Initial Conditions

We need to initialise 3 vectors $x^{\mu}, p^{\mu}, s^{\mu}$ for position, momentum and spin respectively. All this takes place in src/universal_constants.jl and src/initial_conditions.jl.

Position, $x^{\mu}$

The initial coordinates are specified straightforwardly. Working in Boyer-Lindquist coordinates, we set $t=0$, $\theta = \pi/2$, $\phi=0.0$ and set the radial coordinate $r$ equal to the value of the semi-major axis $\alpha$ specified in src/system_parameters.jl.

Momentum, $p^{\mu}$

The specification of the initial momentum proceeds via two steps.

1. Map from the user-specified Keplerian orbital parameters to the conserved quantities
2. Define the momenta from the first-order equations of motion derived from the Hamilton-Jacobi function for the Kerr metric.

Both of these steps are well-described in Schmidt 2002. The general overview is as follows:

For the Kerr spacetime we have 3 conserved quantities, the energy $E$, the angular momentum $L_z$ and the Carter constant $Q$. We want to be able to determine the value of these quantities, given the Keplerian orbital parameters, $\alpha, e, \iota$.

Given the first order ODEs for $r$ and $\theta$ from the Kerr Hamiltonian, solve

\[\frac{dr}{d\lambda} = 0 ; \, \, \frac{d\theta}{d\lambda} = 0\]

i.e. the turning points of the radial and polar motion. One can solve these equations to find $E,L,Q$ given $\alpha, e, \iota$.

With the conserved quantities in hand, the 4-velocity is defined from the Kerr Hamiltonian,

\[\Sigma \frac{dt}{d\lambda} = \frac{r^2 + a^2}{\Delta} P - a(aE\sin^2 \theta -L_z)\]

\[\Sigma \frac{dr}{d\lambda} = \pm \sqrt{R}\]

\[\Sigma \frac{d\theta}{d\lambda} = \pm \sqrt{\Theta}\]

\[\Sigma \frac{d\phi}{d\lambda} = \frac{a}{\Delta} - aE + \frac{L_z}{\sin^2 \theta}\]

where $R,\Theta$ and $P$ are again given in Schmidt 2002. We always take the positive square root for the initialisation, such that the initial motion of the pulsar is "outwards and upwards" (increasing $r$ and $\theta$).

The covariant 4-velocity can then be translated into a contravariant 4-momentum as $p^{\alpha} = m g^{\alpha \beta} u_{\beta}$ for metric $g^{\alpha \beta}$.

Spin, $s^{\mu}$

In order to determine the spin vector we must first specify the moment of inertia of the pulsar. We model the pulsar as a solid sphere such that

\[I = \frac{2}{5} m_{\rm PSR} r_{\rm PSR}^2\]

The angular momentum/spin magnitude is,

\[s_0 = 2 \pi I / P_{\rm PSR}\]

where $P_{\rm PSR}$ is the spin period of the pulsar. The spatial components of the spin vector are then,

\[s^r = s_0 \sin(S_{\theta}) \cos(S_{\phi})\]

\[s^{\theta} = -s_0 \cos(S_{\theta})/r\]

\[s^{\phi} = s_0 \sin(S_{\theta}) \sin(S_{\phi})/r \sin(\theta)\]

where $S_{\theta, \phi}$ are the latitude and azimuthal angles of the spin axis, see e.g. Mashhoon & Singh, 2006. The temporal component $s^{t}$ is enforced by the spin condition. Throughout this package we take the Tulczyjew-Dixon (TD) condition

$s^{\mu}p_{\mu} = 0$ see e.g. Costa & Natário, 2015 for discussion of the TD condition and other options