The Spinning Effective One-Body Hamiltonian: "v4P"¶

Author: Tyler Knowles¶

This module documents the reduced spinning effective one-body Hamiltonian as numerically implemented in LALSuite's SEOBNRv4P gravitational waveform approximant.¶

Notebook Status: Validated

Validation Notes: This module has been validated against the LALSuite SEOBNRv4P code that was reviewed and approved for LIGO parameter estimation by the LIGO Scientific Collaboration. That is, the value $H_{\rm real}$ output from this notebook agrees to roundoff error with the value of $H_{\rm real}$ computed by the LALSuite function XLALSimIMRSpinPrecEOBHamiltonian().

NRPy+ Source Code for this module: SEOBNR_v4P_Hamiltonian.py¶

Introduction¶

$$\label{intro}$$

The Physical System of Interest¶

Consider two black holes with masses $m_{1}$, $m_{2}$ and spins ${\bf S}_{1}$, ${\bf S}_{2}$ in a binary system. The spinning effective one-body ("SEOB") Hamiltonian $H_{\rm real}$ (defined in this cell) describes the dynamics of this system; we will define $H_{\rm real}$ as in Barausse and Buonanno (2010) Section VE. There, $H_{\rm real}$ is canonically transformed and mapped to an effective Hamiltonian $H_{\rm eff}$ (defined in this cell) describing the motion of a test particle of mass $\mu$ (defined in this cell) and spin ${\bf S}^{*}$ (defined in this cell) moving in a defomred Kerr background. Here we seek to break up $H_{\rm real}$ and document the terms in such a way that the resulting Python code can be used to numerically evaluate $H_{\rm real}$.

We write $H_{\rm real}$ in terms of Cartesian quasi-isotropic coordinates $x$, $y$, and $z$ (see Barausse and Buonanno (2010) Section III). The spatial coordinates $r$, $\theta$, and $\phi$ referenced throughout are Boyer-Lindquist coordinates (see Barausse and Buonanno (2010) Section IV).

Please note that throughout this notebook we adpot the following conventions:

1. $c = 1$ where $c$ is the speed of light in a vacuum,
2. spacial tensor indicies are denoted by lowercase Latin letters,
3. repeated indices indicate Einstein summation notation, and
4. we normalize $M=1$ in all terms except for $\eta$ and $\mu$ for agreement with LALSuite. Nonetheless, $M$ appears in other text cells for comparison with the cited literature.

Running this notebook to completion will generate a file called v4P_Hreal_on_bottom.py. This file contains the Python function v4P_compute_Hreal(), which takes as input m1, m2 (each in solar masses), the value of the Euler-Mascheroni constant, a tortoise coordinate, and values for all twelve dynamic variables (3 components of the separation vector, three components of the momentum vector, and three spin components of each compact object). Note that the spin components should be dimensionless.

Citations¶

Throughout this module, we will refer to

• Barausse and Buonanno (2010) as BB2010,
• Barausse and Buonanno (2011) as BB2011,
• Ossokine, Buonanno, Marsat, et al (2020) as OB2020,
• Steinhoff, Hinderer, Buonanno, et al (2016) as SH2016,
• Bohe, Shao, Taracchini, et al (2017) as BL2017,
• Pan, Buonanno, Buchman, et. al (2010) as P2010,
• Taracchini, Buonanno, Pan, et al (2014) as T2014,
• Taracchini, Pan, Buonanno, et al (2012) as T2012, and
• Damour, Jaranowski, and Schaefer (2000) as D2000.

Table of Contents¶

$$\label{toc}$$

This notebook is organized as follows:

1. Step 0: Creating the output directory for SEOBNR
2. Step 1: The Real Hamiltonian $H_{\rm real}$
3. Step 2: The Effective Hamiltonian $H_{\rm eff}$
4. Step 3: Terms of $H_{\rm eff}$
   1. Step 3.a: Leading Order Spin Effects $H_{\rm S}$
   2. Step 3.b: The Nonspinning Hamiltonian $H_{\rm NS}$
   3. Step 3.c: The Quadrupole Deformation $H_{\rm D}$
5. Step 4: The Spin-Orbit Term $H_{\rm SO}$
   1. Step 4.a: $H_{\rm SO}$ Term 1
   2. Step 4.b: $H_{\rm SO}$ Term 2 Coefficient
   3. Step 4.c: $H_{\rm SO}$ Term 2
      1. Step 4.c.i: $H_{\rm SO}$ Term 2a
      2. Step 4.c.ii: $H_{\rm SO}$ Term 2b
      3. Step 4.c.iii: $H_{\rm SO}$ Term 2c
6. Step 5: The Spin-Spin Term $H_{\rm SS}$
   1. Step 5.a: $H_{\rm SS}$ Term 1
   2. Step 5.b: $H_{\rm SS}$ Term 2 coefficient
   3. Step 5.c: $H_{\rm SS}$ Term 2
   4. Step 5.d: $H_{\rm SS}$ Term 3 coefficient
   5. Step 5.e: $H_{\rm SS}$ Term 3
7. Step 6: The $H_{\rm NS}$ Terms
   1. Step 6.a: $\beta p$ Sum
   2. Step 6.b: $\alpha$
   3. Step 6.c: $H_{\rm NS}$ Radicand
      1. Step 6.c.i: $\gamma p$ Sum
      2. Step 6.c.ii: ${\cal Q}_{4}$
8. Step 7: The $H_{\rm D}$ Terms
   1. Step 7.a: $H_{\rm D}$ Coefficient
   2. Step 7.b: $H_{\rm D}$ Sum
      1. Step 7.b.i: $H_{\rm D}$ Sum Term 1
      2. Step 7.b.ii: $H_{\rm D}$ Sum Term 2
9. Step 8: Common Dot Products
   1. Step 8.a: ${\bf S} \cdot \boldsymbol{\xi}$
   2. Step 8.b: ${\bf S} \cdot {\bf v}$
   3. Step 8.c: ${\bf S} \cdot {\bf n}$
   4. Step 8.d: ${\bf S} \cdot \hat{\bf S}_{\rm Kerr}$
   5. Step 8.e: ${\bf S}^{*} \cdot {\bf n}$
10. Step 9: $H_{\rm real}$ Spin Combination ${\bf S}^{*}$
    1. Step 9a: ${\bf S}^{*}$
    2. Step 9b: $\Delta_{\sigma^{*}}$
    3. Step 9c: $\sigma^{*}$ Coefficient
       1. Step 9c i: $\sigma^{*}$ Coefficient Term 1
       2. Step 9c ii: $\sigma^{*}$ Coefficient Term 2
    4. Step 9d: $\sigma$ Coefficient
       1. Step 9d i: $\sigma$ Coefficient Term 1
       2. Step 9d ii: $\sigma$ Coefficient Term 2
       3. Step 9d iii: $\sigma$ Coefficient Term 3
11. Step 10: Derivatives of the Metric Potential
    1. Step 10.a: $\omega_{r}$
    2. Step 10.b: $\nu_{r}$
    3. Step 10.c: $\mu_{r}$
    4. Step 10.d: $\omega_{\cos\theta}$
    5. Step 10.e: $\nu_{\cos\theta}$
    6. Step 10.f: $\mu_{\cos\theta}$
    7. Step 10.g: $\Lambda_{t}^{\prime}$
    8. Step 10.h: $\tilde{\omega}_{\rm fd}^{\prime}$
12. Step 11: The Deformed and Rescaled Metric Potentials
    1. Step 11.a: $\omega$
    2. Step 11.b: $e^{2 \nu}$
    3. Step 11.c: $\tilde{B}$
    4. Step 11.d: $\tilde{B}_{r}$
    5. Step 11.e: $e^{2 \tilde{\mu}}$
    6. Step 11.f: $\tilde{J}$
    7. Step 11.g: $Q$
       1. Step 11.g.i: $\frac{ \Delta_{r} }{ \Sigma } \left( \hat{\bf p} \cdot {\bf n} \right)^{2}$
       2. Step 11.g.ii: Q Coefficient 1
       3. Step 11.g.iii: Q Coefficient 2
13. Step 12: Tortoise terms
    1. Step 12.a: $p_{\phi}$
    2. Step 12.b: $\hat{\bf p} \cdot {\bf v} r$
    3. Step 12.c: $\hat{\bf p} \cdot {\bf n}$
    4. Step 12.d: $\hat{\bf p} \cdot \boldsymbol{\xi} r$
    5. Step 12.e: $\hat{\bf p}$
    6. Step 12.f: prT
    7. Step 12.g: csi2
    8. Step 12.h: csi1
    9. Step 12.i: csi
14. Step 13: Metric Terms
    1. Step 13.a: $\Lambda_{t}$
    2. Step 13.b: $\Delta_{r}$
    3. Step 13.c: $\Delta_{t}$
    4. Step 13.d: $\Delta_{t}^{\prime}$
    5. Step 13.e: $\Delta_{u}$
       1. Step 13.e.i: $\bar{\Delta}_{u}$
       2. Step 13.e.ii: $\Delta_{u}$ Calibration Term
       3. Step 13.e.iii: Calibration Coefficients
       4. Step 13.e.iv: $K$
       5. Step 13.e.v: $\chi$
    6. Step 13.f: $\tilde{\omega}_{\rm fd}$
    7. Step 13.g: $D^{-1}$
15. Step 14: Terms Dependent on Coordinates
    1. Step 14.a: $\Sigma$
    2. Step 14.b: $\varpi^{2}$
    3. Step 14.d: $\sin^{2}\theta$
    4. Step 14.e: $\cos\theta$
16. Step 15: Important Vectors
    1. Step 15.a: ${\bf v}$
    2. Step 15.b: $\boldsymbol{\xi}$
    3. Step 15.c: ${\bf e}_{3}$
    4. Step 15.d: ${\bf n}$
    5. Step 15.e: ${\bf S}^{\perp}$
    6. Step 15.f: ${\bf L}$
17. Step 16: Spin Combinations $\boldsymbol{\sigma}$, $\boldsymbol{\sigma}^{*}$, and ${\bf S}_{\rm Kerr}$
    1. Step 16.a: $a$
    2. Step 16.b: $\hat{\bf S}_{\rm Kerr}$
    3. Step 16.c: $\left\lvert {\bf S}_{\rm Kerr} \right\rvert$
    4. Step 16.d: ${\bf S}_{\rm Kerr}$
    5. Step 16.e: $\boldsymbol{\sigma}$
    6. Step 16.f: $\boldsymbol{\sigma}^{*}$
18. Step 17: Fundamental Quantities
    1. Step 17.a: $u$
    2. Step 17.b: $r$
    3. Step 17.c: $\eta$
    4. Step 17.d: $\mu$
    5. Step 17.e: $M$
19. Step 18: Validation
20. Step 19: Output this notebook to $\LaTeX$-formatted PDF file

Step 0: Creating the output directory for SEOBNR [Back to top]¶

$$\label{outputcreation}$$

First we create the output directory for SEOBNR (if it does not already exist):

Step 1: The real Hamiltonian $H_{\textrm{real}}$ [Back to top]¶

$$\label{hreal}$$

The SEOB Hamiltonian $H_{\rm real}$ is given by BB2010 Equation (5.69):

$$\begin{equation*} H_{\rm real} = M \sqrt{ 1 + 2 \eta \left( \frac{ H_{\rm eff} }{ \mu } - 1 \right) }. \end{equation*}$$

Here $H_{\rm eff}$ (defined in this cell) is an effective Hamiltonian (see this cell) and $M$ (defined in this cell), $\mu$ (defined in this cell), and $\eta$ (defined in this cell) are constants determined by $m_{1}$ and $m_{2}$.

Step 2: The Effective Hamiltonian $H_{\rm eff}$ [Back to top]¶

$$\label{heff}$$

The effective Hamiltonian $H_{\rm eff}$ is given by BB2010 Equation (5.70):

$$\begin{equation*} H_{\rm eff} = H_{\rm S} + \underbrace{ \beta^{i} p_{i} + \alpha \sqrt{ \mu^{2} + \gamma^{ij} p_{i} p_{j} + {\cal Q}_{4} } }_{ H_{\rm NS} } - \underbrace{ \frac{ \mu }{ 2 M r^{3} } \left( \delta^{ij} - 3 n^{i} n^{j} \right) S^{*}_{i} S^{*}_{j} }_{ H_{\rm D} }. \end{equation*}$$

Here $H_{\rm S}$ (considered further in this cell) denotes leading order effects of spin-spin and spin-orbit coupling, $H_{\rm NS}$ (considered further in this cell) is the Hamiltonian for a nonspinning test particle, and $H_{\rm D}$ (considered further in this cell) describes quadrupole deformation of the coupling of the particle's spin with itself to leading order. T2014 adds to $H_{\rm eff}$ a 3PN spin-spin term given by

$$\begin{equation*} \frac{d_{\rm SS} \eta }{ r^{4} } \left( {\bf S}_{1}^{2} + {\bf S}_{2}^{2} \right) \end{equation*}$$

where $d_{\rm SS}$ is an adjustable parameter determined by fitting to numerical relativity results. Equation (4.13) of BL2017 gives

$$\begin{equation*} d_{\rm SS} = 528.511252 \chi^{3} \eta^{2} - 41.000256 \chi^{3} \eta + 1161.780126 \chi^{2} \eta^{3} - 326.324859 \chi^{2} \eta^{2} + 37.196389 \chi \eta + 706.958312 \eta^{3} - 36.027203 \eta + 6.068071. \end{equation*}$$

We take $u \equiv \frac{1}{r}$ (as described in this cell), and define $\eta$ in this cell $\chi$ in this cell. Note that the coefficients for $d_{\rm SS}$ have been rounded to coincide with the LALSuite implementation (see the file LALSimIMRSpinEOBHamiltonian.h).

Step 3: Terms of $H_{\rm eff}$ [Back to top]¶

$$\label{heff_terms}$$

In this step, we break down each of the terms $H_{\rm S}$ (defined in this cell), $H_{\rm NS}$ (defined in this cell), and $H_{\rm D}$ (defined in this cell) in $H_{\rm eff}$ (defined in this cell).

Step 3.a: Leading Order Spin Effects $H_{\rm S}$ [Back to top]¶

$$\label{hs}$$

From BB2010 Equation (4.17),

$$\begin{equation*} H_{\rm S} = H_{\rm SO} + H_{\rm SS} \end{equation*}$$

where $H_{\rm SO}$ (defined in this cell) includes spin-orbit terms and $H_{\rm SS}$ (defined in this cell) includes spin-spin terms.

Step 3.b: The Nonspinning Hamiltonian $H_{\rm NS}$ [Back to top]¶

$$\label{hns}$$

We defined $H_{\rm NS}$ in this cell as

$$\begin{equation*} H_{\rm NS} = \underbrace{ \beta^{i} p_{i} }_{ \beta\ p\ \rm sum } + \alpha \sqrt{ \smash[b]{ \underbrace{ \mu^{2} + \gamma^{ij} p_{i} p_{j} + {\cal Q}_{4} }_{ H_{\rm NS}\ \rm radicand } } }. \end{equation*}$$

We compute $\beta\ p$ sum in this cell, $\alpha$ in this cell, and $H_{\rm NS}$ radicand in this cell.

Step 3.c: The Quadrupole Deformation $H_{\rm D}$ [Back to top]¶

$$\label{hd}$$

We defined $H_{\rm D}$ in this cell as:

$$\begin{equation*} H_{\rm D} = \underbrace{ \frac{ \mu }{ 2 M r^{3} } }_{H_{\rm D}\ {\rm coefficient}} \underbrace{ \left( \delta^{ij} - 3 n^{i} n^{j} \right) S^{*}_{i} S^{*}_{j} }_{H_{\rm D}\ {\rm sum}} \end{equation*}$$

We compute $H_{\rm D}$ coefficient in this cell and $H_{\rm D}$ sum in this cell.

Step 4: The Spin-Orbit Term $H_{\rm SO}$ [Back to top]¶

$$\label{hso}$$

We will write BB2010 Equation (4.18) as:

$$\begin{align*} H_{\rm SO} = H_{\rm SO}\ {\rm Term\ 1} + H_{\rm SO}\ {\rm Term\ 2\ coefficient} * H_{\rm SO}\ {\rm Term\ 2}. \end{align*}$$

We define and consider $H_{\rm SO}$ Term 1 in this cell, $H_{\rm SO}$ Term 2 coefficient in this cell, and $H_{\rm SO}$ Term 2 in this cell.

Step 4.a: $H_{\rm SO}$ Term 1 [Back to top]¶

$$\label{hsoterm1}$$

Combining our notation $H_{\rm SO}$ (defined in this cell) with BB2010 Equation (4.18), we have

$$\begin{equation*} H_{\rm SO}\ {\rm Term\ 1} = \frac{ e^{2 \nu - \tilde{\mu} } \left( e^{\tilde{\mu} + \nu} - \tilde{B} \right) \left( \hat{\bf p} \cdot \boldsymbol{\xi} r \right) \left( {\bf S} \cdot \hat{\bf S}_{\rm Kerr} \right) }{ \tilde{B}^{2} \sqrt{Q} \xi^{2} }. \end{equation*}$$

We will write

$$\begin{equation*} H_{\rm SO}\ {\rm Term\ 1} = \frac{ e^{2 \nu} \left( e^{\tilde{\mu}} e^{\nu} - \tilde{B} \right) \left( \hat{\bf p} \cdot \boldsymbol{\xi} r \right) \left( {\bf S} \cdot \hat{\bf S}_{\rm Kerr} \right) }{ e^{ \tilde{\mu} } \tilde{B}^{2} \sqrt{Q} \xi^{2} }. \end{equation*}$$

We define $e^{\tilde{\mu}}$ in this cell, $e^{\nu}$ in this cell, $\tilde{B}$ in this cell, $\hat{\bf p} \cdot \boldsymbol{\xi} r$ in this cell, ${\bf S} \cdot \hat{\bf S}_{\rm Kerr}$ in this cell, $Q$ in this cell, and $\boldsymbol{\xi}^{2}$ in this cell.

Step 4.b: $H_{\rm SO}$ Term 2 Coefficient [Back to top]¶

$$\label{hsoterm2coeff}$$

Combining our notation $H_{\rm SO}$ (defined in this cell) with BB2010 Equation (4.18), we have

$$\begin{equation*} H_{\rm SO}\ {\rm Term\ 2\ coefficient} = \frac{ e^{\nu - 2 \tilde{\mu}} }{ \tilde{B}^{2} \left( \sqrt{Q} + 1 \right) \sqrt{Q} \xi^{2} } \end{equation*}$$

which we write in the form

$$\begin{equation*} H_{\rm SO}\ {\rm Term\ 2\ coefficient} = \frac{ e^{\nu} }{ e^{2 \tilde{\mu}} \tilde{B}^{2} \left( Q + \sqrt{Q} \right) \xi^{2} }. \end{equation*}$$

We define and consider $e^{\nu}$ in this cell, $e^{\tilde{\mu}}$ in this cell, $\tilde{B}$ in this cell, $Q$ in this cell, and $\xi^{2}$ in this cell.

Step 4.c: $H_{\rm SO}$ Term 2 [Back to top]¶

$$\label{hsoterm2}$$

Combining our notation $H_{\rm SO}$ (defined in this cell) with BB2010 Equation (4.18), we have

$$\begin{align*} H_{\rm SO}\ {\rm Term\ 2} &= \underbrace{ \left( {\bf S} \cdot \boldsymbol{\xi} \right) \tilde{J} \left[ \mu_r \left( \hat{\bf p} \cdot {\bf v} r \right) \left( \sqrt{Q} + 1 \right) - \mu_{\cos \theta} \left( \hat{\bf p} \cdot {\bf n} \right) \xi^{2} -\sqrt{Q} \left( \nu_r \left( \hat{\bf p} \cdot {\bf v} r \right) + \left( \mu_{\cos \theta} - \nu_{\cos \theta} \right) \left( \hat{\bf p} \cdot {\bf n} \right) \xi^{2} \right) \right] \tilde{B}^{2} }_{H_{\rm SO}\ {\rm Term\ 2a}} \\ &\ \ \ \ \ + \underbrace{ e^{\tilde{\mu} + \nu} \left( \hat{\bf p} \cdot \boldsymbol{\xi} r \right) \left( 2 \sqrt{Q} + 1 \right) \left[ \tilde{J} \nu_r \left( {\bf S} \cdot {\bf v} \right) - \nu_{\cos \theta} \left( {\bf S} \cdot {\bf n} \right) \xi^{2} \right] \tilde{B} }_{H_{\rm SO}\ {\rm Term\ 2b}} - \underbrace{ \tilde{J} \tilde{B}_{r} e^{\tilde{\mu} + \nu} \left( \hat{\bf p} \cdot \boldsymbol{\xi} r \right) \left( \sqrt{Q} + 1 \right) \left( {\bf S} \cdot {\bf v} \right) }_{H_{\rm SO}\ {\rm Term\ 2c}} \end{align*}$$

We compute $H_{\rm SO}$ Term 2a in this cell, $H_{\rm SO}$ Term 2b in this cell, and $H_{\rm SO}$ Term 2c in this cell.

Step 4.c.i: $H_{\rm SO}$ Term 2a [Back to top]¶

$$\label{hsoterm2a}$$

We defined $H_{\rm S0}$ Term 2a in this cell as

$$\begin{equation*} H_{\rm SO}\ {\rm Term\ 2a} = \left( {\bf S} \cdot \boldsymbol{\xi} \right) \tilde{J} \left[ \mu_r \left( \hat{\bf p} \cdot {\bf v} r \right) \left( \sqrt{Q} + 1 \right) - \mu_{\cos \theta} \left( \hat{\bf p} \cdot {\bf n} \right) \xi^{2} -\sqrt{Q} \left( \nu_r \left( \hat{\bf p} \cdot {\bf v} r \right) + \left( \mu_{\cos \theta} - \nu_{\cos \theta} \right) \left( \hat{\bf p} \cdot {\bf n} \right) \xi^{2} \right) \right] \tilde{B}^{2}. \end{equation*}$$

We define ${\bf S} \cdot \boldsymbol{\xi}$ in this cell, $\tilde{J}$ in this cell, $\mu_{r}$ in this cell, $\hat{\bf p} \cdot {\bf v} r$ in this cell, $Q$ in this cell, $\mu_{\cos \theta}$ in this cell, $\hat{\bf p} \cdot {\bf n}$ in this cell, $\xi^{2}$ in this cell, $\nu_{r}$ in this cell, $\nu_{\cos\theta}$ in this cell, and $\tilde{B}$ in this cell.

Step 4.c.ii: $H_{\rm SO}$ Term 2b [Back to top]¶

$$\label{hsoterm2b}$$

We defined $H_{\rm S0}$ Term 2b in this cell as

$$\begin{equation*} H_{\rm SO}\ {\rm Term\ 2b} = e^{\tilde{\mu} + \nu} \left( \hat{\bf p} \cdot \boldsymbol{\xi} r \right) \left( 2 \sqrt{Q} + 1 \right) \left[ \tilde{J} \nu_r \left( {\bf S} \cdot {\bf v} \right) - \nu_{\cos \theta} \left( {\bf S} \cdot {\bf n} \right) \xi^{2} \right] \tilde{B}. \end{equation*}$$

We define $e^{\tilde{\mu}}$ in this cell, $e^{\nu}$ in this cell, $\hat{\bf p} \cdot \xi r$ in this cell, $Q$ in this cell, $\tilde{J}$ in this cell, $\nu_{r}$ in this cell, ${\bf S} \cdot {\bf v}$ in this cell, $\nu_{\cos\theta}$ in this cell, ${\bf S} \cdot {\bf n}$ in this cell, $\xi^{2}$ in this cell, and $\tilde{B}$ in this cell.

Step 4.c.iii: $H_{\rm SO}$ Term 2c [Back to top]¶

$$\label{hsoterm2c}$$

We defined $H_{\rm S0}$ Term 2c in this cell as

$$\begin{equation*} H_{\rm SO}\ {\rm Term\ 2c} = \tilde{J} \tilde{B}_{r} e^{\tilde{\mu} + \nu} \left( \hat{\bf p} \cdot \boldsymbol{\xi} r \right) \left( \sqrt{Q} + 1 \right) \left( {\bf S} \cdot {\bf v} \right) \end{equation*}$$

We define $\tilde{J}$ in this cell, $\tilde{B}_{r}$ in this cell, $e^{\tilde{\mu}}$ in this cell, $e^{\nu}$ in this cell, $\hat{\bf p} \cdot \xi r$ in this cell, $Q$ in this cell, and ${\bf S} \cdot {\bf v}$ in this cell.

Step 5: The Spin-Spin Term $H_{\rm SS}$ [Back to top]¶

$$\label{hss}$$

We will write BB2010 Equation (4.19) as

$$\begin{equation*} H_{\rm SS} = H_{\rm SS}\ {\rm Term\ 1} + H_{\rm SS}\ {\rm Term\ 2\ coefficient} * H_{\rm SS}\ {\rm Term\ 2} + H_{\rm SS}\ {\rm Term\ 3\ coefficient} * H_{\rm SS}\ {\rm Term\ 3}. \end{equation*}$$

We define $H_{\rm SS}$ Term 1 in this cell, $H_{\rm SS}$ Term 2 coefficient in this cell, $H_{\rm SS}$ Term 2 in this cell, $H_{\rm SS}$ Term 3 coefficient in this cell, and $H_{\rm SS}$ Term 3 in this cell.

Step 5.a: $H_{\rm SS}$ Term 1 [Back to top]¶

$$\label{hssterm1}$$

Combining BB2010 Equation (4.19) with our definition of $H_{\rm SS}$ Term 1 in this cell, we have

$$\begin{equation*} H_{\rm SS}\ {\rm Term\ 1} = \omega \left( {\bf S} \cdot \hat{\bf S}_{\rm Kerr} \right). \end{equation*}$$

We define $\omega$ in this cell and ${\bf S} \cdot \hat{\bf S}_{\rm Kerr}$ in this cell.

Step 5.b: $H_{\rm SS}$ Term 2 Coefficient [Back to top]¶

$$\label{hssterm2coeff}$$

Combining BB2010 Equation (4.19) with ore definition of $H_{\rm SS}$ Term 2 coefficient in this cell, we have

$$\begin{equation*} H_{\rm SS}\ {\rm Term\ 2\ coefficient} = \frac{ e^{-3 \tilde{\mu} -\nu} \tilde{J} \omega_{r} }{ 2 \tilde{B} \left( \sqrt{Q} + 1 \right) \sqrt{Q} \xi^{2} } \end{equation*}$$

which we write as

$$\begin{equation*} H_{\rm SS}\ {\rm Term\ 2\ coefficient} = \frac{ \tilde{J} \omega_{r} }{ 2 e^{2 \tilde{\mu}} e^{\tilde{\mu}} e^{\nu} \tilde{B} \left( Q + \sqrt{Q} \right) \xi^{2} }. \end{equation*}$$

We define $\tilde{J}$ in this cell, $\omega_{r}$ in this cell, $e^{\tilde{\mu}}$ in this cell, $e^{\nu}$ in this cell, $\tilde{B}$ in this cell, $Q$ in this cell, and $\xi^{2}$ in this cell.

Step 5.c: $H_{\rm SS}$ Term 2 [Back to top]¶

$$\label{hssterm2}$$

Combining BB2010 Equation (4.19) with our definition of $H_{\rm SS}$ Term 2 in this cell, we have

$$\begin{equation*} H_{\rm SS}\ {\rm Term\ 2} = -e^{\tilde{\mu} + \nu} \left( {\bf \hat{p}} \cdot {\bf v} r \right) \left( {\bf \hat{p}} \cdot {\bf \xi} r \right) \left( {\bf S} \cdot {\bf \xi} \right) \tilde{B} + e^{2 \left( \tilde{\mu} + \nu \right)} \left( {\bf \hat{p}} \cdot {\bf \xi} r \right)^2 \left( {\bf S} \cdot {\bf v} \right) + e^{2 \tilde{\mu}} \left( 1 + \sqrt{Q} \right) \sqrt{Q} \left( {\bf S} \cdot {\bf v} \right)\xi^2 \tilde{B}^{2} + \tilde{J} \left( {\bf \hat{p}} \cdot {\bf n} \right) \left[ \left( {\bf \hat{p}} \cdot {\bf v} r \right) \left( {\bf S} \cdot {\bf n}\right) - \tilde{J} \left( {\bf \hat{p}} \cdot {\bf n} \right) \left( {\bf S} \cdot {\bf v} \right)\right] \xi^{2} \tilde{B}^{2} \end{equation*}$$

which we write as

$$\begin{align*} H_{\rm SS}\ {\rm Term\ 2} &= e^{\tilde{\mu}} \left( {\bf \hat{p}} \cdot {\bf \xi} r \right) \left[ e^{\tilde{\mu}} e^{2 \nu} \left( {\bf \hat{p}} \cdot {\bf \xi} r \right) \left( {\bf S} \cdot {\bf v} \right) - e^{\nu} \left( {\bf \hat{p}} \cdot {\bf v} r \right) \left( {\bf S} \cdot {\bf \xi} \right) \tilde{B} \right] \\ &\ \ \ \ \ + \xi^2 \tilde{B}^{2} \left\{ e^{2 \tilde{\mu}} \left( \sqrt{Q} + Q \right) \left( {\bf S} \cdot {\bf v} \right) + \tilde{J} \left( {\bf \hat{p}} \cdot {\bf n} \right) \left[ \left( {\bf \hat{p}} \cdot {\bf v} r \right) \left( {\bf S} \cdot {\bf n}\right) - \tilde{J} \left( {\bf \hat{p}} \cdot {\bf n} \right) \left( {\bf S} \cdot {\bf v} \right)\right] \right\} \end{align*}$$

We define $e^{\tilde{\mu}}$ in this cell, $\hat{\bf p} \cdot \boldsymbol{\xi} r$ in this cell, $e^{\nu}$ in this cell, ${\bf S} \cdot {\bf v}$ in this cell, $\hat{\bf p} \cdot {\bf v} r$ in this cell, ${\bf S} \cdot \boldsymbol{\xi}$ in this cell, $\tilde{B}$ in this cell, $Q$ in this cell, $\tilde{J}$ in this cell, $\hat{\bf p} \cdot {\bf n}$ in this cell, ${\bf S} \cdot {\bf n}$ in this cell, and $\xi^{2}$ in this cell.

Step 5.d: $H_{\rm SS}$ Term 3 Coefficient [Back to top]¶

$$\label{hssterm3coeff}$$

Combining BB2010 Equation (4.19) with our definition of $H_{\rm SS}$ Term 3 coefficient in this cell, we have

$$\begin{equation*} H_{\rm SS}\ {\rm Term\ 3\ coefficient} = \frac{ e^{-3 \tilde{\mu} - \nu} \omega_{\cos\theta} }{ 2 \tilde{B} \left( \sqrt{Q} + 1 \right) \sqrt{Q} } \end{equation*}$$

which we write as

$$\begin{equation*} H_{\rm SS}\ {\rm Term\ 3\ coefficient} = \frac{ \omega_{\cos\theta} }{ 2 e^{2 \tilde{\mu}} e^{\tilde{\mu}} e^{\nu} \tilde{B} \left( Q + \sqrt{Q} \right) }. \end{equation*}$$

We define $\omega_{\cos\theta}$ in this cell, $e^{\tilde{\mu}}$ in this cell, $e^{\nu}$ in this cell, and $\tilde{B}$ in this cell, $Q$ in this cell.

Step 5.e: $H_{\rm SS}$ Term 3 [Back to top]¶

$$\label{hssterm3}$$

Combining BB2010 Equation (4.19) with our definition of $H_{\rm SS}$ Term 3 in this cell, we have

$$\begin{align*} H_{\rm SS}\ {\rm Term\ 3} &= -e^{2 \left( \tilde{\mu} + \nu \right)} \left( \hat{\bf p} \cdot {\bf \xi} r \right)^{2} \left( {\bf S} \cdot {\bf n} \right) + e^{\tilde{\mu} +\nu} \tilde{J} \left( {\bf \hat{p}} \cdot {\bf n} \right) \left( {\bf \hat{p}} \cdot {\bf \xi} r \right) \left( {\bf S} \cdot {\bf \xi} \right) \tilde{B} \\ &\ \ \ \ \ + \left[ \left( {\bf S} \cdot {\bf n} \right) \left( {\bf \hat{p}} \cdot {\bf v} r \right)^{2} - \tilde{J} \left( {\bf \hat{p}} \cdot {\bf n} \right) \left( {\bf S} \cdot {\bf v} \right) \left( {\bf \hat{p}} \cdot {\bf v} r\right) - e^{2 \tilde{\mu}} \left( 1 + \sqrt{Q} \right) \sqrt{Q} \left( {\bf S} \cdot {\bf n} \right) \xi^{2} \right] \tilde{B}^{2} \end{align*}$$

which we write as

$$\begin{align*} H_{\rm SS}\ {\rm Term\ 3} &= e^{\tilde{\mu}} e^{\nu} \left( \hat{\bf p} \cdot {\bf \xi} r \right) \left[ \tilde{J} \left( {\bf \hat{p}} \cdot {\bf n} \right) \left( {\bf S} \cdot {\bf \xi} \right) \tilde{B} - e^{\tilde{\mu}} e^{\nu} \left( \hat{\bf p} \cdot {\bf \xi} r \right) \left( {\bf S} \cdot {\bf n} \right) \right] \\ &\ \ \ \ \ + \left\{ \left( {\bf \hat{p}} \cdot {\bf v} r \right) \left[ \left( {\bf S} \cdot {\bf n} \right) \left( {\bf \hat{p}} \cdot {\bf v} r \right) - \tilde{J} \left( {\bf \hat{p}} \cdot {\bf n} \right) \left( {\bf S} \cdot {\bf v} \right) \right] - e^{2 \tilde{\mu}} \left( \sqrt{Q} + Q \right) \left( {\bf S} \cdot {\bf n} \right) \xi^{2} \right\} \tilde{B}^{2} \end{align*}$$

We define $e^{\tilde{\mu}}$ in this cell, $e^{\nu}$ in this cell, $\hat{\bf p} \cdot \boldsymbol{\xi} r$ in this cell, $\tilde{J}$ in this cell, $\hat{\bf p} \cdot {\bf n}$ in this cell, ${\bf S} \cdot \boldsymbol{\xi}$ in this cell, $\tilde{B}$ in this cell, ${\bf S} \cdot {\bf n}$ in this cell, $\hat{\bf p} \cdot {\bf v} r$ in this cell, ${\bf S} \cdot {\bf v}$ in this cell, $Q$ in this cell, and $\xi^{2}$ in this cell.

Step 6: $H_{\rm NS}$ Terms [Back to top]¶

$$\label{hnsterms}$$

We collect here the terms in $H_{\rm NS}$ (defined in this cell).

Step 6.a: $\beta p$ sum [Back to top]¶

$$\label{betapsum}$$

We defined the term $\beta p$ sum in this cell as

$$\begin{equation*} \beta p\ {\rm sum} = \beta^{i} p_{i}. \end{equation*}$$

From BB2010 Equation (5.45), we have

$$\begin{equation*} \beta^{i} = \frac{ g^{ti} }{ g^{tt} }, \end{equation*}$$

but from BB2010 Equations (5.36) we see that $g^{tr} = g^{t \theta} = 0$. Thus only $\beta^{\phi}$ is nonzero. Combining BB2010 Equations (5.45), (5.36e), and (5.36a), we find

$$\begin{equation*} \beta^{\phi} = \frac{ -\frac{ \tilde{\omega}_{\rm fd} }{ \Delta_{t} \Sigma } }{ -\frac{ \Lambda_{t} }{ \Delta_{t} \Sigma } } = \frac{ \tilde{\omega}_{\rm fd} }{ \Lambda_{t} } \end{equation*}$$

Therefore

$$\begin{equation*} \beta^{i} p_{i} = \frac{ \tilde{\omega}_{\rm fd} }{ \Lambda_{t} } p_{\phi}. \end{equation*}$$

We define $\tilde{\omega}_{\rm fd}$ in this cell, $\Lambda_{t}$ in this cell, and $p_{\phi}$ in this cell.

Step 6.b: $\alpha$ [Back to top]¶

$$\label{alpha}$$

From BB2010 Equation (5.44), we have $$\begin{equation*} \alpha = \frac{ 1 }{ \sqrt{ -g^{tt}} }, \end{equation*}$$

and from BB2010 Equation (5.36a) we have

$$\begin{equation*} g^{tt} = -\frac{ \Lambda_{t} }{ \Delta_{t} \Sigma }. \end{equation*}$$

Therefore

$$\begin{equation*} \alpha = \sqrt{ \frac{ \Delta_{t} \Sigma }{ \Lambda_{t} } }. \end{equation*}$$

We define $\Delta_{t}$ in this cell, $\Sigma$ in this cell, and $\Lambda_{t}$ in this cell.

Step 6.c: $H_{\rm NS}$ radicand [Back to top]¶

$$\label{hnsradicand}$$

Recall that we defined $H_{\rm NS}$ radicand in this cell as

$$\begin{equation*} H_{\rm NS}\ {\rm radicand} = \mu^{2} + \underbrace{\gamma^{ij} p_{i} p_{j}}_{\gamma p\ \rm sum} + {\cal Q}_{4} \end{equation*}$$

We define $\mu$ in this cell, $\gamma p$ sum in this cell, and ${\cal Q}_{4}$ in this cell.

Step 6.c.i: $\gamma^{ij} p_{i} p_{j}$ [Back to top]¶

$$\label{gammappsum}$$

From BB2010 Equation (5.46), we have

$$\begin{equation*} \gamma^{ij} = g^{ij} - \frac{ g^{ti} g^{tj} }{ g^{tt} }. \end{equation*}$$

Combining this result with BB2010 Equations 5.36, we have

$$\begin{equation*} \gamma^{r\theta} = \gamma^{r\phi} = \gamma^{\theta r} = \gamma^{\theta\phi} = \gamma^{\phi r} = \gamma^{\phi\theta} = 0 \end{equation*}$$

and

$$\begin{align*} \gamma^{rr} &= g^{rr} = \frac{ \Delta_{r} }{ \Sigma } \\ \gamma^{\theta\theta} &= g^{\theta\theta} = \frac{ 1 }{ \Sigma } \\ \gamma^{\phi\phi} &= \frac{ \Sigma }{ \Lambda_{t} \sin^{2} \theta }. \end{align*}$$

Therefore

$$\begin{align*} \gamma^{ij} p_{i} p_{j} &= \gamma^{rr} p_{r} p_{r} + \gamma^{\theta\theta} p_{\theta} p_{\theta} + \gamma^{\phi\phi} p_{\phi} p_{\phi} \\ &= \frac{ \Delta_{r} }{ \Sigma } p_{r}^{2} + \frac{ 1 }{ \Sigma } p_{\theta}^{2} + \frac{ \Sigma }{ \Lambda_{t} \sin^{2} \theta } p_{\phi}^{2}. \end{align*}$$

Converting Boyer-Lindquist coordinates to tortoise coordinates (the transformation for which is found in the Appendix of P2010), we have

$$\begin{align*} p_{r} &= \hat{\bf p} \cdot {\bf n} \\ p_{\theta} &= \hat{\bf p} \cdot {\bf v} \frac{ r }{ \sin \theta } \\ p_{\phi} &= \hat{\bf p} \cdot \boldsymbol{\xi} r. \end{align*}$$

Therefore

$$\begin{equation*} \gamma^{ij} p_{i} p_{j} = \frac{ \Delta_{r} }{ \Sigma } \left( \hat{\bf p} \cdot {\bf n} \right)^{2} + \Sigma^{-1} \left( \hat{\bf p} \cdot {\bf v} \frac{ r }{ \sin \theta } \right)^{2} + \frac{ \Sigma }{ \Lambda_{t} \sin^{2} \theta } \left( \hat{\bf p} \cdot \boldsymbol{\xi} r \right)^{2}. \end{equation*}$$

We define $\Delta_{r}$ in this cell, $\Sigma$ in this cell, $\hat{\bf p} \cdot {\bf n}$ in this cell, $\hat{\bf p} \cdot {\bf v} r$ in this cell, $\sin^{2} \theta$ in this cell, $\Lambda_{t}$ in this cell, and $\hat{\bf p} \cdot \boldsymbol{\xi} r$ in this cell.

Step 6.c.ii: ${\cal Q}_{4}$ [Back to top]¶

$$\label{q4}$$

From T2012 Equation (15),

$$\begin{equation*} {\cal Q}_{4} \propto \frac{ p_{r^{*}}^{4} }{ r^{2} } \left( r^{2} + \chi_{\rm Kerr}^{2} \right)^{4}. \end{equation*}$$

We denote $p_{r^{*}}$ by prT. Converting from tortoise coordinates to physical coordinates(the transformation for which is found in the Appendix of P2010), we find

$$\begin{equation*} {\cal Q}_{4} = \frac{ prT^{4} }{ r^{2} } z_{3} \end{equation*}$$

where $z_{3}$ is found in D2000 Equation (4.34):

$$\begin{equation*} z_{3} = 2 \left( 4 - 3 \nu \right) \nu. \end{equation*}$$

In the notation of BB2010, $\nu = \eta$ (see discussion after T2012 Equation (2)). Thus

$$\begin{equation*} {\cal Q}_{4} = 2 prT^{4} u^{2} \left( 4 - 3 \eta \right) \eta. \end{equation*}$$

We define prT in this cell, $u$ in this cell, and $\eta$ in this cell below.

Step 7: The $H_{\rm D}$ Terms [Back to top]¶

$$\label{hdterms}$$

Recall we defined $H_{\rm D}$ in this cell as

$$\begin{equation*} H_{\rm D} = H_{\rm D}\ {\rm coeffecient} * H_{\rm D}\ {\rm sum}. \end{equation*}$$

In this step we break down each of $H_{\rm D}$ coefficient (defined in this cell) and $H_{\rm D}$ sum (defined in this cell).

Step 7.a: $H_{\rm D}$ Coefficient [Back to top]¶

$$\label{hdcoeff}$$

From our definition of $H_{\rm D}$ in this cell, we have

$$\begin{equation*} H_{\rm D}\ {\rm coefficient} = \frac{ \mu }{ 2 M r^{3} }, \end{equation*}$$

and recalling the definition of $\eta$ we'll write

$$\begin{equation*} H_{\rm D}\ {\rm coefficient} = \frac{ \eta }{ 2 r^{3} }. \end{equation*}$$

We define $\eta$ in this cell and $r$ in this cell.

Step 7.b: $H_{\rm D}$ Sum [Back to top]¶

$$\label{hdsum}$$

From our definition of $H_{\rm D}$ in this cell, we have

$$\begin{align*} H_{\rm D}\ {\rm sum} &= \left( \delta^{ij} - 3 n^{i} n^{j} \right) S^{*}_{i} S^{*}_{j} \\ &= \underbrace{\delta^{ij} S^{*}_{i} S^{*}_{j}}_{\rm Term\ 1} - \underbrace{3 n^{i} n^{j} S^{*}_{i} S^{*}_{j}}_{\rm Term\ 2}. \end{align*}$$

We compute $H_{\rm D}$ Term 1 in this cell and $H_{\rm D}$ Term 2 in this cell.

Step 7.b.i: $H_{\rm D}$ Sum Term 1 [Back to top]¶

$$\label{hdsumterm1}$$

From our definition of $H_{\rm D}$ sum Term 1 in this cell, we have

$$\begin{equation*} H_{\rm D}\ {\rm sum\ Term\ 1} = \delta^{ij} S^{*}_{i} S^{*}_{j} \end{equation*}$$

where $\delta^{ij}$ is the Kronecker delta:

$$\begin{equation*} \delta_{ij} = \left\{ \begin{array}{cc} 0, & i \not= j \\ 1, & i = j. \end{array} \right. \end{equation*}$$

Thus we have

$$\begin{equation*} H_{\rm D}\ {\rm sum\ Term\ 1} = S^{*}_{1} S^{*}_{1} + S^{*}_{2} S^{*}_{2} + S^{*}_{3} S^{*}_{3} \end{equation*}$$

We define ${\bf S}^{*}$ in this cell.

Step 7.b.ii: $H_{\rm D}$ Sum Term 2 [Back to top]¶

$$\label{hdsumterm2}$$

From our definition of $H_{\rm D}$ sum Term 2 in this cell, we have

$$\begin{align*} H_{\rm D}\ {\rm sum\ Term\ 2} &= 3 n^{i} n^{j} S^{*}_{i} S^{*}_{j} \\ &= 3 \left( {\bf S}^{*} \cdot {\bf n} \right)^{2} \\ \end{align*}$$

We define ${\bf S}^{*} \cdot {\bf n}$ in this cell.

Step 8: Common Dot Products [Back to top]¶

$$\label{dotproducts}$$

What follows are definitions of many common dot products.

Step 8.a: ${\bf S} \cdot \boldsymbol{\xi}$ [Back to top]¶

$$\label{sdotxi}$$

We have

$$\begin{equation*} {\bf S} \cdot \boldsymbol{\xi} = S^{1} \xi^{1} + S^{2} \xi^{2} + S^{3} \xi^{3} \end{equation*}$$

We define $\xi$ in this cell.

Step 8.b: ${\bf S} \cdot {\bf v}$ [Back to top]¶

$$\label{sdotv}$$

We have

$$\begin{equation*} {\bf S} \cdot {\bf v} = S^{1} v^{1} + S^{2} v^{2} + S^{3} v^{3}. \end{equation*}$$

We define ${\bf v}$ in this cell.

Step 8.c: ${\bf S} \cdot {\bf n}$ [Back to top]¶

$$\label{sdotn}$$

We have

$$\begin{equation*} {\bf S} \cdot {\bf n} = S^{1} n^{1} + S^{2} n^{2} + S^{3} n^{3}. \end{equation*}$$

We define ${\bf n}$ in this cell.

Step 8.d: ${\bf S} \cdot \hat{\bf S}_{\rm Kerr}$ [Back to top]¶

$$\label{sdotskerrhat}$$

We have

$$\begin{equation*} {\bf S} \cdot \hat{\bf S}_{\rm Kerr} = S^{1} \hat{S}_{\rm Kerr}^{1} + S^{2} \hat{S}_{\rm Kerr}^{2} + S^{3} \hat{S}_{\rm Kerr}^{3}. \end{equation*}$$

We define $\hat{\bf S}_{\rm Kerr}$ in this cell.

Step 8.e: ${\bf S}^{*} \cdot {\bf n}$ [Back to top]¶

$$\label{sstardotn}$$

We have

$$\begin{equation*} {\bf S}^{*} \cdot {\bf n} = {\bf S}^{*}_{1} n_{1} + {\bf S}^{*}_{2} n_{2} + {\bf S}^{*}_{3} n_{3}. \end{equation*}$$

We define ${\bf S}^{*}$ in this cell and ${\bf n}$ in this cell.

Step 9: $H_{\rm real}$ Spin Combination ${\bf S}^{*}$ [Back to top]¶

$$\label{hreal_spin_combos}$$

We collect here terms defining and containing ${\bf S}^{*}$.

Step 9.a: ${\bf S}^{*}$ [Back to top]¶

$$\label{sstar}$$

From BB2010 Equation (5.63):

$$\begin{equation*} {\bf S}^{*} = \boldsymbol{\sigma}^{*} + \frac{ 1 }{ c^{2} } \boldsymbol{\Delta}_{\sigma^{*}}. \end{equation*}$$

We define $\boldsymbol{\sigma}^{*}$ in this cell and $\boldsymbol{\Delta}_{\sigma^{*}}$ in this cell.

Please note: after normalization, ${\bf S} = {\bf S}^{*}$. See BB2010 Equation (4.26).

Step 9.b: $\boldsymbol{\Delta}_{\sigma^{*}}$ [Back to top]¶

$$\label{deltasigmastar}$$

We can write $\boldsymbol{\Delta}_{\sigma^{*}}$ as

$$\begin{equation*} \boldsymbol{\Delta}_{\sigma^{*}} = \boldsymbol{\sigma}^{*} \left( \boldsymbol{\sigma}^{*}\ {\rm coefficient} \right) + \boldsymbol{\sigma} \left( \boldsymbol{\sigma}\ {\rm coefficient} \right) \end{equation*}$$

For further dissection, see $\boldsymbol{\sigma}^{*}$ in this cell, $\boldsymbol{\sigma}^{*}$ coefficient in this cell, $\boldsymbol{\sigma}$ in this cell, and $\boldsymbol{\sigma}$ coefficient in this cell.

Step 9.c: $\boldsymbol{\sigma}^{*}$ coefficient [Back to top]¶

$$\label{sigmastarcoeff}$$

We will break down $\boldsymbol{\sigma}^{*}\ {\rm coefficient}$ into two terms:

$$\begin{equation*} \boldsymbol{\sigma}^{*}\ {\rm coefficient} = \boldsymbol{\sigma}^{*}\ {\rm coefficient\ Term\ 1} + \boldsymbol{\sigma}^{*}\ {\rm coefficient\ Term\ 2} \end{equation*}$$

We compute $\boldsymbol{\sigma}^{*}$ coefficient Term 1 in this cell and $\boldsymbol{\sigma}^{*}$ coefficient Term 2 in this cell.

Step 9.c.i: $\boldsymbol{\sigma}^{*}$ Coefficient Term 1 [Back to top]¶

$$\label{sigmastarcoeffterm1}$$

We build this term from BB2011 Equation (51) with $b_{0} = 0$ (see discussion preceeding T2012 Equation (4)), where what is listed below is the coefficient on $\boldsymbol{\sigma}^{*}$:

$$\begin{align*} \boldsymbol{\sigma}^{*}\ {\rm coefficient\ Term\ 1} &= \frac{7}{6} \eta \frac{M}{r} + \frac{1}{3} \eta \left( Q - 1 \right) - \frac{5}{2} \eta \frac{ \Delta_r }{ \Sigma } \left( {\bf n} \cdot \hat{\bf p} \right)^{2} \\ &= \frac{ \eta }{ 12 } \left( 14 \frac{ M }{ r } + 4 \left( Q - 1 \right) - 30 \frac{ \Delta_r }{ \Sigma } \left( {\bf n} \cdot \hat{\bf p} \right)^{2} \right) \end{align*}$$

We group together and compute $Q-1$ in this cell and $\frac{ \Delta_r }{ \Sigma } \left( {\bf n} \cdot \hat{\bf p} \right)^{2}$ in this cell; we define $r$ in this cell, $\eta$ in this cell, and $M$ in this cell below.

Step 9.c.ii: $\boldsymbol{\sigma}^{*}$ Coefficient Term 2 [Back to top]¶

$$\label{sigmastarcoeffterm2}$$

We build this term from BB2011 Equation (52) with all $b_{i} = 0$, $i \in \left\{0, 1, 2, 3\right\}$ (see discussion preceeding T2012 Equation (4)), and just the coefficient on $\boldsymbol{\sigma}^{*}$. In the LALSuite code this is the variable 'sMultiplier1':

$$\begin{align*} \boldsymbol{\sigma}^{*}\ {\rm coefficient\ Term\ 2} &= \frac{1}{36} \left( 353 \eta - 27 \eta^2 \right) \left( \frac{M}{r} \right)^{2} + \frac{5}{3} \left( 3 \eta^2 \right) \frac{ \Delta_{r}^{2} }{ \Sigma^{2} } \left( {\bf n} \cdot \hat{\bf p} \right)^{4} \\ &\ \ \ \ \ + \frac{1}{72} \left( -23 \eta -3 \eta^{2} \right) \left( Q - 1 \right)^{2} + \frac{1}{36} \left( -103 \eta + 60 \eta^{2} \right) \frac{M}{r} \left( Q - 1 \right) \\ &\ \ \ \ \ + \frac{1}{12} \left( 16 \eta - 21 \eta^{2} \right) \frac{ \Delta_{r} }{ \Sigma } \left( {\bf n} \cdot \hat{\bf p} \right)^{2} \left( Q - 1 \right) + \frac{1}{12} \left( 47 \eta - 54 \eta^{2} \right) \frac{M}{r} \frac{ \Delta_{r} }{ \Sigma } \left( {\bf n} \cdot \hat{\bf p} \right)^{2} \\ &= \frac{ \eta }{ 72 r^{2} } \left[ \left( 706 - 54 \eta \right) M^{2} + 360 \eta r^{2} \frac{ \Delta_{r}^{2} }{ \Sigma^{2} } \left( {\bf n} \cdot \hat{\bf p} \right)^{4} + r^{2} \left( -23 - 3 \eta \right) \left( Q - 1 \right)^{2} + \left( -206 + 120 \eta \right) M r \left( Q - 1 \right) \right. \\ &\ \ \ \ \ + \left. \left( 96 - 126 \eta \right) r^{2} \frac{ \Delta_{r} }{ \Sigma } \left( {\bf n} \cdot \hat{\bf p} \right)^{2} \left( Q - 1 \right) + \left( 282 - 324 \eta \right) M r \frac{ \Delta_{r} }{ \Sigma } \left( {\bf n} \cdot \hat{\bf p} \right)^{2} \right] \\ &= \frac{ \eta }{ 72 r^{2} } \left[ 706 + r \left( -206 M \left( Q - 1 \right) + 282 M \frac{ \Delta_{r} }{ \Sigma } \left( {\bf n} \cdot \hat{\bf p} \right)^{2} + r \left( Q -1 \right) \left( 96 \frac{ \Delta_{r} }{ \Sigma } \left( {\bf n} \cdot \hat{\bf p} \right)^{2} - 23 \left( Q - 1 \right) \right) \right) \right. \\ &\ \ \ \ \ + \left. \eta \left( -54 M^{2} + r \left( 120 M \left( Q -1 \right) - 324 M \frac{ \Delta_{r} }{ \Sigma } \left( {\bf n} \cdot \hat{\bf p} \right)^{2} \right.\right.\right. \\ &\ \ \ \ \ + \left.\left.\left. r \left( 360 \frac{ \Delta_{r}^{2} }{ \Sigma^{2} } \left( {\bf n} \cdot \hat{\bf p} \right)^{4} + \left( Q - 1 \right) \left( -126 \frac{ \Delta_{r} }{ \Sigma } \left( {\bf n} \cdot \hat{\bf p} \right)^{2} - 3 \left( Q - 1 \right) \right) \right)\right) \right) \right] \end{align*}$$

We define $r$ in this cell, $\eta$ in this cell, and $M$ in this cell; we group together and define $Q - 1$ in this cell, and $\frac{ \Delta_{r} }{ \Sigma } \left( {\bf n} \cdot \hat{\bf p} \right)^{2}$ in this cell.

Step 9.d: $\boldsymbol{\sigma}$ coefficient [Back to top]¶

$$\label{sigmacoeff}$$

We will break down $\boldsymbol{\sigma}\ {\rm coefficient}$ into three terms:

$$\begin{equation*} \boldsymbol{\sigma}\ {\rm coefficient} = \boldsymbol{\sigma}\ {\rm coefficient\ Term\ 1} + \boldsymbol{\sigma}\ {\rm coefficient\ Term\ 2} + \boldsymbol{\sigma}\ {\rm coefficient\ Term\ 3} \end{equation*}$$

We compute $\boldsymbol{\sigma}$ coefficient Term 1 in this cell, $\boldsymbol{\sigma}$ coefficient Term 2 in this cell, and $\boldsymbol{\sigma}$ coefficient Term 3 in this cell.

Step 9.d.i: $\boldsymbol{\sigma}$ Coefficient Term 1 [Back to top]¶

$$\label{sigmacoeffterm1}$$

We build this term from BB2011 Equation (51) with $a_{0} = 0$ (see discussion preceeding T2012 Equation (4)), where what is listed below is the coefficient on $\boldsymbol{\sigma}$:

$$\begin{align*} \boldsymbol{\sigma}\ {\rm coefficient\ Term\ 1} &= -\frac{2}{3} \eta \frac{ M }{ r } + \frac{1}{4} \eta \left( Q - 1 \right) - 3 \eta \frac{ \Delta_r }{ \Sigma } \left( {\bf n} \cdot \hat{\bf p} \right)^{2} \\ &= \frac{ \eta }{ 12 } \left( -8 \frac{ M }{ r } + 3 \left( Q - 1 \right) - 36 \smash[b]{\underbrace{ \frac{ \Delta_r }{ \Sigma } \left( {\bf n} \cdot \hat{\bf p} \right)^{2} }_{\rm DrSipn2}} \vphantom{\underbrace{a}_{b}} \right) \end{align*}$$

We define $\eta$ in this cell, $M$ in this cell, $Q-1$ in this cell, and $\frac{ \Delta_r }{ \Sigma } \left( {\bf n} \cdot \hat{\bf p} \right)^{2}$ in this cell.

Step 9.d.ii: $\boldsymbol{\sigma}$ Coefficient Term 2 [Back to top]¶

$$\label{sigmacoeffterm2}$$

We build this term from BB2011 Equation (52) with all $a_{i} = 0$, $i \in \left\{0, 1, 2, 3\right\}$ (see discussion preceeding T2012 Equation (4)), and just the coefficient on $\boldsymbol{\sigma}$:

$$\begin{align*} \boldsymbol{\sigma}\ {\rm coefficient\ Term\ 2} &= \frac{1}{9} \left( -56 \eta -21 \eta^{2} \right) \left( \frac{ M }{ r } \right)^{2} + \frac{5}{24} \left( 27 \eta^{2} \right) \frac{ \Delta_r^{2} }{ \Sigma^{2} } \left( {\bf n} \cdot \hat{\bf p} \right)^{4} \\ &\ \ \ \ \ + \frac{1}{144} \left(-45 \eta \right) \left( Q - 1 \right)^{2} + \frac{1}{36} \left( -109 \eta + 51 \eta^{2} \right) \frac{ M }{ r } \left( Q - 1 \right) \\ &\ \ \ \ \ + \frac{1}{24} \left( 6 \eta - 39\eta^{2} \right) \frac{ \Delta_{r} }{ \Sigma } \left( {\bf n} \cdot \hat{\bf p} \right)^{2} \left( Q - 1 \right) + \frac{1}{24} \left( -16 \eta - 147 \eta^{2} \right) \frac{ M }{ r } \frac{ \Delta_{r} }{ \Sigma } \left( {\bf n} \cdot \hat{\bf p} \right)^{2} \\ &= \frac{ \eta }{ 144 r^{2} } \left[ -896 M^{2} + r \left( -436 M \left( Q - 1 \right) - 96 M \frac{ \Delta_{r} }{ \Sigma } \left( {\bf n} \cdot \hat{\bf p} \right)^{2} \right.\right. \\ &\ \ \ \ \ \left.\left. + r \left( -45 \left( Q - 1 \right)^{2} + 36 \left( Q - 1 \right) \frac{ \Delta_{r} }{ \Sigma } \left( {\bf n} \cdot \hat{\bf p} \right)^{2} \right) \right) + \eta \left( -336 M^{2} + r \left( 204 M \left( Q -1 \right) - 882 M \frac{ \Delta_{r} }{ \Sigma } \left( {\bf n} \cdot \hat{\bf p} \right)^{2} \right.\right.\right. \\ &\ \ \ \ \ \left.\left.\left. + r \left( 810 \frac{ \Delta_{r}^{2} }{ \Sigma^{2} } \left( {\bf n} \cdot \hat{\bf p} \right)^{4} - 234 \left( Q - 1 \right) \frac{ \Delta_{r} }{ \Sigma } \left( {\bf n} \cdot \hat{\bf p} \right)^{2} \right) \right) \right) \right] \end{align*}$$

We define $\eta$ in this cell, $M$ in this cell, $Q - 1$ in this cell, and $\frac{ \Delta_{r} }{ \Sigma } \left( {\bf n} \cdot \hat{\bf p} \right)^{2}$ in this cell.

Step 9.d.iii: $\boldsymbol{\sigma}$ Coefficient Term 3 [Back to top]¶

$$\label{sigmacoeffterm3}$$

From Section II of T2014),

$$\begin{equation*} \boldsymbol{\sigma}\ {\rm coefficient\ Term\ 3} = \eta d_{\rm SO} u^{3}. \end{equation*}$$

where $d_{\rm SO}$ is a fitting parameter. Equation (4.13) of BL2017 gives

$$\begin{equation*} d_{\rm SO} = 147.481449 \chi^{3} \eta^{2} - 568.651115 \chi^3 \eta + 66.198703 \chi^{3} - 343.313058 \chi^{2} \eta + 2495.293427 \chi \eta^{2} - 44.532373 \end{equation*}$$

We define $\eta$ in this cell, $u$ in this cell, and $\chi$ in this cell. Note that the values have been rounded to agree with those in the LALSuite implementation (see the file LALSimIMRSpinEOBHamiltonian.h).

Step 10: Derivatives of the Metric Potential [Back to top]¶

$$\label{metpotderivs}$$

We collect here terms dependent on derivatives of the metric potential (see BB2010 Equations (5.47)).

Step 10.a: $\omega_{r}$ [Back to top]¶

$$\label{omegar}$$

From BB2010 Equation (5.47b) we have

$$\begin{equation*} \omega_{r} = \frac{ \Lambda_{t} \tilde{\omega}_{\rm fd}^{\prime} - \Lambda_{t}^{\prime} \tilde{\omega}_{\rm fd} }{ \Lambda_{t}^{2} }. \end{equation*}$$

We define $\Lambda_{t}$ in this cell, $\tilde{\omega}_{\rm fd}^{\prime}$ in this cell, $\Lambda_{t}^{\prime}$ in this cell, and $\tilde{\omega}_{\rm fd}$ in this cell.

Step 10.b: $\nu_{r}$ [Back to top]¶

$$\label{nur}$$

From BB2010 Equation (5.47c) we have

$$\begin{equation*} \nu_{r} = \frac{ r }{ \Sigma } + \frac{ \varpi^{2} \left( \varpi^{2} \Delta^{\prime}_{t} - 4 r \Delta_{t} \right) }{ 2 \Lambda_{t} \Delta_{t} }. \end{equation*}$$

We define $r$ in this cell, $\Sigma$ in this cell, $\varpi^{2}$ in this cell, $\Delta_{t}^{\prime}$ in this cell, $\Delta_{t}$ in this cell, and $\Lambda_{t}$ in this cell.

Step 10.c: $\mu_{r}$ [Back to top]¶

$$\label{mur}$$

From BB2010 Equation (5.47d) we have

$$\begin{equation*} \mu_{r} = \frac{ r }{ \Sigma } - \frac{ 1 }{ \sqrt{ \Delta_{r} } }. \end{equation*}$$

We define $r$ in this cell, $\Sigma$ in this cell, and $\Delta_{r}$ in this cell.

Step 10.d: $\omega_{\cos\theta}$ [Back to top]¶

$$\label{omegacostheta}$$

From BB2010 Equation (5.47f), we have

$$\begin{equation*} \omega_{\cos\theta} = -\frac{ 2 a^{2} \cos\theta \Delta_{t} \tilde{\omega}_{\rm fd} }{ \Lambda_{t}^{2} }. \end{equation*}$$

We define $a$ in this cell, $\cos\theta$ in this cell, $\Delta_{t}$ in this cell, $\tilde{\omega}_{\rm fd}$ in this cell, and $\Lambda_{t}$ in this cell.

Step 10.e: $\nu_{\cos\theta}$ [Back to top]¶

$$\label{nucostheta}$$

From BB2010 Equation (5.47g) we have

$$\begin{equation*} \nu_{\cos\theta} = \frac{ a^{2} \varpi^{2} \cos\theta \left( \varpi^{2} - \Delta_{t} \right) }{ \Lambda_{t} \Sigma }. \end{equation*}$$

We define $a$ in this cell, $\varpi^{2}$ in this cell, $\cos\theta$ in this cell, $\Delta_{t}$ in this cell, $\Lambda_{t}$ in this cell, and $\Sigma$ in this cell.

Step 10.f: $\mu_{\cos \theta}$ [Back to top]¶

$$\label{mucostheta}$$

From BB2010 Equation (5.47h) we have

$$\begin{equation*} \mu_{\cos \theta} = \frac{ a^{2} \cos \theta }{ \Sigma }. \end{equation*}$$

We define $a$ in this cell, $\cos \theta$ in this cell, and $\Sigma$ in this cell below.

Step 10.g: $\Lambda_{t}^{\prime}$ [Back to top]¶

$$\label{lambdatprm}$$

From the discussion after BB2010 Equations (5.47), we know that the prime notation indicates a derivative with respect to $r$. Using the definiton of $\Lambda_{t}$ in this cell, we have

$$\begin{equation*} \Lambda_{t}^{\prime} = 4 \left( a^{2} + r^{2} \right) r - a^{2} \Delta_{t}^{\prime} \sin^{2} \theta. \end{equation*}$$

We define $a$ in this cell, $r$ in this cell, $\Delta_{u}$ in this cell, and $\sin^{2}\theta$ in this cell.

Step 10.h: $\tilde{\omega}_{\rm fd}^{\prime}$ [Back to top]¶

$$\label{omegatildeprm}$$

From the discussion after BB2010 Equation (5.47), we know that the prime notation indicates a derivative with respect to $r$. Using the definiton of $\tilde{\omega}_{\rm fd}$ in this cell, we have

$$\begin{equation*} \tilde{\omega}_{\rm fd}^{\prime} = 2 a M. \end{equation*}$$

We define $a$ in this cell and $M$ in this cell.

Step 11: The Deformed and Rescaled Metric Potentials [Back to top]¶

$$\label{metpots}$$

We collect here terms of the deformed and scaled metric potentils. See BB2010 Equations (5.30)--(5.34) and (5.48)--(5.52).

Step 11.a: $\omega$ [Back to top]¶

$$\label{omega}$$

From BB2010 Equation (5.31) we have

$$\begin{equation*} \omega = \frac{ \tilde{\omega}_{\rm fd} }{ \Lambda_{t} }. \end{equation*}$$

We define $\tilde{\omega}_{\rm fd}$ in this cell and $\Lambda_{t}$ in this cell.

Step 11.b: $e^{2\nu}$ and $e^{\nu}$ [Back to top]¶

$$\label{exp2nu}$$

From BB2010 Equation (5.32), we have

$$\begin{equation*} e^{2 \nu} = \frac{ \Delta_{t} \Sigma }{ \Lambda_t }. \end{equation*}$$

It follows that

$$\begin{equation*} e^{\nu} = \sqrt{ \frac{ \Delta_{t} \Sigma }{ \Lambda_t } }. \end{equation*}$$

We define $\Delta_{t}$ in this cell, $\Sigma$ in this cell, and $\Lambda_{t}$ in this cell.

Step 11.c: $\tilde{B}$ [Back to top]¶

$$\label{btilde}$$

From BB2010 Equation (5.48), we have

$$\begin{equation*} \tilde{B} = \sqrt{ \Delta_{t} }. \end{equation*}$$

We define $\Delta_{t}$ in this cell.

Step 11.d: $\tilde{B}_{r}$ [Back to top]¶

$$\label{brtilde}$$

From BB2010 Equation (5.49), we have

$$\begin{equation*} \tilde{B}_{r} = \frac{ \sqrt{ \Delta_{r} } \Delta_{t}^{\prime} - 2 \Delta_{t} }{ 2 \sqrt{ \Delta_{r} \Delta_{t} } }. \end{equation*}$$

We define $\Delta_{r}$ in this cell, $\Delta_{t}^{\prime}$ in this cell, and $\Delta_{t}$ in this cell.

Step 11.e: $e^{2\tilde{\mu}}$ and $e^{\tilde{\mu}}$ [Back to top]¶

$$\label{exp2mu}$$

From BB2010 Equation (5.50), we have

$$\begin{equation*} e^{2 \tilde{\mu}} = \Sigma. \end{equation*}$$

It follows that

$$\begin{equation*} e^{\tilde{\mu}} = \sqrt{ \Sigma }. \end{equation*}$$

We define $\Sigma$ in this cell.

Step 11.f: $\tilde{J}$ [Back to top]¶

$$\label{jtilde}$$

From BB2010 Equation (5.51) we have

$$\begin{equation*} \tilde{J} = \sqrt{ \Delta_{r} }. \end{equation*}$$

We define $\Delta_{r}$ in this cell below.

Step 11.g: $Q$ [Back to top]¶

$$\label{q}$$

From BB2010 Equation (5.52),

$$\begin{equation*} Q = 1 + \underbrace{ \frac{ \Delta_{r} }{ \Sigma } \left( \hat{\bf p} \cdot {\bf n} \right)^{2} }_{\rm DrSipn2} + \underbrace{ \frac{ \Sigma }{ \Lambda_t \sin^{2} \theta } }_{\rm Q\ coefficient\ 1} \left( \smash[b]{ \underbrace{ \hat{\bf p} \cdot \boldsymbol{\xi} r }_{\rm pdotxir} } \right)^{2} + \underbrace{ \frac{ 1 }{ \Sigma \sin^{2} \theta } }_{\rm Q\ coefficient\ 2} \left( \smash[b]{ \underbrace{ \hat{\bf p} \cdot {\bf v} r }_{\rm pdotvr} } \right)^{2}; \end{equation*}$$

We group togther and compute $\frac{ \Delta_{r} }{ \Sigma } \left( \hat{\bf p} \cdot {\bf n} \right)^{2}$ in this cell, $\frac{ \Sigma }{ \Lambda_t \sin^{2} \theta }$ in this cell, $\hat{\bf p} \cdot \boldsymbol{\xi} r$ in this cell, $\frac{ 1 }{ \Sigma \sin^{2} \theta }$ in this cell, and $\hat{\bf p} \cdot {\bf v} r$ in this cell.

Step 11.g.i: $\frac{ \Delta_{r} }{ \Sigma } \left( \hat{\bf p} \cdot {\bf n} \right)^{2}$ [Back to top]¶

$$\label{drsipn2}$$

We define $\Delta_{r}$ in this cell, $\Sigma$ in this cell, and $\hat{\bf p} \cdot {\bf n}$ in this cell.

Step 11.g.ii: Q Coefficient 1 [Back to top]¶

$$\label{qcoeff1}$$

We defined $Q$ coefficient 1 in this cell as

$$\begin{equation*} Q\ {\rm coefficient\ 1} = \frac{ \Sigma }{ \Lambda_t \sin^{2} \theta } \end{equation*}$$

We define $\Sigma$ in this cell, $\Lambda_{t}$ in this cell, and $\sin^{2} \theta$ in this cell.

Step 11.g.iii: Q Coefficient 2 [Back to top]¶

$$\label{qcoeff2}$$

We defined $Q$ coefficient 2 in this cell as

$$\begin{equation*} Q\ {\rm coefficient\ 2} = \frac{ 1 }{ \Sigma \sin^{2} \theta } \end{equation*}$$

We define $\Sigma$ in this cell and $\sin^{2} \theta$ in this cell.

Step 12: Tortoise Terms [Back to top]¶

$$\label{tort}$$

We collect here terms related to the conversion from Boyer-Lindquist coordinates to tortoise coordinates. Details of the converstion are given in the appendix of P2010.

Step 12.a: $p_{\phi}$ [Back to top]¶

$$\label{pphi}$$

From the discussion preceding BB2010 Equation (3.41), the phi component of the tortoise momentum $p_{\phi}$ is given by

$$\begin{equation*} p_{\phi} = \hat{\bf p} \cdot \boldsymbol{\xi} r. \end{equation*}$$

We define $\hat{\bf p} \cdot \boldsymbol{\xi} r$ in this cell.

Step 12.b: $\hat{\bf p} \cdot {\bf v} r$ [Back to top]¶

$$\label{pdotvr}$$

We have

$$\begin{equation*} \hat{\bf p} \cdot {\bf v} r = \left( \hat{p}_{1} v_{1} + \hat{p}_{2} v_{2} + \hat{p}_{3} v_{3} \right) r \end{equation*}$$

We define $\hat{\bf p}$ in this cell, ${\bf v}$ in this cell, and $r$ in this cell.

Step 12.c: $\hat{\bf p} \cdot {\bf n}$ [Back to top]¶

$$\label{pdotn}$$

We have

$$\begin{equation*} \hat{\bf p} \cdot {\bf n} = \hat{p}_{1} n_{1} + \hat{p}_{2} n_{2} + \hat{p}_{3} n_{3} \end{equation*}$$

We define $\hat{\bf p}$ in this cell and ${\bf n}$ in this cell.

Step 12.d: $\hat{\bf p} \cdot \boldsymbol{\xi} r$ [Back to top]¶

$$\label{pdotxir}$$

We have

$$\begin{equation*} \hat{\bf p} \cdot \boldsymbol{\xi} r = \left( \hat{p}_{1} \xi_{1} + \hat{p}_{2} \xi_{2} + \hat{p}_{3} \xi_{3} \right) r \end{equation*}$$

We define $\hat{\bf p}$ in this cell, $\boldsymbol{\xi}$ in this cell, and $r$ in this cell.

Step 12.e: $\hat{\bf p}$ [Back to top]¶

$$\label{hatp}$$

From the discussion after BB2010 Equation (3.41), we have $\hat{\bf p} = {\bf p}/m$ where $m$ is the mass of a nonspinning test particle and ${\bf p}$ is conjugate momentum. Following Lines 319--321 of LALSimIMRSpinEOBHamiltonianPrec.c, we convert the Boyer-Lindquist momentum ${\bf p}$ to the tortoise momentum (see the appendix of P2010) via

$$\begin{align*} \hat{\bf p} = {\bf p} + {\rm prT} \left( 1 - \frac{1}{\rm csi1} \right) {\bf n} \end{align*}$$

We define prT in this cell, csi1 in this cell, and ${\bf n}$ in this cell.

Step 12.f: prT [Back to top]¶

$$\label{prt}$$

The first component of the momenum vector, after conversion to tortoise coordinates (see the Appendix of P2010), is

$$\begin{align*} {\rm prT} = {\rm csi2}\left( p_{1} n_{1} + p_{2} n_{2} + p_{3} n_{3} \right) \end{align*}$$

We define csi2 in this cell and ${\bf n}$ in this cell.

Step 12.g: csi2 [Back to top]¶

$$\label{csi2}$$

From the transformation to tortoise coordinates in the Appendix of P2010,

$$\begin{equation*} {\rm csi2} = 1 + \left( \frac{1}{2} - \frac{1}{2}{\rm sign}\left( \frac{3}{2} - \tau \right) \right) \left( {\rm csi} - 1 \right) \end{equation*}$$

We define csi in this cell; $\tau$ is a tortoise coordinate ($\tau \in \left\{ 0, 1 ,2 \right\}$).

Step 12.h: csi1 [Back to top]¶

$$\label{csi1}$$

From the transformation to tortoise coordinates in the Appendix of P2010,

$$\begin{equation*} {\rm csi1} = 1 + \left( 1 - \left\lvert 1 - \tau \right\rvert \right) \left( {\rm csi} - 1 \right) \end{equation*}$$

We define csi in this cell; $\tau$ is a tortoise coordinate ($\tau \in \left\{ 0, 1 ,2 \right\}$).

Step 12.i: csi [Back to top]¶

$$\label{csi}$$

From the transformation to tortoise coordinates in the Appendix of P2010,

$$\begin{equation*} {\rm csi} = \frac{ \sqrt{ \Delta_{t} \Delta_{r} } }{ \varpi^{2} }. \end{equation*}$$

We define $\Delta_{t}$ in this cell, $\Delta_{r}$ in this cell, and $\varpi^{2}$ in this cell.

Step 13: Metric Terms [Back to top]¶

$$\label{metric}$$

We collect here terms used to define the deformed Kerr metric. See BB2010 Equations (5.38)--(5.40) and (5.71)--(5.75).

Step 13.a: $\Lambda_{t}$ [Back to top]¶

$$\label{lambdat}$$

From BB2010 Equation (5.39),

$$\begin{equation*} \Lambda_{t} = \varpi^{4} - a^{2} \Delta_{t} \sin^{2} \theta. \end{equation*}$$

We define $\varpi^{2}$ in this cell, $a$ in this cell, $\Delta_{t}$ in this cell, and $\sin^{2}\theta$ in this cell.

Step 13.b: $\Delta_{r}$ [Back to top]¶

$$\label{deltar}$$

From BB2010 Equation (5.38),

$$\begin{equation*} \Delta_{r} = \Delta_{t} D^{-1}. \end{equation*}$$

We define $\Delta_{t}$ in this cell and $D^{-1}$ in this cell.

Step 13.c: $\Delta_{t}$ [Back to top]¶

$$\label{deltat}$$

From BB2010 Equation (5.71), we have

$$\begin{equation*} \Delta_{t} = r^{2} \Delta_{u}. \end{equation*}$$

We define $\Delta_{u}$ in this cell and $r$ in this cell.

Step 13.d: $\Delta_{t}^{\prime}$ [Back to top]¶

$$\label{deltatprm}$$

From the discussion after BB2010 Equation (5.47), we know that the prime notation indicates a derivative with respect to $r$. Using the definition of $\Delta_{t}$, we have

$$\begin{equation*} \Delta_{t}^{\prime} = 2 r \Delta_{u} + r^{2} \Delta_{u}^{\prime}. \end{equation*}$$

We define $\Delta_{u}$ in this cell and $r$ in this cell.

Step 13.e: $\Delta_{u}$ [Back to top]¶

$$\label{deltau}$$

From BB2010 Equation (5.73), we have

$$\begin{equation*} \Delta_u = \bar{\Delta}_{u} \left[ \smash[b]{\underbrace{ 1 + \eta \Delta_{0} + \eta \log \left( 1 + {\rm logarg} \right) }_{\Delta_{u}\ {\rm calibration\ term}}} \vphantom{\underbrace{1}_{n}} \right] \end{equation*}$$

We compute $\bar{\Delta}_{u}$ in this cell and $\Delta_{u}$ calibration term and logarg in this cell. From the discussion after BB2010 Equation (5.47), we know that primes denote derivatives with respect to $r$. We have

$$\begin{equation*} \Delta_u = \bar{\Delta}^{\prime}_{u} \left( \Delta_{u}\ {\rm calibration\ term} \right) + \bar{\Delta}_{u} \left( \Delta_{u}\ {\rm calibration\ term} \right)^{\prime} \end{equation*}$$

Step 13.e.i: $\bar{\Delta}_{u}$ [Back to top]¶

$$\label{deltaubar}$$

From BB2010 Equation (5.75), we have

$$\begin{equation*} \bar{\Delta}_u = \frac{ a^{2} u^{2} }{ M^{2} } + \frac{ 1 }{ \eta K - 1 } \left( 2 u + \frac{ 1 }{ \eta K - 1 } \right). \end{equation*}$$

We define $a$ in this cell, $u$ in this cell, $M$ in this cell, $\eta$ in this cell, and $K$ in this cell. From the discussion after BB2010 Equation (5.47), we know that primes denote derivatives with respect to $r$. We have

$$\begin{equation*} \bar{\Delta}^{\prime}_u = \frac{ -2 a^{2} u^{3} }{ M^{2} } - \frac{ 2 u^{2} }{ \eta K - 1 }. \end{equation*}$$

Step 13.e.ii: $\Delta_{u}$ Calibration Term [Back to top]¶

$$\label{deltaucalib}$$

From BB2010 Equation (5.73), we have

$$\begin{align*} \Delta_u\ {\rm calibration\ term} &= 1 + \eta \Delta_{0} + \eta \log \left( 1 + \Delta_{1} u + \Delta_{2} u^{2} + \Delta_{3} u^{3} + \Delta_{4} u^{4} \right) \\ &= 1 + \eta \left[ \Delta_{0} + \log \left( 1 + \Delta_{1} u + \Delta_{2} u^{2} + \Delta_{3} u^{3} + \Delta_{4} u^{4} \right) \right]. \end{align*}$$

In T2014 Equation (2) an additional term is and is defined in Equation (A2) of this paper. We then have

$$\begin{equation*} \Delta_u\ {\rm calibration\ term} = 1 + \eta \left[ \Delta_{0} + \log \left( 1 + \Delta_{1} u + \Delta_{2} u^{2} + \Delta_{3} u^{3} + \Delta_{4} u^{4} + \Delta_{5} u^{5} \right) \right]. \end{equation*}$$

In the LALSuite code itself (see LALSimIMRSpinEOBHamiltonianPrec.c line 274 on Git commit a70b43d), there's one more term ($\Delta_{5\ell}$), for which documentation is elusive. That bring us to

$$\begin{equation*} \Delta_u\ {\rm calibration\ term} = 1 + \eta \left[ \Delta_{0} + \log \left( 1 + \underbrace{ \Delta_{1} u + \Delta_{2} u^{2} + \Delta_{3} u^{3} + \Delta_{4} u^{4} + \Delta_{5} u^{5} + \Delta_{5\ell} u^{5} \ln\left(u\right) }_{ \rm logarg } \right) \right]. \end{equation*}$$

Note our notation for logarg. We define $u$ in this cell, $\eta$ in this cell, and the calibration coefficients $\Delta_{i}$, $i \in \left\{0, 1, 2, 3, 4\right\}$, in this cell.

From the discussion after BB2010 Equation (5.47), we know that primes denote derivatives with respect to $r$. We have $$\begin{equation*} \left( \Delta_u\ {\rm calibration\ term} \right)^{\prime} = \frac{ -\eta u^{2} \left( \Delta_{1} + 2 \Delta_{2} u + 3 \Delta_{3} u^{2} + 4 \Delta_{4} u^{3} + 5 \Delta_{5} u^{4} + 5 \Delta_{5\ell} u^{4} \ln\left( u \right) + \Delta_{5\ell} u^{5} u^{-1} \right) }{ 1 + {\rm logarg} }. \end{equation*}$$

Step 13.e.iii: Calibration Coefficients $\Delta_{i}$, $i \in \left\{0, 1, 2, 3, 4\right\}$ [Back to top]¶

$$\label{calib_coeffs}$$

The first term in the brackets of SH2016 Equation (A2c) is

$$\begin{equation*} \Delta_{5\ell} = \frac{64}{5} \left( \eta K - 1 \right)^{2}. \end{equation*}$$

We combine SH2016 Equation (A2c) with BL2017 Equation (2.3) (the last term in our expression) to find

$$\begin{align*} \Delta_{5} &= \left( \eta K - 1 \right)^{2} \left( -\frac{4237}{60} + \frac{128}{5}\gamma + \frac{2275}{512} \pi^{2} - \frac{1}{3} a^{2} \left\{ \Delta_{1}^{3} - 3 \Delta_{1} \Delta_{2} + 3 \Delta_{3} \right\} \right. \\ &\ \ \ \ \ - \frac{ \Delta_{1}^{5} - 5 \Delta_{1}^{3} \Delta_{2} + 5 \Delta_{1} \Delta_{2}^{2} + 5 \Delta_{1}^{2} \Delta_{3} - 5 \Delta_{2} \Delta_{3} - 5 \Delta_{1} \Delta_{4} }{ 5 \left( \eta K - 1 \right)^{2} } \\ &\left.\ \ \ \ \ + \frac{ \Delta_{1}^{4} - 4 \Delta_{1}^{2} \Delta_{2} + 2 \Delta_{2}^{2} + 4 \Delta_{1} \Delta_{3} - 4 \Delta_{4} }{ 2\left( \eta K - 1 \right) } + \frac{256}{5} \log(2) + \left\{ \frac{41\pi^2}{32} - \frac{221}{6} \right\} \eta \right) \end{align*}$$

Note that we have exlcuded the first term in the brackets in (A2c); this is the term $\Delta_{5\ell}$ which we defined in this cell.

From BB2010 Equations (5.81), we have

$$\begin{align*} \Delta_{4} &= \frac{1}{12} \left\{ 6 \frac{ a^{2} }{ M^{2} } \left( \Delta_{1}^{2} - 2 \Delta_{2} \right) \left( \eta K - 1 \right)^{2} + 3 \Delta_{1}^{4} - 8 \left( \eta K - 1 \right) \Delta_{1}^{3} - 12 \Delta_{2} \Delta_{1}^{2} + 12 \left[ 2 \left( \eta K - 1 \right) \Delta_{2} + \Delta_{3} \right] \Delta_{1} \right.\\ &\left.\ \ \ \ \ + 12 \left( \frac{94}{3} - \frac{41}{32} \pi^{2} \right) \left( \eta K - 1 \right)^{2} + 6 \left[ \Delta_{2}^{2} - 4 \Delta_{3} \left( \eta K - 1 \right) \right] \right\} \\ \end{align*}$$

From BB2010 Equations (5.80), we have

$$\begin{align*} \Delta_{3} &= \frac{1}{3} \left[ -\Delta_{1}^{3} + 3 \left( \eta K - 1 \right) \Delta_{1}^{2} + 3 \Delta_{2} \Delta_{1} - 6 \left( \eta K - 1 \right) \left( -\eta K + \Delta_{2} + 1 \right) - 3 \frac{ a^{2} }{ M^{2} } \left( \eta K - 1 \right)^{2} \Delta_{1} \right] \\ &= -\frac{1}{3}\Delta_{1}^{3} + \left( \eta K - 1 \right) \Delta_{1}^{2} + \Delta_{2} \Delta_{1} - 2 \left( \eta K - 1 \right) \left( \Delta_{2}- \left( \eta K - 1 \right) \right) - \frac{ a^{2} }{ M^{2} } \left( \eta K - 1 \right)^{2} \Delta_{1} \end{align*}$$

From BB2010 Equations (5.79, we have

$$\begin{equation*} \Delta_{2} = \frac{1}{2} \Delta_{1} \left( -4 \eta K + \Delta_{1} + 4 \right) - \frac{ a^{2} }{ M^{2} } \left( \eta K - 1 \right)^{2} \Delta_{0}\\ \end{equation*}$$

From BB2010 Equations (5.78), we have

$$\begin{equation*} \Delta_{1} = -2 \left( \eta K - 1 \right) \left( K + \Delta_{0} \right) \end{equation*}$$

From BB2010 Equations (5.77), we have

$$\begin{equation*} \Delta_{0} = K \left( \eta K - 2 \right) \end{equation*}$$

We define $K$ and $\eta K-1$ in this cell, $\eta$ in this cell, $a$ in this cell, and $M$ in this cell. Note that the constant $\gamma$ is the Euler-Mascheroni, and the value is taken from the LALSuite documentation. In the Python code we donote $\gamma$ by EMgamma.

Step 13.e.iv: $K$ [Back to top]¶

$$\label{k}$$

The calibration constant $K$ is defined in BL2017 Section IV, Equations (4.8) and (4.12) (equations copied from BL2017 source code):

$$\begin{equation*} \left.K\right|_{\chi=0} = 267.788247\, \nu^3 -126.686734\, \nu^2 + 10.257281\,\nu + 1.733598, \end{equation*}$$ $$\begin{align*} K =& - 59.165806\,\chi^3\nu^3 - 0.426958\,\chi^3\nu + 1.436589\,\chi^3 + 31.17459\,\chi^2\nu^3 + 6.164663\,\chi^2\nu^2 - 1.380863\,\chi^2 \\ & - 27.520106\,\chi \nu^3 + 17.373601\,\chi\nu^2 + 2.268313\,\chi\nu - 1.62045\,\chi +\left.K\right|_{\chi=0} \end{align*}$$

Here $\left.K\right|_{\chi=0}$ denotes the nonspining fits and $\nu = \eta$ (see discussion after BL2017 Equation (2.1)). Furthermore, most coefficients are rounded in the LALSuite code; below we round to match the values therein (see the file LALSimIMRSpinEOBHamiltonian.h). The term $\eta K - 1$ is sufficiently common that we also define it:

$$\begin{equation*} {\rm etaKminus1} = \eta K - 1. \end{equation*}$$

We define $\eta$ in this cell.

Step 13.e.v: $\chi$ [Back to top]¶

$$\label{chi}$$

The augmented spin $\chi$ is defined in OB2020 Equation (3.7) (where it is denoted $\tilde{\chi}$; note that $\nu = \eta$ from the first paragraph of Section II):

$$\begin{equation*} \chi = \frac{{\bf S}_{\rm Kerr} \cdot \hat{\bf L}}{1 - 2\eta} + \alpha\frac{({\bf S}_{1}^{\perp} + {\bf S}_{2}^{\perp}) \cdot {\bf S}_{\rm Kerr}}{ \left\lvert{\bf S}_{\rm Kerr} \right\rvert (1 - 2\eta)}. \end{equation*}$$

From the discussion after this equation we take $\alpha = \frac{1}{2}$. We define ${\bf S}^{\perp} = {\bf S}_{1}^{\perp} + {\bf S}_{2}^{\perp}$ in this cell, ${\bf L}$ in this cell, $\left\lvert{\bf S}_{\rm Kerr} \right\rvert$ in this cell, and $\eta$ in this cell.

Step 13.f: $\tilde{\omega}_{\rm fd}$ [Back to top]¶

$$\label{omegatilde}$$

From BB2010 Equation (5.40), we have

$$\begin{equation*} \tilde{\omega}_{\rm fd} = 2 a M r + \omega_{1}^{\rm fd} \eta \frac{ a M^{3} }{ r } + \omega_{2}^{\rm fd} \eta \frac{ M a^{3} }{ r }. \end{equation*}$$

From discussion after BB2010 Equation (6.7), we set $\omega_{1}^{\rm fd} = \omega_{2}^{\rm fd} = 0$. Thus

$$\begin{equation*} \tilde{\omega}_{\rm fd} = 2 a M r. \end{equation*}$$

We define $a$ in this cell, $M$ in this cell, and $r$ in this cell below.

Step 13.g: $D^{-1}$ [Back to top]¶

$$\label{dinv}$$

From BB2010 Equation (5.83),

$$\begin{equation*} D^{-1} = 1 + \log \left[ 1 + 6 \eta u^{2} + 2 \left( 26 - 3 \eta \right) \eta u^{3} \right]. \end{equation*}$$

We define $\eta$ in this cell and $u$ in this cell.

Step 14: Terms Dependent on Coordinates [Back to top]¶

$$\label{coord}$$

We collect here terms directly dependend on the coordinates. See BB2010 Equations (4.5) and (4.6).

Step 14.a: $\Sigma$ [Back to top]¶

$$\label{usigma}$$

From BB2010 Equation (4.5), we have

$$\begin{equation*} \Sigma = r^{2} + a^{2} \cos^{2} \theta. \end{equation*}$$

We define $r$ in this cell, $a$ in this cell, and $\cos \theta$ in this cell.

Step 14.b: $\varpi^{2}$ [Back to top]¶

$$\label{w2}$$

From BB2010 Equation (4.7),

$$\begin{equation*} \varpi^{2} = a^{2} + r^{2}. \end{equation*}$$

We define $a$ in this cell and $r$ in this cell.

Step 14.d: $\sin^{2} \theta$ [Back to top]¶

$$\label{sin2theta}$$

Using a common trigonometric idenitity,

$$\begin{equation*} \sin^{2} \theta = 1 - \cos^{2} \theta. \end{equation*}$$

We define $\cos \theta$ in this cell. Note that by construction (from discussion after BB2010 Equation (5.52))

$$\begin{equation*} \xi^{2} = \sin^{2} \theta. \end{equation*}$$

Step 14.e: $\cos \theta$ [Back to top]¶

$$\label{costheta}$$

From the discussion in BB2010 after equation (5.52) (noting that ${\bf e}_{3} = \hat{\bf S}_{\rm Kerr}$),

$$\begin{equation*} \cos \theta = {\bf e}_{3} \cdot {\bf n} = {\bf e}_{3}^{1} n^{1} + {\bf e}_{3}^{2} n^{2} + {\bf e}_{3}^{3} n^{3}. \end{equation*}$$

We define ${\bf e}_{3}$ in this cell and ${\bf n}$ in this cell.

Step 15: Important Vectors [Back to top]¶

$$\label{vectors}$$

We collect the vectors common for computing $H_{\rm real}$ (defined in this cell) below.

Step 15.a: ${\bf v}$ [Back to top]¶

$$\label{v}$$

From BB2010 Equation (3.39), we have

$$\begin{equation*} {\bf v} = {\bf n} \times \boldsymbol{\xi}. \end{equation*}$$

We define ${\bf n}$ in this cell and $\boldsymbol{\xi}$ in this cell.

Step 15.b: $\boldsymbol{\xi}$ [Back to top]¶

$$\label{xi}$$

From BB2010 Equation (3.38), we have

$$\begin{equation*} \boldsymbol{\xi} = {\bf e}_{3} \times {\bf n}. \end{equation*}$$

We define ${\bf e}_{3}$ in this cell and ${\bf n}$ in this cell.

Step 15.c: ${\bf e}_{3}$ [Back to top]¶

$$\label{e3}$$

From the discussion in BB2010 after equation (5.52),

$$\begin{equation*} {\bf e}_{3} = \hat{\bf S}_{\rm Kerr}. \end{equation*}$$

We define $\hat{\bf S}_{\rm Kerr}$ in this cell.

Step 15.d: ${\bf n}$ [Back to top]¶

$$\label{n}$$

From BB2010 Equation (3.37), we have

$$\begin{equation*} {\bf n} = \frac{\bf x }{ r } \end{equation*}$$

where ${\bf x} = (x, y, z)$. We define $r$ in this cell.

Step 15.e: ${\bf S}_{\rm perp}$ [Back to top]¶

$$\label{sperp}$$

In the definition of $\chi$ we denoted ${\bf S}^{\perp} = {\bf S}_{1}^{\perp} + {\bf S}_{2}^{\perp}$.

From the discussion after OB2020 Equation (3.7), we find

$$\begin{align*} {\bf S}_{1}^{\perp} &= {\bf S}_{1} - \left( {\bf S}_{1} \cdot \hat{\bf L} \right) \hat{\bf L} \\ {\bf S}_{2}^{\perp} &= {\bf S}_{2} - \left( {\bf S}_{2} \cdot \hat{\bf L} \right) \hat{\bf L} \end{align*}$$

For convenience, we here compute ${\bf S}_{1} \cdot \hat{\bf L}$ and ${\bf S}_{2} \cdot \hat{\bf L}$. We define $\hat{\bf L}$ in this cell.

Step 15.f: ${\bf L}$ [Back to top]¶

$$\label{orb_momentum}$$

From the discussion after P2010, the orbital angular momentum ${\bf L}$ of the system is given by

$$\begin{equation*} {\bf L} = {\bf x }\times{\bf p } \end{equation*}$$

where ${\bf x} = (x, y, z)$ is the position vector and ${\bf p} = (p_{1}, p_{2}, p_{3})$ is the momentum vector. We denote by $\hat{\bf L}$ the normed orbital angular momentum.

Step 16: Spin Combinations $\boldsymbol{\sigma}$, $\boldsymbol{\sigma}^{*}$, and ${\bf S}_{\rm Kerr}$ [Back to top]¶

$$\label{spin_combos}$$

We collect here various combinations of the spins.

Step 16.a: $a$ [Back to top]¶

$$\label{a}$$

From BB2010 Equation (4.9), we have

$$\begin{equation*} a = \frac{ \left\lvert {\bf S}_{\rm Kerr} \right\rvert }{ M }. \end{equation*}$$

We define $\left\lvert{\bf S}_{\rm Kerr}\right\rvert$ in this cell and $M$ in this cell.

Step 16.b: $\hat{\bf S}_{\rm Kerr}$ [Back to top]¶

$$\label{skerrhat}$$

From BB2010 Equation (4.24), we have

$$\begin{equation*} \hat{\bf S}_{\rm Kerr} = \frac{ {\bf S}_{\rm Kerr} }{ \left\lvert {\bf S}_{\rm Kerr} \right\rvert }. \end{equation*}$$

We define ${\bf S}_{\rm Kerr}$ in this cell.

Step 16.c: $\left\lvert {\bf S}_{\rm Kerr} \right\rvert$ [Back to top]¶

$$\label{skerrmag}$$

We have

$$\begin{equation*} \left\lvert {\bf S}_{\rm Kerr} \right\rvert = \sqrt{ {\bf S}_{\rm Kerr}^{1} {\bf S}_{\rm Kerr}^{1} + {\bf S}_{\rm Kerr}^{2} {\bf S}_{\rm Kerr}^{2} + {\bf S}_{\rm Kerr}^{3} {\bf S}_{\rm Kerr}^{3} }. \end{equation*}$$

We define ${\bf S}_{\rm Kerr}$ in this cell.

Step 16.d: ${\bf S}_{\rm Kerr}$ [Back to top]¶

$$\label{skerr}$$

From BB2010 Equation (5.64):

$$\begin{equation*} {\bf S}_{\rm Kerr} = \boldsymbol{\sigma} + \frac{ 1 }{ c^{2} } \boldsymbol{\Delta}_{\sigma}. \end{equation*}$$

In BB2010 Equation (5.67), $\boldsymbol{\Delta}_{\sigma} = 0$. Thus

$$\begin{equation*} {\bf S}_{\rm Kerr} = \boldsymbol{\sigma}. \end{equation*}$$

We define $\boldsymbol{\sigma}$ in this cell.

Step 16.e: $\boldsymbol{\sigma}$ [Back to top]¶

$$\label{sigma}$$

From BB2010 Equation (5.2):

$$\begin{equation*} \boldsymbol{\sigma} = {\bf S}_{1} + {\bf S}_{2}. \end{equation*}$$

Step 16.f: $\boldsymbol{\sigma}^{*}$ [Back to top]¶

$$\label{sigmastar}$$

From BB2010 Equation (5.3):

$$\begin{equation*} \boldsymbol{\sigma}^{*} = \frac{ m_{2} }{ m_{1} } {\bf S}_{1} + \frac{ m_{1} }{ m_{2} }{\bf S}_{2}. \end{equation*}$$

Step 17: Fundamental Quantities [Back to top]¶

$$\label{fundquant}$$

We collect here fundamental quantities from which we build $H_{\rm real}$ (defined in this cell).

Step 17.a: $u$ [Back to top]¶

$$\label{u}$$

From the discussion after BB2010 Equation (5.40),

$$\begin{equation*} u = \frac{ M }{ r }. \end{equation*}$$

We define $M$ in this cell and $r$ in this cell.

Step 17.b: $r$ [Back to top]¶

$$\label{r}$$

From the discussion after BB2010 Equation (5.52),

$$\begin{equation*} r = \sqrt{ x^{2} + y^{2} + z^{2} }. \end{equation*}$$

Step 17.c: $\eta$ [Back to top]¶

$$\label{eta}$$

From the discussion preceding BB2010 Equation (5.1),

$$\begin{equation*} \eta = \frac{ \mu }{ M }. \end{equation*}$$

We define $\mu$ in this cell.

Step 17.d: $\mu$ [Back to top]¶

$$\label{mu}$$

From the discussion preceding BB2010 Equation (5.1),

$$\begin{equation*} \mu = \frac{ m_{1} m_{2} }{ M }. \end{equation*}$$

We define $M$ in this cell.

Step 17.e: $M$ [Back to top]¶

$$\label{m}$$

From the discussion preceding BB2010 Equation (5.1),

$$\begin{equation*} M = m_{1} + m_{2}. \end{equation*}$$

Step 18: Validation [Back to top]¶

$$\label{validation}$$

The following code cell reverses the order of the expressions output to SEOBNR/Hamiltonian_on_top.txt and creates a Python function to validate the value of $H_{\rm real}$ against the SEOBNRv3 Hamiltonian value computed in LALSuite git commit bba40f21e9 for command-line input parameters

-M 23 -m 10 -f 20 -X 0.01 -Y 0.02 -Z -0.03 -x 0.04 -y -0.05 -z 0.06.

Step 19: Output this notebook to $\LaTeX$-formatted PDF file [Back to top]¶

$$\label{latex_pdf_output}$$

The following code cell converts this Jupyter notebook into a proper, clickable $\LaTeX$-formatted PDF file. After the cell is successfully run, the generated PDF may be found in the root NRPy+ tutorial directory, with filename Tutorial-SEOBNR_Documentation.pdf (Note that clicking on this link may not work; you may need to open the PDF file through another means.)