GrSelfForce::FirstOrderSystem Struct Reference

Gravitational self-force of a small gravitating body in a Kerr background. More...

#include <FirstOrderSystem.hpp>

Public Types
using	primal_fields = tmpl::list<Tags::MMode>
using	primal_fluxes
using	background_fields
using	inv_metric_tag = void
using	fluxes_computer = Fluxes
using	sources_computer = Sources
using	modify_boundary_data = ModifyBoundaryData
using	boundary_conditions_base

Static Public Attributes
static constexpr size_t	volume_dim = 2

Detailed Description

Gravitational self-force of a small gravitating body in a Kerr background.

Extension of the ScalarSelfForce::FirstOrderSystem to the gravitational case. We solve a 2D elliptic equation for the 10 independent components of the symmetric m-mode field \((\Psi_m)_{ab}\). The two dimensions are a radial and angular coordinate, specifically the tortoise radius \(r_\star\) and polar angle \(\theta\). We parametrize the 2D elliptic equations as:

\begin{equation}-\Delta_m (\Psi_m)_{ab} = -\partial_i F^i_{ab} + \beta_{ab}^{cd} (\Psi_m)_{cd} + \gamma_{iab}^{cd} F^i_{cd} = 0 \end{equation}

with the flux

\begin{equation}F^i_{ab} = \{\partial_{r_\star}, \alpha \partial_\theta\} (\Psi_m)_{ab} \text{,} \end{equation}

where \(\alpha\), \(\beta_{ab}^{cd}\), and \(\gamma_{iab}^{cd}\) are coefficients that define the elliptic equations. The particular coefficients for a circular equatorial orbit in Kerr are implemented in GrSelfForce::AnalyticData::CircularOrbit.

See ScalarSelfForce::FirstOrderSystem for a description of the regularization and modified boundary data.

Once the 2D elliptic equations are solved for a given m-mode, the contribution to the self-force can be extracted from the gradient of \((\Psi_m)_{ab}\) at the location of the small body. These self-force contributions are then summed over all m-modes up to some cutoff. The resulting self-force can be used to drive a quasi-adiabatic inspiral to generate waveforms. For details and more references see [160] for now (more references will be added when they are published).

Member Typedef Documentation

background_fields

using GrSelfForce::FirstOrderSystem::background_fields

Initial value:

 
      tmpl::list<Tags::Alpha, Tags::Beta, Tags::GammaRstar, Tags::GammaTheta>

boundary_conditions_base

using GrSelfForce::FirstOrderSystem::boundary_conditions_base

Initial value:

 
      elliptic::BoundaryConditions::BoundaryCondition<2>

primal_fluxes

using GrSelfForce::FirstOrderSystem::primal_fluxes

Initial value:

 
      tmpl::list<::Tags::Flux<Tags::MMode, tmpl::size_t<2>, Frame::Inertial>>

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The documentation for this struct was generated from the following file:

• src/Elliptic/Systems/SelfForce/GeneralRelativity/FirstOrderSystem.hpp