Self-force of a scalar charge on a Kerr background. More...

#include <FirstOrderSystem.hpp>

Public Types
using	primal_fields = tmpl::list<Tags::MMode>
using	primal_fluxes
using	background_fields = tmpl::list<Tags::Alpha, Tags::Beta, Tags::Gamma>
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

Self-force of a scalar charge on a Kerr background.

In this formulation we solve a 2D elliptic equation for the m-mode field \(\Psi_m\), following [160] . The two dimensions are a radial and angular coordinate, specifically the tortoise radius \(r_\star\) and \(\cos\theta\) (note that [160] uses \(\theta\)).

We write the equations in generic first-order flux form

\begin{equation}-\Delta_m \Psi_m = -\partial_i F^i + \beta \Psi_m + \gamma_i \partial_i \Psi_m = 0 \end{equation}

with the flux

\begin{equation}F^i = \{\partial_{r_\star}, \alpha \partial_{\cos\theta}\} \Psi_m \text{,} \end{equation}

where \(\alpha\), \(\beta\), and \(\gamma_i\) are coefficients that define the elliptic equations. The particular coefficients that match Eq. (2.9) of [160] for a circular equatorial orbit in Kerr are implemented in ScalarSelfForce::AnalyticData::CircularOrbit.

Regularization and modified boundary data

As described in [160] Sec. III, the field \(\Psi_m\) is singular at the location of the scalar point charge ("puncture"). Therefore, in a region near the puncture we split the field into a regular and a singular part,

\begin{equation} \Psi_m = \Psi_m^R + \Psi_m^P \text{,} \end{equation}

where \(\Psi_m^P\) is the singular part that we can (approximately) compute analytically and \(\Psi_m^R\) is the regular part that we solve for. Therefore, we need to transform variables between the full field \(\Psi_m\) and the regularized field \(\Psi_m^R\) when data moves across the boundary of the regularized region. We do this at element boundaries using the modify_boundary_data mechanism detailed in elliptic::protocols::FirstOrderSystem by just adding or subtracting the singular field to/from the received data (see also ScalarSelfForce::ModifyBoundaryData). In this regularized region the equations transform to

\begin{equation}-\Delta_m \Psi_m^R = \Delta_m \Psi_m^P = S_m^\mathrm{eff} \text{,} \end{equation}

so they gain an effective source \(S_m^\mathrm{eff}\) from the singular field \(\Psi_m^P\). The singular field and the effective source for a circular equatorial orbit in Kerr is implemented in ScalarSelfForce::AnalyticData::CircularOrbit.