Some Notes on Misner Thorne and Wheeler, Chapters 14 and 27

I was motivated to examine MTW 14.7, the subject of my previous post, because I wanted to understand the calculation of the Einstein Tensor in MTW Box 14.5. Box 14.5 presents a calculation of the curvature components for the Friedmann metric based on the Cartan Structure Equations. This is used in Chapter 27 to write the Friedmann Equations for an idealized homogenious isotropic cosmological model, aka the Friedmann Lemaitre Robertson Walker model. What follows are just some small details concerning the last bit of Box 14.5 where the contraction is taken in order to form the Einstein Tensor. I do not address the Cartan Equations at this time because cosmology beckons. Plus those calculations are tedious but also presented in pretty good detail in MTW. Anyhow, you have to contract to get the Friedmann equations so that is the topic of this post.

Plug into Einstein Field Equations...

OK. So that was like totally easy. Now these two Friedmann equations can be combined to get a relationship between the acceleration in the scale factor, the density and pressure, and the cosmological constant.