Mathisson-Papetrou-Dixon Equations

The MPD equations are derived from the conservation of the energy-momentum tensor:

\[{T^{\mu \nu}}_{;\nu} = 0\]

Taking the multipole expansion leads to a description of the momentum vector $p^{\mu}$ (0th moment) and the spin tensor $s^{\mu \nu}$ (dipole moment)

\[\frac{Dp^{\mu}}{d \lambda} = -\frac{1}{2}{R^{\mu}}_{\nu \alpha \beta} u^{\nu} s^{\alpha \beta}\]

\[\frac{Ds^{\mu \nu}}{d \lambda} =p^{\mu}u^{\nu} - p^{\nu}u^{\mu}\]

for affine parameter $\lambda$, 4-velocity $u^{\nu}$ and Riemann curvature tensor ${R^{\mu}}_{\nu \alpha \beta}$, with $\frac{D}{d\lambda}$ denoting a covariant derivative. We ignore the higher order moments since for pulsar mass ($m$) << BH mass ($M$) and pulsar radius << BH radius, the motion is dominated by the lowest order moments.

The system is closed by providing a spin supplementary condition. Throughout this package we take the Tulczyjew-Dixon (TD) condition

s^{\mu \nu} p_{\nu} = 0$ see e.g. Costa & Natário, 2015 for discussion of the spin conditions.

In the extreme mass ratio limit m << M, the pulsar Möller radius is much less than the gravitational lengthscale. This means that the pole-dipole interaction is much stronger than the dipole-dipole interaction. Within this approximation, the MPD equations reduced to a set of ODEs (see e.g. Mashoon & Singh, 2006, Singh, Wu & Sarty, 2014)

\[\frac{dp^{\alpha}}{d\lambda} = - \Gamma_{\mu\nu}^{\alpha} p^{\mu}u^{\nu} + \left( \frac{1}{2m} {R^{\alpha}}_{\beta \rho \sigma} \epsilon^{\rho \sigma}_{\quad \mu \nu} s^{\mu} p^{\nu} u^{\beta}\right) \ ,\]

\[\frac{ds^{\alpha}}{d \lambda} = - \Gamma^{\alpha}_{\mu \nu} s^{\mu}u^{\nu} + \left(\frac{1}{2m^3}R_{\gamma \beta \rho \sigma} \epsilon^{\rho \sigma}_{\quad \mu \nu} s^{\mu} p^{\nu} s^{\gamma} u^{\beta}\right)p^{\alpha} \ ,\]

\[\frac{dx^{\alpha}}{d\lambda} = -\frac{p^{\delta}u_{\delta}}{m^2} \left[ p^{\alpha} + \frac{1}{2} \frac{ (s^{\alpha \beta} R_{\beta \gamma \mu \nu} p^{\gamma} s^{\mu \nu})}{m^2 + (R_{\mu \nu \rho \sigma} s^{\mu \nu} s^{\rho \sigma}/4)}\right]\]

Whilst $\lambda$ has the freedom to be any affine parameter, we take it to be the proper time $\tau$ such that $g_{\mu \nu}u^{\mu} u^{\nu}=-1$. These equations are generally covariant - we take the astrophysically motivated Kerr metric for a spinning BH and work in Boyer-Lindquist coordinates.