326 8 Conclusion of Volume 1
urdo = (∂coordi[[f ]]Gam[[a,g,b]]− ∂coordi[[g]]Gam[[a,f,b]]);
urdo = Simplify[urdo, Trig → True];
mdim
= [ −
weggio Simplify Gam[[a,f,z]] Gam[[z,g,b]]
z=1
mdim
→ ];
Gam[[a,g,z]] Gam[[z,f,b]], Trig True
z=1
Rie[[a, b, f, g]] = Simplify[ 12 (urdo + weggio)
, Trig → True]; }, {f, 1, mdim}], {g, 1, mdim}]; }, {b, 1, mdim}]; }, {a, 1, mdim}]; Print["Finished"];
Print["————————-"];
Print["Now I evaluate the curvature 2-form of your space"]; RR = Table[0, {i, 1, mdim}, {j, 1, mdim}];
mdim mdim
[ [{ [[ ]] = }
Do Do RR i, j 2 Rie[[i,j,a,b]] (holviel[[a]]**holviel[[b]]) ,
a=1 b=a+1
{i, 1, mdim}], {j, 1, mdim}];
Print["I find the following answer"];
Do[Do[Print["R[", i, j, "] = ", RR[[i, j]]], {j, 1, mdim}], {i, 1, mdim}]; Print["The result is encoded in a tensor RR[i,j]"];
Print["Its components are encoded in a tensor Rie[i,j,a,b]"]; Print["—————————"];
Print[" Now I calculate the Ricci tensor"]; ricten = Table[0, {a, 1, mdim}, {b, 1, mdim}];
Do[ricten[[b, e]] = Simplify[Sum[Rie[[xx, b, xx, e]], {xx, mdim}]]; ulla = 0;
If[ricten[[b, e]]=!=0, {
Print[b, " ", e, " ", " non-zero"]; Print["Ricci[", b, e, "]= ", ricten[[b, e]]];
ulla = ulla + 1; }],
{b, mdim}, {e, mdim}];
Print["I have finished the calculation"];
If[ulla == 0, {Print["The Ricci tensor is zero"]; }, { Print[" The tensor ricten[a,b]] giving the Ricci tensor "]; Print[" is ready for storing on hard disk"]; }]; Print["—————————-"];
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Calculation of the Ricci Tensor of the Reissner Nordstrom Metric Using Metrigrav
mainmetric
OK I calculate your space, Give me the data
Give me the dimension of your space
B Mathematica Packages |
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Your space has dimension n = 4
Now I stop and you give me two vectors of dimension 4 vector coordi = vector of coordinates
vector diffe = vector of differentials Next you give me the metric as ds2 =
Then to resume calculation you print metricresume {Null}
coordi = {t, r, θ, ϕ}; |
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diffe = {dt, dr, dθ, dϕ}; |
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ds2 = − 1 − Ar + rQ2 |
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dr2 + r2 |
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dϕ2 + r2 dθ2; |
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metricresume |
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I resume the calculation |
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First I extract the metric coefficients from your data |
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Then I calculate the inverse metric |
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Done! |
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and I calculate also the metric determinant |
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Done |
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I perform the calculation of the Christoffel symbols |
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—————– |
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I finished |
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the Levi Civita connection is given by: |
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Γ [11] = |
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dr(−2Q+Ar) |
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2r(Q+r(−A+r)) |
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Γ [12] = |
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dt(−2Q+Ar) |
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2r(Q+r(−A+r)) |
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Γ [13] = 0 |
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Γ [14] = 0 |
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r( A |
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Γ [21] = dt(−2Q+Ar 2r5 + |
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Ar) |
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Γ [22] = |
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3 |
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2Qr−2Ar |
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Γ [23] = dθ A − Q+r r2 |
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Γ [24] = |
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r |
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Γ[31] = 0
Γ[32] = drθ
Γ[33] = drr
Γ[34] = −dϕCos[θ ]Sin[θ ]
Γ[41] = 0
Γ[42] = drϕ
Γ[43] = dϕCot[θ ]
Γ[44] = drr + dθ Cot[θ ] Task finished
The result is encoded in a tensor Gam[a,b,c]
—————–
