Action of Einstein equations

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In summary, the conversation discusses a problem with deriving Einstein equations and using Euler-Lagrange equations to solve it. The resulting equation has a coefficient of twice the cosmological constant compared to the original equation.
  • #1
astros
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Hi,
I have a problem with deriving Einstein equations :

[tex]\epsilon_{IJKL}(e^{I} \wedge R^{JK} + \lambda e^{I} \wedge e^{J} \wedge e^{K}) = 0[/tex]

[tex]de^{I} + \omega^{I}_{J} \wedge e^{J} = 0[/tex]

From the action :

[tex]S[e , \omega] = \frac{1}{16 \pi G} \int \epsilon_{IJKL} (e^{I} \wedge e^{J} \wedge R^{KL} + e^{I} \wedge e^{J} \wedge e^{K} \wedge e^{L})[/tex]

Using Euler-Lagrange equations, for example for the first one I found:

[tex]\epsilon_{IJKL}(e^{I} \wedge R^{JK} + 2 \lambda e^{I} \wedge e^{J} \wedge e^{K}) = 0[/tex]

I know that my problem is surely simple :confused: but I'm back to calculus after a long time of absence :cry: thx2
 
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  • #2
help me The Euler-Lagrange equation you found is correct. The difference between that and the equation you stated is that the coefficient of $e^I \wedge e^J \wedge e^K$ is twice the cosmological constant $\lambda$. This is because when you apply the Euler-Lagrange equations, the coefficient of the variation terms is twice the coefficient of the original term. So the equation you need to solve is: \epsilon_{IJKL}(e^{I} \wedge R^{JK} + 2\lambda e^{I} \wedge e^{J} \wedge e^{K}) = 0
 

1. What are the Einstein equations?

The Einstein equations are a set of ten related equations in the field of general relativity. They describe the relationship between the curvature of spacetime and the distribution of matter and energy within it.

2. What is the significance of the Einstein equations?

The Einstein equations are significant because they provide a mathematical framework for understanding the effects of gravity and the behavior of matter in the universe. They also led to the prediction of phenomena such as black holes and gravitational waves.

3. How were the Einstein equations developed?

The Einstein equations were developed by Albert Einstein in 1915 as part of his theory of general relativity. He derived them using the principles of differential geometry and the equivalence principle, which states that gravity is indistinguishable from acceleration.

4. What is the role of the Einstein equations in modern physics?

The Einstein equations continue to be a fundamental part of modern physics and are used to study a wide range of phenomena, from the behavior of massive objects in space to the structure of the universe on a large scale. They also play a crucial role in the development of theories such as quantum gravity.

5. Are there any limitations to the Einstein equations?

While the Einstein equations have been incredibly successful in explaining many observations and phenomena, they are not a complete theory of gravity. They do not take into account the principles of quantum mechanics and do not provide a unified explanation for all physical interactions. As such, scientists continue to work on developing a theory of everything that can reconcile the Einstein equations with quantum mechanics.

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