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

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

  Métrica lg (l, lower indices) y su inversa ug (u, upper indices).

Se simplifica con la notación en base al radio de Schwarzschild:

--> a=2·GM/c^2;
lg:ident(4)$
lg[1,1]:−(1−a/r)$
lg[2,2]:1/(1−a/r)$
lg[3,3]:r^2$
lg[4,4]:r^2·sin(theta)^2$
lg;
\[\tag{%o5} a=\frac{2 \mathit{GM}}{{{c}^{2}}}\] \[\tag{%o11} \begin{bmatrix}\frac{a}{r}-1 & 0 & 0 & 0\\ 0 & \frac{1}{1-\frac{a}{r}} & 0 & 0\\ 0 & 0 & {{r}^{2}} & 0\\ 0 & 0 & 0 & {{r}^{2}}\, {{\sin{(theta)}}^{2}}\end{bmatrix}\]

  Cálculo de la métrica inversa:

--> cmetric()$
ug;
\[\tag{%o13} \begin{bmatrix}\frac{1}{\frac{a}{r}-1} & 0 & 0 & 0\\ 0 & 1-\frac{a}{r} & 0 & 0\\ 0 & 0 & \frac{1}{{{r}^{2}}} & 0\\ 0 & 0 & 0 & \frac{1}{{{r}^{2}}\, {{\sin{(theta)}}^{2}}}\end{bmatrix}\]

 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)$
\[\mbox{}\\\mbox{ARRSTORE: use\_ fast\_ arrays=false; allocate a new property hash table for \$ LCS}\mbox{}\\\mbox{ARRSTORE: use\_ fast\_ arrays=false; allocate a new property hash table for \$ MCS}\] \[\tag{%t14} {{\mathit{lcs}}_{1,1,2}}=\frac{a}{2 {{r}^{2}}}\] \[\tag{%t15} {{\mathit{lcs}}_{1,2,1}}=-\frac{a}{2 {{r}^{2}}}\] \[\tag{%t16} {{\mathit{lcs}}_{2,2,2}}=-\frac{a}{2 {{\left( 1-\frac{a}{r}\right) }^{2}}\, {{r}^{2}}}\] \[\tag{%t17} {{\mathit{lcs}}_{2,3,3}}=r\] \[\tag{%t18} {{\mathit{lcs}}_{2,4,4}}=r\, {{\sin{(theta)}}^{2}}\] \[\tag{%t19} {{\mathit{lcs}}_{3,3,2}}=-r\] \[\tag{%t20} {{\mathit{lcs}}_{3,4,4}}={{r}^{2}} \cos{(theta)} \sin{(theta)}\] \[\tag{%t21} {{\mathit{lcs}}_{4,4,2}}=-r\, {{\sin{(theta)}}^{2}}\] \[\tag{%t22} {{\mathit{lcs}}_{4,4,3}}=-{{r}^{2}} \cos{(theta)} \sin{(theta)}\] \[\tag{%t23} {{\mathit{mcs}}_{1,1,2}}=\frac{a r-{{a}^{2}}}{2 {{r}^{3}}}\] \[\tag{%t24} {{\mathit{mcs}}_{1,2,1}}=\frac{a}{2 {{r}^{2}}-2 a r}\] \[\tag{%t25} {{\mathit{mcs}}_{2,2,2}}=-\frac{a}{2 {{r}^{2}}-2 a r}\] \[\tag{%t26} {{\mathit{mcs}}_{2,3,3}}=\frac{1}{r}\] \[\tag{%t27} {{\mathit{mcs}}_{2,4,4}}=\frac{1}{r}\] \[\tag{%t28} {{\mathit{mcs}}_{3,3,2}}=a-r\] \[\tag{%t29} {{\mathit{mcs}}_{3,4,4}}=\frac{\cos{(theta)}}{\sin{(theta)}}\] \[\tag{%t30} {{\mathit{mcs}}_{4,4,2}}=\left( a-r\right) \, {{\sin{(theta)}}^{2}}\] \[\tag{%t31} {{\mathit{mcs}}_{4,4,3}}=-\cos{(theta)} \sin{(theta)}\]

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

  x = 1 + 1er subíndice de Christoffel (?)

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

  z = 1 + superíndice de Christoffel

 4 Tensor de Riemann:

--> riemann(true),ratriemann=true$ratsimp(%);
\[\mbox{}\\\mbox{ARRSTORE: use\_ fast\_ arrays=false; allocate a new property hash table for \$ RIEM}\] \[\tag{%t32} {{\mathit{riem}}_{1,2,1,2}}=\frac{a\, \left( a r-{{a}^{2}}\right) }{{{r}^{3}}\, \left( 2 {{r}^{2}}-2 a r\right) }+\frac{3 \left( a r-{{a}^{2}}\right) }{2 {{r}^{4}}}-\frac{a}{2 {{r}^{3}}}\] \[\tag{%t33} {{\mathit{riem}}_{1,3,1,3}}=-\frac{a r-{{a}^{2}}}{2 {{r}^{4}}}\] \[\tag{%t34} {{\mathit{riem}}_{1,4,1,4}}=-\frac{a r-{{a}^{2}}}{2 {{r}^{4}}}\] \[\tag{%t35} {{\mathit{riem}}_{2,2,1,1}}=\frac{a}{{{r}^{3}}-a\, {{r}^{2}}}\] \[\tag{%t36} {{\mathit{riem}}_{2,3,2,3}}=\frac{a}{r\, \left( 2 {{r}^{2}}-2 a r\right) }\] \[\tag{%t37} {{\mathit{riem}}_{2,4,2,4}}=\frac{a}{r\, \left( 2 {{r}^{2}}-2 a r\right) }\] \[\tag{%t38} {{\mathit{riem}}_{3,3,1,1}}=-\frac{a}{2 r}\] \[\tag{%t39} {{\mathit{riem}}_{3,3,2,2}}=-\frac{a}{2 r}\] \[\tag{%t40} {{\mathit{riem}}_{3,4,3,4}}=-\frac{a-r}{r}-1\] \[\tag{%t41} {{\mathit{riem}}_{4,4,1,1}}=-\frac{a\, {{\sin{(theta)}}^{2}}}{2 r}\] \[\tag{%t42} {{\mathit{riem}}_{4,4,2,2}}=-\frac{a\, {{\sin{(theta)}}^{2}}}{2 r}\] \[\tag{%t43} {{\mathit{riem}}_{4,4,3,3}}=\frac{a\, {{\sin{(theta)}}^{2}}}{r}\] \[\tag{%o44} \mathit{done}\]

  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];
\[\tag{%o45} -\frac{a\, \left( a r-{{a}^{2}}\right) }{{{r}^{3}}\, \left( 2 {{r}^{2}}-2 a r\right) }-\frac{3 \left( a r-{{a}^{2}}\right) }{2 {{r}^{4}}}+\frac{a}{2 {{r}^{3}}}\] \[\tag{%o46} \frac{a r-{{a}^{2}}}{2 {{r}^{4}}}\] \[\tag{%o47} \frac{a r-{{a}^{2}}}{2 {{r}^{4}}}\] \[\tag{%o48} -\frac{a}{{{r}^{3}}-a\, {{r}^{2}}}\] \[\tag{%o49} \frac{a}{2 {{r}^{3}}-2 a\, {{r}^{2}}}\] \[\tag{%o50} \frac{a}{2 {{r}^{3}}-2 a\, {{r}^{2}}}\] \[\tag{%o51} \frac{a}{2 r}\] \[\tag{%o52} -\frac{a}{2 r}\] \[\tag{%o53} -\frac{a}{r}\] \[\tag{%o54} \frac{a\, {{\sin{(theta)}}^{2}}}{2 r}\] \[\tag{%o55} -\frac{a\, {{\sin{(theta)}}^{2}}}{2 r}\] \[\tag{%o56} \frac{a\, {{\sin{(theta)}}^{2}}}{r}\]

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

--> riem[1,1,2,2]$ratsimp(%);
\[\tag{%o59} -\frac{a r-{{a}^{2}}}{{{r}^{4}}}\]

 5 Tensor de Ricci

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

--> ricci(true)$
\[\mbox{}\\\mbox{ARRSTORE: use\_ fast\_ arrays=false; allocate a new property hash table for \$ RIC}\] \[\tag{%t66} {{\mathit{ric}}_{1,1}}=-\frac{{r_s} \left( {{{r_s}}^{2}}-r\, {r_s}\right) }{{{r}^{3}}\, \left( 2 r\, {r_s}-2 {{r}^{2}}\right) }+\frac{{{{r_s}}^{2}}-r\, {r_s}}{2 {{r}^{4}}}+\frac{{r_s}}{2 {{r}^{3}}}\] \[\tag{%t67} {{\mathit{ric}}_{2,2}}=\frac{2 {r_s}}{r\, \left( 2 r\, {r_s}-2 {{r}^{2}}\right) }-\frac{2 {{{r_s}}^{2}}}{{{\left( 2 r\, {r_s}-2 {{r}^{2}}\right) }^{2}}}-\frac{{r_s} \left( 2 {r_s}-4 r\right) }{{{\left( 2 r\, {r_s}-2 {{r}^{2}}\right) }^{2}}}\]
--> ric[1,1]; ratsimp(%);
\[\tag{%o68} -\frac{{r_s} \left( {{{r_s}}^{2}}-r\, {r_s}\right) }{{{r}^{3}}\, \left( 2 r\, {r_s}-2 {{r}^{2}}\right) }+\frac{{{{r_s}}^{2}}-r\, {r_s}}{2 {{r}^{4}}}+\frac{{r_s}}{2 {{r}^{3}}}\] \[\tag{%o69} 0\]
--> ric[2,2]; ratsimp(%);
\[\tag{%o70} \frac{2 {r_s}}{r\, \left( 2 r\, {r_s}-2 {{r}^{2}}\right) }-\frac{2 {{{r_s}}^{2}}}{{{\left( 2 r\, {r_s}-2 {{r}^{2}}\right) }^{2}}}-\frac{{r_s} \left( 2 {r_s}-4 r\right) }{{{\left( 2 r\, {r_s}-2 {{r}^{2}}\right) }^{2}}}\] \[\tag{%o71} 0\]

  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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