Classes | Enumerations | Functions
Xcts::Solutions Namespace Reference

Analytic solutions to the XCTS equations. More...

Classes

class  ConstantDensityStar
 A constant density star in general relativity. More...
 
class  Schwarzschild
 Schwarzschild spacetime in general relativity. More...
 

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
 
std::ostreamoperator<< (std::ostream &os, SchwarzschildCoordinates coords) noexcept
 
template<SchwarzschildCoordinates Coords>
bool operator== (const Schwarzschild< Coords > &lhs, const Schwarzschild< Coords > &rhs) noexcept
 
template<SchwarzschildCoordinates Coords>
bool operator!= (const Schwarzschild< Coords > &lhs, const Schwarzschild< Coords > &rhs) noexcept
 

Detailed Description

Analytic solutions to 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 \(r=2M\).