Help in understanding the derivation of Einstein equations

In summary, the conversation discusses two parts related to the variation of the Reimann tensor and the term ##\int\nabla_{\rho}A^{\rho}\sqrt{-g}\,\mathrm{d}^4x=0##. The first part involves showing the equation ##\Gamma^{\lambda}_{\phantom{\lambda}\nu\mu}\delta\Gamma^{\rho}_{\phantom{\rho}\lambda\sigma}-\Gamma^{\lambda}_{\phantom{\lambda}\mu\nu}\delta\Gamma^{\rho}_{\phantom{\rho}\lambda\sigma}=0##, while the second part involves the integral being zero by the divergence theorem.
  • #1
user1139
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Homework Statement
I am working through the derivation of the Einstein field equations by varying the Einstein-Hilbert action. I need some help in understanding certain steps
Relevant Equations
Given below
There are two parts to my question.

The first is concerns the variation of the Reimann tensor. I am trying to show

$$\delta R^{\rho}_{\phantom{\rho}\sigma\mu\nu}=\nabla_{\mu}\left(\delta\Gamma^{\rho}_{\phantom{\rho}\nu\sigma}\right)-\nabla_{\nu}\left(\delta\Gamma^{\rho}_{\phantom{\rho}\mu\sigma}\right)$$

In order to show the above, it is necessary that ##\Gamma^{\lambda}_{\phantom{\lambda}\nu\mu}\delta\Gamma^{\rho}_{\phantom{\rho}\lambda\sigma}-\Gamma^{\lambda}_{\phantom{\lambda}\mu\nu}\delta\Gamma^{\rho}_{\phantom{\rho}\lambda\sigma}=0##. Why is this true?

The second part concerns the term ##\int\nabla_{\rho}A^{\rho}\sqrt{-g}\,\mathrm{d}^4x=0## where ##A^{\rho}=g^{\sigma\nu}\delta\Gamma^{\rho}_{\phantom{\rho}\nu\sigma}-g^{\sigma\rho}\delta\Gamma^{\mu}_{\phantom{\mu}\mu\sigma}##. Why is the integral zero?
 
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  • #2
Thomas1 said:
In order to show the above, it is necessary that ##\Gamma^{\lambda}_{\phantom{\lambda}\nu\mu}\delta\Gamma^{\rho}_{\phantom{\rho}\lambda\sigma}-\Gamma^{\lambda}_{\phantom{\lambda}\mu\nu}\delta\Gamma^{\rho}_{\phantom{\rho}\lambda\sigma}=0##. Why is this true?
It's been a long time since I took GR, so I may be misremembering. But aren't Christoffel symbols symmetric in the bottom two indices?
 
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  • #3
Thomas1 said:
The second part concerns the term ##\int\nabla_{\rho}A^{\rho}\sqrt{-g}\,\mathrm{d}^4x=0## where ##A^{\rho}=g^{\sigma\nu}\delta\Gamma^{\rho}_{\phantom{\rho}\nu\sigma}-g^{\sigma\rho}\delta\Gamma^{\mu}_{\phantom{\mu}\mu\sigma}##. Why is the integral zero?
It's a total derivative so vanishes by the divergence theorem upon integration over the whole spacetime. First show that\begin{align*}
\int d^4 x \sqrt{-g} \nabla_{\rho} A^{\rho} = \int d^4 x \partial_{\rho} (\sqrt{-g} A^{\rho})
\end{align*}Then use the divergence theorem of ordinary calculus\begin{align*}
\int d^4 x \partial_{\rho} (\sqrt{-g} A^{\rho}) = \oint d^3 x \sqrt{-\gamma} n_{\rho} A^{\rho}
\end{align*}where the integral on the rhs is taken over the boundary 3-surface of induced metric ##\gamma_{ab}## and normal ##n_{\rho}##
 
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Related to Help in understanding the derivation of Einstein equations

1. What are the Einstein equations?

The Einstein equations, also known as the Einstein field equations, are a set of ten partial differential equations that describe the relationship between the curvature of spacetime and the distribution of matter and energy within it. They are a cornerstone of Einstein's theory of general relativity.

2. How were the Einstein equations derived?

The Einstein equations were derived by Albert Einstein in 1915 through a series of mathematical calculations and thought experiments. He used the principles of special relativity and the equivalence principle to develop his theory of general relativity, which forms the basis for the equations.

3. What is the significance of the Einstein equations?

The Einstein equations are significant because they provide a mathematical framework for understanding how gravity works and how it affects the shape of the universe. They have been extensively tested and have been shown to accurately predict the behavior of objects in the presence of massive bodies, such as planets and stars.

4. Are the Einstein equations difficult to understand?

The Einstein equations can be challenging to understand, as they involve complex mathematical concepts such as tensors and differential geometry. However, with a basic understanding of calculus and physics, it is possible to gain a general understanding of the equations and their implications.

5. Can the Einstein equations be applied to everyday situations?

While the Einstein equations were originally developed to explain the behavior of large-scale objects in the universe, they have also been successfully applied to everyday situations. For example, GPS technology relies on the principles of general relativity to accurately determine the location of objects on Earth's surface.

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