General Space time Admitting A G6 Being A Cross product
Exact Solutions of Einstein's Field Equations
Description
General space-time admitting a G6 being a cross-product
References
Kramer et al. (10.8) p118
Object definitions
Coframe
- TeX
- MathML
- Mathematica input
- Maple input
\left\{e_1\to d(t),e_2\to d(x),e_3\to d(y),e_4\to d(z)\right\}
<math>
<mrow>
<mo>{</mo>
<mrow>
<mrow>
<msub>
<mi>e</mi>
<mn>1</mn>
</msub>
<semantics>
<mo>→</mo>
<annotation encoding='Mathematica'>"\[Rule]"</annotation>
</semantics>
<mrow>
<mi>d</mi>
<mo>⁡</mo>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</mrow>
<mo>,</mo>
<mrow>
<msub>
<mi>e</mi>
<mn>2</mn>
</msub>
<semantics>
<mo>→</mo>
<annotation encoding='Mathematica'>"\[Rule]"</annotation>
</semantics>
<mrow>
<mi>d</mi>
<mo>⁡</mo>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</mrow>
<mo>,</mo>
<mrow>
<msub>
<mi>e</mi>
<mn>3</mn>
</msub>
<semantics>
<mo>→</mo>
<annotation encoding='Mathematica'>"\[Rule]"</annotation>
</semantics>
<mrow>
<mi>d</mi>
<mo>⁡</mo>
<mo>(</mo>
<mi>y</mi>
<mo>)</mo>
</mrow>
</mrow>
<mo>,</mo>
<mrow>
<msub>
<mi>e</mi>
<mn>4</mn>
</msub>
<semantics>
<mo>→</mo>
<annotation encoding='Mathematica'>"\[Rule]"</annotation>
</semantics>
<mrow>
<mi>d</mi>
<mo>⁡</mo>
<mo>(</mo>
<mi>z</mi>
<mo>)</mo>
</mrow>
</mrow>
</mrow>
<mo>}</mo>
</mrow>
</math>
{Subscript[e, 1] -> d[t], Subscript[e, 2] -> d[x], Subscript[e, 3] -> d[y], Subscript[e, 4] -> d[z]}
[e[1] = d(t), e[2] = d(x), e[3] = d(y), e[4] = d(z)]
Metric
- TeX
- MathML
- Mathematica input
- Maple input
\left\{g\to A^2 e_3\otimes e_3 \text{$\Sigma $1}(x)^2+A^2 e_2\otimes e_2-B^2 e_1\otimes e_1 \text{$\Sigma $2}(z)^2+B^2 e_4\otimes e_4\right\}
<math>
<mrow>
<mo>{</mo>
<mrow>
<mi>g</mi>
<semantics>
<mo>→</mo>
<annotation encoding='Mathematica'>"\[Rule]"</annotation>
</semantics>
<mrow>
<mrow>
<msup>
<mi>A</mi>
<mn>2</mn>
</msup>
<mo>⁢</mo>
<mrow>
<msub>
<mi>e</mi>
<mn>3</mn>
</msub>
<mo>⊗</mo>
<msub>
<mi>e</mi>
<mn>3</mn>
</msub>
</mrow>
<mo>⁢</mo>
<msup>
<mrow>
<mi>Σ1</mi>
<mo>⁡</mo>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>A</mi>
<mn>2</mn>
</msup>
<mo>⁢</mo>
<mrow>
<msub>
<mi>e</mi>
<mn>2</mn>
</msub>
<mo>⊗</mo>
<msub>
<mi>e</mi>
<mn>2</mn>
</msub>
</mrow>
</mrow>
<mo>-</mo>
<mrow>
<msup>
<mi>B</mi>
<mn>2</mn>
</msup>
<mo>⁢</mo>
<mrow>
<msub>
<mi>e</mi>
<mn>1</mn>
</msub>
<mo>⊗</mo>
<msub>
<mi>e</mi>
<mn>1</mn>
</msub>
</mrow>
<mo>⁢</mo>
<msup>
<mrow>
<mi>Σ2</mi>
<mo>⁡</mo>
<mo>(</mo>
<mi>z</mi>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>B</mi>
<mn>2</mn>
</msup>
<mo>⁢</mo>
<mrow>
<msub>
<mi>e</mi>
<mn>4</mn>
</msub>
<mo>⊗</mo>
<msub>
<mi>e</mi>
<mn>4</mn>
</msub>
</mrow>
</mrow>
</mrow>
</mrow>
<mo>}</mo>
</mrow>
</math>
{g -> A^2*Subscript[e, 2] \[CircleTimes] Subscript[e, 2] + B^2*Subscript[e, 4] \[CircleTimes] Subscript[e, 4] + A^2*Subscript[e, 3] \[CircleTimes] Subscript[e, 3]*\[CapitalSigma]1[x]^2 - B^2*Subscript[e, 1] \[CircleTimes] Subscript[e, 1]*\[CapitalSigma]2[z]^2}
[g = A^2*`&.`(e[2],e[2])+B^2*`&.`(e[4],e[4])+A^2*`&.`(e[3],e[3])*Sigma1(x)^2-B^2*`&.`(e[1],e[1])*Sigma2(z)^2]
Constraints
- TeX
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- Mathematica input
- Maple input
\left\{\frac{\partial ^2\text{$\Sigma $1}(x)}{\partial x^2}\to \text{k1} \text{$\Sigma $1}(x),\frac{\partial ^2\text{$\Sigma $2}(z)}{\partial z^2}\to \text{k2} \text{$\Sigma $2}(z)\right\}
<math>
<mrow>
<mo>{</mo>
<mrow>
<mrow>
<semantics>
<mfrac>
<mrow>
<msup>
<mo>∂</mo>
<mn>2</mn>
</msup>
<mrow>
<mi>Σ1</mi>
<mo>⁡</mo>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mo>∂</mo>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mfrac>
<annotation encoding='Mathematica'>TagBox[FractionBox[RowBox[List[SuperscriptBox["\[PartialD]", "2"], RowBox[List["\[CapitalSigma]1", "(", "x", ")"]]]], RowBox[List["\[PartialD]", SuperscriptBox["x", "2"]]], Rule[MultilineFunction, None]], HoldForm]</annotation>
</semantics>
<semantics>
<mo>→</mo>
<annotation encoding='Mathematica'>"\[Rule]"</annotation>
</semantics>
<mrow>
<mi>k1</mi>
<mo>⁢</mo>
<mrow>
<mi>Σ1</mi>
<mo>⁡</mo>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</mrow>
</mrow>
<mo>,</mo>
<mrow>
<semantics>
<mfrac>
<mrow>
<msup>
<mo>∂</mo>
<mn>2</mn>
</msup>
<mrow>
<mi>Σ2</mi>
<mo>⁡</mo>
<mo>(</mo>
<mi>z</mi>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mo>∂</mo>
<msup>
<mi>z</mi>
<mn>2</mn>
</msup>
</mrow>
</mfrac>
<annotation encoding='Mathematica'>TagBox[FractionBox[RowBox[List[SuperscriptBox["\[PartialD]", "2"], RowBox[List["\[CapitalSigma]2", "(", "z", ")"]]]], RowBox[List["\[PartialD]", SuperscriptBox["z", "2"]]], Rule[MultilineFunction, None]], HoldForm]</annotation>
</semantics>
<semantics>
<mo>→</mo>
<annotation encoding='Mathematica'>"\[Rule]"</annotation>
</semantics>
<mrow>
<mi>k2</mi>
<mo>⁢</mo>
<mrow>
<mi>Σ2</mi>
<mo>⁡</mo>
<mo>(</mo>
<mi>z</mi>
<mo>)</mo>
</mrow>
</mrow>
</mrow>
</mrow>
<mo>}</mo>
</mrow>
</math>
{Derivative[2][\[CapitalSigma]1][x] -> k1*\[CapitalSigma]1[x], Derivative[2][\[CapitalSigma]2][z] -> k2*\[CapitalSigma]2[z]}
[diff(diff(Sigma1(x),x),x) = k1*Sigma1(x), diff(diff(Sigma2(z),z),z) = k2*Sigma2(z)]
Constants
- TeX
- MathML
- Mathematica input
- Maple input
\{A,B,\text{k1},\text{k2}\}
<math>
<mrow>
<mo>{</mo>
<mrow>
<mi>A</mi>
<mo>,</mo>
<mi>B</mi>
<mo>,</mo>
<mi>k1</mi>
<mo>,</mo>
<mi>k2</mi>
</mrow>
<mo>}</mo>
</mrow>
</math>
{A, B, k1, k2}
[A, B, k1, k2]
Cite this as:
Exact Solutions of Einstein's Field Equations: General Space time Admitting A G6 Being A Cross product from Differential Geometry Library. http://digi-area.com/DifferentialGeometryLibrary/ExactSolutionsofEinsteinsFieldEquations/General-Space-time-Admitting-A-G6-Being-A-Cross-product.php
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