Derivation of Einstein Field Equations

tensor33

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 [tex]R_{\mu\nu}-\frac{1}{2} Rg_{\mu\nu}=\kappa T_{\mu\nu}[/tex] Wher [itex]R_{\mu\nu}[/itex] is the Ricci tensor, [itex]\kappa[/itex] is some constant, and [itex]T_{\mu\nu}[/itex] is the stress-energy tensor. Then he sates that by contracting both sides it becomes,[tex]R=-\kappa T[/tex] where [itex]R=R_{\mu\nu}g^{\mu\nu}[/itex] and [itex]T=T_{\mu\nu}g^{\mu\nu}[/itex]. I don't see how he came to this answer. When I tried to work it out myself I got [tex]R=2\kappa T[/tex] This obviously isn't the right answer, and I can't see what I'm missing. I'm drawing a blank here.

robphy

[tex]g_{\mu\nu}g^{\mu\nu}=4[/tex]

WannabeNewton

[itex]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 [/itex]
EDIT: robphy beat me to it =D

bcrowell

I made the same mistake as you the first time I did it. Although [itex]g^\mu_\nu[/itex] 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 [itex]g^\mu_\mu=4[/itex].

tensor33

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.

WannabeNewton

In the future however, make sure you check his errata page because the book does have a ton of errata and in the event that he did make a typo, you don't want to rip your hair out when your result is correct but the thing in the book says something else.

tensor33

Thanks I'll make sure to keep that in mind.