Eddington Finkelstein Form Of The Schwarzschild Exterior Metric
Exact Solutions of Einstein's Field Equations
Description
Eddington-Finkelstein form of the Schwarzschild exterior metric (epsilon=+/-1;0<=c<=1;c=0)
Object definitions
Coframe
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- Maple input
\left\{e_1\to d(t),e_2\to d(r),e_3\to d(\theta ),e_4\to d(\phi )\right\}
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{Subscript[e, 1] -> d[t], Subscript[e, 2] -> d[r], Subscript[e, 3] -> d[\[Theta]], Subscript[e, 4] -> d[\[Phi]]}
[e[1] = d(t), e[2] = d(r), e[3] = d(theta), e[4] = d(phi)]
Metric
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\left\{g\to \epsilon e_1\otimes e_2 \sqrt{c \left(\frac{2 m}{r}-1\right)+1}+\epsilon e_2\otimes e_1 \sqrt{c \left(\frac{2 m}{r}-1\right)+1}+c e_2\otimes e_2+e_1\otimes e_1 \left(\frac{2 m}{r}-1\right)+r^2 e_4\otimes e_4 \sin ^2(\theta )+r^2 e_3\otimes e_3\right\}
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{g -> (-1 + (2*m)/r)*Subscript[e, 1] \[CircleTimes] Subscript[e, 1] + Sqrt[1 + c*(-1 + (2*m)/r)]*\[Epsilon]*Subscript[e, 1] \[CircleTimes] Subscript[e, 2] + Sqrt[1 + c*(-1 + (2*m)/r)]*\[Epsilon]*Subscript[e, 2] \[CircleTimes] Subscript[e, 1] + c*Subscript[e, 2] \[CircleTimes] Subscript[e, 2] + r^2*Subscript[e, 3] \[CircleTimes] Subscript[e, 3] + r^2*Subscript[e, 4] \[CircleTimes] Subscript[e, 4]*Sin[\[Theta]]^2}
[g = (-1+2*m/r)*`&.`(e[1],e[1])+(1+c*(-1+2*m/r))^(1/2)*epsilon*`&.`(e[1],e[2])+(1+c*(-1+2*m/r))^(1/2)*epsilon*`&.`(e[2],e[1])+c*`&.`(e[2],e[2])+r^2*`&.`(e[3],e[3])+r^2*`&.`(e[4],e[4])*sin(theta)^2]
Constraints
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\left\{\epsilon ^2\to 1,\epsilon ^4\to 1,\epsilon ^6\to 1,\epsilon ^8\to 1\right\}
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{\[Epsilon]^2 -> 1, \[Epsilon]^4 -> 1, \[Epsilon]^6 -> 1, \[Epsilon]^8 -> 1}
[epsilon^2 = 1, epsilon^4 = 1, epsilon^6 = 1, epsilon^8 = 1]
Constants
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\{c,m,\epsilon \}
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{c, m, \[Epsilon]}
[c, m, epsilon]
Cite this as:
Exact Solutions of Einstein's Field Equations: Eddington Finkelstein Form Of The Schwarzschild Exterior Metric from Differential Geometry Library. http://digi-area.com/DifferentialGeometryLibrary/ExactSolutionsofEinsteinsFieldEquations/Eddington-Finkelstein-Form-Of-The-Schwarzschild-Exterior-Metric.php
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