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VecAl::SphericalSymmetric Struct Reference

A generic, spherically symmetric spacetime that is determined by two scalar functions $g_{tt} = A(r,t)$ and $g_{rr}=B(r,t)$. More...

#include <SphericalSymmetric.hpp>

Public Types

enum  {
  t , r , h , p ,
  theta =h , phi = p
}

Public Member Functions

Scalar_t A (Scalar_t r, Scalar_t t)
Scalar_t B (Scalar_t r, Scalar_t t)
void metric (Metric< Scalar_t, 4 > &g, const Point_t &P) const
void cometric (Metric< Scalar_t, 4 > &g, const Point_t &P) const
void getChristoffel (Christoffel< Scalar_t, 4 > &G, const Point_t &P) const
 Levi-Civita-Connection (Christoffelsymbols):
void Ricci (Metric< Scalar_t, 4 > &g, const Point_t &P) const
 Ricci - Tensor:

Detailed Description

A generic, spherically symmetric spacetime that is determined by two scalar functions $g_{tt} = A(r,t)$ and $g_{rr}=B(r,t)$.

Member Function Documentation

◆ cometric()

void VecAl::SphericalSymmetric::cometric ( Metric< Scalar_t, 4 > & g,
const Point_t & P ) const
inline

\‍[ \frac{1}{{A(r,t)}}\, {\partial_{{t}}^2} -\frac{1}{{B(r,t)}}\, {\partial_{{r}}^2} - {r}^{2}\, {\partial_{{\vartheta}}^2} -\sin^2\left({\vartheta}\right) \cdot {r}^{2}\, {\partial_{{\varphi}}^2} \‍]

◆ getChristoffel()

void VecAl::SphericalSymmetric::getChristoffel ( Christoffel< Scalar_t, 4 > & G,
const Point_t & P ) const
inline

Levi-Civita-Connection (Christoffelsymbols):

\‍[\begin{array}{ll} \nabla^{t }_{t t } = \frac{\dot{{A}}}{2\cdot {A}}& \nabla^{t }_{r t } = \frac{{A}'}{2\cdot {A}}\\ \nabla^{t }_{t r } = \frac{{A}'}{2\cdot {A}}& \nabla^{t }_{r r } = \frac{\dot{{B}}}{2\cdot {A}}\\ \nabla^{r }_{t t } = \frac{{A}'}{2\cdot {B}}& \nabla^{r }_{r t } = \frac{\dot{{B}}}{2\cdot {B}}\\ \nabla^{r }_{t r } = \frac{\dot{{B}}}{2\cdot {B}}& \nabla^{r }_{r r } = \frac{{B}'}{2\cdot {B}}\\ \nabla^{r }_{\vartheta \vartheta } = -\frac{{r}}{{B}}& \nabla^{r }_{\varphi \varphi } = -\frac{{r}\cdot \sin^2 {\vartheta} }{{B}}\\ \nabla^{\vartheta }_{\vartheta r } = \frac{1}{{r}}& \nabla^{\vartheta }_{r \vartheta } = \frac{1}{{r}}\\ \nabla^{\vartheta }_{\varphi \varphi } = -\sin {\vartheta} \cdot \cos {\vartheta} & \nabla^{\varphi }_{\varphi r } = \frac{1}{{r}}\\ \nabla^{\varphi }_{\varphi \vartheta } = \frac{1}{\tan {\vartheta} }& \nabla^{\varphi }_{r \varphi } = \frac{1}{{r}}\\ \nabla^{\varphi }_{\vartheta \varphi } = \frac{1}{\tan {\vartheta} }& \end{array} \‍]

◆ metric()

void VecAl::SphericalSymmetric::metric ( Metric< Scalar_t, 4 > & g,
const Point_t & P ) const
inline

\‍[ ds^2 = {A(r,t)}\, { d{t} ^2} -{B(r,t)}\, { d{r} ^2} - {r}^{2}\, { d{\vartheta} ^2} -\sin^2\left({\vartheta}\right) \cdot {r}^{2}\, { d{\varphi} ^2} \‍]

◆ Ricci()

void VecAl::SphericalSymmetric::Ricci ( Metric< Scalar_t, 4 > & g,
const Point_t & P ) const
inline

Ricci - Tensor:

\‍[\begin{array}{l} R_{t t } = \frac{4\cdot {A}\cdot {B}\cdot {A}'+{r}\cdot \left({A}\cdot \left\lbrack \left\lbrace {\dot{{B}}}\right\rbrace ^{2}+2\cdot {B}\cdot \left\lbrace {A}''-\ddot{{B}}\right\rbrace \right\rbrack +{B}\cdot \left\lbrack \dot{{A}}\cdot \dot{{B}}-\left\lbrace {{A}'}\right\rbrace ^{2}\right\rbrack \right)-{A}\cdot {r}\cdot {A}'\cdot {B}'}{4\cdot {A}\cdot {r}\cdot {B}^{2}}\\ R_{r t } = \frac{\dot{{B}}}{{B}\cdot {r}}\\ R_{t r } = \frac{\dot{{B}}}{{B}\cdot {r}}\\ R_{r r } = \frac{4\cdot {A}^{2}\cdot {B}'+{r}\cdot \left({A}\cdot \left\lbrack {A}'\cdot {B}'-\left\lbrace {\dot{{B}}}\right\rbrace ^{2}\right\rbrack +{B}\cdot \left\lbrack \left\lbrace {{A}'}\right\rbrace ^{2}+2\cdot {A}\cdot \left\lbrace \ddot{{B}}-{A}''\right\rbrace \right\rbrack \right)-{B}\cdot {r}\cdot \dot{{A}}\cdot \dot{{B}}}{4\cdot {B}\cdot {r}\cdot {A}^{2}}\\ R_{\vartheta \vartheta } = \frac{2\cdot {A}\cdot {B}\cdot \left({B}-1\right)+{r}\cdot \left({A}\cdot {B}'-{B}\cdot {A}'\right)}{2\cdot {A}\cdot {B}^{2}}\\ R_{\varphi \varphi } = \frac{\sin^2 {\vartheta} \cdot \left(2\cdot {A}\cdot {B}\cdot \left\lbrack {B}-1\right\rbrack +{r}\cdot \left\lbrack {A}\cdot {B}'-{B}\cdot {A}'\right\rbrack \right)}{2\cdot {A}\cdot {B}^{2}}\\ \end{array} \‍]

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