While studying the Einstein Equation, I noticed something curious, at least to me with little experience in General Relativity. Start with the usual formulation of the equation:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R + g_{\mu\nu}\Lambda = \frac{8{\pi}G}{c^2}T_{\mu\nu}[/tex]

Then, apply the definition of R, the scalar curvature:

[tex]R = g^{ij}R_{ij}[/tex]

and one can rewrite the Einstein Equation as follows:

[tex]R_{\mu\nu} + g_{\mu\nu}\Lambda - \frac{1}{2}g_{\mu\nu}g^{ij}R_{ij} = \frac{8{\pi}G}{c^2}T_{\mu\nu}[/tex]

Intuitively, this looks a lot like some sort of second order approximation in g. One can imagine there might be higher order terms, insignificant except in the most extreme environments. Something like this:

[tex]R_{\mu\nu} + g_{\mu\nu}\Lambda - \frac{1}{2}g_{\mu\nu}g^{ij}R_{ij} + \frac{1}{3!}g_{\mu\pi}g^{\pi\sigma}g_{\sigma\nu}\Lambda_3 - \frac{1}{4!}g_{\mu\pi}g^{\pi\sigma}g_{\sigma\rho}g^{\rho\tau}R_{\tau\nu} + ... = \frac{8{\pi}G}{c^2}T_{\mu\nu}[/tex]

The above is completely arbitrary and made-up, just trying to paint an intuitive picture of the idea. Is the possibility something we can rule out on mathematical grounds? Has this kind of extension to GR been explored?

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# Is GR a 2nd order approximation in g?

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