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.

## ◆ SchwarzschildCoordinates

 strong

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

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

(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:

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

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

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

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 $$r=2M$$.