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Derivation of Einstein Field Equations

  1. Jan 24, 2013 #1
    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.
     
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  3. Jan 24, 2013 #2

    robphy

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    [tex]g_{\mu\nu}g^{\mu\nu}=4[/tex]
     
  4. Jan 24, 2013 #3

    WannabeNewton

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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
     
  5. Jan 24, 2013 #4

    bcrowell

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    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].
     
  6. Jan 24, 2013 #5
    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.
     
  7. Jan 24, 2013 #6

    WannabeNewton

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    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.
     
  8. Jan 24, 2013 #7
    Thanks I'll make sure to keep that in mind.
     
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