4.3. InvToRiemann

There is a simple function which combines the action of the previous two. It also has a second argument:

In a single line:

In[87]:=

InvToRiemann[RInv[metric][{0, 0, 0, 0, 0, 0, 0}, 1000]]

Out[87]=

It is possible to give a list of invariants:

In[88]:=

RInvs[metric][5, {0, 0, 2}]

Out[88]=

{I_002,1, I_002,2, I_002,5, I_002,6, I_002,7, I_002,8, I_002,14, I_002,41, I_002,52, I_002,55}

In[89]:=

Map[InvToRiemann, %]

Out[89]=

In[90]:=

Transpose[{%%, %}]//TableForm

Out[90]//TableForm=

I_002,1 Scalar[R_ (  a )^(ab c) R_ (b d )^( d e) R_ (  fg ; c ; e)^fg      ]
I_002,2 Scalar[R_ (  a )^(ab c) R_ (b d )^( d e) R_ (c ef   ; g)^( f   ; g  )]
I_002,5 Scalar[R_ (  a )^(ab c) R_ (  d )^(de f) R_ (b cg ; e ; f)^( g      )]
I_002,6 Scalar[R_ (  a )^(ab c) R_ (  d )^(de f) R_ (b eg ; c ; f)^( g      )]
I_002,7 Scalar[R_ (  a )^(ab c) R_ (b c )^( d e) R_ (  fg ; d ; e)^fg      ]
I_002,8 Scalar[R_ (  a )^(ab c) R_ (b c )^( d e) R_ (d ef   ; g)^( f   ; g  )]
I_002,14 Scalar[R_ (  a )^(ab c) R_b   ^( def) R_ (c eg ; d ; f)^( g      )]
I_002,41 Scalar[R_ (  a )^(ab c) R_    ^defg R_ (bdcf ; e ; g)^        ]
I_002,52 Scalar[R_ab  ^(  ef) R_    ^abcd R_ (c eg ; d ; f)^( g      )]
I_002,55 Scalar[R_ (a c )^( e f) R_    ^abcd R_ (b dg ; e ; f)^( g      )]

With contracted Riemanns substituted by Riccis,

In[91]:=

RInvs[metric][5, {0, 0, 2}]

Out[91]=

{I_002,1, I_002,2, I_002,5, I_002,6, I_002,7, I_002,8, I_002,14, I_002,41, I_002,52, I_002,55}

In[92]:=

Map[InvToRiemann[#, True] &, %]

Out[92]=

In[93]:=

Transpose[{%%, %}]//TableForm

Out[93]//TableForm=

I_002,1 -Scalar[R_ ( a)^b  R_c ^( a) R_ (   ; b)^(; c  )]
I_002,2 -Scalar[R_ca^   R_d ^( a) R_ (     ; b)^(dc ; b  )]
I_002,5 Scalar[R_ ( d)^a  R_bc^   R_ (     ; a)^(cb ; d  )]
I_002,6 Scalar[R_ ( d)^a  R_bc^   R_ (     ; a)^(cd ; b  )]
I_002,7 Scalar[R_  ^ba R_ (a b )^( c d) R_ (; c ; d)^    ]
I_002,8 Scalar[R_  ^ba R_ (a b )^( c d) R_ (cd   ; e)^(   ; e  )]
I_002,14 Scalar[R_e ^( a) R_a   ^( bcd) R_ ( c ; b ; d)^e     ]
I_002,41 Scalar[R_ef^   R_    ^abcd R_ ( a c ; b ; d)^(f e     )]
I_002,52 Scalar[R_ab  ^(  ef) R_    ^abcd R_ (ce ; d ; f)^      ]
I_002,55 Scalar[R_ (a c )^( e f) R_    ^abcd R_ (bd ; e ; f)^      ]

Note that we have got a minus sign. This is because the invariants are sorted with respect to their Riemann-only expression:


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