gr::Solutions::TrumpetSchwarzschild Class Reference

Trumpet Schwarzschild solution in isotropic coordinates. More...

#include <TrumpetSchwarzschild.hpp>

Classes
struct	Mass
struct	N

Public Types
using	options = tmpl::list<Mass, N>
template<typename DataType>
using	DerivLapse
template<typename DataType>
using	DerivShift
template<typename DataType>
using	DerivSpatialMetric
template<typename DataType>
using	tags
Public Types inherited from gr::AnalyticSolution< 3_st >
using	DerivLapse
using	DerivShift
using	DerivSpatialMetric
using	tags

Public Member Functions
	TrumpetSchwarzschild (double mass, double n, const Options::Context &context={})
	TrumpetSchwarzschild (const TrumpetSchwarzschild &)=default
TrumpetSchwarzschild &	operator= (const TrumpetSchwarzschild &)=default
	TrumpetSchwarzschild (TrumpetSchwarzschild &&)=default
TrumpetSchwarzschild &	operator= (TrumpetSchwarzschild &&)=default
	TrumpetSchwarzschild (CkMigrateMessage *)
template<typename DataType, typename... Tags>
tuples::TaggedTuple< Tags... >	variables (const tnsr::I< DataType, volume_dim, Frame::Inertial > &x, double t, tmpl::list< Tags... >) const
template<typename DataType, typename... Tags>
tuples::TaggedTuple< Tags... >	variables (const tnsr::I< DataType, volume_dim, Frame::Inertial > &x, double t, const IntermediateVars< DataType > &vars, tmpl::list< Tags... >) const
void	pup (PUP::er &p)
double	mass () const
double	n () const

Static Public Attributes
static constexpr size_t	volume_dim = 3
static constexpr Options::String	help
Static Public Attributes inherited from gr::AnalyticSolution< 3_st >
static constexpr size_t	volume_dim

Detailed Description

Trumpet Schwarzschild solution in isotropic coordinates.

Details

This solution is a trumpet Schwarzschild black hole in the isotropic coordinates. It is a time-independent puncture solution in 1+log slicing. The solution cannot be written down analytically in Schwarzschild coordinates or isotropic coordinates. Refer to [94] for equations and details.

We first set up a source grid in isotropic radial coordinate r, on which we compute the lapse and Schwarzschild radial coordinate R. The lapse is computed by solving eq. (54)-(56) using the toms748 algorithm, where the integrals in eq. (54)-(56) are evaluated by a tanh_sinh integrator.

Eq. (54) is for \(\alpha < \alpha_s=0.1\):

\begin{equation*}r(\alpha) = R(\alpha_s)^{(1/\alpha_s)} \exp \left[ - \int_{\alpha}^{\alpha_s} \frac{1}{\bar{\alpha} R(\bar{\alpha})} \frac{dR}{d\alpha}(\bar{\alpha}) \: d\bar{\alpha} - C_0 \right], \end{equation*}

where eq. (55) gives

\begin{equation*}C_0 = \int_{\alpha_s}^{1} \frac{\ln R(\alpha)}{\alpha^2} d\alpha. \end{equation*}

Eq. (56) is for \(\alpha > \alpha_s=0.1\):

\begin{equation*}r(\alpha) = R(\alpha)^{(1/\alpha)} \exp \left[ \int_{\alpha_s}^{\alpha} \frac{\ln R(\bar{\alpha})}{\bar{\alpha}^2} d\bar{\alpha} - C_0 \right]. \end{equation*}

A standard Gauss quadratures will fail due to singular behaviors of the integrands near the integration limits. The Schwarzschild R is then computed by solving eq. (39) using again the toms748 algorithm.

\begin{equation*}\alpha^2 = 1 - \frac{2M}{R} + \frac{C(n)^2 e^{2 \alpha / n} }{R^4}. \end{equation*}

Note that for a value of n near 2 (corresponding to standard 1+log slicing), eq. (39) has two positive roots for a fixed lapse in [0., 1.). See the following plot by Mathematica. We choose

the physical root, i.e. with the correct asymptotic behaviors: R diverges as \(\alpha\) tends to 1, and R tends to the smaller solution as \(\alpha\) tends to 0.

The physical root is numerically selected by finding the critical lapse and critical Schwarzschild R, below which we tell the toms748 solver to find a root \(R\in\) [min_schwarzschild_r, crit_schwarzschild_r] or else we find a root \(R\in\) [crit_schwarzschild_r, max_schwarzschild_r]. min_schwarzschild_r is currently selected to be 0., and max_schwarzschild_r is at least as large as max_isotropic_r (currently 5000M), the maximum coordinate radius we support for this initial data. In case the solver is asked to find a Schwarzschild R greater than the max_isotropic_r, we use double the asymptotic solution from eq. (39), i.e. \(4/(1-\alpha^2)\), as solver upper bound. This latter upper bound is necessary since the integrator in eq. (55) needs lapse very close to 1 to converge.

After acquiring the lapse and Schwarzschild R, we can assemble all the 3+1 quantities in the isotropic coordinates on the source grid.

Since the user supplies grid points in the Cartesian version of the isotropic coordinates, we compute a user grid in the isotropic radial coordinate based on the Cartesian grid, compute the lapse and Schwarzschild R on the source grid, interpolate to the user grid, and then assemble all 3+1 quantities on the user grid and transform them back to the Cartesian grid.

To insulate our implementation from different mass parameters, we nondimensionalize the above process using the black hole mass until the final step of computing 3+1 quantities on the Cartesian grid, where we restore the correct unit.

The following are input file options that can be specified:

• Mass
• N (the parameter n in the slicing condition eq. (36))

Note

N=2. is strongly suggested, as this gives a stationary solution in the standard 1+log slicing. The other values of N near 2. should work but have not been tested thoroughly. N outside of [2., 3.] has not been tested at all.

Some quantities very close to the puncture (<1.e-4 in isotropic radius) may have larger truncation errors. This is expected since some quantities such as the determiant of the spatial metric diverges at the puncture.

Member Typedef Documentation

DerivLapse

template<typename DataType>

using gr::Solutions::TrumpetSchwarzschild::DerivLapse

Initial value:

 ::Tags::deriv<gr::Tags::Lapse<DataType>,
                                   tmpl::size_t<volume_dim>, Frame::Inertial>

DerivShift

template<typename DataType>

using gr::Solutions::TrumpetSchwarzschild::DerivShift

Initial value:

 ::Tags::deriv<gr::Tags::Shift<DataType, volume_dim>,
                                   tmpl::size_t<volume_dim>, Frame::Inertial>

DerivSpatialMetric

template<typename DataType>

using gr::Solutions::TrumpetSchwarzschild::DerivSpatialMetric

Initial value:

 
      ::Tags::deriv<gr::Tags::SpatialMetric<DataType, volume_dim>,
                    tmpl::size_t<volume_dim>, Frame::Inertial>

tags

template<typename DataType>

using gr::Solutions::TrumpetSchwarzschild::tags

Initial value:

 tmpl::list<
      gr::Tags::Lapse<DataType>, ::Tags::dt<gr::Tags::Lapse<DataType>>,
      DerivLapse<DataType>, gr::Tags::Shift<DataType, volume_dim>,
      ::Tags::dt<gr::Tags::Shift<DataType, volume_dim>>, DerivShift<DataType>,
      gr::Tags::SpatialMetric<DataType, volume_dim>,
      ::Tags::dt<gr::Tags::SpatialMetric<DataType, volume_dim>>,
      DerivSpatialMetric<DataType>, gr::Tags::SqrtDetSpatialMetric<DataType>,
      gr::Tags::ExtrinsicCurvature<DataType, volume_dim>,
      gr::Tags::InverseSpatialMetric<DataType, volume_dim>>

Member Data Documentation

help

Options::String gr::Solutions::TrumpetSchwarzschild::help

staticconstexpr

Initial value:

{
      "Schwarzschild solution in trumpet isotropic coordinates"}

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

• src/PointwiseFunctions/AnalyticSolutions/GeneralRelativity/TrumpetSchwarzschild.hpp