Alternative Goode Wainwright Representation Of The Szekeres Models
Exact Solutions of Einstein's Field Equations
Description
Alternative Goode-Wainwright representation of the Szekeres models
References
Goode, S.W., Wainwright, J., Phys. Rev. D, v26, p3315, (1982)
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 \frac{e_4\otimes e_4 S(t)^2 \left(a(z) \left(k \left(x^2+y^2\right)-4\right)-4 (x b(z)+y c(z))\right)^2}{\left(k \left(x^2+y^2\right)+4\right)^2}+\frac{16 e_2\otimes e_2 S(t)^2}{\left(k \left(x^2+y^2\right)+4\right)^2}+\frac{16 e_3\otimes e_3 S(t)^2}{\left(k \left(x^2+y^2\right)+4\right)^2}-e_1\otimes e_1\right\}
<math>
<mrow>
<mo>{</mo>
<mrow>
<mi>g</mi>
<semantics>
<mo>→</mo>
<annotation encoding='Mathematica'>"\[Rule]"</annotation>
</semantics>
<mrow>
<mfrac>
<mrow>
<mrow>
<msub>
<mi>e</mi>
<mn>4</mn>
</msub>
<mo>⊗</mo>
<msub>
<mi>e</mi>
<mn>4</mn>
</msub>
</mrow>
<mo>⁢</mo>
<msup>
<mrow>
<mi>S</mi>
<mo>⁡</mo>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
<mo>⁢</mo>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mrow>
<mi>a</mi>
<mo>⁡</mo>
<mo>(</mo>
<mi>z</mi>
<mo>)</mo>
</mrow>
<mo>⁢</mo>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mi>k</mi>
<mo>⁢</mo>
<mrow>
<mo>(</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
<mo>+</mo>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>-</mo>
<mn>4</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>-</mo>
<mrow>
<mn>4</mn>
<mo>⁢</mo>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mi>x</mi>
<mo>⁢</mo>
<mrow>
<mi>b</mi>
<mo>⁡</mo>
<mo>(</mo>
<mi>z</mi>
<mo>)</mo>
</mrow>
</mrow>
<mo>+</mo>
<mrow>
<mi>y</mi>
<mo>⁢</mo>
<mrow>
<mi>c</mi>
<mo>⁡</mo>
<mo>(</mo>
<mi>z</mi>
<mo>)</mo>
</mrow>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mi>k</mi>
<mo>⁢</mo>
<mrow>
<mo>(</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
<mo>+</mo>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>+</mo>
<mn>4</mn>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mfrac>
<mo>+</mo>
<mfrac>
<mrow>
<mn>16</mn>
<mo>⁢</mo>
<mrow>
<msub>
<mi>e</mi>
<mn>2</mn>
</msub>
<mo>⊗</mo>
<msub>
<mi>e</mi>
<mn>2</mn>
</msub>
</mrow>
<mo>⁢</mo>
<msup>
<mrow>
<mi>S</mi>
<mo>⁡</mo>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mi>k</mi>
<mo>⁢</mo>
<mrow>
<mo>(</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
<mo>+</mo>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>+</mo>
<mn>4</mn>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mfrac>
<mo>+</mo>
<mfrac>
<mrow>
<mn>16</mn>
<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>S</mi>
<mo>⁡</mo>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mi>k</mi>
<mo>⁢</mo>
<mrow>
<mo>(</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
<mo>+</mo>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>+</mo>
<mn>4</mn>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mfrac>
<mo>-</mo>
<mrow>
<msub>
<mi>e</mi>
<mn>1</mn>
</msub>
<mo>⊗</mo>
<msub>
<mi>e</mi>
<mn>1</mn>
</msub>
</mrow>
</mrow>
</mrow>
<mo>}</mo>
</mrow>
</math>
{g -> -(Subscript[e, 1] \[CircleTimes] Subscript[e, 1]) + (16*Subscript[e, 2] \[CircleTimes] Subscript[e, 2]*S[t]^2)/(4 + k*(x^2 + y^2))^2 + (16*Subscript[e, 3] \[CircleTimes] Subscript[e, 3]*S[t]^2)/(4 + k*(x^2 + y^2))^2 + (((-4 + k*(x^2 + y^2))*a[z] - 4*(x*b[z] + y*c[z]))^2*Subscript[e, 4] \[CircleTimes] Subscript[e, 4]*S[t]^2)/(4 + k*(x^2 + y^2))^2}
[g = -`&.`(e[1],e[1])+16/(4+k*(x^2+y^2))^2*`&.`(e[2],e[2])*S(t)^2+16/(4+k*(x^2+y^2))^2*`&.`(e[3],e[3])*S(t)^2+1/(4+k*(x^2+y^2))^2*((-4+k*(x^2+y^2))*a(z)-4*x*b(z)-4*y*c(z))^2*`&.`(e[4],e[4])*S(t)^2]
Constants
- TeX
- MathML
- Mathematica input
- Maple input
\{k\}
<math>
<mrow>
<mo>{</mo>
<mi>k</mi>
<mo>}</mo>
</mrow>
</math>
{k}
[k]
Cite this as:
Exact Solutions of Einstein's Field Equations: Alternative Goode Wainwright Representation Of The Szekeres Models from Differential Geometry Library. http://digi-area.com/DifferentialGeometryLibrary/ExactSolutionsofEinsteinsFieldEquations/Alternative-Goode-Wainwright-Representation-Of-The-Szekeres-Models.php
Terms of use:
You can use the contents of DG Library pages for education, research, professional needs and enjoyment.If you use this library, please, cite DG Library as the source of the data.
Pages of DG Library may not be copied, mirrored, redistributed, printed, or reproduced in bulk without permission. Usage for any commercial purposes without permission is prohibited.
DG Library is database of over 580 objects for differential geometry and its applications. Read more...
