Lewis Papapetrou Metric Contravariant NP Tetrad
Exact Solutions of Einstein's Field Equations
Description
Contravariant NP tetrad for Lewis Papapetrou metric
References
Ernst, Phys. Rev., v167, p1175, (1968)
Object definitions
Frame
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- Maple input
\left\{E_1\to \frac{\text{Nat}(z) e^{s(x,y)}}{\sqrt{2} r(x,y)}+\text{Nat}(t),E_2\to \frac{\text{Nat}(t) \left(\frac{e^{s(x,y)} w(x,y)}{r(x,y)}+e^{-s(x,y)}\right)}{\sqrt{2}}-\frac{\text{Nat}(z) e^{s(x,y)}}{\sqrt{2} r(x,y)},E_3\to \sqrt{2} \text{Nat}(x) e^{s(x,y)-k(x,y)},E_4\to \sqrt{2} \text{Nat}(y) e^{s(x,y)-k(x,y)}\right\}
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{Subscript[\[CapitalEpsilon], 1] -> Nat[t] + (E^s[x, y]*Nat[z])/(Sqrt[2]*r[x, y]), Subscript[\[CapitalEpsilon], 2] -> -((E^s[x, y]*Nat[z])/(Sqrt[2]*r[x, y])) + (Nat[t]*(E^(-s[x, y]) + (E^s[x, y]*w[x, y])/r[x, y]))/Sqrt[2], Subscript[\[CapitalEpsilon], 3] -> Sqrt[2]*E^(-k[x, y] + s[x, y])*Nat[x], Subscript[\[CapitalEpsilon], 4] -> Sqrt[2]*E^(-k[x, y] + s[x, y])*Nat[y]}
[Epsilon[1] = `&@`(t)+1/2*2^(1/2)*exp(s(x,y))*`&@`(z)/r(x,y), Epsilon[2] = -1/2*2^(1/2)*exp(s(x,y))*`&@`(z)/r(x,y)+1/2*2^(1/2)*`&@`(t)*(exp(-s(x,y))+exp(s(x,y))/r(x,y)*w(x,y)), Epsilon[3] = 2^(1/2)*exp(-k(x,y)+s(x,y))*`&@`(x), Epsilon[4] = 2^(1/2)*exp(-k(x,y)+s(x,y))*`&@`(y)]
Metric
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\left\{g\to E_1\otimes E_2+E_2\otimes E_1-E_3\otimes E_4-E_4\otimes E_3\right\}
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{g -> Subscript[\[CapitalEpsilon], 1] \[CircleTimes] Subscript[\[CapitalEpsilon], 2] + Subscript[\[CapitalEpsilon], 2] \[CircleTimes] Subscript[\[CapitalEpsilon], 1] - Subscript[\[CapitalEpsilon], 3] \[CircleTimes] Subscript[\[CapitalEpsilon], 4] - Subscript[\[CapitalEpsilon], 4] \[CircleTimes] Subscript[\[CapitalEpsilon], 3]}
[g = `&.`(Epsilon[1],Epsilon[2])+`&.`(Epsilon[2],Epsilon[1])-`&.`(Epsilon[3],Epsilon[4])-`&.`(Epsilon[4],Epsilon[3])]
Cite this as:
Exact Solutions of Einstein's Field Equations: Lewis Papapetrou Metric Contravariant NP Tetrad from Differential Geometry Library. http://digi-area.com/DifferentialGeometryLibrary/ExactSolutionsofEinsteinsFieldEquations/Lewis-Papapetrou-Metric-Contravariant-NP-Tetrad.php
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