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SpECTRE
v2024.04.12
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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... | |
| class | WrappedGr< GrSolution, HasMhd, tmpl::list< GrSolutionOptions... > > |
Enumerations | |
| enum class | SchwarzschildCoordinates { Isotropic , PainleveGullstrand , KerrSchildIsotropic } |
| 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) |
Analytic solutions of the XCTS equations.
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Various coordinate systems in which to express the Schwarzschild solution.
| Enumerator | |
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| 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 [13]) 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 [13]). The lapse in the conformal radius is \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 \(r=2M\). |
| 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{align} \gamma_{ij} &= \eta_{ij} \\ \alpha &= 1 \\ \beta^i &= \sqrt{\frac{2M}{r}} \frac{x^i}{r} \\ K &= \frac{3}{2}\sqrt{\frac{2M}{r^3}} \end{align} (Table 2.1 in [13]). |
| 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{align} \gamma_{ij} &= \eta_{ij} + \frac{2M}{r}\frac{x^i x^j}{r^2} \\ \alpha &= \sqrt{1 + \frac{2M}{r}}^{-1} \\ \beta^i &= \frac{2M\alpha^2}{r} \frac{x^i}{r} \\ K &= \frac{2M\alpha^3}{r^2} \left(1 + \frac{3M}{r}\right) \text{.} \end{align} (Table 2.1 in [13]). Since the Schwarzschild spacetime is spherically symmetric we can transform to a radial coordinate \(\bar{r}\) in which it is conformally flat (see, e.g., Sec. 7.4.1 in [151] for details): \begin{equation} {}^{(3)}\mathrm{d}s^2 = \left(1 + \frac{2M}{r}\right)\mathrm{d}r^2 + r^2 \mathrm{d}\Omega^2 = \psi^4 \left(\mathrm{d}\bar{r}^2 + \bar{r}^2 \mathrm{d}\Omega^2\right) \end{equation} Therefore, the conformal factor is \(\psi^2 = r / \bar{r}\) and \begin{equation} \frac{\mathrm{d}\bar{r}}{\mathrm{d}r} = \frac{\bar{r}}{r} \sqrt{1 + \frac{2M}{r}} = \frac{\bar{r}}{r} \frac{1}{\alpha} \text{,} \end{equation} which has the solution \begin{equation} \bar{r} = \frac{r}{4} \left(1 + \sqrt{1 + \frac{2M}{r}}\right)^2 e^{2 - 2\sqrt{1 + 2M / r}} \end{equation} when we impose \(\bar{r} \rightarrow r\) as \(r \rightarrow \infty\). We can invert this transformation law with a numerical root find to obtain the areal radius \(r\) for any isotropic radius \(\bar{r}\). In the isotropic radial coordinate \(\bar{r}\) the solution is then: \begin{align} \gamma_{ij} &= \psi^4 \eta_{ij} \\ \psi &= \sqrt{\frac{r}{\bar{r}}} = \frac{2e^{\sqrt{1 + 2M / r} - 1}}{1 + \sqrt{1 + 2M / r}} \\ \alpha &= \sqrt{1 + \frac{2M}{r}}^{-1} \\ \beta^i &= \frac{\mathrm{d}\bar{r}}{\mathrm{d}r} \beta^r \frac{x^i}{\bar{r}} = \frac{2M\alpha}{r^2} x^i \\ K &= \frac{2M\alpha^3}{r^2} \left(1 + \frac{3M}{r}\right) \end{align} Here, \(x^i\) are the (isotropic) Cartesian coordinates from which we compute the isotropic radius \(\bar{r}\), \(r\) 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 [151]): \begin{equation} \bar{r}_\mathrm{AH} / M \approx 1.2727410334221052 \end{equation} |