8.1. Definitions

This section deals only with algebraic invariants. It is well known that there are 14 independent scalars of the Riemann tensor. Narlikar and Karmarkar gave 14 such scalars in 1947. We express them in terms of our basis of invariants.

Definitions taken from [Alex Harvey, Class. Quantum Grav. 7 (1990) 715-716]. First define six intermediate tensors:

In[234]:=

IndexSet[Atensor[h_, i_, j_, k_], WeylCD[h, i, -a, -b] WeylCD[a, b, j, k]] ;

IndexSet[Btensor[h_, i_, j_, k_], WeylCD[h, i, -a, -b] Atensor[a, b, j, k]] ;

IndexSet[Dtensor[h_, i_, j_, k_], Btensor[h, i, j, k] - 1/12J2 (metric[h, j] metric[i, k] - metric[h, k] metric[i, j]) - 1/4J1 WeylCD[h, i, j, k]] ;

IndexSet[Etensor[h_, i_, j_, k_], WeylCD[h, i, -a, -b] Dtensor[a, b, j, k]] ;

IndexSet[Ftensor[h_, i_, j_, k_], WeylCD[h, i, -a, -b] Etensor[a, b, j, k]] ;

IndexSet[Qtensor[a_, b_], RicciCD[-c, a] RicciCD[c, b]] ;

These are the 14 scalars:

In[240]:=

I1 := RicciScalarCD[] ;

IndexSet[I2, RicciCD[-a, b] RicciCD[-b, a]] ;

IndexSet[I3, RicciCD[-a, b] RicciCD[-b, c] RicciCD[-c, a]] ;

IndexSet[I4, RicciCD[-a, b] RicciCD[-b, c] RicciCD[-c, d] RicciCD[-d, a]] ;

IndexSet[J1, Atensor[-i, -j, i, j]] ;

IndexSet[J2, Btensor[-i, -j, i, j]] ;

IndexSet[J3, Etensor[-i, -j, i, j]] ;

IndexSet[J4, Ftensor[-i, -j, i, j]] ;

IndexSet[K1, WeylCD[-h, -i, -j, -k] RicciCD[h, j] RicciCD[i, k]] ;

IndexSet[K2, Atensor[-h, -i, -j, -k] RicciCD[h, j] RicciCD[i, k]] ;

IndexSet[K3, Dtensor[-h, -i, -j, -k] RicciCD[h, j] RicciCD[i, k]] ;

IndexSet[K4, WeylCD[-h, -i, -j, -k] Qtensor[h, j] Qtensor[i, k]] ;

IndexSet[K5, Atensor[-h, -i, -j, -k] Qtensor[h, j] Qtensor[i, k]] ;

IndexSet[K6, Dtensor[-h, -i, -j, -k] Qtensor[h, j] Qtensor[i, k]] ;

Pure Ricci:

In[254]:=

{I1, I2, I3, I4}

Out[254]=

{R_^, R_a ^( b) R_b ^( a), R_a ^( b) R_b ^( c) R_c ^( a), R_a ^( b) R_b ^( c) R_c ^( d) R_d ^( a)}

Pure Weyl:

In[255]:=

{J1, J2, J3, J4}//ContractMetric

Out[255]=

Mixed invariants:

In[256]:=

{K1, K2, K3, K4, K5, K6}//ContractMetric

Out[256]=

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