ScalarSelfForce::AnalyticData::CircularOrbit Class Reference

Scalar self force for a scalar charge on a circular equatorial orbit. More...

#include <CircularOrbit.hpp>

Classes
struct	BlackHoleMass
struct	BlackHoleSpin
struct	HyperboloidalSlicingTransitions
struct	ImposeEquatorialSymmetry
struct	MModeNumber
struct	OrbitalRadius

Public Types
using	options = tmpl::list< BlackHoleMass, BlackHoleSpin, OrbitalRadius, MModeNumber, HyperboloidalSlicingTransitions, ImposeEquatorialSymmetry >
using	background_tags = tmpl::list< Tags::Alpha, Tags::Beta, Tags::Gamma >
using	source_tags = tmpl::list< ::Tags::FixedSource< Tags::MMode >, Tags::SingularField, ::Tags::deriv< Tags::SingularField, tmpl::size_t< 2 >, Frame::Inertial >, Tags::BoyerLindquistRadius >

Public Member Functions
	CircularOrbit (const CircularOrbit &)=default
CircularOrbit &	operator= (const CircularOrbit &)=default
	CircularOrbit (CircularOrbit &&)=default
CircularOrbit &	operator= (CircularOrbit &&)=default
	CircularOrbit (double black_hole_mass, double black_hole_spin, double orbital_radius, int m_mode_number, std::optional< std::array< double, 4 > > hyperboloidal_slicing_transitions, bool impose_equatorial_symmetry)
	CircularOrbit (CkMigrateMessage *m)
	WRAPPED_PUPable_decl_template (CircularOrbit)
tnsr::I< double, 2 >	puncture_position () const
double	black_hole_mass () const
double	black_hole_spin () const
double	orbital_radius () const
int	m_mode_number () const
std::optional< std::array< double, 4 > >	hyperboloidal_slicing_transitions () const
bool	impose_equatorial_symmetry () const
tuples::tagged_tuple_from_typelist< background_tags >	variables (const tnsr::I< DataVector, 2 > &x, background_tags) const
tuples::tagged_tuple_from_typelist< source_tags >	variables (const tnsr::I< DataVector, 2 > &x, source_tags) const
template<typename... RequestedTags>
tuples::TaggedTuple< RequestedTags... >	variables (const tnsr::I< DataVector, 2 > &x, const Mesh< 2 > &, const InverseJacobian< DataVector, 2, Frame::ElementLogical, Frame::Inertial > &, tmpl::list< RequestedTags... >) const
void	pup (PUP::er &p) override

Static Public Member Functions
static tuples::TaggedTuple< Tags::MMode >	variables (const tnsr::I< DataVector, 2 > &x, tmpl::list< Tags::MMode >)

Static Public Attributes
static constexpr Options::String	help

Friends
bool	operator== (const CircularOrbit &lhs, const CircularOrbit &rhs)

Detailed Description

Scalar self force for a scalar charge on a circular equatorial orbit.

This class implements Eq. (2.9) in [160] . It does so by defining the background fields \(\alpha\), \(\beta\), and \(\gamma_i\) in the general form of the equations

\begin{equation} -\partial_i F^i + \beta \Psi_m + \gamma_i \partial_i \Psi_m = S_m \text{.} \end{equation}

with the flux

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

We make the following changes compared to [160] :

• Multiply by the factor \(\Sigma^2 / (r^2 + a^2)^2\) so that we can easily write the equations in first-order flux form.
• Use \(\cos\theta\) as angular coordinate instead of \(\theta\). This avoids the \(\cot\theta\) term by rewriting the angular derivatives as:

  \[ \partial_\theta^2+\cot\theta\partial_\theta = \partial_{\cos\theta}\sin^2\theta\partial_{\cos\theta} \]

• Decompose \(\Psi_m = \sin(\theta)^m u_m(r_\star, \theta)\). This avoids the \(m^2/\sin^2\theta\) term by factoring out the singular behavior at the poles. The equations transform as:

  \[ -\partial_{\cos\theta}\sin^2\theta\partial_{\cos\theta} \Psi_m + \frac{m^2}{\sin^2\theta}\Psi_m = \sin(\theta)^m \left( m(m+1) + 2m \cos\theta \partial_{\cos\theta} - \partial_{\cos\theta}\sin^2\theta\partial_{\cos\theta} \right) u_m \]

  We divide by \(\sin(\theta)^m\) to get the equations for \(u_m\).

Written this way, the equations are regular at the poles and converge exponentially. We also don't have to apply angular boundary conditions because regularity at the poles is automatically enforced by the \(\sin^2\theta\) factor in the flux.

The resulting factors in the equation are:

\begin{align} &\alpha = \frac{\Delta}{(r^2 + a^2)^2} \sin^2\theta \\ &\beta = \left(-m^2\Omega^2 \Sigma^2 + 4a m^2 \Omega M r + \Delta \left[ m (m + 1) + \frac{2M}{r}(1-\frac{a^2}{Mr}) + \frac{2iam}{r} \right]\right) \frac{1}{(r^2 + a^2)^2} \\ &\gamma_{r_\star} = -\frac{2iam}{r^2+a^2} + \frac{2a^2}{r} \frac{\alpha}{\sin^2\theta} \\ &\gamma_{\cos\theta} = 2 m \cos(\theta) \frac{\Delta}{(r^2 + a^2)^2} \end{align}

This class also provides the effective source \(S_m^\mathrm{eff} = \Delta_m \Psi_m^P\) and the singular field \(\Psi_m^P\) in the regularized region (see ScalarSelfForce::FirstOrderSystem and Sec. III in [160] ). The effective source is computed using the scalar EffectiveSource code by Wardell et. al. (https://github.com/barrywardell/EffectiveSource and [213] ). It is then transformed to correspond to the m-mode decomposition used in [160] Eq. (2.8) as:

\begin{align} &\Psi_m^P = \frac{r}{2 \pi} e^{-i m \Delta\phi} \Phi_m^\mathrm{Wardell} \\ &S_m^\mathrm{eff} = \frac{r}{2 \pi} e^{-i m \Delta\phi} \frac{\Delta\,(r^2 + a^2\cos^2\theta)}{(r^2 + a^2)^2} S_m^\mathrm{Wardell} \text{,} \end{align}

where \(\Delta\phi = \frac{a}{r_+ - r_-} \ln(\frac{r-r_+}{r-r_-})\) (Eq. (2.7) in [160] ). We also divide by \(\sin(\theta)^m\) to account for the change of variable from \(\Psi_m\) to \(u_m\) described above.

Impose equatorial symmetry

Since this work is restricted to equatorial orbits, we can enforce equatorial symmetry by reformulating the equations in terms of the coordinate \(z^2 = \cos^2\theta\) instead of \(z = \cos\theta\). This transform the factors of first-order flux formulation as:

\begin{equation} F^{\cos^2\theta} = 4 \cos^2\theta F^{\cos\theta} \text{.} \end{equation}

\begin{equation} \gamma_{\cos^2\theta} = 2 \cos\theta \gamma_{\cos\theta} + 2 \alpha \text{.} \end{equation}

Hyperboloidal slicing

Transforming to a hyperboloidal time coordinate \(s = t - h(r_*)\) can simplify the solution a lot because the slice will only intersect a finite number of wave fronts. In frequency domain this transformation means that partial derivatives transform as:

\begin{align} &\partial_{r_*} \rightarrow \partial_{r_*} + i m\Omega H(r_*) \\ &\partial_{r_*}^2 \rightarrow \partial_{r_*}^2 + 2 i m\Omega H(r_*) \partial_{r_*} + i m\Omega H'(r_*) -m^2\Omega^2 H(r_*)^2 \end{align}

where \(H(r_*) = h'(r_*)\) is the boost function that asymptotes to \(H(r_* \rightarrow \pm \infty) = \pm 1\). This maps to the following additional terms in \(\beta\) and \(\gamma_{r_*}\):

\begin{align} &\beta \rightarrow \beta + i m\Omega \gamma_{r_*} H(r_*) - i m\Omega H'(r_*) + m^2\Omega^2 H(r_*)^2 \\ &\gamma_{r_*} \rightarrow \gamma_{r_*} - 2 i m\Omega H(r_*) \end{align}

For the boost function we choose here a simple sigmoid so that it is exactly zero in a finite region around the puncture where we apply the effective source, and transitions to -1 and 1 outside of this region. We choose the \(C^1\) continuous smoothstep<1> for this. Reasonable transition points are from the boundaries of the effective source region to about \(20M\) away from it, though this hasn't been tested very much yet. We may also want to try jumping from \(H=0\) to \(H=\pm 1\) discontinuously at the boundary of the effective source region, which is supported by the DG scheme (see fluxes_computer::is_discontinuous in elliptic::protocols::FirstOrderSystem), so (1) we don't have to resolve the transition, and (2) we can more easily support the 2nd order self-force source.

Member Data Documentation

help

constexpr Options::String ScalarSelfForce::AnalyticData::CircularOrbit::help

staticconstexpr

Initial value:

=
      "Quasicircular orbit of a scalar point charge in Kerr spacetime"

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

• src/Elliptic/Systems/SelfForce/Scalar/AnalyticData/CircularOrbit.hpp