"Einstein field equation"의 두 판 사이의 차이
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imported>Pythagoras0 |
imported>Pythagoras0 |
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* prerequiste : Riemann tensor, the Ricci tensor, and the Ricci scalar | * prerequiste : Riemann tensor, the Ricci tensor, and the Ricci scalar | ||
| − | * relativistic matter field equation | + | * relativistic matter field equation |
| + | :<math>R_{\mu \nu} - {1 \over 2}g_{\mu \nu}\,R + g_{\mu \nu} \Lambda = {8 \pi G \over c^4} T_{\mu \nu}</math> where <math>R_{\mu \nu}</math> is the Ricci curvature tensor, <math>R</math> the Ricci scalar curvature, <math>\Lambda</math> is the [[cosmological constant]], <math>T_{\mu \nu}</math> momentum-energy tensor | ||
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==steps to solve an Einstein equation== | ==steps to solve an Einstein equation== | ||
| − | * consider a metric, for example | + | * consider a metric, for example<math>ds^2=-dt^2+e^{2b(t,r)}dr^2+R(t,r)d\phi^2</math> where b, R are unknown functions |
| − | * find the components of the curvature tensor | + | * find the components of the curvature tensor |
| − | * find the components of the Einstein tensor | + | * find the components of the Einstein tensor |
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==Einstein-Hilbert action== | ==Einstein-Hilbert action== | ||
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:<math>\kappa = {8 \pi G \over c^4} </math> | :<math>\kappa = {8 \pi G \over c^4} </math> | ||
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==equation of motion== | ==equation of motion== | ||
| − | * http://en.wikipedia.org/wiki/Einstein%E2%80%93Hilbert_action#Derivation_of_Einstein.27s_field_equations | + | * http://en.wikipedia.org/wiki/Einstein%E2%80%93Hilbert_action#Derivation_of_Einstein.27s_field_equations |
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==solutions example : Schwarzschild black hole== | ==solutions example : Schwarzschild black hole== | ||
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==solutions example : gravitational wave== | ==solutions example : gravitational wave== | ||
| − | * Einstein | + | * Einstein |
| − | * eventually led to the graviton idea\ | + | * eventually led to the graviton idea\ |
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==history== | ==history== | ||
| 55번째 줄: | 56번째 줄: | ||
* http://www.google.com/search?hl=en&tbs=tl:1&q= | * http://www.google.com/search?hl=en&tbs=tl:1&q= | ||
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==related items== | ==related items== | ||
| 66번째 줄: | 67번째 줄: | ||
* [[string theory and Einstein equations]] | * [[string theory and Einstein equations]] | ||
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==encyclopedia== | ==encyclopedia== | ||
2014년 1월 9일 (목) 02:19 판
introduction
- prerequiste : Riemann tensor, the Ricci tensor, and the Ricci scalar
- relativistic matter field equation
\[R_{\mu \nu} - {1 \over 2}g_{\mu \nu}\,R + g_{\mu \nu} \Lambda = {8 \pi G \over c^4} T_{\mu \nu}\] where \(R_{\mu \nu}\) is the Ricci curvature tensor, \(R\) the Ricci scalar curvature, \(\Lambda\) is the cosmological constant, \(T_{\mu \nu}\) momentum-energy tensor
steps to solve an Einstein equation
- consider a metric, for example\(ds^2=-dt^2+e^{2b(t,r)}dr^2+R(t,r)d\phi^2\) where b, R are unknown functions
- find the components of the curvature tensor
- find the components of the Einstein tensor
Einstein-Hilbert action
- Einstein-Hilbert action
\[S= - {1 \over 2\kappa}\int R \sqrt{-g} \, d^4x \] \[\kappa = {8 \pi G \over c^4} \]
equation of motion
solutions example : Schwarzschild black hole
solutions example : gravitational wave
- Einstein
- eventually led to the graviton idea\
history
- cosmological constant
- differential geometry and topology
- Yang-Mills Theory(Non-Abelian gauge theory)
- string theory and Einstein equations