Xcts::Solutions::ConstantDensityStar Class Reference

A constant density star in general relativity. More...

#include <ConstantDensityStar.hpp>

Classes
struct	Density
struct	Radius

Public Types
using	options = implementation defined

Public Member Functions
	ConstantDensityStar (const ConstantDensityStar &)=default
ConstantDensityStar &	operator= (const ConstantDensityStar &)=default
	ConstantDensityStar (ConstantDensityStar &&)=default
ConstantDensityStar &	operator= (ConstantDensityStar &&)=default
std::unique_ptr< elliptic::analytic_data::AnalyticSolution >	get_clone () const override
	ConstantDensityStar (double density, double radius, const Options::Context &context={})
double	density () const
double	radius () const
template<typename DataType , typename... Tags>
tuples::TaggedTuple< Tags... >	variables (const tnsr::I< DataType, 3, Frame::Inertial > &x, tmpl::list< Tags... >) const
	Retrieve a collection of variables at coordinates x
void	pup (PUP::er &p) override
template<typename DataType >
auto	variables (const tnsr::I< DataType, 3, Frame::Inertial > &x, tmpl::list< Xcts::Tags::ConformalFactor< DataType > >) const -> tuples::TaggedTuple< Xcts::Tags::ConformalFactor< DataType > >
	Retrieve variable at coordinates x
template<typename DataType >
auto	variables (const tnsr::I< DataType, 3, Frame::Inertial > &x, tmpl::list<::Tags::Initial< Xcts::Tags::ConformalFactor< DataType > > >) const -> tuples::TaggedTuple< ::Tags::Initial< Xcts::Tags::ConformalFactor< DataType > > >
	Retrieve variable at coordinates x
template<typename DataType >
auto	variables (const tnsr::I< DataType, 3, Frame::Inertial > &x, tmpl::list<::Tags::Initial< ::Tags::deriv< Xcts::Tags::ConformalFactor< DataType >, tmpl::size_t< 3 >, Frame::Inertial > > >) const -> tuples::TaggedTuple< ::Tags::Initial<::Tags::deriv< Xcts::Tags::ConformalFactor< DataType >, tmpl::size_t< 3 >, Frame::Inertial > > >
	Retrieve variable at coordinates x
template<typename DataType >
auto	variables (const tnsr::I< DataType, 3, Frame::Inertial > &x, tmpl::list<::Tags::FixedSource< Xcts::Tags::ConformalFactor< DataType > > >) const -> tuples::TaggedTuple< ::Tags::FixedSource< Xcts::Tags::ConformalFactor< DataType > > >
	Retrieve variable at coordinates x
template<typename DataType >
auto	variables (const tnsr::I< DataType, 3, Frame::Inertial > &x, tmpl::list< gr::Tags::EnergyDensity< DataType > >) const -> tuples::TaggedTuple< gr::Tags::EnergyDensity< DataType > >
	Retrieve variable at coordinates x
virtual std::unique_ptr< AnalyticSolution >	get_clone () const =0

Static Public Attributes
static constexpr Options::String	help

Detailed Description

A constant density star in general relativity.

Details

This solution describes a star with constant density ${\rho }_0$ that extends to a (conformal) radius $R$. It solves the XCTS Hamiltonian constraint that reduces to the non-linear elliptic equation

$${\mathrm{\Delta }}^2\psi +2\pi \rho {\psi }^5=0$$

for the conformal factor $\psi$ (see Xcts) under the following assumptions [16] :

• Time-symmetry $K_{ij}=0$
• Conformal flatness $\overline{\gamma }=\delta$, so $\mathrm{\Delta }$ is the flat-space Laplacian
• Spherical symmetry

Imposing boundary conditions

$$\begin{gathered} \frac{\partial \psi }{\partial r}=0\text{for}r=0 \\ \psi \to 1\text{for}r\to \mathrm{\infty } \end{gathered}$$

and considering the energy density

$$\rho (r\le R)={\rho }_0\text{and}\rho (r>R)=0$$

of the star the authors of [16] find the solution

$$\psi (r\le R)=C{u}_{\alpha }(r)\text{and}\psi (r>R)=\frac{\beta }{r}+1$$

with $C=(2\pi {\rho }_0/3{)}^{-1/4}$, the Sobolev functions

$$u_{\alpha }(r)=\sqrt{\frac{\alpha R}{r^2+(\alpha R{)}^2}}$$

and real parameters $\alpha$ and $\beta$ that are determined by the following relations:

$$\begin{gathered} {\rho }_0{R}^2=\frac{3}{2\pi }{f}^2(\alpha )\text{with}f(\alpha )=\frac{{\alpha }^5}{(1+{\alpha }^2{)}^3} \\ \frac{\beta }{R}+1=C{u}_{\alpha }(R) \end{gathered}$$

This solution is described in detail in [16] , and also in Exercise 3.8 in [14] , since it exhibits the non-uniqueness properties that are typical for the XCTS system. In the simple case of the constant-density star the non-uniqueness is apparent from the function $f(\alpha )$, which has two solutions for any ${\rho }_0$ smaller than a critical density

$${\rho }_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}=\frac{3}{2\pi R^2}\frac{5^2}{6^6}\approx \frac{0.0320}{R^2}\text{,}$$

a unique solution for ${\rho }_0={\rho }_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}$ and no solutions above the critical density [16] . The authors identify the $\alpha <{\alpha }_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}=\sqrt{5}$ and $\alpha >{\alpha }_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}$ branches of solutions with the strong-field and weak-field regimes, respectively (see [16] for details). In this implementation we compute the weak-field solution by choosing the $\alpha >{\alpha }_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}$ that corresponds to the density ${\rho }_0$ of the star. Therefore, we supply initial data ${\psi }_{\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}}=1$ so that a nonlinear iterative numerical solver will converge to the same weak-field solution.

Member Function Documentation

get_clone()

std::unique_ptr< elliptic::analytic_data::AnalyticSolution > Xcts::Solutions::ConstantDensityStar::get_clone ( ) const

inlineoverridevirtual

Implements elliptic::analytic_data::AnalyticSolution.

Member Data Documentation

help

constexpr Options::String Xcts::Solutions::ConstantDensityStar::help

staticconstexpr

Initial value:

{
      "A constant density star in general relativity"}

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

• src/PointwiseFunctions/AnalyticSolutions/Xcts/ConstantDensityStar.hpp