<< EDCRGTCcode.m

$RGTCversion

Consider the modified Schwarzschild metric of Example 10a in section 6.2 of RGTC.nb. It depends on a parameter "b" and becomes Schwarzschild when b=0.

xCoord = {t, r, θ, φ}; 
    simpRules = TrigRules;

A[r] = 1 - 2  m  r/(r^2 - b^2); B[r] = Exp[2  b/r];

gPERT = 
      DiagonalMatrix[{A[r], -1/A[r], -r^2/B[r], -r^2 B[r] 
            Sin[θ]^2}]

The exact metric is Petrov type I:

RGtensors[gPERT, xCoord]

Classify[Wdddd]

We can use the Mathematica function Series to expand the metric around b=0 to any order:

gSERb0[n_] := FacSimp[Series[gPERT, {b, 0, n}]]

gSERb0[3]

Now, depending on the order of approximation, Classify[Wdddd] gives different results:

RGtensors[gSERb0[1], xCoord]

Classify[Wdddd]

RGtensors[gSERb0[2], xCoord]

Classify[Wdddd]

RGtensors[gSERb0[3], xCoord]

Classify[Wdddd]

RGtensors[gSERb0[4], xCoord]

Classify[Wdddd]

This behavior is due to the fact that the result of classification depends on quantities of order up to 6 in the components of the Weyl tensor. The quantity I^3-27J^2, where I, J are the quadratic and cubic complex invariants of the Weyl tensor, is of order 4 in the expansion parameter "b".

Same calculations in an appropriate NP frame

Here one can see that the NP-components of the Weyl tensor in this frame, to any order, satisfy  Ψ[0]=0=Ψ[4], Ψ[1]+Ψ[3]=0, which implies that the space is type I. However, the quantity I^3-27J^2=4Ψ[1]^4(16Ψ[1]^2+9Ψ[2]^2) is of order 4 in "b" and thus vanishes at lower orders.

rt1 = Sqrt[(r^2) - 2 m r - (b^2)]; rt2 = Sqrt[(r^2) - (b^2)];

NPfr = {rt1/rt2 d[t] - rt2/rt1 d[r], 
          rt1/rt2 d[t] + 
            rt2/rt1 d[
                r], (E^(-(( b)/r))) r d[θ] - 
            I (E^((( b)/r))) r Sin[θ] 
              d[φ], (E^(-(( b)/r))) r d[θ] + 
            I (E^((( b)/r))) r Sin[θ] 
              d[φ]}/Sqrt[2]

gNP = {{0, 1, 0, 0}, {1, 0, 0, 0}, {0, 0, 0, -1}, {0, 0, -1, 
          0}}

Factor[gPERT . d[xCoord] . d[xCoord] - gNP . NPfr . NPfr]

invI2 := 
      FacSimp[Ψ[0] Ψ[4] - 
          4  Ψ[1] Ψ[3] + 
          3  Ψ[2]^2]

invJ3 := 
      FacSimp[Ψ[
              2] (Ψ[0] Ψ[4] + 
                2  Ψ[1] Ψ[3] - Ψ[
                    2]^2) - Ψ[
              0] Ψ[3]^2 - Ψ[
              4] Ψ[1]^2]

Exact calculation:

RGtensors[gNP, xCoord, NPfr]

Classify[Wdddd]

Ψ /@ {0, 1, 2, 3, 4}

invI2

invJ3

FacSimp[invI2^3 - 27  invJ3^2]

FacSimp[
      invI2^3 - 27  invJ3^2 - 
        4  Ψ[1]^4 (16  Ψ[1]^2 + 
              9  Ψ[2]^2)]

simpRules = Join[TrigRules, RootRules]

serNPfr[n_] := PowerExpand[FacSimp[Series[NPfr, {b, 0, n}]]]

serNPfr[2]

Approximate calculations:

Clear$dx

RGtensors[gNP, xCoord, serNPfr[1]]

Classify[Wdddd]

Ψ /@ {0, 1, 2, 3, 4}

invI2

invJ3

FacSimp[invI2^3 - 27  invJ3^2]

Clear$dx

RGtensors[gNP, xCoord, serNPfr[2]]

Classify[Wdddd]

Ψ /@ {0, 1, 2, 3, 4}

invI2

invJ3

FacSimp[invI2^3 - 27  invJ3^2]

Clear$dx

RGtensors[gNP, xCoord, serNPfr[3]]

Classify[Wdddd]

Ψ /@ {0, 1, 2, 3, 4}

invI2

invJ3

FacSimp[invI2^3 - 27  invJ3^2]

Clear$dx

RGtensors[gNP, xCoord, serNPfr[4]]

Classify[Wdddd]

Ψ /@ {0, 1, 2, 3, 4}

invI2

invJ3

FacSimp[invI2^3 - 27  invJ3^2]

FacSimp[
      invI2^3 - 27  invJ3^2 - 
        4  Ψ[1]^4 (16  Ψ[1]^2 + 
              9  Ψ[2]^2)]