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equations:einstein_equation [2018/03/13 11:13]
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-====== Einstein Equation ======+<WRAP lag>$ G_{\mu \nu} 8 \pi G T_{\mu \nu}$</​WRAP>​
  
-<note tip>$$ G_{\mu \nu} 8 \pi G T_{\mu \nu}$$ ​+====== Einstein Equation ​ ======
  
--->​Definitions#​ +//see also [[models:​general_relativity]] //
-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 $\Gamma^\mu_{\nu \rho}$ 
  
-\begin{equation} +<tabbox Intuitive> ​
-    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} +
-<--+
  
-</​note>​+Einstein'​s equation describes how spacetime gets curved through mass and energy. ​
  
-<tabbox Why is it interesting?>​  +-----
- +
-The Einstien equation is the fundamental equation of general relativity. It describes how spacetime is curved through the presence of matter and energy. +
- +
-<tabbox Layman> ​+
  
  
   * [[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 ​
   ​   ​
-<​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 explains the Einstein equations perfectly. ​   * [[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. ​
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 ---- ----
  
-The static limit of the Einstein equation is known as [[equations:​newtons_law|Newton'​s law]]. ​+The static limit of the Einstein equation is known as [[formulas:​newtons_law|Newton'​s law]]. ​
  
-<​tabbox ​Researcher+<​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>​
  
  
equations/einstein_equation.1520936010.txt.gz · Last modified: 2018/03/13 10:13 (external edit)