ell}^{3} m n^{2}}{{\left(2 \, {\ell}^{2} {r_H}^{2} + 1\right)} \sqrt{{\ell}^{2} {r_H}^{2} + 1}} + 3 \, {\ell}^{2} n^{2} r + \frac{4 \, C_{1} {\ell}^{2} n^{2} {r_H}^{2}}{{\left(2 \, {\ell}^{2} {r_H}^{2} + 1\right)} \sin\left({\Theta_2}\right)} + \frac{\pi {\ell} n^{2} {\mathfrak{q}}^{2}}{2 \, {\left(2 \, {\ell}^{2} {r_H}^{2} + 1\right)} \sqrt{{\ell}^{2} {r_H}^{2} + 1}} + \frac{3 \, \pi {\ell}^{3} {r_H}^{2}}{2 \, {\left(2 \, {\ell}^{2} {r_H}^{2} + 1\right)} \sqrt{{\ell}^{2} {r_H}^{2} + 1}} + \frac{8 \, C_{1} {\ell}^{2} n {r_H}^{2}}{{\left(2 \, {\ell}^{2} {r_H}^{2} + 1\right)} \sin\left({\Theta_2}\right)} + \frac{\pi {\ell}^{3} m}{{\left(2 \, {\ell}^{2} {r_H}^{2} + 1\right)} \sqrt{{\ell}^{2} {r_H}^{2} + 1}} - 3 \, {\ell}^{2} r + \frac{4 \, C_{1} {\ell}^{2} {r_H}^{2}}{{\left(2 \, {\ell}^{2} {r_H}^{2} + 1\right)} \sin\left({\Theta_2}\right)} - \frac{2 \, \pi {\ell} n^{2}}{{\left(2 \, {\ell}^{2} {r_H}^{2} + 1\right)} \sqrt{{\ell}^{2} {r_H}^{2} + 1}} - \frac{\pi {\ell} {\mathfrak{p}}^{2}}{2 \, {\left(2 \, {\ell}^{2} {r_H}^{2} + 1\right)} \sqrt{{\ell}^{2} {r_H}^{2} + 1}} + \frac{3 \, n^{2}}{r} + \frac{2 \, C_{1} n^{2}}{{\left(2 \, {\ell}^{2} {r_H}^{2} + 1\right)} \sin\left({\Theta_2}\right)} + \frac{2 \, \pi {\ell}}{{\left(2 \, {\ell}^{2} {r_H}^{2} + 1\right)} \sqrt{{\ell}^{2} {r_H}^{2} + 1}} + \frac{4 \, C_{1} n}{{\left(2 \, {\ell}^{2} {r_H}^{2} + 1\right)} \sin\left({\Theta_2}\right)} - \frac{3}{r} + \frac{2 \, C_{1}}{{\left(2 \, {\ell}^{2} {r_H}^{2} + 1\right)} \sin\left({\Theta_2}\right)}

The term in factor of $C_1$ is

In [119]:

s1.coefficient(C_1)

\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{4 \, {\ell}^{2} n^{2} {r_H}^{2}}{{\left(2 \, {\ell}^{2} {r_H}^{2} + 1\right)} \sin\left({\Theta_2}\right)} + \frac{8 \, {\ell}^{2} n {r_H}^{2}}{{\left(2 \, {\ell}^{2} {r_H}^{2} + 1\right)} \sin\left({\Theta_2}\right)} + \frac{4 \, {\ell}^{2} {r_H}^{2}}{{\left(2 \, {\ell}^{2} {r_H}^{2} + 1\right)} \sin\left({\Theta_2}\right)} + \frac{2 \, n^{2}}{{\left(2 \, {\ell}^{2} {r_H}^{2} + 1\right)} \sin\left({\Theta_2}\right)} + \frac{4 \, n}{{\left(2 \, {\ell}^{2} {r_H}^{2} + 1\right)} \sin\left({\Theta_2}\right)} + \frac{2}{{\left(2 \, {\ell}^{2} {r_H}^{2} + 1\right)} \sin\left({\Theta_2}\right)}

It is a constant term, of the type $\tilde{C}_1/\sin(2\Theta_0)$ , in agreement with Eq. (4.14). We remove it from the main term:

In [120]:

s2 = (s1 - s1.coefficient(C_1)*C_1).simplify_full()
s2

\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{12 \, {\left({\ell}^{4} n^{2} - {\ell}^{4} + {\left({\ell}^{6} n^{2} - {\ell}^{6}\right)} r^{2}\right)} {r_H}^{4} + 6 \, {\left({\ell}^{2} n^{2} - {\ell}^{2}\right)} r^{2} + 18 \, {\left({\ell}^{2} n^{2} + {\left({\ell}^{4} n^{2} - {\ell}^{4}\right)} r^{2} - {\ell}^{2}\right)} {r_H}^{2} + 6 \, n^{2} - \sqrt{{\ell}^{2} {r_H}^{2} + 1} {\left(3 \, {\left(\pi {\ell}^{3} n^{2} - \pi {\ell}^{3}\right)} r {r_H}^{2} - {\left(\pi {\ell} n^{2} {\mathfrak{q}}^{2} + 2 \, \pi {\ell}^{3} m - \pi {\ell} {\mathfrak{p}}^{2} - 2 \, {\left(\pi {\ell}^{3} m + 2 \, \pi {\ell}\right)} n^{2} + 4 \, \pi {\ell}\right)} r\right)} - 6}{2 \, {\left(2 \, {\ell}^{4} r {r_H}^{4} + 3 \, {\ell}^{2} r {r_H}^{2} + r\right)}}

In [121]:

s2.numerator().simplify_full()

\renewcommand{\Bold}[1]{\mathbf{#1}}12 \, {\left({\ell}^{4} n^{2} - {\ell}^{4} + {\left({\ell}^{6} n^{2} - {\ell}^{6}\right)} r^{2}\right)} {r_H}^{4} + 6 \, {\left({\ell}^{2} n^{2} - {\ell}^{2}\right)} r^{2} + 18 \, {\left({\ell}^{2} n^{2} + {\left({\ell}^{4} n^{2} - {\ell}^{4}\right)} r^{2} - {\ell}^{2}\right)} {r_H}^{2} + 6 \, n^{2} - \sqrt{{\ell}^{2} {r_H}^{2} + 1} {\left(3 \, {\left(\pi {\ell}^{3} n^{2} - \pi {\ell}^{3}\right)} r {r_H}^{2} - {\left(\pi {\ell} n^{2} {\mathfrak{q}}^{2} + 2 \, \pi {\ell}^{3} m - \pi {\ell} {\mathfrak{p}}^{2} - 2 \, {\left(\pi {\ell}^{3} m + 2 \, \pi {\ell}\right)} n^{2} + 4 \, \pi {\ell}\right)} r\right)} - 6

In [122]:

s2.denominator().factor()

\renewcommand{\Bold}[1]{\mathbf{#1}}2 \, {\left(2 \, {\ell}^{2} {r_H}^{2} + 1\right)} {\left({\ell}^{2} {r_H}^{2} + 1\right)} r

Let divide both the numerator and denominator by $r$ :

In [123]:

s2n = (s2.numerator()/r).expand()
s2n

\renewcommand{\Bold}[1]{\mathbf{#1}}12 \, {\ell}^{6} n^{2} r {r_H}^{4} - 12 \, {\ell}^{6} r {r_H}^{4} + 18 \, {\ell}^{4} n^{2} r {r_H}^{2} + \frac{12 \, {\ell}^{4} n^{2} {r_H}^{4}}{r} - 3 \, \pi \sqrt{{\ell}^{2} {r_H}^{2} + 1} {\ell}^{3} n^{2} {r_H}^{2} - 2 \, \pi \sqrt{{\ell}^{2} {r_H}^{2} + 1} {\ell}^{3} m n^{2} - 18 \, {\ell}^{4} r {r_H}^{2} - \frac{12 \, {\ell}^{4} {r_H}^{4}}{r} + \pi \sqrt{{\ell}^{2} {r_H}^{2} + 1} {\ell} n^{2} {\mathfrak{q}}^{2} + 3 \, \pi \sqrt{{\ell}^{2} {r_H}^{2} + 1} {\ell}^{3} {r_H}^{2} + 2 \, \pi \sqrt{{\ell}^{2} {r_H}^{2} + 1} {\ell}^{3} m + 6 \, {\ell}^{2} n^{2} r + \frac{18 \, {\ell}^{2} n^{2} {r_H}^{2}}{r} - 4 \, \pi \sqrt{{\ell}^{2} {r_H}^{2} + 1} {\ell} n^{2} - \pi \sqrt{{\ell}^{2} {r_H}^{2} + 1} {\ell} {\mathfrak{p}}^{2} - 6 \, {\ell}^{2} r - \frac{18 \, {\ell}^{2} {r_H}^{2}}{r} + 4 \, \pi \sqrt{{\ell}^{2} {r_H}^{2} + 1} {\ell} + \frac{6 \, n^{2}}{r} - \frac{6}{r}

In [124]:

s2d = (s2.denominator()/r).factor()
s2d

\renewcommand{\Bold}[1]{\mathbf{#1}}2 \, {\left(2 \, {\ell}^{2} {r_H}^{2} + 1\right)} {\left({\ell}^{2} {r_H}^{2} + 1\right)}

The coefficient of the term in $r$ is

In [125]:

s = s2n.coefficient(r).factor()
s/s2d

\renewcommand{\Bold}[1]{\mathbf{#1}}3 \, {\ell}^{2} {\left(n + 1\right)} {\left(n - 1\right)}

This is in agreement with Eq. (4.14).

We remove it:

In [126]:

s3n = (s2n - s*r).simplify_full().expand()
s3n

\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{12 \, {\ell}^{4} n^{2} {r_H}^{4}}{r} - 3 \, \pi \sqrt{{\ell}^{2} {r_H}^{2} + 1} {\ell}^{3} n^{2} {r_H}^{2} - 2 \, \pi \sqrt{{\ell}^{2} {r_H}^{2} + 1} {\ell}^{3} m n^{2} - \frac{12 \, {\ell}^{4} {r_H}^{4}}{r} + \pi \sqrt{{\ell}^{2} {r_H}^{2} + 1} {\ell} n^{2} {\mathfrak{q}}^{2} + 3 \, \pi \sqrt{{\ell}^{2} {r_H}^{2} + 1} {\ell}^{3} {r_H}^{2} + 2 \, \pi \sqrt{{\ell}^{2} {r_H}^{2} + 1} {\ell}^{3} m + \frac{18 \, {\ell}^{2} n^{2} {r_H}^{2}}{r} - 4 \, \pi \sqrt{{\ell}^{2} {r_H}^{2} + 1} {\ell} n^{2} - \pi \sqrt{{\ell}^{2} {r_H}^{2} + 1} {\ell} {\mathfrak{p}}^{2} - \frac{18 \, {\ell}^{2} {r_H}^{2}}{r} + 4 \, \pi \sqrt{{\ell}^{2} {r_H}^{2} + 1} {\ell} + \frac{6 \, n^{2}}{r} - \frac{6}{r}

The coefficient of the term in $1/r$ is

In [127]:

s = s3n.coefficient(r^(-1)).factor()
s/s2d

\renewcommand{\Bold}[1]{\mathbf{#1}}3 \, {\left(n + 1\right)} {\left(n - 1\right)}

This is in agreement with Eq. (4.14).

Finally the remaining term is

In [128]:

s4n = (s3n - s/r).simplify_full().expand()
s4n

\renewcommand{\Bold}[1]{\mathbf{#1}}-3 \, \pi \sqrt{{\ell}^{2} {r_H}^{2} + 1} {\ell}^{3} n^{2} {r_H}^{2} - 2 \, \pi \sqrt{{\ell}^{2} {r_H}^{2} + 1} {\ell}^{3} m n^{2} + \pi \sqrt{{\ell}^{2} {r_H}^{2} + 1} {\ell} n^{2} {\mathfrak{q}}^{2} + 3 \, \pi \sqrt{{\ell}^{2} {r_H}^{2} + 1} {\ell}^{3} {r_H}^{2} + 2 \, \pi \sqrt{{\ell}^{2} {r_H}^{2} + 1} {\ell}^{3} m - 4 \, \pi \sqrt{{\ell}^{2} {r_H}^{2} + 1} {\ell} n^{2} - \pi \sqrt{{\ell}^{2} {r_H}^{2} + 1} {\ell} {\mathfrak{p}}^{2} + 4 \, \pi \sqrt{{\ell}^{2} {r_H}^{2} + 1} {\ell}

In [129]:

s4n.factor()

\renewcommand{\Bold}[1]{\mathbf{#1}}-\pi {\left(3 \, {\ell}^{2} n^{2} {r_H}^{2} + 2 \, {\ell}^{2} m n^{2} - n^{2} {\mathfrak{q}}^{2} - 3 \, {\ell}^{2} {r_H}^{2} - 2 \, {\ell}^{2} m + 4 \, n^{2} + {\mathfrak{p}}^{2} - 4\right)} \sqrt{{\ell}^{2} {r_H}^{2} + 1} {\ell}

In [130]:

s = (s4n.factor()/s2d).subs({m: m_rH}).factor()
s

\renewcommand{\Bold}[1]{\mathbf{#1}}-\frac{\pi {\left({\ell}^{4} n^{2} {r_H}^{4} - {\ell}^{4} {r_H}^{4} + 4 \, {\ell}^{2} n^{2} {r_H}^{2} - n^{2} {\mathfrak{q}}^{2} - 4 \, {\ell}^{2} {r_H}^{2} + 4 \, n^{2} + {\mathfrak{p}}^{2} - 4\right)} {\ell}}{2 \, {\left(2 \, {\ell}^{2} {r_H}^{2} + 1\right)} \sqrt{{\ell}^{2} {r_H}^{2} + 1}}

The denominator clearly agrees with Eq. (4.14); the numerator agrees as well:

In [131]:

bool(s.numerator()/(pi*l) == (1 - n^2)*(l^2*rH^2 + 2)^2 - pf^2 + n^2*qf^2)

\renewcommand{\Bold}[1]{\mathbf{#1}}\mathrm{True}

Conclusion: we have full agreement with Eq. (4.14).

In [0]:

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

Case b = n a

Spacetime manifold

Metric tensor

Check of Eq. (2.9)

Einstein equation

String worldsheet

Induced metric on the string worldsheet

Nambu-Goto action

Euler-Lagrange equations

Solving the equation for ϕ1\phi_1ϕ1​ (Eq. (4.8))

Solving the equation for ψ1\psi_1ψ1​ (Eq. (4.8))

Nambu-Goto Lagrangian at fourth order in aaa

The equation for μ1\mu_1μ1​

Check of Eq. (4.9)

Solving the equation for Υ:=θ1′\Upsilon := \theta_1'Υ:=θ1′​

Check of Eq. (4.10) (expression of θ1′=Υ\theta'_1 = \Upsilonθ1′​=Υ)

Conjugate momenta

Check of Eq. (4.15):

Check of Eq. (4.14):

Solving the equation for $\phi_1$ (Eq. (4.8))

Solving the equation for $\psi_1$ (Eq. (4.8))

Nambu-Goto Lagrangian at fourth order in $a$

The equation for $\mu_1$

Solving the equation for $\Upsilon := \theta_1'$

Check of Eq. (4.10) (expression of $\theta'_1 = \Upsilon$ )