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equations:einstein_equation [2017/12/04 08:01] 127.0.0.1 external edit |
equations:einstein_equation [2018/05/13 09:17] jakobadmin ↷ Links adapted because of a move operation |
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- | ====== Einstein Equation ====== | + | <WRAP lag>$ G_{\mu \nu} = 8 \pi G T_{\mu \nu}$</WRAP> |
- | <tabbox Why is it interesting?> | + | ====== Einstein Equation ====== |
- | It's the fundamental equation of general relativity. | + | //see also [[theories:general_relativity]] // |
- | <tabbox Layman> | + | |
+ | <tabbox Intuitive> | ||
+ | |||
+ | Einstein's equation describes how spacetime gets curved through mass and energy. | ||
+ | |||
+ | ----- | ||
* [[http://jakobschwichtenberg.com/how-to-invent-general-relativity/|How to Invent General Relativity]] by J. Schwichtenberg | * [[http://jakobschwichtenberg.com/how-to-invent-general-relativity/|How to Invent General Relativity]] by J. Schwichtenberg | ||
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- | <tabbox Student> | + | <tabbox Concrete> |
- | * [[http://math.ucr.edu/home/baez/einstein/einstein.pdf|The Meaning of Einstein’s Equation]] by John C. Baez and Emory F. Bunn | + | * [[http://math.ucr.edu/home/baez/einstein/einstein.pdf|The Meaning of Einstein’s Equation]] by John C. Baez and Emory F. Bunn explains the Einstein equations perfectly. |
- | + | ||
- | <tabbox Researcher> | + | ---- |
+ | |||
+ | The static limit of the Einstein equation is known as [[formulas:newtons_law|Newton's law]]. | ||
+ | |||
+ | <tabbox Abstract> | ||
<note tip> | <note tip> | ||
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</note> | </note> | ||
- | --> Common Question 1# | + | <tabbox Why is it interesting?> |
- | + | The Einstein equation is the fundamental equation of general relativity. It describes how spacetime is curved through the presence of matter and energy. | |
- | <-- | + | |
- | --> Common Question 2# | + | <tabbox Definitions> |
- | |||
- | <-- | ||
- | | ||
- | <tabbox Examples> | ||
- | --> Example1# | + | On the __right-hand side__, Newton's gravitational constant $G$, the speed of light $c$ and the stress-energy tensor $T_{\mu \nu}$. |
- | + | On the __left-hand side__, the Einstein tensor $G_{\mu \nu}$ is defined as a sum of the Ricci Tensor $R_{\mu\nu}$ and the trace of the Ricci tensor, called Ricci scalar $R =R_{\nu}^\nu$ | |
- | <-- | + | \begin{equation} G_{\mu \nu} = R_{\mu\nu}-\frac{1}{2}Rg_{\mu \nu} \end{equation} |
+ | where the Ricci Tensor $R_{\mu\nu}$ is defined in terms of the Christoffel symbols $\Gamma^\mu_{\nu \rho}$ | ||
- | --> Example2:# | + | \begin{equation} |
- | + | R_{\alpha\beta} = \partial_{\rho}{\Gamma^\rho_{\beta\alpha}} - \partial_{\beta}\Gamma^\rho_{\rho\alpha} + \Gamma^\rho_{\rho\lambda} \Gamma^\lambda_{\beta\alpha} - \Gamma^\rho_{\beta\lambda}\Gamma^\lambda_{\rho\alpha} \end{equation} | |
- | + | and the Christoffel Symbols are defined in terms of the metric | |
- | <-- | + | \begin{equation} |
- | + | \Gamma_{\alpha \beta \rho} =\frac12 \left(\frac{\partial g_{\alpha \beta}}{\partial x^\rho} + \frac{\partial g_{\alpha \rho}}{\partial x^\beta} - \frac{\partial g_{\beta \rho}}{\partial x^\alpha} \right) = \frac12\, \left(\partial_{\rho}g_{\alpha \beta} + \partial_{\beta}g_{\alpha \rho} - \partial_{\alpha}g_{\beta \rho}\right). \end{equation} | |
- | <tabbox History> | + | |
</tabbox> | </tabbox> | ||