Classes | Typedefs | Enumerations | Functions
Xcts::Solutions Namespace Reference

Analytic solutions of the XCTS equations. More...

Classes

class  AnalyticSolution
 Base class for analytic solutions of the XCTS equations. More...
 
struct  CommonVariables
 Implementations for variables that solutions can share. More...
 
class  ConstantDensityStar
 A constant density star in general relativity. More...
 
class  Flatness
 Flat spacetime in general relativity. Useful as initial guess. More...
 
class  Schwarzschild
 Schwarzschild spacetime in general relativity. More...
 

Typedefs

template<typename DataType >
using common_tags = tmpl::push_back< AnalyticData::common_tags< DataType >, ::Tags::Flux< Tags::ConformalFactor< DataType >, tmpl::size_t< 3 >, Frame::Inertial >, ::Tags::Flux< Tags::LapseTimesConformalFactor< DataType >, tmpl::size_t< 3 >, Frame::Inertial >, Tags::LongitudinalShiftExcess< DataType, 3, Frame::Inertial >, gr::Tags::Shift< 3, Frame::Inertial, DataType >, Tags::LongitudinalShiftMinusDtConformalMetricSquare< DataType >, Tags::LongitudinalShiftMinusDtConformalMetricOverLapseSquare< DataType >, Tags::ShiftDotDerivExtrinsicCurvatureTrace< DataType > >
 Tags for variables that solutions can share.
 

Enumerations

enum  SchwarzschildCoordinates { SchwarzschildCoordinates::Isotropic }
 Various coordinate systems in which to express the Schwarzschild solution. More...
 

Functions

bool operator== (const ConstantDensityStar &, const ConstantDensityStar &) noexcept
 
bool operator!= (const ConstantDensityStar &lhs, const ConstantDensityStar &rhs) noexcept
 
template<typename Registrars >
bool operator== (const Flatness< Registrars > &, const Flatness< Registrars > &) noexcept
 
template<typename Registrars >
bool operator!= (const Flatness< Registrars > &lhs, const Flatness< Registrars > &rhs) noexcept
 
std::ostreamoperator<< (std::ostream &os, SchwarzschildCoordinates coords) noexcept
 

Detailed Description

Analytic solutions of the XCTS equations.

Enumeration Type Documentation

◆ SchwarzschildCoordinates

Various coordinate systems in which to express the Schwarzschild solution.

Enumerator
Isotropic 

Isotropic Schwarzschild coordinates.

These arise from the canonical Schwarzschild coordinates by the radial transformation

\begin{equation} r = \bar{r}\left(1+\frac{M}{2\bar{r}}\right)^2 \end{equation}

(Eq. (1.61) in [6]) where \(r\) is the canonical Schwarzschild radius, also referred to as "areal" radius because it is defined such that spheres with constant \(r\) have the area \(4\pi r^2\), and \(\bar{r}\) is the "isotropic" radius. In the isotropic radius the Schwarzschild spatial metric is conformally flat:

\begin{equation} \gamma_{ij}=\psi^4\eta_{ij} \quad \text{with conformal factor} \quad \psi=1+\frac{M}{2\bar{r}} \end{equation}

(Table 2.1 in [6]). Its lapse transforms to

\begin{equation} \alpha=\frac{1-M/(2\bar{r})}{1+M/(2\bar{r})} \end{equation}

and the shift vanishes ( \(\beta^i=0\)) as it does in areal Schwarzschild coordinates. The solution also remains maximally sliced, i.e. \(K=0\).

The Schwarzschild horizon in these coordinates is at \(\bar{r}=\frac{M}{2}\) due to the radial transformation from \(r=2M\).