8.2. Relation with our basis

The basis is given in terms of Weyl and Ricci. We need to convert the Weyl tensors into Riemann tensors, what takes quite a long time. The results form appendix B of the first paper.

Show timings over 1 second:

In[257]:=

<<xAct`ShowTime1`

The function do1 performs the translation from Weyl to Riemann (do1 has a very strange form in order to make it faster). The function do2 changes to invariants and simplify them:

In[258]:=

do1[expr_] := ToCanonical @ ContractMetric[ToCanonical @ ContractMetric @ WeylToRiemann @ ToCanonical[expr]/.CurvatureRelations[CD]] ;

do2[expr_] := expr//RiemannToInv//InvSimplify//Simplify ;

In[260]:=

$InvSimplifyLevel = 6

Out[260]=

6

Pure Ricci:

In[261]:=

I1//do2

Out[261]=

I_0,1

In[262]:=

I2//do2

Out[262]=

I_00,1

In[263]:=

I3//do2

Out[263]=

-I_000,1

In[264]:=

I4//do2

Out[264]=

I_0000,1

Pure Weyl:

In[265]:=

J1//do1//do2

Out[265]=

I_0,1^2/3 - 2 I_00,1 + I_00,2

In[266]:=

J2//do1//do1

1.13607 Second

Out[266]=

In[267]:=

%//do2

Out[267]=

-(17 I_0,1^3)/18 + I_0,1 (7 I_00,1 - I_00,2/2) + 6 I_000,1 - 6 I_000,2 + I_000,5

In[268]:=

J3//do1//do1

20.7773 Second

Out[268]=

In[269]:=

%//do2

Out[269]=

-D_00,2^2/16

In[270]:=

J4//do1//do1

182.151 Second

Out[270]=

In[271]:=

%//do2

Out[271]=

5/96 D_00,2 (-4 D_000,2 - 2 D_000,13 + D_00,2 I_0,1)

Mixed invariants:

In[272]:=

K1//do1//do1

Out[272]=

R_a ^( c) R_  ^ab R_bc^   - 7/6 R_ab^   R_  ^ab R_^ + R_^^3/6 + R_  ^ab R_  ^cd R_acbd^    

In[273]:=

%//do2

Out[273]=

1/6 (I_0,1^3 - 7 I_0,1 I_00,1 - 6 I_000,1 + 6 I_000,2)

In[274]:=

K2//do1//do1

Out[274]=

In[275]:=

%//do2

Out[275]=

-(5 I_0,1^4)/18 + 41/18 I_0,1^2 I_00,1 - I_00,1^2 + I_0,1 (3 I_000,1 - (4 I_000,2)/3) + 2 I_0000,1 + I_0000,5

In[276]:=

K3//do1//do1

8.20851 Second

Out[276]=

In[277]:=

%//do2

Out[277]=

(D_00,2 D_000,2)/16

In[278]:=

K4//do1//do1

Out[278]=

In[279]:=

%//do2

Out[279]=

1/6 (I_0,1^5 - 8 I_0,1^3 I_00,1 + I_0,1^2 (-11 I_000,1 + 3 I_000,2) + I_00,1 (I_000,1 + 3 I_000,2) + I_0,1 (4 I_00,1^2 - 7 I_0000,1) - 12 I_00000,2)

In[280]:=

K5//do1//do1

Out[280]=

In[281]:=

%//do2

Out[281]=

In[282]:=

K6//do1//do1

12.3368 Second

Out[282]=

In[283]:=

%//do2

1.16007 Second

Out[283]=

1/32 D_00,2 (-4 D_00000,2 + D_000,2 (I_0,1^2 + I_00,1))

Created by Mathematica  (May 16, 2008) Valid XHTML 1.1!