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Derivation of Einstein Field Equations 
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#1
Jan2413, 10:12 PM

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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 stressenergy 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.



#3
Jan2413, 10:21 PM

C. Spirit
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[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 


#4
Jan2413, 10:22 PM

Emeritus
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Derivation of Einstein Field Equations
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].



#5
Jan2413, 10:36 PM

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.



#6
Jan2413, 10:41 PM

C. Spirit
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#7
Jan2413, 10:43 PM

P: 52




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