Diagonal Form Of Kruskal Szekeres Metric Covariant NP Tetrad
Exact Solutions of Einstein's Field Equations
Description
Covariant NP tetrad for diagonal form of Kruskal Szekeres metric
References
Kruskal, Phys. Rev., v119, p1743, (1960)
Szekeres, Publ Mat Debrecen, v7, p285, (1960)
Kramer et al. (13.25) p158
Object definitions
Coframe
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\left\{e_1\to \frac{16 m^2 d(u) (r(u,v)-2 m)}{(v-u) (u+v) r(u,v)}+\frac{16 m^2 d(v) (r(u,v)-2 m)}{(v-u) (u+v) r(u,v)},e_2\to \frac{d(u)}{2}-\frac{d(v)}{2},e_3\to -\frac{d(\Phi ) \sin (\Theta ) r(u,v)}{\sqrt{2}}-\frac{i d(\Theta ) r(u,v)}{\sqrt{2}},e_4\to -\frac{d(\Phi ) \sin (\Theta ) r(u,v)}{\sqrt{2}}+\frac{i d(\Theta ) r(u,v)}{\sqrt{2}}\right\}
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{Subscript[e, 1] -> (16*m^2*d[u]*(-2*m + r[u, v]))/((-u + v)*(u + v)*r[u, v]) + (16*m^2*d[v]*(-2*m + r[u, v]))/((-u + v)*(u + v)*r[u, v]), Subscript[e, 2] -> d[u]/2 - d[v]/2, Subscript[e, 3] -> ((-I)*d[\[CapitalTheta]]*r[u, v])/Sqrt[2] - (d[\[CapitalPhi]]*r[u, v]*Sin[\[CapitalTheta]])/Sqrt[2], Subscript[e, 4] -> (I*d[\[CapitalTheta]]*r[u, v])/Sqrt[2] - (d[\[CapitalPhi]]*r[u, v]*Sin[\[CapitalTheta]])/Sqrt[2]}
[e[1] = 16*m^2/(-u+v)/(u+v)*d(u)/r(u,v)*(-2*m+r(u,v))+16*m^2/(-u+v)/(u+v)*d(v)/r(u,v)*(-2*m+r(u,v)), e[2] = 1/2*d(u)-1/2*d(v), e[3] = -1/2*I*2^(1/2)*d(Theta)*r(u,v)-1/2*2^(1/2)*d(Phi)*r(u,v)*sin(Theta), e[4] = 1/2*I*2^(1/2)*d(Theta)*r(u,v)-1/2*2^(1/2)*d(Phi)*r(u,v)*sin(Theta)]
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[e, 1] \[CircleTimes] Subscript[e, 2] + Subscript[e, 2] \[CircleTimes] Subscript[e, 1] - Subscript[e, 3] \[CircleTimes] Subscript[e, 4] - Subscript[e, 4] \[CircleTimes] Subscript[e, 3]}
[g = `&.`(e[1],e[2])+`&.`(e[2],e[1])-`&.`(e[3],e[4])-`&.`(e[4],e[3])]
Constraints
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\left\{\frac{\partial r(u,v)}{\partial u}\to \frac{4 m u (2 m-r(u,v))}{\left(v^2-u^2\right) r(u,v)},\frac{\partial r(u,v)}{\partial v}\to -\frac{4 m v (2 m-r(u,v))}{\left(v^2-u^2\right) r(u,v)}\right\}
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{Derivative[1, 0][r][u, v] -> (4*m*u*(2*m - r[u, v]))/((-u^2 + v^2)*r[u, v]), Derivative[0, 1][r][u, v] -> (-4*m*v*(2*m - r[u, v]))/((-u^2 + v^2)*r[u, v])}
[diff(r(u,v),u) = 4*m*u/(-u^2+v^2)*(2*m-r(u,v))/r(u,v), diff(r(u,v),v) = -4*m*v/(-u^2+v^2)*(2*m-r(u,v))/r(u,v)]
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Cite this as:
Exact Solutions of Einstein's Field Equations: Diagonal Form Of Kruskal Szekeres Metric Covariant NP Tetrad from Differential Geometry Library. http://digi-area.com/DifferentialGeometryLibrary/ExactSolutionsofEinsteinsFieldEquations/Diagonal-Form-Of-Kruskal-Szekeres-Metric-Covariant-NP-Tetrad.php
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