I Einstein Field Equations: PDEs or ODEs? - Thomas

[Thomas Moore]

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:

Thanks!
Thomas

 

[PeterDonis]

Are they PDEs or ODEs?

Since in general the metric is a function of multiple variables, and the Einstein Field Equations are equations in the metric and its derivatives, they would have to be PDEs.

However, in a situation where the metric is a function of just one variable--as is the case in the FRW spacetimes standardly used in cosmology (everything is a function of the time coordinate $t$ only)--then the Einstein Field Equations will be ODEs. I believe that is the case being discussed in the video.