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This past semester, I just took an introductory course on G.R., which translates to a lot of differential geometry and then concluding with Schwarzschild's solution. We really didn't do any cosmology. However, one of the themes that kept creeping up again and again is that in 4-dimensions, Einstein's field equations are partial differential equations, that are coupled, and generally, very difficult to solve. So, I was all fine with that. However, searching on YouTube, I came across these "newer" lectures that now claim around 6:00 that Einstein's equations are ODEs. Further, the lecturer discusses solutions of them. Can anyone elaborate? Are they PDEs or ODEs?

The lecture can be found here: https://www.youtube.com/watch?v=SpCwDGKZKB0

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    $\begingroup$ They are PDEs. $\!\!\!\!$ $\endgroup$ – Ryan Unger May 6 '17 at 21:30
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    $\begingroup$ Could you provide the timestamp of his first discussion of ODEs? In general the field equations are PDEs, although some problems may be reducible to ODEs by exploiting the constants of motion. $\endgroup$ – J.G. May 6 '17 at 21:30
  • $\begingroup$ @J.G. It seems to be at 6:20, when he says the "Einstein field equations can be written", but I'm not sure as to how the rest of the lectures go from here. Thanks. $\endgroup$ – Thomas Moore May 6 '17 at 21:51
  • $\begingroup$ I'd like to add that there are solutions to the equations too, these days generally found using computers and numerical methods $\endgroup$ – astronat May 6 '17 at 22:03
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The speaker at 6:00 is essentially referring to the fact that for a spatially homogeneous (and isotropic) universe, the Einstein PDEs reduce to the Friedmann ODEs.

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  • $\begingroup$ Hi. Okay, that makes sense. But, in the video, the Friedmann equation is not written as an ODE, if you see it is an algebraic equation. Then there is a dynamical equation called the Raychaudhuri equation and a conservation of energy equation. Together, he terms these as Einstein's equations! .... $\endgroup$ – Thomas Moore May 7 '17 at 1:31
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The Einstein field equations,

$$R_{\mu\nu}-\frac12 g_{\mu\nu}R = 8\pi G \, T_{\mu\nu}$$

are a set of differential equations relating components of the metric $g_{\mu\nu}$ and their derivatives. Since the components of the metric can depend in general on all coordinates, they are multi-variable functions and thus the field equations are partial differential equations.


Nevertheless, suppose we make an ansatz such as,

$$ds^2 =dt^2 - a(t)^2 \left( dx^2+dy^2+dz^2\right)$$

eliminating any dependence on the coordinates, except for time, $t$. Then all components $g_{\mu\nu}$ can be seen as dependent on only a single variable and the field equations become an ODE.

However, one could argue if one has say a function $f(x,y)$ and say make it so that $f(x,y) = g(x)$, we can still consider $f$ to be a multi-variable function, in the sense that $f(x) = \mathrm{const}$ is still a function.

Thus, formally, one may argue that even though we make an ansatz, the equations are still partial differential equations, as it is the formal domain of the functions that matter.

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