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Copy pathElectricMagneticEquations_modified.txt
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ElectricMagneticEquations_modified.txt
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%%%%%%%% The wave equation for the electric part %%%%%%%%%%
+ ginv[d,c]*Electric[a,b]_[c,d]
-Gamma[d,c,e]*ginv[e,c]*Electric[a,b]_[d]
-Gamma[e,c,b]*ginv[d,c]*Electric[a,e]_[d]
-Gamma[e,c,a]*ginv[d,c]*Electric[e,b]_[d]
-Electric[e,b]*ginv[d,c]*Gamma[e,d,a]_[c]
-Electric[a,e]*ginv[d,c]*Gamma[e,d,b]_[c]
-Gamma[e,d,b]*ginv[d,c]*Electric[a,e]_[c]
-Gamma[e,d,a]*ginv[d,c]*Electric[e,b]_[c] !run10
+Gamma[e,d,f]*Gamma[f,c,b]*Electric[a,e]*ginv[d,c] !run4
+Gamma[e,d,f]*Gamma[f,c,a]*Electric[e,b]*ginv[d,c]
+Gamma[e,c,a]*Gamma[f,d,b]*Electric[e,f]*ginv[d,c]
+Gamma[e,d,a]*Gamma[f,c,b]*Electric[e,f]*ginv[d,c]
+Gamma[d,e,f]*Gamma[c,d,b]*Electric[a,c]*ginv[f,e]
+Gamma[d,e,f]*Gamma[c,d,a]*Electric[c,b]*ginv[f,e] !run5
-(Normal[d]*Normal[e]*Weyl[a,f,b,c]*RicciScalar*ginv[f,d]*ginv[c,e])/2 !run9
- 2*Normal[c]*ginv[d,c]*ginv[e,f]*ginv[m,n]*Weyl[a,d,b,e]_[m]*Normal[f]_[n]
- 2*Normal[c]*ginv[d,c]*ginv[e,f]*ginv[m,n]*Weyl[a,e,b,d]_[m]*Normal[f]_[n]
- Normal[c]*Weyl[a,d,b,e]*ginv[d,c]*ginv[e,f]*ginv[m,n]*Normal[f]_[m,n]
- Normal[c]*Weyl[a,d,b,e]*ginv[e,c]*ginv[d,f]*ginv[m,n]*Normal[f]_[m,n]
- 2*Weyl[a,c,b,e]*ginv[c,d]*ginv[f,m]*ginv[e,n]*Normal[n]_[f]*Normal[d]_[m] !run11
+ Normal[c]*Normal[d]*Weyl[a,e,b,f]*ginv[e,d]*ginv[f,m]*ginv[n,p]*Gamma[c,p,m]_[n]
+ Normal[c]*Normal[d]*Weyl[a,e,b,f]*ginv[f,d]*ginv[e,n]*ginv[m,p]*Gamma[c,p,n]_[m]
+ 2*Gamma[c,d,e]*Normal[c]*Normal[f]*ginv[m,f]*ginv[n,e]*ginv[p,d]*Weyl[a,m,b,n]_[p]
+ 2*Gamma[c,d,e]*Normal[c]*Normal[f]*ginv[m,f]*ginv[n,e]*ginv[p,d]*Weyl[a,n,b,m]_[p]
+ 2*Gamma[c,d,e]*Normal[c]*Weyl[a,f,b,m]*ginv[f,n]*ginv[d,p]*ginv[m,e]*Normal[n]_[p] !run12
+ Gamma[b,c,d]*Normal[e]*Weyl[a,f,b,m]*ginv[f,e]*ginv[m,d]*ginv[n,c]*Normal[b]_[n] !TOO MANY Bs!
+ Gamma[c,d,e]*Normal[f]*Weyl[a,n,b,p]*ginv[p,f]*ginv[n,e]*ginv[m,d]*Normal[c]_[m]
+ Gamma[c,d,e]*Normal[f]*Weyl[a,m,b,n]*ginv[m,f]*ginv[n,e]*ginv[d,p]*Normal[c]_[p]
+ Gamma[c,d,e]*Normal[f]*Weyl[a,m,b,n]*ginv[m,f]*ginv[n,p]*ginv[d,e]*Normal[p]_[c]
+ Gamma[c,m,d]*Normal[e]*Weyl[a,f,b,n]*ginv[n,e]*ginv[f,d]*ginv[m,p]*Normal[c]_[p]
+ Gamma[c,m,d]*Normal[e]*Weyl[a,f,b,n]*ginv[n,e]*ginv[f,p]*ginv[m,d]*Normal[p]_[c] !run13
+ 2*Gamma[c,d,e]*Normal[c]*Weyl[a,f,b,m]*ginv[f,e]*ginv[n,d]*ginv[m,p]*Normal[p]_[n]
+ 2*Gamma[c,d,e]*Normal[f]*Weyl[a,c,b,m]*ginv[e,f]*ginv[m,n]*ginv[d,p]*Normal[n]_[p]
+ 2*Gamma[c,d,e]*Normal[f]*Weyl[a,m,b,c]*ginv[m,f]*ginv[e,n]*ginv[d,p]*Normal[n]_[p]
+ 2*Gamma[c,d,b]*Normal[e]*Weyl[a,f,c,m]*ginv[f,e]*ginv[m,n]*ginv[d,p]*Normal[n]_[p]
+ 2*Gamma[c,d,a]*Normal[e]*Weyl[c,f,b,m]*ginv[f,e]*ginv[m,n]*ginv[d,p]*Normal[n]_[p]
+ 2*Gamma[c,d,e]*Normal[f]*Weyl[a,m,b,c]*ginv[e,f]*ginv[m,n]*ginv[d,p]*Normal[n]_[p]
+ 2*Gamma[c,d,e]*Normal[f]*Weyl[a,c,b,m]*ginv[m,f]*ginv[e,n]*ginv[d,p]*Normal[n]_[p]
+ 2*Gamma[c,d,b]*Normal[e]*Weyl[a,f,c,m]*ginv[m,e]*ginv[n,p]*ginv[d,f]*Normal[p]_[n]
+ 2*Gamma[c,d,a]*Normal[e]*Weyl[c,f,b,m]*ginv[m,e]*ginv[f,n]*ginv[d,p]*Normal[n]_[p] !run14
-Gamma[c,d,e]*Gamma[e,f,m]*Normal[c]*Normal[n]*Weyl[a,p,b,q]*ginv[p,n]*ginv[q,m]*ginv[f,d]
-Gamma[b,c,d]*Gamma[c,e,f]*Normal[b]*Normal[m]*Weyl[a,f,b,p]*ginv[n,m]*ginv[p,d]*ginv[e,n]
-Gamma[c,d,e]*Gamma[e,m,f]*Normal[c]*Normal[n]*Weyl[a,p,b,q]*ginv[q,n]*ginv[p,f]*ginv[m,d]
-Gamma[c,d,e]*Gamma[d,m,f]*Normal[c]*Normal[n]*Weyl[a,p,b,q]*ginv[q,n]*ginv[p,e]*ginv[m,f]
-2*Gamma[c,d,e]*Gamma[f,m,n]*Normal[f]*Normal[p]*Weyl[a,c,b,q]*ginv[e,p]*ginv[q,n]*ginv[d,m]
-2*Gamma[c,d,e]*Gamma[f,m,n]*Normal[f]*Normal[p]*Weyl[a,q,b,c]*ginv[q,p]*ginv[e,n]*ginv[d,m]
-2*Gamma[c,d,e]*Gamma[f,m,b]*Normal[c]*Normal[n]*Weyl[a,p,f,q]*ginv[p,n]*ginv[q,e]*ginv[m,d]
-2*Gamma[c,d,e]*Gamma[f,m,a]*Normal[c]*Normal[n]*Weyl[f,p,b,q]*ginv[p,n]*ginv[q,e]*ginv[m,d]
-2*Gamma[c,d,e]*Gamma[f,m,n]*Normal[f]*Normal[p]*Weyl[a,q,b,c]*ginv[e,p]*ginv[q,n]*ginv[d,m]
-2*Gamma[c,d,e]*Gamma[f,m,n]*Normal[f]*Normal[p]*Weyl[a,c,b,q]*ginv[q,p]*ginv[e,n]*ginv[d,m]
-2*Gamma[c,d,e]*Gamma[f,m,b]*Normal[c]*Normal[n]*Weyl[a,p,f,q]*ginv[q,n]*ginv[p,e]*ginv[m,d]
-2*Gamma[c,d,e]*Gamma[f,m,a]*Normal[c]*Normal[n]*Weyl[f,p,b,q]*ginv[q,n]*ginv[p,e]*ginv[m,d]
-2*Gamma[c,d,e]*Gamma[f,m,n]*Normal[c]*Normal[f]*Weyl[a,p,b,q]*ginv[p,n]*ginv[d,m]*ginv[q,e] !run15
+2*ConformalFactor*Normal[c]*Normal[d]*Weyl[a,e,b,f]*Weyl[m,n,p,q]*ginv[m,c]*ginv[p,d]*ginv[e,n]*ginv[f,q]
+2*ConformalFactor*Normal[c]*Normal[d]*Weyl[a,e,f,m]*Weyl[b,n,p,q]*ginv[f,c]*ginv[n,d]*ginv[e,p]*ginv[m,q]
- 2*ConformalFactor*Normal[c]*Normal[d]*Weyl[a,f,m,n]*Weyl[b,e,p,q]*ginv[m,f]*ginv[p,d]*ginv[c,q]*ginv[n,e] !run16
== 0
%%%%%%%%%%% The wave equation for the magnetic part %%%%%%%%%%%%%%%%%%%
+ ginv[c,d]*Magnetic[a,b]_[d,c]
- Gamma[c,e,d]*ginv[d,e]*Magnetic[a,b]_[c]
- Gamma[e,d,b]*ginv[c,d]*Magnetic[a,e]_[c]
- Gamma[e,d,a]*ginv[c,d]*Magnetic[e,b]_[c]
- Magnetic[e,b]*ginv[c,d]*Gamma[e,c,a]_[d]
- Magnetic[a,e]*ginv[c,d]*Gamma[e,c,b]_[d]
- Gamma[e,c,b]*ginv[c,d]*Magnetic[a,e]_[d]
- Gamma[e,c,a]*ginv[c,d]*Magnetic[e,b]_[d]
+ Gamma[e,c,f]*Gamma[f,d,b]*Magnetic[a,e]*ginv[c,d]
+ Gamma[e,c,f]*Gamma[f,d,a]*Magnetic[e,b]*ginv[c,d]
+ Gamma[e,d,a]*Gamma[f,c,b]*Magnetic[e,f]*ginv[c,d]
+ Gamma[e,c,a]*Gamma[f,d,b]*Magnetic[e,f]*ginv[c,d]
+ Gamma[c,e,d]*Gamma[f,c,b]*Magnetic[a,f]*ginv[d,e]
+ Gamma[c,e,d]*Gamma[f,c,a]*Magnetic[f,b]*ginv[d,e]
- (Normal[c]*Normal[d]*WeylDual[a,e,b,f]*RicciScalar*ginv[c,e]*ginv[d,f])/2
- 2*Normal[c]*ginv[c,f]*ginv[d,p]*ginv[e,q]*Normal[d]_[e]*WeylDual[a,f,b,p]_[q]
- 2*Normal[c]*ginv[c,f]*ginv[d,p]*ginv[e,q]*Normal[d]_[e]*WeylDual[a,p,b,f]_[q]
- Normal[c]*WeylDual[a,p,b,q]*ginv[c,p]*ginv[d,q]*ginv[e,f]*Normal[d]_[e,f]
- Normal[c]*WeylDual[a,q,b,p]*ginv[c,p]*ginv[d,q]*ginv[e,f]*Normal[d]_[e,f]
- 2*WeylDual[a,m,b,n]*ginv[c,m]*ginv[d,f]*ginv[e,n]*Normal[c]_[d]*Normal[e]_[f]
+ Normal[c]*Normal[d]*WeylDual[a,q,b,m]*ginv[c,q]*ginv[e,m]*ginv[f,p]*Gamma[d,f,e]_[p]
+ Normal[c]*Normal[d]*WeylDual[a,m,b,q]*ginv[c,q]*ginv[e,m]*ginv[f,p]*Gamma[d,f,e]_[p]
+ 2*Gamma[d,f,e]*Normal[c]*Normal[d]*ginv[c,p]*ginv[e,q]*ginv[f,m]*WeylDual[a,p,b,q]_[m]
+ 2*Gamma[d,f,e]*Normal[c]*Normal[d]*ginv[c,p]*ginv[e,q]*ginv[f,m]*WeylDual[a,q,b,p]_[m]
+ 2*Gamma[c,e,d]*Normal[c]*WeylDual[a,q,b,m]*ginv[d,q]*ginv[e,p]*ginv[f,m]*Normal[f]_[p]
+ Gamma[e,p,f]*Normal[c]*WeylDual[a,n,b,m]*ginv[c,n]*ginv[d,m]*ginv[f,p]*Normal[d]_[e]
+ Gamma[e,p,f]*Normal[c]*WeylDual[a,m,b,n]*ginv[c,n]*ginv[d,m]*ginv[f,p]*Normal[d]_[e]
+ Gamma[d,p,e]*Normal[c]*WeylDual[a,n,b,m]*ginv[c,n]*ginv[e,m]*ginv[f,p]*Normal[d]_[f]
+ Gamma[d,p,e]*Normal[c]*WeylDual[a,m,b,n]*ginv[c,n]*ginv[e,m]*ginv[f,p]*Normal[d]_[f]
+ Gamma[d,f,e]*Normal[c]*WeylDual[a,n,b,m]*ginv[c,n]*ginv[e,m]*ginv[f,p]*Normal[d]_[p]
+ Gamma[d,f,e]*Normal[c]*WeylDual[a,m,b,n]*ginv[c,n]*ginv[e,m]*ginv[f,p]*Normal[d]_[p]
+ 2*Gamma[e,p,f]*Normal[e]*WeylDual[a,q,b,m]*ginv[c,q]*ginv[d,p]*ginv[f,m]*Normal[c]_[d]
+ 2*Gamma[m,q,b]*Normal[c]*WeylDual[a,f,m,p]*ginv[c,f]*ginv[d,p]*ginv[e,q]*Normal[d]_[e]
+ 2*Gamma[m,q,b]*Normal[c]*WeylDual[a,p,m,f]*ginv[c,f]*ginv[d,p]*ginv[e,q]*Normal[d]_[e]
+ 2*Gamma[m,q,a]*Normal[c]*WeylDual[m,f,b,p]*ginv[c,f]*ginv[d,p]*ginv[e,q]*Normal[d]_[e]
+ 2*Gamma[m,q,a]*Normal[c]*WeylDual[m,p,b,f]*ginv[c,f]*ginv[d,p]*ginv[e,q]*Normal[d]_[e]
+ 2*Gamma[p,m,q]*Normal[c]*WeylDual[a,f,b,p]*ginv[c,f]*ginv[d,q]*ginv[e,m]*Normal[d]_[e]
+ 2*Gamma[p,m,q]*Normal[c]*WeylDual[a,p,b,f]*ginv[c,f]*ginv[d,q]*ginv[e,m]*Normal[d]_[e]
+ 2*Gamma[f,m,p]*Normal[c]*WeylDual[a,f,b,q]*ginv[c,p]*ginv[d,q]*ginv[e,m]*Normal[d]_[e]
+ 2*Gamma[f,m,p]*Normal[c]*WeylDual[a,q,b,f]*ginv[c,p]*ginv[d,q]*ginv[e,m]*Normal[d]_[e]
- Gamma[d,f,n]*Gamma[n,p,e]*Normal[c]*Normal[d]*WeylDual[a,q,b,m]*ginv[c,q]*ginv[e,m]*ginv[f,p]
- Gamma[d,f,n]*Gamma[n,p,e]*Normal[c]*Normal[d]*WeylDual[a,m,b,q]*ginv[c,q]*ginv[e,m]*ginv[f,p]
- Gamma[d,f,e]*Gamma[f,q,p]*Normal[c]*Normal[d]*WeylDual[a,m,b,n]*ginv[c,m]*ginv[e,n]*ginv[p,q]
- Gamma[d,f,e]*Gamma[f,q,p]*Normal[c]*Normal[d]*WeylDual[a,n,b,m]*ginv[c,m]*ginv[e,n]*ginv[p,q]
- 2*Gamma[d,f,e]*Gamma[n,m,b]*Normal[c]*Normal[d]*WeylDual[a,p,n,q]*ginv[c,p]*ginv[e,q]*ginv[f,m]
- 2*Gamma[d,f,e]*Gamma[n,m,b]*Normal[c]*Normal[d]*WeylDual[a,q,n,p]*ginv[c,p]*ginv[e,q]*ginv[f,m]
- 2*Gamma[d,f,e]*Gamma[n,m,a]*Normal[c]*Normal[d]*WeylDual[n,p,b,q]*ginv[c,p]*ginv[e,q]*ginv[f,m]
- 2*Gamma[d,f,e]*Gamma[n,m,a]*Normal[c]*Normal[d]*WeylDual[n,q,b,p]*ginv[c,p]*ginv[e,q]*ginv[f,m]
- 2*Gamma[d,f,e]*Gamma[q,n,m]*Normal[c]*Normal[d]*WeylDual[a,p,b,q]*ginv[c,p]*ginv[e,m]*ginv[f,n]
- 2*Gamma[d,f,e]*Gamma[q,n,m]*Normal[c]*Normal[d]*WeylDual[a,q,b,p]*ginv[c,p]*ginv[e,m]*ginv[f,n]
- 2*Gamma[d,f,e]*Gamma[p,n,q]*Normal[c]*Normal[d]*WeylDual[a,p,b,m]*ginv[c,q]*ginv[e,m]*ginv[f,n]
- 2*Gamma[d,f,e]*Gamma[p,n,q]*Normal[c]*Normal[d]*WeylDual[a,m,b,p]*ginv[c,q]*ginv[e,m]*ginv[f,n]
- 2*Gamma[c,e,d]*Gamma[f,q,p]*Normal[c]*Normal[f]*WeylDual[a,m,b,n]*ginv[d,m]*ginv[e,q]*ginv[p,n]
+ 2*ConformalFactor*Normal[c]*Normal[d]*Weyl[b,e,p,m]*WeylDual[a,f,q,n]*ginv[c,e]*ginv[d,f]*ginv[p,q]*ginv[m,n]
+ 2*ConformalFactor*Normal[c]*Normal[d]*Weyl[b,e,p,m]*WeylDual[a,q,f,n]*ginv[c,e]*ginv[d,f]*ginv[p,q]*ginv[m,n]
+ ConformalFactor*Normal[c]*Normal[b]*Weyl[d,e,p,m]*WeylDual[a,f,q,n]*ginv[c,d]*ginv[e,f]*ginv[p,q]*ginv[m,n]
+ ConformalFactor*Weyl[b,c,e,p]*WeylDual[a,d,f,q]*ginv[c,d]*ginv[e,f]*ginv[p,q]
== 0
%%%%%%%%%%% Expression for the Weyl tensor in terms of the electric and magnetic parts %%%%%%%%
Weyl[a,b,c,d] ==
2*Electric[b,d]*Normal[a]*Normal[c]
-2*Electric[a,d]*Normal[b]*Normal[c]
-2*Electric[b,c]*Normal[a]*Normal[d]
+2*Electric[a,c]*Normal[b]*Normal[d]
+Electric[b,d]*g[a, c]
-Electric[b,c]*g[a,d]
-Electric[a,d]*g[b,c]
+Electric[a,c]*g[b,d]
-epsilon[c,d,e,f]*Magnetic[b,m]*Normal[a]*Normal[n]*ginv[e,n]*ginv[f,m]
+epsilon[c,d,e,f]*Magnetic[a,m]*Normal[b]*Normal[n]*ginv[e,n]*ginv[f,m]
-epsilon[a,b,e,f]*Magnetic[d,m]*Normal[c]*Normal[n]*ginv[e,n]*ginv[f,m]
+epsilon[a,b,e,f]*Magnetic[c,m]*Normal[d]*Normal[n]*ginv[e,n]*ginv[f,m]
WeylDual[a,b,c,d] ==
2*Magnetic[b,d]*Normal[a]*Normal[c]
-2*Magnetic[a,d]*Normal[b]*Normal[c]
-2*Magnetic[b,c]*Normal[a]*Normal[d]
+2*Magnetic[a,c]*Normal[b]*Normal[d]
+Magnetic[b,d]*g[a,c]
-Magnetic[b,c]*g[a,d]
-Magnetic[a,d]*g[b,c]
+Magnetic[a,c]*g[b,d]
+Electric[d,f]*epsilon[a,b,c,e]*ginv[f,e]
-Electric[c,f]*epsilon[a,b,d,e]*ginv[f,e]
+2*Electric[d,n]*epsilon[a,b,e,m]*Normal[c]*Normal[f]*ginv[f,e]*ginv[n,m]
-2*Electric[c,n]*epsilon[a,b,e,m]*Normal[d]*Normal[f]*ginv[f,e]*ginv[n,m]
%%%%%%%%%%%%%%%% Replacement table
Independent components
Electric[1,1], Electric[1,2], Electric[1,3], Electric[2,2], Electric[2,3], Electric[3,3]
Dependent components
Electric[0,0]==0
Electric[0,1]==0
Electric[0,2]==0
Electric[0,3]==0
Electric[2,1]==Electric[1,2]
Electric[3,1]==Electric[1,3]
Electric[3,2]==Electric[2,3]
%%%%%%%%%%%%%%% Exact initial data for the electric part
Electric[1,1]==0
Electric[1,2]==0
Electric[1,3]== -4y
Electric[2,2]==0
Electric[2,3]== 4x
Electric[3,3]== 0
Electric[0,0]==0
Electric[0,1]==0
Electric[0,2]==0
Electric[0,3]==0
Electric[2,1]==0
Electric[3,1]== -4y
Electric[3,2]== 4x
%%%%%% The wave equation for the Schouten tensor
Gamma[e,d,f]*Gamma[f,a,c]*Schouten[b,e]*ginv[a,d] - 4*Schouten[b,d]*Schouten[c, a]*ginv[a,d]
+ Gamma[e,d,f]*Gamma[f, a, b]*Schouten[e, c]*ginv[a,d] + Gamma[e,d,b]*Gamma[f, a, c]*Schouten[e,f]*ginv[a,d]
+ Gamma[e, a, b]*Gamma[f,d,c]*Schouten[e,f]*ginv[a,d]+Gamma[d,a,e]*Gamma[f,d,c]*Schouten[b,f]*ginv[a,e]
+ Gamma[d,a,e]*Gamma[f,d,b]*Schouten[f, c]*ginv[a,e] +2*ConformalFactor*Weyl[b,a, c,f]*Schouten[d,e]*ginv[a,d]*ginv[f,e]
+ Schouten[a,f]*Schouten[d,e]*ginv[a,d]*ginv[b,c]*ginv[f,e]-Schouten[e,c]*ginv[a,d]*Gamma[e,d,b]_[a]
- Schouten[b,e]*ginv[a,d]*Gamma[e, d, c]_[a] + (Gamma[a, b, c]*RicciScalar_[a])/6
- Gamma[e, d, c]*ginv[a,d]*Schouten[b,e]_[a] - Gamma[e,d,b]*ginv[a,d]*Schouten[e, c]_[a]
+ ginv[a,d]*Schouten[b, c]_[a,d] - RicciScalar]_[b,c]/6 - Gamma[d,a,e]*ginv[a,e]*Schouten[b, c]_[d]
- Gamma[e,a,c]*ginv[a,d]*Schouten[b,e]_[d] - Gamma[e,a,b]*ginv[a,d]*Schouten[e, c]_[d] == 0
%%%%%%% The wave equation for the Friedrich scalar
(FriedrichScalar*RicciScalar)/6 - ConformalFactor*Schouten[a,d]*Schouten[b,c]*ginv[a,b]*ginv[d,c]
- Gamma[a,c,b]*ginv[b,c]*FriedrichScalar_[a] +ginv[a,b]*RicciScalar_[a]*ConformalFactor_[b]/6
+ ginv[a,b]*FriedrichScalar_[a,b] == 0
%%%%%%% The wave equation for the conformal factor
-4*FriedrichScalar+ ConformalFactor*RicciScalar/6 - Gamma[a,c,b]*ginv[b,c]*ConformalFactor_[a]
+ ginv[a,b]*ConformalFactor_[a,b] == 0