equations:einstein_equation
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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)
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