Examples of Special Relativity

Renan Cabrera 
Academia Nacional de Ciencias de Bolivia

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

Abstract
This Mathematica Notebook shows  basic applications of the "tensor.m" package in special relativity. In the first part we define the Metric Tensor for the Minkovsky space and later we define the Jacobian Matrix for system's transformation.

"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\  = {\ {\ \ 1\ \ \ \ , \ \ \ 0\ \ \ \ \ , \ 0\ \ \ \ \ , \ \ \ \ \ \ 0
          \ }, 
		\ \ \ \ \ \ \ \ \ {\ \ 0\ \ \ \ , \ \ \(-1\)\ \ \ , \ 0
          \ \ \ \ \ \ , \ \ \ \ \ 0\ \ }, 

        \ \ 		\ \ \ \ \ \ \ {\ \ 0\ \ \ \ , \ \ \ 0\ \ \ \ \ , \ 
          \(-1\)\ \ \ \ , \ \ \ \ 0\ \ }, 

        		\ \ \ \ \ \ \ \ \ {\ \ 0\ \ \ \ , \ \ \ \ 0\ \ \ \ , \ \ \ 0
          \ \ \ \ \ , \ \ \(-1\)\ }\ }; 

SetSystem[x\ , g, \ GM\ ]\)

Arc element in Minkovsky space

\(dS\  = \ \ Tensor[g, {i, j}, {Void, Void}]\ x[i] x[j]
	\)ExpandDoubleIndex[dS, {i, j}]

Velocity

Tensor[V, {Void}, {i_?IntegerQ}]  =   AbsoluteDerivative[x[i], s]

Acceleration

Tensor[A, {Void}, {i_?IntegerQ}] =  
      IntrinsecDerivative[ Tensor[V, {Void}, {i}],  s, {x, g} ]

Momentum Energy

\(Tensor[\ P\ , {Void}, {1}]\  = \ En/c; 

    Tensor[\ P\ , {Void}, {2}]\  = \ Px; 

    Tensor[\ P\ , {Void}, {3}]\  = \ Py; 

    Tensor[\ P\ , {Void}, {4}]\  = \ Pz; \)

Tensor[P, {Void}, {i}]

Transformations

Lets define a system transformation with Jacobian (Subscript[L, "  "j]^(i"  "))

\(\(
\(γRule\ \  = \ \ \ \ \ 
      γ -> \ 1/Sqrt[1 - β^2\ ]; \)
\)\)

\(\(MT
       = {
{\ \ \ γ\ \ \ \ , \ \(-γ\)\ \ β\ \ \ \ \ , 
          \ \ 0\ \ , \ \ \ 0\ \ \ \ \ \ }, 
        
{\(-γ\)\ β\ , \ \ \ \ \ γ\ \ \ \ \ \ \ \ , \ \ 0
          \ \ , \ \ \ 0\ \ \ \ \ \ }, 
        
{\ \ \ \ \ 0\ , \ \ \ \ 0\ \ \ \ \ \ \ \ \ , \ \ 1\ \ \ , \ \ 0
          \ \ \ \ }, 
        
{\ \ \ \ 0\ \ , \ \ \ 0\ \ \ \ \ \ \ \ \ \ , \ \ 0\ \ \ \ , 1
          \ \ \ \ \ \ }}; \)\)

Assigning the values to (Subscript[L, "  "j]^(i"  "))

\(Do[
	Tensor[L, {Void, j}, {i, Void}]\  = \ MT[\([i, j]\)]

      	, {i, 1, 4}, {j, 1, 4}]\)

\(Tensor[L, {Void, j}, {i, Void}]
\)\(ExpandIndex[Tensor[L, {Void, i}, {j, Void}], {i, j}]
\)

Defining   (Subscript[y, "  "]^i)

\(Tensor[y, {Void}, {i}]
\)\(\(Tensor[y, {Void}, {i_?NumberQ}]\  = \ 

      ExpandDoubleIndex[
        Tensor[L, {Void, j}, {i, Void}]\ Tensor[x, {Void}, {j}]\ , {j}]; 
    \)\)

As a particular case:

\(ExpandIndex[
	Tensor[y, {Void}, {i}]
, {i}]\)

The arc must be conserved    (ds^2)  =  (Subscript[g, ij]^("  ""  ")) (Subscript[y, "  "]^i) (Subscript[y, "  "]^j) =      (Subscript[g, ij]^("  ""  ")) (Subscript[x, "  "]^i) (Subscript[x, "  "]^j)

\(dS2\  = \ 
      Tensor[g, {i, j}, {Void, Void}]\ Tensor[y, {Void}, {i}]\ 
        Tensor[y, {Void}, {j}]
\)\(dum\  = 
      ExpandDoubleIndex[dS2, {i, j}]\  /. \ γRule\  // Simplify
\)

Electromagnetic Tensor

Lets Define the Electromagnetic Tensor (Subscript[F, ij]^("  ""  "))

\(\(FTV
       = {
{\ \ \ 0\ \ \ \ , \ Ex\ \ \ , \ Ey\ \ , \ \ \ Ez\ \ \ \ \ \ }, 
        
{\ \(-Ex\)\ \ \ , \ \ \ \ 0\ \ \ \ \ \ \ , \ \ 0\ \ , \ \ \ 0
          \ \ \ \ \ \ }, 
        
{\ \(-Ey\)\ \ \ , \ \ \ \ 0\ \ \ \ \ \ \ , \ \ 0\ \ \ , \ \ 0
          \ \ \ \ }, 
        
{\ \(-Ez\)\ \ \ , \ \ \ 0\ \ \ \ \ \ \ , \ \ 0\ \ \ \ , 0
          \ \ \ \ \ \ }}; 


    Do[
Tensor[F, {Void, Void}, {i, j}]\  = \ FTV[\([i, j]\)]

      	, {i, 1, 4}, {j, 1, 4}]
\)\)

\(ExpandIndex[
	\ Tensor[F, {Void, Void}, {μ, ν}]

      \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , {μ, ν}]\)

Calculating the transformation of the Electromagnetic Tensor (Subscript[F, ij]^("  ""  "))
                                   (Subscript[F, "  ""  "]^(μν))  =   (Subscript[F, "  ""  "]^(ij)) (Subscript[L, "  "i]^(μ"  ")) (Subscript[L, "  "j]^(ν"  "))

\(\(	Tensor[F\&_, {Void, Void}, {μ_?IntegerQ, ν_?IntegerQ}] := 
      ExpandDoubleIndex[
		\ \ \ 
        Tensor[F, {Void, Void}, {i, j}]*\ Tensor[L, {Void, i}, {μ, Void}]*
          Tensor[L, {Void, j}, {ν, Void}], {i, j}]\)\)

\(\(ExpandIndex[
	
        Tensor[F\&_, {Void, Void}, {μ, ν}], {μ, ν}] // 
      Simplify
\)\)