\( \DeclareMathOperator{\abs}{abs} \newcommand{\ensuremath}[1]{\mbox{$#1$}} \)

Métrica de Schwarzschild

 1 Inicializar Coordenadas esféricas para el espacio-tiempo:

--> load(ctensor)$
ct_coords:[ct,r,theta,phi]$
ct_coords;
dim;
\[\tag{%o3} [\mathit{ct},r,theta,phi]\] \[\tag{%o4} 4\]

 2 Introducir métrica de Schwarzschild

--> lg:ident(4)$
lg[1,1]:−(1−(2·GM/c^2)/r)$
lg[2,2]:1/(1−(2·GM/(r·c^2)))$
lg[3,3]:r^2$
lg[4,4]:r^2·sin(theta)^2$
lg;
\[\tag{%o154} \begin{bmatrix}\frac{2 \mathit{GM}}{{{c}^{2}} r}-1 & 0 & 0 & 0\\ 0 & \frac{1}{1-\frac{2 \mathit{GM}}{{{c}^{2}} r}} & 0 & 0\\ 0 & 0 & {{r}^{2}} & 0\\ 0 & 0 & 0 & {{r}^{2}}\, {{\sin{(theta)}}^{2}}\end{bmatrix}\]

  Cálculo de la métrica inversa:

--> cmetric()$
ug;
\[\tag{%o156} \begin{bmatrix}\frac{1}{\frac{2 \mathit{GM}}{{{c}^{2}} r}-1} & 0 & 0 & 0\\ 0 & 1-\frac{2 \mathit{GM}}{{{c}^{2}} r} & 0 & 0\\ 0 & 0 & \frac{1}{{{r}^{2}}} & 0\\ 0 & 0 & 0 & \frac{1}{{{r}^{2}}\, {{\sin{(theta)}}^{2}}}\end{bmatrix}\]

 3 Símbolos de Christoffel

  Cálculo de símbolos de Christoffel de primera especie lcs y de segunda especie mcs (m, mixed indices, el tercer índice es el superior). Se muestran los términos no nulos y omitiendo los simétricos:

--> christof(all)$
\[\tag{%t157} {{\mathit{lcs}}_{1,1,2}}=\frac{\mathit{GM}}{{{c}^{2}}\, {{r}^{2}}}\] \[\tag{%t158} {{\mathit{lcs}}_{1,2,1}}=-\frac{\mathit{GM}}{{{c}^{2}}\, {{r}^{2}}}\] \[\tag{%t159} {{\mathit{lcs}}_{2,2,2}}=-\frac{\mathit{GM}}{{{c}^{2}}\, {{\left( 1-\frac{2 \mathit{GM}}{{{c}^{2}} r}\right) }^{2}}\, {{r}^{2}}}\] \[\tag{%t160} {{\mathit{lcs}}_{2,3,3}}=r\] \[\tag{%t161} {{\mathit{lcs}}_{2,4,4}}=r\, {{\sin{(theta)}}^{2}}\] \[\tag{%t162} {{\mathit{lcs}}_{3,3,2}}=-r\] \[\tag{%t163} {{\mathit{lcs}}_{3,4,4}}={{r}^{2}} \cos{(theta)} \sin{(theta)}\] \[\tag{%t164} {{\mathit{lcs}}_{4,4,2}}=-r\, {{\sin{(theta)}}^{2}}\] \[\tag{%t165} {{\mathit{lcs}}_{4,4,3}}=-{{r}^{2}} \cos{(theta)} \sin{(theta)}\] \[\tag{%t166} {{\mathit{mcs}}_{1,1,2}}=\frac{\mathit{GM}\, {{c}^{2}} r-2 {{\mathit{GM}}^{2}}}{{{c}^{4}}\, {{r}^{3}}}\] \[\tag{%t167} {{\mathit{mcs}}_{1,2,1}}=\frac{\mathit{GM}}{{{c}^{2}}\, {{r}^{2}}-2 \mathit{GM} r}\] \[\tag{%t168} {{\mathit{mcs}}_{2,2,2}}=-\frac{\mathit{GM}}{{{c}^{2}}\, {{r}^{2}}-2 \mathit{GM} r}\] \[\tag{%t169} {{\mathit{mcs}}_{2,3,3}}=\frac{1}{r}\] \[\tag{%t170} {{\mathit{mcs}}_{2,4,4}}=\frac{1}{r}\] \[\tag{%t171} {{\mathit{mcs}}_{3,3,2}}=-\frac{{{c}^{2}} r-2 \mathit{GM}}{{{c}^{2}}}\] \[\tag{%t172} {{\mathit{mcs}}_{3,4,4}}=\frac{\cos{(theta)}}{\sin{(theta)}}\] \[\tag{%t173} {{\mathit{mcs}}_{4,4,2}}=-\frac{\left( {{c}^{2}} r-2 \mathit{GM}\right) \, {{\sin{(theta)}}^{2}}}{{{c}^{2}}}\] \[\tag{%t174} {{\mathit{mcs}}_{4,4,3}}=-\cos{(theta)} \sin{(theta)}\]

  La correspondencia de subíndices es, dado mcs x,y,z es:

  x = 1 + 1er (¿o 2º?) subíndice de Christoffel

  y = 1 + 2o (¿o 1er?) subíndice de Christoffel

  z = 1 + superíndice de Christoffel

 4 Tensor de Riemann:

--> riemann(true),ratriemann=true$ratsimp(%);
\[\tag{%t175} {{\mathit{riem}}_{1,2,1,2}}=\frac{2 \mathit{GM}\, \left( \mathit{GM}\, {{c}^{2}} r-2 {{\mathit{GM}}^{2}}\right) }{{{c}^{4}}\, {{r}^{3}}\, \left( {{c}^{2}}\, {{r}^{2}}-2 \mathit{GM} r\right) }+\frac{3 \left( \mathit{GM}\, {{c}^{2}} r-2 {{\mathit{GM}}^{2}}\right) }{{{c}^{4}}\, {{r}^{4}}}-\frac{\mathit{GM}}{{{c}^{2}}\, {{r}^{3}}}\] \[\tag{%t176} {{\mathit{riem}}_{1,3,1,3}}=-\frac{\mathit{GM}\, {{c}^{2}} r-2 {{\mathit{GM}}^{2}}}{{{c}^{4}}\, {{r}^{4}}}\] \[\tag{%t177} {{\mathit{riem}}_{1,4,1,4}}=-\frac{\mathit{GM}\, {{c}^{2}} r-2 {{\mathit{GM}}^{2}}}{{{c}^{4}}\, {{r}^{4}}}\] \[\tag{%t178} {{\mathit{riem}}_{2,2,1,1}}=\frac{2 \mathit{GM}}{{{c}^{2}}\, {{r}^{3}}-2 \mathit{GM}\, {{r}^{2}}}\] \[\tag{%t179} {{\mathit{riem}}_{2,3,2,3}}=\frac{\mathit{GM}}{r\, \left( {{c}^{2}}\, {{r}^{2}}-2 \mathit{GM} r\right) }\] \[\tag{%t180} {{\mathit{riem}}_{2,4,2,4}}=\frac{\mathit{GM}}{r\, \left( {{c}^{2}}\, {{r}^{2}}-2 \mathit{GM} r\right) }\] \[\tag{%t181} {{\mathit{riem}}_{3,3,1,1}}=-\frac{\mathit{GM}}{{{c}^{2}} r}\] \[\tag{%t182} {{\mathit{riem}}_{3,3,2,2}}=-\frac{\mathit{GM}}{{{c}^{2}} r}\] \[\tag{%t183} {{\mathit{riem}}_{3,4,3,4}}=\frac{{{c}^{2}} r-2 \mathit{GM}}{{{c}^{2}} r}-1\] \[\tag{%t184} {{\mathit{riem}}_{4,4,1,1}}=-\frac{\mathit{GM}\, {{\sin{(theta)}}^{2}}}{{{c}^{2}} r}\] \[\tag{%t185} {{\mathit{riem}}_{4,4,2,2}}=-\frac{\mathit{GM}\, {{\sin{(theta)}}^{2}}}{{{c}^{2}} r}\] \[\tag{%t186} {{\mathit{riem}}_{4,4,3,3}}=\frac{2 \mathit{GM}\, {{\sin{(theta)}}^{2}}}{{{c}^{2}} r}\] \[\tag{%o187} \mathit{done}\]

  Usando los índices de los Riemann del Capítulo 33:

(ver fórmulas 16.4)

--> riem[1,1,2,2];
riem[1,1,3,3];
riem[1,1,4,4];
riem[2,1,2,1];
riem[2,3,2,3];
riem[2,4,2,4];
riem[3,1,3,1];
riem[3,3,2,2];
riem[3,4,3,4];
riem[4,1,4,1];
riem[4,4,2,2];
riem[4,4,3,3];
\[\tag{%o188} -\frac{2 \mathit{GM}\, \left( \mathit{GM}\, {{c}^{2}} r-2 {{\mathit{GM}}^{2}}\right) }{{{c}^{4}}\, {{r}^{3}}\, \left( {{c}^{2}}\, {{r}^{2}}-2 \mathit{GM} r\right) }-\frac{3 \left( \mathit{GM}\, {{c}^{2}} r-2 {{\mathit{GM}}^{2}}\right) }{{{c}^{4}}\, {{r}^{4}}}+\frac{\mathit{GM}}{{{c}^{2}}\, {{r}^{3}}}\] \[\tag{%o189} \frac{\mathit{GM}\, {{c}^{2}} r-2 {{\mathit{GM}}^{2}}}{{{c}^{4}}\, {{r}^{4}}}\] \[\tag{%o190} \frac{\mathit{GM}\, {{c}^{2}} r-2 {{\mathit{GM}}^{2}}}{{{c}^{4}}\, {{r}^{4}}}\] \[\tag{%o191} -\frac{2 \mathit{GM}}{{{c}^{2}}\, {{r}^{3}}-2 \mathit{GM}\, {{r}^{2}}}\] \[\tag{%o192} \frac{\mathit{GM}}{{{c}^{2}}\, {{r}^{3}}-2 \mathit{GM}\, {{r}^{2}}}\] \[\tag{%o193} \frac{\mathit{GM}}{{{c}^{2}}\, {{r}^{3}}-2 \mathit{GM}\, {{r}^{2}}}\] \[\tag{%o194} \frac{\mathit{GM}}{{{c}^{2}} r}\] \[\tag{%o195} -\frac{\mathit{GM}}{{{c}^{2}} r}\] \[\tag{%o196} -\frac{2 \mathit{GM}}{{{c}^{2}} r}\] \[\tag{%o197} \frac{\mathit{GM}\, {{\sin{(theta)}}^{2}}}{{{c}^{2}} r}\] \[\tag{%o198} -\frac{\mathit{GM}\, {{\sin{(theta)}}^{2}}}{{{c}^{2}} r}\] \[\tag{%o199} \frac{2 \mathit{GM}\, {{\sin{(theta)}}^{2}}}{{{c}^{2}} r}\]

  Para algún caso hay que indicar ratsimp(%) si queremos ver la versión simplificada:

--> riem[1,1,2,2]$ratsimp(%);
\[\tag{%o201} -\frac{2 \mathit{GM}\, {{c}^{2}} r-4 {{\mathit{GM}}^{2}}}{{{c}^{4}}\, {{r}^{4}}}\]

 5 Tensor de Ricci

  La solución de Schwarzschild es para el vacío, debe anularse:

--> ricci(true)$
\[\mbox{}\\\mbox{ARRSTORE: use\_ fast\_ arrays=false; allocate a new property hash table for \$ RIC}\] \[\tag{%t202} {{\mathit{ric}}_{1,1}}=-\frac{2 \mathit{GM}\, \left( \mathit{GM}\, {{c}^{2}} r-2 {{\mathit{GM}}^{2}}\right) }{{{c}^{4}}\, {{r}^{3}}\, \left( {{c}^{2}}\, {{r}^{2}}-2 \mathit{GM} r\right) }-\frac{\mathit{GM}\, {{c}^{2}} r-2 {{\mathit{GM}}^{2}}}{{{c}^{4}}\, {{r}^{4}}}+\frac{\mathit{GM}}{{{c}^{2}}\, {{r}^{3}}}\] \[\tag{%t203} {{\mathit{ric}}_{2,2}}=-\frac{2 \mathit{GM}}{r\, \left( {{c}^{2}}\, {{r}^{2}}-2 \mathit{GM} r\right) }+\frac{\mathit{GM}\, \left( 2 {{c}^{2}} r-2 \mathit{GM}\right) }{{{\left( {{c}^{2}}\, {{r}^{2}}-2 \mathit{GM} r\right) }^{2}}}-\frac{2 {{\mathit{GM}}^{2}}}{{{\left( {{c}^{2}}\, {{r}^{2}}-2 \mathit{GM} r\right) }^{2}}}\]
--> ric[1,1]; ratsimp(%);
\[\tag{%o204} -\frac{2 \mathit{GM}\, \left( \mathit{GM}\, {{c}^{2}} r-2 {{\mathit{GM}}^{2}}\right) }{{{c}^{4}}\, {{r}^{3}}\, \left( {{c}^{2}}\, {{r}^{2}}-2 \mathit{GM} r\right) }-\frac{\mathit{GM}\, {{c}^{2}} r-2 {{\mathit{GM}}^{2}}}{{{c}^{4}}\, {{r}^{4}}}+\frac{\mathit{GM}}{{{c}^{2}}\, {{r}^{3}}}\] \[\tag{%o205} 0\]
--> ric[2,2]; ratsimp(%);
\[\tag{%o206} -\frac{2 \mathit{GM}}{r\, \left( {{c}^{2}}\, {{r}^{2}}-2 \mathit{GM} r\right) }+\frac{\mathit{GM}\, \left( 2 {{c}^{2}} r-2 \mathit{GM}\right) }{{{\left( {{c}^{2}}\, {{r}^{2}}-2 \mathit{GM} r\right) }^{2}}}-\frac{2 {{\mathit{GM}}^{2}}}{{{\left( {{c}^{2}}\, {{r}^{2}}-2 \mathit{GM} r\right) }^{2}}}\] \[\tag{%o207} 0\]

  Con parámetro "ratfac" para simplificar directamente:

--> ricci(true),ratfac=true$
\[\mbox{}\\THIS SPACETIME IS EMPTY AND/OR FLAT \]

  (Dice que es un espacio-tiempo plano o vacío. En este caso está vacío, fuera de la distribución con simetría esférica de masa se entiende, que es donde es aplicable la métrica de Schwarzschild, pero sólo es plano para M=0)

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