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SpECTRE
2021.08.02
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The radial variables needed to compute the full TOVStar solution.
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#include <TovStar.hpp>
Public Member Functions | |
| RadialVariables (DataType radial_coordinate_in) | |
Public Attributes | |
| DataType | radial_coordinate {} |
| Scalar< DataType > | rest_mass_density {} |
| Scalar< DataType > | pressure {} |
| Scalar< DataType > | specific_internal_energy {} |
| Scalar< DataType > | specific_enthalpy {} |
| DataType | dr_pressure {} |
| DataType | metric_time_potential {} |
| DataType | dr_metric_time_potential {} |
| DataType | metric_radial_potential {} |
| DataType | dr_metric_radial_potential {} |
| DataType | metric_angular_potential {} |
| DataType | dr_metric_angular_potential {} |
The radial variables needed to compute the full TOVStar solution.
RadialSolution must provide a method that fills the radial variables given the equation of state and the Cartesian coordinates \(x^i\)
If the spherically symmetric metric is written as
\[ ds^2 = - e^{2 \Phi_t} dt^2 + e^{2 \Phi_r} dr^2 + e^{2 \Phi_\Omega} r^2 d\Omega^2 \]
where \(r = \delta_{mn} x^m x^n\) is the radial_coordinate and \(\Phi_t\), \(\Phi_r\), and \(\Phi_\Omega\) are respectvely the metric_time_potential, metric_radial_potential, and metric_angular_potential, then the lapse, shift, and spatial metric in the Cartesian coordinates are
\begin{align*} \alpha &= e^{\Phi_t} \\ \beta^i &= 0 \\ \gamma_{ij} &= \delta_{ij} e^{2 \Phi_r} + \delta_{im} \delta_{jn} \frac{x^m x^n}{r^2} \left( e^{2 \Phi_r} - e^{2 \Phi_\Omega} \right) \end{align*}