Schawarzschild Equations

Lic. Renan Cabrera 
Academia Nacional de Ciencias de Bolivia

renanbo@hotmail.com
rencabla@ceibo.entelnet.bo
La Paz Bolivia -2000

Abstract
This work shows an application of the "tensor.m" package in the obtaining of 
the Schawarzschild differential equations that describe the space curvature 
produced by a point mass. The program is completely general, so that it could 
deal with any Metric Tensor proposed.

"Tensorial.m" is a package that can be downloaded from Mat Source 

http://www.mathsource.com/Content/Applications/Physics/0211-419

Please write me to get the last version of it.

Needs["tensorial`", "F:\\Renan\\Fiscom\\Book\\tensorial.m"]

\(x[p_]\  = \ Tensor[x, {Void}, {p}]; 


    GM\  = {\ {\ Exp[μ[x[2]]]\ , \ \ \ 0
          \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , \ 0
          \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , \ \ \ \ \ \ \ \ \ \ \ 0\ }, 
        
		\ \ \ \ \ \ \ \ \ {\ \ 0\ \ \ \ \ \ \ \ \ \ \ \ \ \ , \ \ 
          \(-Exp[λ[x[2]]]\)\ \ \ , \ 0
          \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , \ \ \ \ \ \ \ \ \ \ 0\ \ }, 

        \ \ 		\ \ \ \ \ \ \ {\ \ 0\ \ \ \ \ \ \ \ \ \ \ \ \ \ , \ \ \ 0
          \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , \(-\ x[2]^2\)\ , 
          \ \ \ \ \ \ \ \ \ \ \ 0\ \ }, 

        		\ \ \ \ \ \ \ \ \ {\ \ 0\ \ \ \ \ \ \ \ \ \ \ , \ \ \ \ 0
          \ \ \ \ \ \ \ , \ \ \ 0\ \ \ , \(-\ \((Sin[\ x[3]\ ] x[2])\)^2\)}\ }
        
\)SetSystem[x , g,  GM ]

Lets define de Riemann Christoffel Tensor

\(\(RC[{i_, j_, k_}, {a_}]\  = \ 

      IndexDerivative[\ \(Christoffel2[g, x]\)[{i, k}, {a}]\ , \ 
          Tensor[x, {Void}, {j}]\ ] - 

        IndexDerivative[\ \(Christoffel2[g, x]\)[{i, j}, {a}]\ , \ 
          Tensor[x, {Void}, {k}]\ ] + 
	
        ExpandDoubleIndex[

          \(Christoffel2[g, x]\)[{i, k}, {β}]\ 
              \(Christoffel2[g, x]\)[{β, j}, {a}]\  - 

            \(Christoffel2[g, x]\)[{i, j}, {β}]\ 
              \(Christoffel2[g, x]\)[{β, k}, {a}]

          	\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , β]\ 	\)\)

Now lets define the Ricci Tensor

Tensor[Ricci, {i_?IntegerQ, j_?IntegerQ}, {Void, Void}]  =  
      ExpandDoubleIndex[RC[{i, j, a}, {a}], {a}];

In a few seconds we have their differential equations

\(EvaluateIndexDerivative[
	Tensor[Ricci, {1, 1}, {Void, Void}]\ 
		]
         // Simplify\)

\(EvaluateIndexDerivative[
	Tensor[Ricci, {2, 2}, {Void, Void}]\ 
		]
         // Simplify\)

\(EvaluateIndexDerivative[
	Tensor[Ricci, {3, 3}, {Void, Void}]\ 
		]
         // Simplify\)

\(EvaluateIndexDerivative[
	Tensor[Ricci, {4, 4}, {Void, Void}]\ 
		]
         // Simplify\)

You can try your own Metric Tensor