User: eschnett
Date: 2011/12/19 03:26 PM
Modified:
/trunk/m/
WeylScal4.m
Log:
[PATCH 2/4] Separate calculation into a part for psi4
Separate calculation into a part for psi4, a part for the other psis
and a part for the invariants. Add parameter for enabling each part.
File Changes:
Directory: /trunk/m/
====================
File [modified]: WeylScal4.m
Delta lines: +182 -162
===================================================================
--- trunk/m/WeylScal4.m 2011-12-19 21:26:03 UTC (rev 97)
+++ trunk/m/WeylScal4.m 2011-12-19 21:26:47 UTC (rev 98)
@@ -105,209 +105,216 @@
realParameters = {{Name -> offset, Default -> 10^(-15)},xorig,yorig,zorig};
-PsisCalc[fdOrder_, PD_] :=
-{
- Name -> "psis_calc_" <> fdOrder,
- Where -> Interior,
- After -> "ADMBase_SetADMVars",
- ConditionalOnKeyword -> {"fd_order", fdOrder},
- Shorthands -> shorthands,
- Equations ->
- {
- detg -> gDet,
- invdetg -> 1 / detg,
- gInv[ua,ub] -> invdetg gDet MatrixInverse[g[ua,ub]],
- gamma[ua, lb, lc] -> 1/2 gInv[ua,ud] (PD[g[lb,ld], lc] +
- PD[g[lc,ld], lb] - PD[g[lb,lc],ld]),
+psi4Eqs[PD_] := Flatten@{
+ detg -> gDet,
+ invdetg -> 1 / detg,
+ gInv[ua,ub] -> invdetg gDet MatrixInverse[g[ua,ub]],
+ gamma[ua, lb, lc] -> 1/2 gInv[ua,ud] (PD[g[lb,ld], lc] + PD[g[lc,ld], lb] - PD[g[lb,lc],ld]),
-(****************************************************************************
- Offset the origin
- ****************************************************************************)
+ (****************************************************************************
+ Offset the origin
+ ****************************************************************************)
- xmoved -> x - xorig,
- ymoved -> y - yorig,
- zmoved -> z - zorig,
+ xmoved -> x - xorig,
+ ymoved -> y - yorig,
+ zmoved -> z - zorig,
-(****************************************************************************
- Compute the local tetrad
- ****************************************************************************)
+ (****************************************************************************
+ Compute the local tetrad
+ ****************************************************************************)
+ (* azmuthal *)
+ va1 -> -ymoved, va2 -> xmoved+offset, va3 -> 0,
- (* azmuthal *)
- va1 -> -ymoved, va2 -> xmoved+offset, va3 -> 0,
+ (* radial *)
+ vb1 -> xmoved+offset, vb2 -> ymoved, vb3 -> zmoved,
- (* radial *)
- vb1 -> xmoved+offset, vb2 -> ymoved, vb3 -> zmoved,
+ (* polar *)
+ vc[ua] -> Sqrt[detg] gInv[ua,ud] Eps[ld,lb,lc] va[ub] vb[uc],
- (* polar *)
- vc[ua] -> Sqrt[detg] gInv[ua,ud] Eps[ld,lb,lc] va[ub] vb[uc],
+ (* Orthonormalize using Gram-Schmidt*)
+ (* Orthonormalize in the order phi, r, theta *)
+ wa[ua] -> va[ua],
+ omega11 -> wa[ua] wa[ub] g[la,lb],
+ ea[ua] -> wa[ua] / Sqrt[omega11],
- (* Orthonormalize using Gram-Schmidt*)
+ omega12 -> ea[ua] vb[ub] g[la,lb],
+ wb[ua] -> vb[ua] - omega12 ea[ua],
+ omega22 -> wb[ua] wb[ub] g[la,lb],
+ eb[ua] -> wb[ua]/Sqrt[omega22],
- (* Orthonormalize in the order phi, r, theta *)
- wa[ua] -> va[ua],
- omega11 -> wa[ua] wa[ub] g[la,lb],
- ea[ua] -> wa[ua] / Sqrt[omega11],
+ omega13 -> ea[ua] vc[ub] g[la,lb],
+ omega23 -> eb[ua] vc[ub] g[la,lb],
+ wc[ua] -> vc[ua] - omega13 ea[ua] - omega23 eb[ua],
+ omega33 -> wc[ua] wc[ub] g[la,lb],
+ ec[ua] -> wc[ua]/Sqrt[omega33],
- omega12 -> ea[ua] vb[ub] g[la,lb],
- wb[ua] -> vb[ua] - omega12 ea[ua],
- omega22 -> wb[ua] wb[ub] g[la,lb],
- eb[ua] -> wb[ua]/Sqrt[omega22],
+ (* Create Spatial Portion of Null Tetrad *)
+ isqrt2 -> 0.7071067811865475244,
+ ltet[ua] -> isqrt2 eb[ua],
+ n[ua] -> - isqrt2 eb[ua],
+ rm[ua] -> isqrt2 ec[ua],
+ im[ua] -> isqrt2 ea[ua],
+ rmbar[ua] -> isqrt2 ec[ua],
+ imbar[ua] -> -isqrt2 ea[ua],
- omega13 -> ea[ua] vc[ub] g[la,lb],
- omega23 -> eb[ua] vc[ub] g[la,lb],
- wc[ua] -> vc[ua] - omega13 ea[ua] - omega23 eb[ua],
- omega33 -> wc[ua] wc[ub] g[la,lb],
- ec[ua] -> wc[ua]/Sqrt[omega33],
+ (* nn here is the projection of both l^a and n^a with u^a (the time-like unit
+ vector normal to the hypersurface). We do NOT save the t component of the
+ tetrads in this code to avoid unnecessary factors of lapse and shift. *)
+ nn -> isqrt2,
- (* Create Spatial Portion of Null Tetrad *)
- isqrt2 -> 0.7071067811865475244,
- ltet[ua] -> isqrt2 eb[ua],
- n[ua] -> - isqrt2 eb[ua],
- rm[ua] -> isqrt2 ec[ua],
- im[ua] -> isqrt2 ea[ua],
- rmbar[ua] -> isqrt2 ec[ua],
- imbar[ua] -> -isqrt2 ea[ua],
- (* nn here is the projection of both l^a and n^a with u^a (the time-like unit
- vector normal to the hypersurface). We do NOT save the t component of the
- tetrads in this code to avoid unnecessary factors of lapse and shift. *)
- nn -> isqrt2,
+ (****************************************************************************
+ Compute the NP pseudoscalars
+ ****************************************************************************)
-
-(****************************************************************************
- Compute the NP pseudoscalars
- ****************************************************************************)
-
- (* Calculate the relevant Riemann Quantities *)
+ (* Calculate the relevant Riemann Quantities *)
- (* The 3-Riemann *)
- R[la,lb,lc,ld] -> 1/2 ( PD[g[la,ld],lc,lb] + PD[g[lb,lc],ld,la] )
- - 1/2 ( PD[g[la,lc],lb,ld] + PD[g[lb,ld],la,lc] )
- + g[lj,le] gamma[uj,lb,lc] gamma[ue,la,ld]
- - g[lj,le] gamma[uj,lb,ld] gamma[ue,la,lc],
+ (* The 3-Riemann *)
+ R[la,lb,lc,ld] -> 1/2 ( PD[g[la,ld],lc,lb] + PD[g[lb,lc],ld,la] )
+ - 1/2 ( PD[g[la,lc],lb,ld] + PD[g[lb,ld],la,lc] )
+ + g[lj,le] gamma[uj,lb,lc] gamma[ue,la,ld]
+ - g[lj,le] gamma[uj,lb,ld] gamma[ue,la,lc],
- (* The 4-Riemann projected into the slice on all its indices.
- The Gauss equation. *)
- R4p[li,lj,lk,ll] -> R[li,lj,lk,ll] +
- 2 AntiSymmetrize[k[li,lk] k[ll,lj], lk, ll],
+ (* The 4-Riemann projected into the slice on all its indices. The Gauss equation. *)
+ R4p[li,lj,lk,ll] -> R[li,lj,lk,ll] + 2 AntiSymmetrize[k[li,lk] k[ll,lj], lk, ll],
- (* The 4-Riemann projected in the unit normal direction on one
- index, then into the slice on the remaining indices. The Codazzi
- equation. *)
- Ro[lj,lk,ll] -> - 2 AntiSymmetrize[ PD[k[lj,lk],ll], lk,ll]
- - 2 AntiSymmetrize[ gamma[up,lj,lk] k[ll,lp], lk,ll],
+ (* The 4-Riemann projected in the unit normal direction on one
+ index, then into the slice on the remaining indices. The Codazzi equation. *)
+ Ro[lj,lk,ll] -> - 2 AntiSymmetrize[ PD[k[lj,lk],ll], lk,ll]
+ - 2 AntiSymmetrize[ gamma[up,lj,lk] k[ll,lp], lk,ll],
- (* The 4-Riemann projected in the unit normal direction on two
- indices, and into the slice on the remaining two. *)
- Rojo[lj,ll] -> gInv[uc,ud] (R[lj,lc,ll,ld] )
- - k[lj,lp] gInv[up,ud] k[ld,ll]
- + k[lc,ld] gInv[uc,ud] k[lj,ll],
+ (* The 4-Riemann projected in the unit normal direction on two
+ indices, and into the slice on the remaining two. *)
+ Rojo[lj,ll] -> gInv[uc,ud] (R[lj,lc,ll,ld] ) - k[lj,lp] gInv[up,ud] k[ld,ll]
+ + k[lc,ld] gInv[uc,ud] k[lj,ll],
- (* Calculate End Quantities
- NOTE: In writing this, I assume m[0]=0!! to save lots of work *)
+ (* Calculate End Quantities
+ NOTE: In writing this, I assume m[0]=0!! to save lots of work *)
- Psi4r -> R4p[li,lj,lk,ll] n[ui] n[uk]
- ( rmbar[uj] rmbar[ul] - imbar[uj] imbar[ul] )
- + 2 Ro[lj,lk,ll] n[uk] nn
- ( rmbar[uj] rmbar[ul] - imbar[uj] imbar[ul] )
+ Psi4r -> R4p[li,lj,lk,ll] n[ui] n[uk] ( rmbar[uj] rmbar[ul] - imbar[uj] imbar[ul] )
+ + 2 Ro[lj,lk,ll] n[uk] nn ( rmbar[uj] rmbar[ul] - imbar[uj] imbar[ul] )
+ Rojo[lj,ll] nn nn ( rmbar[uj] rmbar[ul] - imbar[uj] imbar[ul]
- (* + terms in mbar^0 == 0*) ),
+ (* + terms in mbar^0 == 0 *) ),
- Psi4i -> R4p[la,lb,lc,ld] n[ua] n[uc] ( - rm[ub] im[ud] - im[ub] rm[ud] )
+ Psi4i -> R4p[la,lb,lc,ld] n[ua] n[uc] ( - rm[ub] im[ud] - im[ub] rm[ud] )
+ 2 Ro[la,lb,lc] n[ub] nn ( - rm[ua] im[uc] - im[ua] rm[uc] )
- + Rojo[la,lb] nn nn ( - rm[ua] im[ub] - im[ua] rm[ub] ) ,
+ + Rojo[la,lb] nn nn ( - rm[ua] im[ub] - im[ua] rm[ub] )
+};
-
- Psi3r -> R4p[la,lb,lc,ld] ltet[ua] n[ub] rm[uc] n[ud]
- + Ro[la,lb,lc] ( nn (n[ua]-ltet[ua]) rm[ub] n[uc]
- - nn rm[ua] ltet[ub] n[uc] )
+otherPsiEqs = {
+ Psi3r -> R4p[la,lb,lc,ld] ltet[ua] n[ub] rm[uc] n[ud]
+ + Ro[la,lb,lc] ( nn (n[ua]-ltet[ua]) rm[ub] n[uc] - nn rm[ua] ltet[ub] n[uc] )
- Rojo[la,lb] nn (n[ua]-ltet[ua]) nn rm[ub],
- Psi3i -> - R4p[la,lb,lc,ld] ltet[ua] n[ub] im[uc] n[ud]
- - Ro[la,lb,lc] ( nn (n[ua]-ltet[ua]) im[ub] n[uc]
- - nn im[ua] ltet[ub] n[uc] )
+ Psi3i -> - R4p[la,lb,lc,ld] ltet[ua] n[ub] im[uc] n[ud]
+ - Ro[la,lb,lc] ( nn (n[ua]-ltet[ua]) im[ub] n[uc] - nn im[ua] ltet[ub] n[uc] )
+ Rojo[la,lb] nn (n[ua]-ltet[ua]) nn im[ub],
- Psi2r -> R4p[la,lb,lc,ld] ltet[ua] n[ud] (rm[ub] rm[uc] + im[ub] im[uc])
- + Ro[la,lb,lc] nn ( n[uc] (rm[ua] rm[ub] + im[ua] im[ub])
- - ltet[ub] (rm[ua] rm[uc] + im[ua] im[uc]) )
+ Psi2r -> R4p[la,lb,lc,ld] ltet[ua] n[ud] (rm[ub] rm[uc] + im[ub] im[uc])
+ + Ro[la,lb,lc] nn ( n[uc] (rm[ua] rm[ub] + im[ua] im[ub]) - ltet[ub] (rm[ua] rm[uc] + im[ua] im[uc]) )
- Rojo[la,lb] nn nn (rm[ua] rm[ub] + im[ua] im[ub]),
- Psi2i -> R4p[la,lb,lc,ld] ltet[ua] n[ud] (im[ub] rm[uc] - rm[ub] im[uc])
- + Ro[la,lb,lc] nn ( n[uc] (im[ua] rm[ub] - rm[ua] im[ub])
- - ltet[ub] (rm[ua] im[uc] - im[ua] rm[uc]) )
+ Psi2i -> R4p[la,lb,lc,ld] ltet[ua] n[ud] (im[ub] rm[uc] - rm[ub] im[uc])
+ + Ro[la,lb,lc] nn ( n[uc] (im[ua] rm[ub] - rm[ua] im[ub]) - ltet[ub] (rm[ua] im[uc] - im[ua] rm[uc]) )
- Rojo[la,lb] nn nn (im[ua] rm[ub] - rm[ua] im[ub]),
- Psi1r -> R4p[la,lb,lc,ld] n[ua] ltet[ub] rm[uc] ltet[ud]
- + Ro[la,lb,lc] ( nn ltet[ua] rm[ub] ltet[uc]
- - nn rm[ua] n[ub] ltet[uc]
- - nn n[ua] rm[ub] ltet[uc] )
+ Psi1r -> R4p[la,lb,lc,ld] n[ua] ltet[ub] rm[uc] ltet[ud]
+ + Ro[la,lb,lc] ( nn ltet[ua] rm[ub] ltet[uc] - nn rm[ua] n[ub] ltet[uc] - nn n[ua] rm[ub] ltet[uc] )
+ Rojo[la,lb] nn nn ( n[ua] rm[ub] - ltet[ua] rm[ub] ),
- Psi1i -> R4p[la,lb,lc,ld] n[ua] ltet[ub] im[uc] ltet[ud]
- + Ro[la,lb,lc] ( nn ltet[ua] im[ub] ltet[uc]
- - nn im[ua] n[ub] ltet[uc]
- - nn n[ua] im[ub] ltet[uc] )
+ Psi1i -> R4p[la,lb,lc,ld] n[ua] ltet[ub] im[uc] ltet[ud]
+ + Ro[la,lb,lc] ( nn ltet[ua] im[ub] ltet[uc] - nn im[ua] n[ub] ltet[uc] - nn n[ua] im[ub] ltet[uc] )
+ Rojo[la,lb] nn nn ( n[ua] im[ub] - ltet[ua] im[ub] ),
- Psi0r -> R4p[la,lb,lc,ld] ltet[ua] ltet[uc] (rm[ub] rm[ud] - im[ub] im[ud])
+ Psi0r -> R4p[la,lb,lc,ld] ltet[ua] ltet[uc] (rm[ub] rm[ud] - im[ub] im[ud])
+ 2 Ro[la,lb,lc] nn ltet[ub] (rm[ua] rm[uc] - im[ua] im[uc])
+ Rojo[la,lb] nn nn (rm[ua] rm[ub] - im[ua] im[ub]),
- Psi0i -> R4p[la,lb,lc,ld] ltet[ua] ltet[uc] (rm[ub] im[ud] + im[ub] rm[ud])
+ Psi0i -> R4p[la,lb,lc,ld] ltet[ua] ltet[uc] (rm[ub] im[ud] + im[ub] rm[ud])
+ 2 Ro[la,lb,lc] nn ltet[ub] (rm[ua] im[uc] + im[ua] rm[uc])
- + Rojo[la,lb] nn nn (rm[ua] im[ub] + im[ua] rm[ub]),
+ + Rojo[la,lb] nn nn (rm[ua] im[ub] + im[ua] rm[ub])
+};
- (* Scalar invariants I and J as defined in (2.2a) and (2.2b) of arXiv:gr-qc/0407013 *)
- curvIr -> ComplexExpand[Re[3 (Psi2r+I Psi2i)^2 - 4 (Psi1r+I Psi1i) (Psi3r + I Psi3i) + (Psi4r + I Psi4i) (Psi0r + I Psi0i)]],
- curvIi -> ComplexExpand[Im[3 (Psi2r+I Psi2i)^2 - 4 (Psi1r+I Psi1i) (Psi3r + I Psi3i) + (Psi4r + I Psi4i) (Psi0r + I Psi0i)]],
- curvJr -> ComplexExpand[Re[Det[{{Psi4r+I Psi4i,Psi3r+I Psi3i,Psi2r+I Psi2i},
- {Psi3r+I Psi3i,Psi2r+I Psi2i,Psi1r+I Psi1i},
- {Psi2r+I Psi2i,Psi1r+I Psi1i,Psi0r+I Psi0i}}]]],
- curvJi -> ComplexExpand[Im[Det[{{Psi4r+I Psi4i,Psi3r+I Psi3i,Psi2r+I Psi2i},
- {Psi3r+I Psi3i,Psi2r+I Psi2i,Psi1r+I Psi1i},
- {Psi2r+I Psi2i,Psi1r+I Psi1i,Psi0r+I Psi0i}}]]],
-
- (* Scalar invariants J1, J2, J3 and J4 of the Narlikar and Karmarkar basis as defined in B5-B8 of arXiv:0704.1756.
- Computed from Weyl tensor expressions using xTensor. *)
- curvJ1 -> -16(3 Psi2i^2-3 Psi2r^2-4 Psi1i Psi3i+4 Psi1r Psi3r+Psi0i Psi4i-Psi0r Psi4r),
- curvJ2 -> 96(-3 Psi2i^2 Psi2r+Psi2r^3+2 Psi1r Psi2i Psi3i+2 Psi1i Psi2r Psi3i-Psi0r Psi3i^2+2 Psi1i Psi2i Psi3r-2 Psi1r Psi2r Psi3r
- -2 Psi0i Psi3i Psi3r+Psi0r Psi3r^2-2 Psi1i Psi1r Psi4i+Psi0r Psi2i Psi4i+Psi0i Psi2r Psi4i-Psi1i^2 Psi4r+Psi1r^2 Psi4r
- +Psi0i Psi2i Psi4r-Psi0r Psi2r Psi4r),
- curvJ3 -> 64(9 Psi2i^4-54 Psi2i^2 Psi2r^2+9 Psi2r^4-24 Psi1i Psi2i^2 Psi3i+48 Psi1r Psi2i Psi2r Psi3i+24 Psi1i Psi2r^2 Psi3i
- +16 Psi1i^2 Psi3i^2-16 Psi1r^2 Psi3i^2+24 Psi1r Psi2i^2 Psi3r+48 Psi1i Psi2i Psi2r Psi3r-24 Psi1r Psi2r^2 Psi3r
- -64 Psi1i Psi1r Psi3i Psi3r-16 Psi1i^2 Psi3r^2+16 Psi1r^2 Psi3r^2+6 Psi0i Psi2i^2 Psi4i-12 Psi0r Psi2i Psi2r Psi4i
- -6 Psi0i Psi2r^2 Psi4i-8 Psi0i Psi1i Psi3i Psi4i+8 Psi0r Psi1r Psi3i Psi4i+8 Psi0r Psi1i Psi3r Psi4i
- +8 Psi0i Psi1r Psi3r Psi4i+Psi0i^2 Psi4i^2-Psi0r^2 Psi4i^2-6 Psi0r Psi2i^2 Psi4r-12 Psi0i Psi2i Psi2r Psi4r+6 Psi0r Psi2r^2 Psi4r
- +8 Psi0r Psi1i Psi3i Psi4r+8 Psi0i Psi1r Psi3i Psi4r+8 Psi0i Psi1i Psi3r Psi4r-8 Psi0r Psi1r Psi3r Psi4r-4 Psi0i Psi0r Psi4i Psi4r-Psi0i^2 Psi4r^2+Psi0r^2 Psi4r^2),
- curvJ4 -> -640(-15 Psi2i^4 Psi2r+30 Psi2i^2 Psi2r^3-3 Psi2r^5+10 Psi1r Psi2i^3 Psi3i+30 Psi1i Psi2i^2 Psi2r Psi3i-30 Psi1r Psi2i Psi2r^2 Psi3i
- -10 Psi1i Psi2r^3 Psi3i-16 Psi1i Psi1r Psi2i Psi3i^2-3 Psi0r Psi2i^2 Psi3i^2-8 Psi1i^2 Psi2r Psi3i^2+8 Psi1r^2 Psi2r Psi3i^2
- -6 Psi0i Psi2i Psi2r Psi3i^2+3 Psi0r Psi2r^2 Psi3i^2+4 Psi0r Psi1i Psi3i^3+4 Psi0i Psi1r Psi3i^3+10 Psi1i Psi2i^3 Psi3r
- -30 Psi1r Psi2i^2 Psi2r Psi3r-30 Psi1i Psi2i Psi2r^2 Psi3r+10 Psi1r Psi2r^3 Psi3r-16 Psi1i^2 Psi2i Psi3i Psi3r
- +16 Psi1r^2 Psi2i Psi3i Psi3r-6 Psi0i Psi2i^2 Psi3i Psi3r+32 Psi1i Psi1r Psi2r Psi3i Psi3r+12 Psi0r Psi2i Psi2r Psi3i Psi3r
- +6 Psi0i Psi2r^2 Psi3i Psi3r+12 Psi0i Psi1i Psi3i^2 Psi3r-12 Psi0r Psi1r Psi3i^2 Psi3r+16 Psi1i Psi1r Psi2i Psi3r^2
- +3 Psi0r Psi2i^2 Psi3r^2+8 Psi1i^2 Psi2r Psi3r^2-8 Psi1r^2 Psi2r Psi3r^2+6 Psi0i Psi2i Psi2r Psi3r^2-3 Psi0r Psi2r^2 Psi3r^2
- -12 Psi0r Psi1i Psi3i Psi3r^2-12 Psi0i Psi1r Psi3i Psi3r^2-4 Psi0i Psi1i Psi3r^3+4 Psi0r Psi1r Psi3r^3-6 Psi1i Psi1r Psi2i^2 Psi4i
- +2 Psi0r Psi2i^3 Psi4i-6 Psi1i^2 Psi2i Psi2r Psi4i+6 Psi1r^2 Psi2i Psi2r Psi4i+6 Psi0i Psi2i^2 Psi2r Psi4i
- +6 Psi1i Psi1r Psi2r^2 Psi4i-6 Psi0r Psi2i Psi2r^2 Psi4i-2 Psi0i Psi2r^3 Psi4i+12 Psi1i^2 Psi1r Psi3i Psi4i-4 Psi1r^3 Psi3i Psi4i
- -2 Psi0r Psi1i Psi2i Psi3i Psi4i-2 Psi0i Psi1r Psi2i Psi3i Psi4i-2 Psi0i Psi1i Psi2r Psi3i Psi4i
- +2 Psi0r Psi1r Psi2r Psi3i Psi4i-2 Psi0i Psi0r Psi3i^2 Psi4i+4 Psi1i^3 Psi3r Psi4i-12 Psi1i Psi1r^2 Psi3r Psi4i
- -2 Psi0i Psi1i Psi2i Psi3r Psi4i+2 Psi0r Psi1r Psi2i Psi3r Psi4i+2 Psi0r Psi1i Psi2r Psi3r Psi4i
- +2 Psi0i Psi1r Psi2r Psi3r Psi4i-2 Psi0i^2 Psi3i Psi3r Psi4i+2 Psi0r^2 Psi3i Psi3r Psi4i+2 Psi0i Psi0r Psi3r^2 Psi4i
- -Psi0r Psi1i^2 Psi4i^2-2 Psi0i Psi1i Psi1r Psi4i^2+Psi0r Psi1r^2 Psi4i^2+2 Psi0i Psi0r Psi2i Psi4i^2+Psi0i^2 Psi2r Psi4i^2
- -Psi0r^2 Psi2r Psi4i^2-3 Psi1i^2 Psi2i^2 Psi4r+3 Psi1r^2 Psi2i^2 Psi4r+2 Psi0i Psi2i^3 Psi4r+12 Psi1i Psi1r Psi2i Psi2r Psi4r
- -6 Psi0r Psi2i^2 Psi2r Psi4r+3 Psi1i^2 Psi2r^2 Psi4r-3 Psi1r^2 Psi2r^2 Psi4r-6 Psi0i Psi2i Psi2r^2 Psi4r+2 Psi0r Psi2r^3 Psi4r
- +4 Psi1i^3 Psi3i Psi4r-12 Psi1i Psi1r^2 Psi3i Psi4r-2 Psi0i Psi1i Psi2i Psi3i Psi4r+2 Psi0r Psi1r Psi2i Psi3i Psi4r
- +2 Psi0r Psi1i Psi2r Psi3i Psi4r+2 Psi0i Psi1r Psi2r Psi3i Psi4r-Psi0i^2 Psi3i^2 Psi4r+Psi0r^2 Psi3i^2 Psi4r
- -12 Psi1i^2 Psi1r Psi3r Psi4r+4 Psi1r^3 Psi3r Psi4r+2 Psi0r Psi1i Psi2i Psi3r Psi4r+2 Psi0i Psi1r Psi2i Psi3r Psi4r
- +2 Psi0i Psi1i Psi2r Psi3r Psi4r-2 Psi0r Psi1r Psi2r Psi3r Psi4r+4 Psi0i Psi0r Psi3i Psi3r Psi4r+Psi0i^2 Psi3r^2 Psi4r-Psi0r^2 Psi3r^2 Psi4r
- -2 Psi0i Psi1i^2 Psi4i Psi4r+4 Psi0r Psi1i Psi1r Psi4i Psi4r+2 Psi0i Psi1r^2 Psi4i Psi4r+2 Psi0i^2 Psi2i Psi4i Psi4r
- -2 Psi0r^2 Psi2i Psi4i Psi4r-4 Psi0i Psi0r Psi2r Psi4i Psi4r+Psi0r Psi1i^2 Psi4r^2+2 Psi0i Psi1i Psi1r Psi4r^2-Psi0r Psi1r^2 Psi4r^2
- -2 Psi0i Psi0r Psi2i Psi4r^2-Psi0i^2 Psi2r Psi4r^2+Psi0r^2 Psi2r Psi4r^2)
- }
+invariantEqs := {
+ (* Scalar invariants I and J as defined in (2.2a) and (2.2b) of arXiv:gr-qc/0407013 *)
+ curvIr -> ComplexExpand[Re[3 (Psi2r+I Psi2i)^2 - 4 (Psi1r+I Psi1i) (Psi3r + I Psi3i) + (Psi4r + I Psi4i) (Psi0r + I Psi0i)]],
+ curvIi -> ComplexExpand[Im[3 (Psi2r+I Psi2i)^2 - 4 (Psi1r+I Psi1i) (Psi3r + I Psi3i) + (Psi4r + I Psi4i) (Psi0r + I Psi0i)]],
+ curvJr -> ComplexExpand[Re[Det[{{Psi4r+I Psi4i,Psi3r+I Psi3i,Psi2r+I Psi2i},
+ {Psi3r+I Psi3i,Psi2r+I Psi2i,Psi1r+I Psi1i},
+ {Psi2r+I Psi2i,Psi1r+I Psi1i,Psi0r+I Psi0i}}]]],
+ curvJi -> ComplexExpand[Im[Det[{{Psi4r+I Psi4i,Psi3r+I Psi3i,Psi2r+I Psi2i},
+ {Psi3r+I Psi3i,Psi2r+I Psi2i,Psi1r+I Psi1i},
+ {Psi2r+I Psi2i,Psi1r+I Psi1i,Psi0r+I Psi0i}}]]],
+
+ (* Scalar invariants J1, J2, J3 and J4 of the Narlikar and Karmarkar basis as defined
+ in B5-B8 of arXiv:0704.1756. Computed from Weyl tensor expressions using xTensor. *)
+ curvJ1 -> -16(3 Psi2i^2-3 Psi2r^2-4 Psi1i Psi3i+4 Psi1r Psi3r+Psi0i Psi4i-Psi0r Psi4r),
+ curvJ2 -> 96(-3 Psi2i^2 Psi2r+Psi2r^3+2 Psi1r Psi2i Psi3i+2 Psi1i Psi2r Psi3i-Psi0r Psi3i^2+2 Psi1i Psi2i Psi3r-2 Psi1r Psi2r Psi3r
+ -2 Psi0i Psi3i Psi3r+Psi0r Psi3r^2-2 Psi1i Psi1r Psi4i+Psi0r Psi2i Psi4i+Psi0i Psi2r Psi4i-Psi1i^2 Psi4r+Psi1r^2 Psi4r
+ +Psi0i Psi2i Psi4r-Psi0r Psi2r Psi4r),
+ curvJ3 -> 64(9 Psi2i^4-54 Psi2i^2 Psi2r^2+9 Psi2r^4-24 Psi1i Psi2i^2 Psi3i+48 Psi1r Psi2i Psi2r Psi3i+24 Psi1i Psi2r^2 Psi3i
+ +16 Psi1i^2 Psi3i^2-16 Psi1r^2 Psi3i^2+24 Psi1r Psi2i^2 Psi3r+48 Psi1i Psi2i Psi2r Psi3r-24 Psi1r Psi2r^2 Psi3r
+ -64 Psi1i Psi1r Psi3i Psi3r-16 Psi1i^2 Psi3r^2+16 Psi1r^2 Psi3r^2+6 Psi0i Psi2i^2 Psi4i-12 Psi0r Psi2i Psi2r Psi4i
+ -6 Psi0i Psi2r^2 Psi4i-8 Psi0i Psi1i Psi3i Psi4i+8 Psi0r Psi1r Psi3i Psi4i+8 Psi0r Psi1i Psi3r Psi4i
+ +8 Psi0i Psi1r Psi3r Psi4i+Psi0i^2 Psi4i^2-Psi0r^2 Psi4i^2-6 Psi0r Psi2i^2 Psi4r-12 Psi0i Psi2i Psi2r Psi4r+6 Psi0r Psi2r^2 Psi4r
+ +8 Psi0r Psi1i Psi3i Psi4r+8 Psi0i Psi1r Psi3i Psi4r+8 Psi0i Psi1i Psi3r Psi4r-8 Psi0r Psi1r Psi3r Psi4r-4 Psi0i Psi0r Psi4i Psi4r-Psi0i^2 Psi4r^2+Psi0r^2 Psi4r^2),
+ curvJ4 -> -640(-15 Psi2i^4 Psi2r+30 Psi2i^2 Psi2r^3-3 Psi2r^5+10 Psi1r Psi2i^3 Psi3i+30 Psi1i Psi2i^2 Psi2r Psi3i-30 Psi1r Psi2i Psi2r^2 Psi3i
+ -10 Psi1i Psi2r^3 Psi3i-16 Psi1i Psi1r Psi2i Psi3i^2-3 Psi0r Psi2i^2 Psi3i^2-8 Psi1i^2 Psi2r Psi3i^2+8 Psi1r^2 Psi2r Psi3i^2
+ -6 Psi0i Psi2i Psi2r Psi3i^2+3 Psi0r Psi2r^2 Psi3i^2+4 Psi0r Psi1i Psi3i^3+4 Psi0i Psi1r Psi3i^3+10 Psi1i Psi2i^3 Psi3r
+ -30 Psi1r Psi2i^2 Psi2r Psi3r-30 Psi1i Psi2i Psi2r^2 Psi3r+10 Psi1r Psi2r^3 Psi3r-16 Psi1i^2 Psi2i Psi3i Psi3r
+ +16 Psi1r^2 Psi2i Psi3i Psi3r-6 Psi0i Psi2i^2 Psi3i Psi3r+32 Psi1i Psi1r Psi2r Psi3i Psi3r+12 Psi0r Psi2i Psi2r Psi3i Psi3r
+ +6 Psi0i Psi2r^2 Psi3i Psi3r+12 Psi0i Psi1i Psi3i^2 Psi3r-12 Psi0r Psi1r Psi3i^2 Psi3r+16 Psi1i Psi1r Psi2i Psi3r^2
+ +3 Psi0r Psi2i^2 Psi3r^2+8 Psi1i^2 Psi2r Psi3r^2-8 Psi1r^2 Psi2r Psi3r^2+6 Psi0i Psi2i Psi2r Psi3r^2-3 Psi0r Psi2r^2 Psi3r^2
+ -12 Psi0r Psi1i Psi3i Psi3r^2-12 Psi0i Psi1r Psi3i Psi3r^2-4 Psi0i Psi1i Psi3r^3+4 Psi0r Psi1r Psi3r^3-6 Psi1i Psi1r Psi2i^2 Psi4i
+ +2 Psi0r Psi2i^3 Psi4i-6 Psi1i^2 Psi2i Psi2r Psi4i+6 Psi1r^2 Psi2i Psi2r Psi4i+6 Psi0i Psi2i^2 Psi2r Psi4i
+ +6 Psi1i Psi1r Psi2r^2 Psi4i-6 Psi0r Psi2i Psi2r^2 Psi4i-2 Psi0i Psi2r^3 Psi4i+12 Psi1i^2 Psi1r Psi3i Psi4i-4 Psi1r^3 Psi3i Psi4i
+ -2 Psi0r Psi1i Psi2i Psi3i Psi4i-2 Psi0i Psi1r Psi2i Psi3i Psi4i-2 Psi0i Psi1i Psi2r Psi3i Psi4i
+ +2 Psi0r Psi1r Psi2r Psi3i Psi4i-2 Psi0i Psi0r Psi3i^2 Psi4i+4 Psi1i^3 Psi3r Psi4i-12 Psi1i Psi1r^2 Psi3r Psi4i
+ -2 Psi0i Psi1i Psi2i Psi3r Psi4i+2 Psi0r Psi1r Psi2i Psi3r Psi4i+2 Psi0r Psi1i Psi2r Psi3r Psi4i
+ +2 Psi0i Psi1r Psi2r Psi3r Psi4i-2 Psi0i^2 Psi3i Psi3r Psi4i+2 Psi0r^2 Psi3i Psi3r Psi4i+2 Psi0i Psi0r Psi3r^2 Psi4i
+ -Psi0r Psi1i^2 Psi4i^2-2 Psi0i Psi1i Psi1r Psi4i^2+Psi0r Psi1r^2 Psi4i^2+2 Psi0i Psi0r Psi2i Psi4i^2+Psi0i^2 Psi2r Psi4i^2
+ -Psi0r^2 Psi2r Psi4i^2-3 Psi1i^2 Psi2i^2 Psi4r+3 Psi1r^2 Psi2i^2 Psi4r+2 Psi0i Psi2i^3 Psi4r+12 Psi1i Psi1r Psi2i Psi2r Psi4r
+ -6 Psi0r Psi2i^2 Psi2r Psi4r+3 Psi1i^2 Psi2r^2 Psi4r-3 Psi1r^2 Psi2r^2 Psi4r-6 Psi0i Psi2i Psi2r^2 Psi4r+2 Psi0r Psi2r^3 Psi4r
+ +4 Psi1i^3 Psi3i Psi4r-12 Psi1i Psi1r^2 Psi3i Psi4r-2 Psi0i Psi1i Psi2i Psi3i Psi4r+2 Psi0r Psi1r Psi2i Psi3i Psi4r
+ +2 Psi0r Psi1i Psi2r Psi3i Psi4r+2 Psi0i Psi1r Psi2r Psi3i Psi4r-Psi0i^2 Psi3i^2 Psi4r+Psi0r^2 Psi3i^2 Psi4r
+ -12 Psi1i^2 Psi1r Psi3r Psi4r+4 Psi1r^3 Psi3r Psi4r+2 Psi0r Psi1i Psi2i Psi3r Psi4r+2 Psi0i Psi1r Psi2i Psi3r Psi4r
+ +2 Psi0i Psi1i Psi2r Psi3r Psi4r-2 Psi0r Psi1r Psi2r Psi3r Psi4r+4 Psi0i Psi0r Psi3i Psi3r Psi4r+Psi0i^2 Psi3r^2 Psi4r-Psi0r^2 Psi3r^2 Psi4r
+ -2 Psi0i Psi1i^2 Psi4i Psi4r+4 Psi0r Psi1i Psi1r Psi4i Psi4r+2 Psi0i Psi1r^2 Psi4i Psi4r+2 Psi0i^2 Psi2i Psi4i Psi4r
+ -2 Psi0r^2 Psi2i Psi4i Psi4r-4 Psi0i Psi0r Psi2r Psi4i Psi4r+Psi0r Psi1i^2 Psi4r^2+2 Psi0i Psi1i Psi1r Psi4r^2-Psi0r Psi1r^2 Psi4r^2
+ -2 Psi0i Psi0r Psi2i Psi4r^2-Psi0i^2 Psi2r Psi4r^2+Psi0r^2 Psi2r Psi4r^2)
};
+Psi4Calc[fdOrder_, PD_] :=
+{
+ Name -> "psi4_calc_" <> fdOrder,
+ Where -> Interior,
+ After -> "ADMBase_SetADMVars",
+ ConditionalOnKeywords -> {{"fd_order", fdOrder}, {"calc_scalars", "psi4"}},
+ Shorthands -> shorthands,
+ Equations -> psi4Eqs[PD]
+};
+PsisCalc[fdOrder_, PD_] :=
+{
+ Name -> "psis_calc_" <> fdOrder,
+ Where -> Interior,
+ After -> "ADMBase_SetADMVars",
+ ConditionalOnKeywords -> {{"fd_order", fdOrder}, {"calc_scalars", "psis"}},
+ Shorthands -> shorthands,
+ Equations -> Join[psi4Eqs[PD], otherPsiEqs]
+};
+
+InvariantsCalc[fdOrder_, PD_] :=
+{
+ Name -> "invars_calc_" <> fdOrder,
+ Where -> Interior,
+ After -> "ADMBase_SetADMVars",
+ ConditionalOnKeywords -> {{"fd_order", fdOrder}, {"calc_scalars", "psis_and_invariants"}},
+ Shorthands -> shorthands,
+ Equations -> Join[psi4Eqs[PD], otherPsiEqs, invariantEqs]
+};
+
+
(****************************************************************************
Construct the thorn
****************************************************************************)
@@ -319,9 +326,16 @@
AllowedValues -> {"Nth", "2nd", "4th"}
};
+calcScalarsParam = {
+ Name -> "calc_scalars",
+ Description -> "Which scalars to calculate",
+ AllowedValues -> {"psi4", "psis", "psis_and_invariants"},
+ Default -> "psi4"
+};
+
keywordParameters =
{
- fdOrderParam
+ fdOrderParam, calcScalarsParam
};
intParameters =
@@ -336,9 +350,15 @@
calculations =
{
+ Psi4Calc["Nth", PDstandard],
+ Psi4Calc["2nd", PDstandard2nd],
+ Psi4Calc["4th", PDstandard4th],
PsisCalc["Nth", PDstandard],
PsisCalc["2nd", PDstandard2nd],
- PsisCalc["4th", PDstandard4th]
+ PsisCalc["4th", PDstandard4th],
+ InvariantsCalc["Nth", PDstandard],
+ InvariantsCalc["2nd", PDstandard2nd],
+ InvariantsCalc["4th", PDstandard4th]
};
CreateKrancThornTT[groups, ".", "WeylScal4",
