We are now on section 4.2

which has some very shady approximations. However we do get to Einstein's equation for general relativity if we tolerate that. The equation is$$

R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=8\pi GT_{\mu\nu}

$$where ##R_{\mu\nu},R## are the Ricci tensor and scalar which tell us about the curvature of spacetime, ##g_{\mu\nu}## is the metric, ##G## is Newton's constant and ##T_{\mu\nu}## is the energy-momentum tensor. So the equation tells us how the curvature of spacetime reacts to the presence of energy-momentum (which includes mass). Newton is not forgotten altogetherðŸ˜Š.

The equation can also be written as $$

R_{\mu\nu}=8\pi G\left(T_{\mu\nu}-\frac{1}{2}Tg_{\mu\nu}\right)

$$where ##T=g_{\mu\nu}T_{\mu\nu}## and in empty space where ##T_{\mu\nu}=0## that gives us$$

R_{\mu\nu}=0

$$The equation is a field equation for the metric and the Newtonian gravity field equation is Poisson's equation$$

\nabla^2\Phi=4\pi G\rho

$$where ##\Phi## is the gravitational potential and ##\rho## the mass density.

The section starts by plausibly guessing that GR field equation must be of the form $$

R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=\kappa T_{\mu\nu}

$$where ##\kappa## is a constant we must find. The GR field equation must be the same as Poisson's equation in almost-flat spacetime. So we use a small perturbation ##h_{\mu\nu}## on the flat metric: $$

g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}

$$and discarding second order terms in ##h_{\mu\nu}## eventually work out that ##\kappa=8\pi G## to bring the two equations into line.

**However!** Carroll's 4.38 is wrong. It says ##T_{00}=\rho## and in fact ##T_{00}=\rho\left(1-h_{00}\right)## and if 4.38 were right then 4.39 would be wrong, but in fact it is right. Carroll is sort of having it both ways and we only get ##\ \kappa\approx8\pi G## at best. It contains first order terms in ##h_00##. Hopefully the next section using the Lagrangian formulation will do better!

See

Commentary 4.2 Einsteins equation.pdf (5 pages) for the details.