Xcts::Solutions Namespace Reference

Analytic solutions of the XCTS equations. More...

Classes
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...
class	TovStar
	TOV solution to the XCTS equations. More...
class	WrappedGr
	XCTS quantities for a solution of the Einstein equations. More...

Typedefs
template<typename DataType >
using	common_tags = implementation defined
	Tags for variables that solutions can share.
template<typename DataType >
using	hydro_tags = AnalyticData::hydro_tags< DataType >
	Tags for hydro variables that are typically retrieved from a hydro solution.
using	all_analytic_solutions = implementation defined
template<typename GrMhdSolution >
using	WrappedGrMhd = WrappedGr< GrMhdSolution, true >

Enumerations
enum class	SchwarzschildCoordinates { Isotropic , PainleveGullstrand , KerrSchildIsotropic , MaximalIsotropic }
	Various coordinate systems in which to express the Schwarzschild solution. More...

Functions
bool	operator== (const ConstantDensityStar &, const ConstantDensityStar &)
bool	operator!= (const ConstantDensityStar &lhs, const ConstantDensityStar &rhs)
bool	operator== (const Flatness &, const Flatness &)
bool	operator!= (const Flatness &lhs, const Flatness &rhs)
std::ostream &	operator<< (std::ostream &os, SchwarzschildCoordinates coords)
bool	operator!= (const TovStar &lhs, const TovStar &rhs)
template<typename GrSolution , bool HasMhd>
bool	operator!= (const WrappedGr< GrSolution, HasMhd > &lhs, const WrappedGr< GrSolution, HasMhd > &rhs)

Detailed Description

Analytic solutions of the XCTS equations.

Enumeration Type Documentation

SchwarzschildCoordinates

enum class Xcts::Solutions::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 $$\begin{array}{}\text{(1)}& r=\overline{r}{(1+\frac{M}{2\overline{r}})}^2\end{array}$$ (Eq. (1.61) in [14]) 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 $\overline{r}$ is the "isotropic" radius. In the isotropic radius the Schwarzschild spatial metric is conformally flat: $$\begin{array}{}\text{(2)}& {\gamma }_{ij}={\psi }^4{\eta }_{ij}\text{with conformal factor}\psi =1+\frac{M}{2\overline{r}}\end{array}$$ (Table 2.1 in [14]). The lapse in the conformal radius is $$\begin{array}{}\text{(3)}& \alpha =\frac{1-M/(2\overline{r})}{1+M/(2\overline{r})}\end{array}$$ 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 $\overline{r}=\frac{M}{2}$ due to the radial transformation from $r=2M$.
PainleveGullstrand	Painlevé-Gullstrand coordinates. In these coordinates the spatial metric is flat and the lapse is trivial, but contrary to (isotropic) Schwarzschild coordinates the shift is nontrivial, $$\begin{array}{}\text{(4)}& {\gamma }_{ij}& ={\eta }_{ij}\text{(5)}& \alpha & =1\text{(6)}& {\beta }^i& =\sqrt{\frac{2M}{r}}\frac{x^i}{r}\text{(7)}& K& =\frac{3}{2}\sqrt{\frac{2M}{r^3}}\end{array}$$ (Table 2.1 in [14]).
KerrSchildIsotropic	Isotropic Kerr-Schild coordinates. Kerr-Schild coordinates with a radial transformation such that the spatial metric is conformally flat. The Schwarzschild spacetime in canonical (areal) Kerr-Schild coordinates is $$\begin{array}{}\text{(8)}& {\gamma }_{ij}& ={\eta }_{ij}+\frac{2M}{r}\frac{x^i{x}^j}{r^2}\text{(9)}& \alpha & ={\sqrt{1+\frac{2M}{r}}}^{-1}\text{(10)}& {\beta }^i& =\frac{2M{\alpha }^2}{r}\frac{x^i}{r}\text{(11)}& K& =\frac{2M{\alpha }^3}{r^2}(1+\frac{3M}{r})\text{.}\end{array}$$ (Table 2.1 in [14]). Since the Schwarzschild spacetime is spherically symmetric we can transform to a radial coordinate $\overline{r}$ in which it is conformally flat (see, e.g., Sec. 7.4.1 in [164] for details): $$\begin{array}{}\text{(12)}& {}^{(3)}\mathrm{d}{s}^2=(1+\frac{2M}{r})\mathrm{d}{r}^2+r^2\mathrm{d}{\mathrm{\Omega }}^2={\psi }^4(\mathrm{d}{\overline{r}}^2+{\overline{r}}^2\mathrm{d}{\mathrm{\Omega }}^2)\end{array}$$ Therefore, the conformal factor is ${\psi }^2=r/\overline{r}$ and $$\begin{array}{}\text{(13)}& \frac{\mathrm{d}\overline{r}}{\mathrm{d}r}=\frac{\overline{r}}{r}\sqrt{1+\frac{2M}{r}}=\frac{\overline{r}}{r}\frac{1}{\alpha }\text{,}\end{array}$$ which has the solution $$\begin{array}{}\text{(14)}& \overline{r}=\frac{r}{4}{(1+\sqrt{1+\frac{2M}{r}})}^2{e}^{2-2\sqrt{1+2M/r}}\end{array}$$ when we impose $\overline{r}\to r$ as $r\to \mathrm{\infty }$. We can invert this transformation law with a numerical root find to obtain the areal radius $r$ for any isotropic radius $\overline{r}$. In the isotropic radial coordinate $\overline{r}$ the solution is then: $$\begin{array}{}\text{(15)}& {\gamma }_{ij}& ={\psi }^4{\eta }_{ij}\text{(16)}& \psi & =\sqrt{\frac{r}{\overline{r}}}=\frac{2e^{\sqrt{1+2M/r}-1}}{1+\sqrt{1+2M/r}}\text{(17)}& \alpha & ={\sqrt{1+\frac{2M}{r}}}^{-1}\text{(18)}& {\beta }^i& =\frac{\mathrm{d}\overline{r}}{\mathrm{d}r}{\beta }^r\frac{x^i}{\overline{r}}=\frac{2M\alpha }{r^2}{x}^i\text{(19)}& K& =\frac{2M{\alpha }^3}{r^2}(1+\frac{3M}{r})\end{array}$$ Here, $x^i$ are the (isotropic) Cartesian coordinates from which we compute the isotropic radius $\overline{r}$, $r$ is the areal radius we can obtain from the isotropic radius by a root find, and ${\beta }^r$ is the magnitude of the shift in areal coordinates, as given above. The horizon in these coordinates is at (Eq. (7.37) in [164]): $$\begin{array}{}\text{(20)}& {\overline{r}}_{\mathrm{A}\mathrm{H}}/M\approx 1.2727410334221052\end{array}$$
MaximalIsotropic	Maximal Isotropic (Horizon Penetrating) Schwarzschild coordinates. Schwarzschild coordinates with a radial transformation such that the radius is isotropic and the coordinates are horizon penetrating. These arise from first choosing a family of time-independent, maximal slicings of the Schwarzschild spacetime and a slicing condition that give a unique solution with a limiting surface at $R=3M/2$ and horizon at $R=2M$ [65]. The latter is then changed by the radial transformation [13] $$\begin{array}{}\text{(21)}& r=\frac{[2R+M+\sqrt{4R^2+4MR+3M^2}]}{4}\times {\left[\frac{(4+3\sqrt{2})(2R-3M)}{8R+6M+3\sqrt{8R^2+8MR+6M^2}}\right]}^{1/\sqrt{2}}\end{array}$$ 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 $r$ is the "isotropic" radius. In the isotropic radius the Schwarzschild spatial metric is conformally flat: $$\begin{array}{}\text{(22)}& {\gamma }_{ij}={\psi }^4{\eta }_{ij}\text{with conformal factor}\psi =\sqrt{\frac{R}{r}}.\end{array}$$ The lapse in the conformal radius is $$\begin{array}{}\text{(23)}& \alpha ={(1-\frac{2M}{R}+\frac{27M^4}{16R^4})}^{1/2}\end{array}$$ and the shift is given by $$\begin{array}{}\text{(24)}& {\beta }^r=\frac{3\sqrt{3}{M}^2}{4}\frac{r}{R^3}\end{array}$$ The solution remains maximally sliced, i.e. $K=0$. And the horizon in these coordinates is at $r\approx 0.7793271080557972M$ due to the radial transformation from $R=2M$.