# Derivation of Einstein Field Equations

by tensor33
Tags: derivation, einstein, equations, field
 P: 52 I'm reading Spacetime and Geometry: An Introduction to General Relativity by Sean M. Carrol and in the chapter on gravitation, he derives the Einstein Field Equations. Here is the part I don't get. He starts with the equation $$R_{\mu\nu}-\frac{1}{2} Rg_{\mu\nu}=\kappa T_{\mu\nu}$$ Wher $R_{\mu\nu}$ is the Ricci tensor, $\kappa$ is some constant, and $T_{\mu\nu}$ is the stress-energy tensor. Then he sates that by contracting both sides it becomes,$$R=-\kappa T$$ where $R=R_{\mu\nu}g^{\mu\nu}$ and $T=T_{\mu\nu}g^{\mu\nu}$. I don't see how he came to this answer. When I tried to work it out myself I got $$R=2\kappa T$$ This obviously isn't the right answer, and I can't see what I'm missing. I'm drawing a blank here.
 Sci Advisor HW Helper PF Gold P: 4,108 $$g_{\mu\nu}g^{\mu\nu}=4$$
 C. Spirit Sci Advisor Thanks P: 4,921 $g^{\mu \nu }R_{\mu \nu } - \frac{1}{2}Rg^{\mu \nu }g_{\mu \nu } = R - \frac{1}{2}R\delta ^{\mu }_{\mu } = R - 2R = - R = \kappa T$ EDIT: robphy beat me to it =D
Emeritus
PF Gold
P: 5,500

## Derivation of Einstein Field Equations

I made the same mistake as you the first time I did it. Although $g^\mu_\nu$ has diagonal *elements* that all equal 1, that doesn't mean you can replace it with 1 when you contract. It has four diagonal elements, so $g^\mu_\mu=4$.
 P: 52 Oh! It all makes sense now! I wish Carroll would've been a little more explicit in that step, it would've saved me a lot of confusion. Oh well. At least I get it now, thanks to all those who replied.
C. Spirit