Tensor de curvatura y relatividad general

Principio:

Ecuación fundamental de la relatividad general: 

H_{jk}+(1/2)·( M_{jj}+M_{kk} ) = k^{2}·( T_{jk}+(1/2)·( T_{jj}+T_{kk} ) )

Contracción tensorial del tensor de curvatura:

H_{jk}·x^{i} = int-int-int[ R_{ijk}^{i} ] d[x^{i}]d[x^{j}]d[x^{k}]

H_{jk} = ( 1/x^{i} )·int-int-int[ R_{ijk}^{i} ] d[x^{i}]d[x^{j}]d[x^{k}]

Tensor métrico:

M_{jj} = int-int[ R_{jjj}^{j} ] d[x^{j}]d[x^{j}]

M_{kk} = int-int[ R_{kkk}^{k} ] d[x^{k}]d[x^{k}]

Tensor de impulsión-energía:

T_{jk} = d_{jk}^{2}[ E(x^{j},x^{k}) ]

Tensor de impulsión-energía Laplaciano:

T_{jj} = d_{jj}^{2}[ E(x^{j}) ]

T_{kk} = d_{kk}^{2}[ E(x^{k}) ]

Exemple:

Curvatura eléctrica:

H_{jk}·x^{i} = ...

... int-int-int[ R_{ijk}^{i} = (x^{i}/x^{i})·( ( x^{j} )^{0}·( x^{k} )^{0} ) ] ...

... d[x^{i}]d[x^{j}]d[x^{k}]

Curvatura gravitatoria:

H_{jk}·x^{i} = ...

... int-int-int[ R_{ijk}^{i} = (x^{i}/x^{i})·( (-1)·( x^{j} )^{0}·( x^{k} )^{0} ) ] ...

... d[x^{i}]d[x^{j}]d[x^{k}]

Contracción tensorial de curvatura:

H_{23}·x = ...

... int-int-int[ R_{123}^{1} = (x/x)·( y^{0}·z^{0} ) ] d[x]d[y]d[z] = x·( yz )

H_{31}·y = ...

... int-int-int[ R_{231}^{2} = (y/y)·( z^{0}·x^{0} ) ] d[x]d[y]d[z] = y·( zx )

H_{12}·z = ...

... int-int-int[ R_{312}^{3} = (z/z)·( x^{0}·y^{0} ) ] d[x]d[y]d[z] = z·( xy )

Campo eléctrico:

E(y,z) = (1/k)·(1/4)·( yz )^{2}

E(z,x) = (1/k)·(1/4)·( zx )^{2}

E(x,y) = (1/k)·(1/4)·( xy )^{2}

Campo gravitatorio:

E(y,z) = (-1)·(1/k)·(1/4)·( yz )^{2}

E(z,x) = (-1)·(1/k)·(1/4)·( zx )^{2}

E(x,y) = (-1)·(1/k)·(1/4)·( xy )^{2}

Métrica eléctrica:

M_{jj} = ...

... int-int-[ R_{jjj}^{j} = (x^{j}/x^{j})·( ( x^{j} )^{0}·( x^{j} )^{0} ) ] d[x^{j}]d[x^{j}]

Métrica gravitatoria:

M_{jj} = ...

... int-int-[ R_{jjj}^{j} = (x^{j}/x^{j})·( (-1)·( x^{j} )^{0}·( x^{j} )^{0} ) ] d[x^{j}]d[x^{j}]

Métrica:

M_{11} = ...

... int-int[ R_{111}^{1} = (x/x)·( x^{0}·x^{0} ) ] d[x]d[x] = (1/2)·x^{2}

M_{22} = ...

... int-int[ R_{222}^{2} = (y/y)·( y^{0}·y^{0} ) ] d[y]d[y] = (1/2)·y^{2}

M_{33} = ...

... int-int[ R_{333}^{3} = (z/z)·( z^{0}·z^{0} ) ] d[z]d[z] = (1/2)·z^{2}

Campo eléctrico:

E(x) = (1/k)·(1/24)·x^{4}

E(y) = (1/k)·(1/24)·y^{4}

E(z) = (1/k)·(1/24)·z^{4}

Campo gravitatorio:

E(x) = (-1)·(1/k)·(1/24)·x^{4}

E(y) = (-1)·(1/k)·(1/24)·y^{4}

E(z) = (-1)·(1/k)·(1/24)·z^{4}

Problema-Ley:

Contracción tensorial de curvatura:

H_{jk}·x^{i} = ...

... int-int-int[ R_{ijk}^{i} = (x^{i}/x^{i})·( x^{j}·x^{k} ) ] d[x^{i}]d[x^{j}]d[x^{k}]

H_{23}·x = ...

... int-int-int[ R_{123}^{1} = (x/x)·( y·z ) ] d[x]d[y]d[z] = x·(1/4)·( yz )^{2}

H_{31}·y = ...

... int-int-int[ R_{231}^{2} = (y/y)·( z·x ) ] d[x]d[y]d[z] = y·(1/4)·( zx )^{2}

H_{12}·z = ...

... int-int-int[ R_{312}^{3} = (z/z)·( x·y ) ] d[x]d[y]d[z] = z·(1/4)·( xy )^{2}

Campo eléctrico:

E(y,z) = (1/k)·(1/36)·( yz )^{3}

E(z,x) = (1/k)·(1/36)·( zx )^{3}

E(x,y) = (1/k)·(1/36)·( xy )^{3}

Métrica:

M_{jj} = ...

... int-int-[ R_{jjj}^{j} = (x^{j}/x^{j})·( x^{j}·x^{j} ) ] d[x^{j}]d[x^{j}]

M_{11} = ...

... int-int[ R_{111}^{1} = (x/x)·( x^{2} ) ] d[x]d[x] = (1/12)·x^{4}

M_{22} = ...

... int-int[ R_{222}^{2} = (y/y)·( y^{2} ) ] d[y]d[y] = (1/12)·y^{4}

M_{33} = ...

... int-int[ R_{333}^{3} = (z/z)·( z^{2} ) ] d[z]d[z] = (1/12)·z^{4}

Campo eléctrico:

E(x) = (1/k)·(1/360)·x^{6}

E(y) = (1/k)·(1/360)·y^{6}

E(z) = (1/k)·(1/360)·z^{6}

Problema-Ley:

Contracción tensorial de curvatura:

H_{jk}·x^{i} = ...

... int-int-int[ R_{ijk}^{i} = (x^{i}/x^{i})·( x^{j}+x^{k} ) ] d[x^{i}]d[x^{j}]d[x^{k}]

H_{23}·x = ...

... int-int-int[ R_{123}^{1} = (x/x)·( y+z ) ] d[x]d[y]d[z] = x·(1/2)·( y^{2}z+yz^{2} )

H_{31}·y = ...

... int-int-int[ R_{231}^{2} = (y/y)·( z+x ) ] d[x]d[y]d[z] = y·(1/2)·( z^{2}x+zx^{2} )

H_{12}·z = ...

... int-int-int[ R_{312}^{3} = (z/z)·( x+y ) ] d[x]d[y]d[z] = z·(1/2)·( x^{2}y+xy^{2} )

Campo eléctrico:

E(y,z) = (1/k)·(1/12)·( y^{3}z^{2}+y^{2}z^{3} )

E(z,x) = (1/k)·(1/12)·( z^{3}x^{2}+z^{2}x^{3} )

E(x,y) = (1/k)·(1/12)·( x^{3}y^{2}+x^{2}y^{3} )

Métrica:

M_{jj} = ...

... int-int-[ R_{jjj}^{j} = (x^{j}/x^{j})·( x^{j}+x^{j} ) ] d[x^{j}]d[x^{j}]

M_{11} = ...

... int-int[ R_{111}^{1} = (x/x)·( 2x ) ] d[x]d[x] = (1/3)·x^{3}

M_{22} = ...

... int-int[ R_{222}^{2} = (y/y)·( 2y ) ] d[y]d[y] = (1/3)·y^{3}

M_{33} = ...

... int-int[ R_{333}^{3} = (z/z)·( 2z ) ] d[z]d[z] = (1/3)·z^{3}

Campo eléctrico:

E(x) = (1/k)·(1/60)·x^{5}

E(y) = (1/k)·(1/60)·y^{5}

E(z) = (1/k)·(1/60)·z^{5}