The classical theory of spacetime geometry that we call gravity consists of the Einstein equation, which relates the curvature of spacetime to the distribution of matter and energy in spacetime. $ds^2=g_{\mu\nu}dx^{\mu}dx^{\nu}$ Mathematically, how do the Einstein's differential equation of the curvature of spacetime come out of the Einstein's field equations, $$G_{\mu\nu}=8{\pi}T_{\mu\nu}$$. ?
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$ds^{2} = g_{ab}dx^{a}dx^{b}$ isn't the Einstein equation. It's just the equation for what arc length is. It's the definition of the metric tensor, pretty much. You're implicitly using it if you've ever done calculus in three dimensions. The Einstein equation and the Einstein field equations are the same thing and they are expressible as: $$R_{ab} - \frac{1}{2}Rg_{ab} = 8\pi T_{ab}$$ Full stop. |
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From Wiki: $G_{\alpha \beta}= (\delta^\gamma_\alpha \delta^\zeta_\beta - \frac{1}{2} g_{\alpha\beta}g^{\gamma\zeta})(\Gamma^\epsilon_{\gamma\zeta,\epsilon} - \Gamma^\epsilon_{\gamma\epsilon,\zeta} + \Gamma^\epsilon_{\epsilon\sigma} \Gamma^\sigma_{\gamma\zeta} - \Gamma^\epsilon_{\zeta\sigma} \Gamma^\sigma_{\epsilon\gamma})$ Where $\Gamma^\alpha_{\beta\gamma} = \frac{1}{2} g^{\alpha\epsilon}(g_{\beta\epsilon,\gamma} + g_{\gamma\epsilon,\beta} - g_{\beta\gamma,\epsilon})$ So, you see, the Einstein tensor, G, is a function of the metric g and the 1st and 2nd derivatives of the metric. The simple looking equation, relating the Einstein tensor to the stress-energy tensor, that you quote in your question is, in fact, a system of differential equations written in compact form. |
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