Derivation of Einstein Field Equations

In summary, Sean Carroll derives the Einstein Field Equations from the equation R-\kappa T=Wher R is the Ricci tensor, \kappa is some constant, and T is the stress-energy tensor. He states that by contracting both sides it becomes R=-\kappa T. However, this answer is incorrect because R=2\kappa T.
  • #1
tensor33
52
0
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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  • #2
[tex]g_{\mu\nu}g^{\mu\nu}=4[/tex]
 
  • #3
[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
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
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
tensor33 said:
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.
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.
 
  • #7
WannabeNewton said:
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.

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

1. What are the Einstein Field Equations?

The Einstein Field Equations are a set of ten equations that describe the relationship between the curvature of spacetime and the distribution of matter and energy within it. They were developed by Albert Einstein as part of his theory of general relativity.

2. Why were the Einstein Field Equations developed?

Einstein developed the field equations in order to provide a more complete and accurate understanding of gravity. They replaced Newton's law of gravitation and provided a way to understand gravity as a curvature of spacetime rather than a force between masses.

3. How are the Einstein Field Equations derived?

The Einstein Field Equations are derived by applying the principles of differential geometry to the theory of general relativity. This involves using the metric tensor, which describes the curvature of spacetime, to calculate the Einstein tensor, which represents the curvature of spacetime caused by the distribution of matter and energy.

4. What is the significance of the Einstein Field Equations?

The Einstein Field Equations have had a profound impact on our understanding of gravity and the universe. They have been used to make predictions about the behavior of objects in extreme environments, such as black holes, and have been confirmed by numerous observations and experiments.

5. Are the Einstein Field Equations still considered valid?

Yes, the Einstein Field Equations are still considered to be valid and are a fundamental part of our current understanding of gravity. They have been extensively tested and have been shown to accurately describe the behavior of matter and energy in the universe.

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