I want to prove that Euler Lagrange equation and Einstein Field equation (and Geodesic equation) are the same thing so I made this calculation.(adsbygoogle = window.adsbygoogle || []).push({});

First, I modified Energy-momentum Tensor (talking about 2 dimension; space+time) :

[itex]T_{\mu\nu}=\begin{pmatrix} \nabla E& \dot{E}\\ \nabla p & \dot{p}\end{pmatrix}=\begin{pmatrix} \nabla K& \dot{K}\\ \nabla p & \dot{p}\end{pmatrix}+\begin{pmatrix} \nabla V& \dot{V}\\ 0 & 0\end{pmatrix}=K_{\mu\nu}+V_{\mu\nu}[/itex]

for kinetic energy K and potential energy V

Then, I defined new tensor that I call Lagrangian-momentum Tensor where

[itex]L_{\mu\nu}=\begin{pmatrix} \nabla L& \dot{L}\\ \nabla p & \dot{p}\end{pmatrix}=K_{\mu\nu}-V_{\mu\nu}=T_{\mu\nu}-2V_{\mu\nu}[/itex]

Substitute this for [itex]T_{\mu\nu}[/itex] in Einstein Field Equation, we have

[a]..... [itex]\frac{1}{\kappa}G_{\mu\nu}-2V_{\mu v}=L_{\mu\nu}[/itex]

for [itex]\kappa=8\pi G[/itex] and set [itex]c=1[/itex]

Now, consider Euler Lagrange Equation

[itex]\frac{\partial}{\partial x}L - \frac{\partial}{\partial t}\frac{\partial}{\partial \dot{x}}L = 0[/itex]

Or written in Lagrangian Tensor form :

[itex]L_{00} - L_{11} = 0 \rightarrow \epsilon^{\mu\nu}L_{\mu\nu}=0; \epsilon^{\mu\nu} = \begin{pmatrix} 1& 0\\ 0 & -1\end{pmatrix}[/itex]

apply this to [a], we have

[itex]\epsilon^{\mu\nu}G_{\mu\nu}=2\kappa\nabla V[/itex]

This is very beautiful equation but I'm not sure that I'm doing it right. So, am I doing it right?

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# Euler Lagrange equation as Einstein Field Equation

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