+ Summary
+ Relativistic binaries composed of a millisecond pulsar (MSP)
+ orbiting a much more massive (
+
+ ≳103M⊙),
+ spinning black hole (BH) are exceptional probes for investigating key
+ questions of fundamental physics and astrophysics. Such systems are
+ natural sources of gravitational waves (GWs) in the mHz regime,
+ expected to be detectable by the next generation of space-based GW
+ detectors such as LISA
+ (Thorpe
+ et al., 2019). The associated radio emission from the companion
+ pulsar raises the possibility of an electromagnetic (EM) counterpart,
+ enabling high precision multimessenger measurements to be made. The
+ description of the orbital dynamics of these systems, and the
+ influence on the resultant observed EM and GW signals, is non-trivial.
+ A proper treatment of the spin-orbital dynamics can be derived from
+ the conservation of the energy-momentum tensor
+
+
+ Tμν;ν=0
+ which when expanded into a set of infinite multipole moments leads to
+ a description of the momentum vector
+
+ pμ
+ and the spin tensor
+
+ sμν
+
+
+ Dpμdλ=−12Rμναβuνsαβ
+
+
+ Dsμνdλ=pμuν−pνuμ
+ for affine parameter
+
+ λ,
+ 4-velocity
+
+ uν
+ and Riemann curvature tensor
+
+ Rμναβ.
+ The system is closed by providing a spin supplementary condition,
+ equivalent to specifying the observer-dependent centre of mass. For
+ this work we take the Tulczyjew-Dixon condition
+ (Dixon,
+ 1964;
+ Tulczyjew,
+ 1959)
+
+
+ sμνpν=0
+ Together, equations 2 -
+ 4 form the
+ Mathisson-Papetrou-Dixon (MPD) equations
+ (Dixon,
+ 1964;
+ Mathisson,
+ 1937;
+ Papapetrou,
+ 1951), and describe the spin-orbital evolution in a fully
+ consistent way that is applicable to strong field regimes.
+
+
+ Statement of need
+ RelativisticDynamics.jl is an open-source
+ Julia package for relativistic spin-orbital dynamics in the
+ gravitational strong field for a Kerr spacetime. Existing codes for
+ modelling the dynamics of spinning objects like pulsars in the
+ strong-field regime are generally lacking, since such systems occupy
+ an intermediate regime that is generally overlooked. At the “low” end
+ of this regime there are post-Newtonian or geodesic descriptions (e.g.
+ Damour
+ & Taylor, 1992) which neglect the influence of the pulsar
+ spin on the underlying spacetime metric (“spin-curvature” coupling).
+ At the “high” end there is the full Numerical Relativity (NR)
+ solutions (e.g.
+ Andrade
+ et al., 2021) which are primarily applicable to two BHs with a
+ mass ratio
+
+ 𝒪(1),
+ and are computationally intractable for these MSP systems which are
+ observed over a large number of orbital cycles.
+ RelativisticDynamics.jl aims to bridge this
+ gap by providing a modern, fast code for accurate numerical evolution
+ of spinning relativistic systems, via the MPD formalism. Julia is a
+ modern language that solves the “two language problem”, enabling fast
+ dynamic typing and JIT compilation in conjunction with petaflop
+ performance, comparable with numerical languages that are better known
+ in the astrophysics community such as C or Fortran. As a modern
+ language, it also provides a dedicated package manager and a large
+ catalogue of composable packages for scientific
+ computing. This enables RelativisticDynamics.jl
+ to easily leverage and interface with other scientific computing
+ packages. The author and collaborators have used the general methods
+ and mathematics described in this package for multiple research
+ projects (e.g.
+ Kimpson
+ et al., 2019,
+ 2020a,
+ 2020b;
+ Li et
+ al., 2019) with a particular focus on the radio signals from
+ spinning pulsar systems. This package represents an attempt to create
+ a documented, well-tested, open source resource for public use in this
+ area, that can also be used as a computational playground for
+ exploring techniques that could be applicable to more advanced
+ numerical models. The package has been formulated in terms of ODE
+ integration, rather than using e.g. action-angle variables
+ (Witzany,
+ 2022), to allow for extension to general spacetime metrics and
+ straightforward computation of quantities relevant for pulsar
+ observations e.g. spin axis orientation.
+ In addition to providing a fast, modern package for strong field
+ spin dynamics, RelativisticDynamics.jl has two
+ additional important features from the perspective of modern
+ relativistic astrophysics. Firstly, it is fully type flexible, being
+ able to support arbitrary number formats. By making use of Julia’s
+ type-flexibility the model is written in such a way so as to be able
+ to support hardware accelerated, low precision arithmetic and
+ alternative rounding methods such as stochastic rounding. This enables
+ rapid prototyping and exploration of reduced precision numerical
+ techniques in astrophysics, an approach common in other numerical
+ fields such as weather and climate modelling (e.g.
+ Váňa et
+ al., 2017). Secondly,
+ RelativisticDynamcis.jl is written to be fully
+ differentiable via automatic differentiation. This enables the package
+ to be used for differentiable physics applications in astrophysics,
+ for example gravitational waveform modelling and parameter estimation
+ or training neural networks based on the model. Automatic
+ differentiation also provides a potential avenue for extension of the
+ package to general (i.e. non-Kerr) spacetimes, whereby a user can
+ specify the metric and the associated Christoffel symbols and Riemann
+ tensors - which are simply linear combinations of the metric
+ derivatives - are calculated automatically.
+ Future potential extensions of this code include taking the
+ dynamics beyond second order in the multipole expansion, and the
+ inclusion of alternative spin conditions and spacetime metrics. The
+ inclusion of a diagnostics tool for extracting gravitational waveforms
+ in the time domain via a numerical kludge method would also be a
+ worthwhile addition. Moreover, we have considered only bound dynamical
+ systems - the ability to also explore hyberbolic systems would also be
+ an interesting development.
+
+
+
+ Example orbital trajectories for a ms-pulsar with
+ eccentricity
+
+ e=0.1
+ (left panels),
+
+ e=0.8
+ (right panels), orbiting a massive BH with extremal spin,
+
+
+ a=0.998.
+ The orbital motion is presented in the
+
+
+ x−y
+ plane (top panels) and
+
+ x−z
+ plane (bottom panels). The pulsar is initialised in the orbital
+ plane with zero inclination. In the absence of spin-curvature
+ coupling the particle would remain in the plane
+ (
+
+ z=0).
+ Note the
+
+ z-motion
+ is on the scale of km, not gravitational radii.
+
+
+