What do I do with these christoffel symbols?

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SUMMARY

The discussion centers on the manipulation of Christoffel symbols in the context of General Relativity (GR). The user is tasked with demonstrating that the derivative of the product of the metric tensor and the derivatives of the coordinates is zero, given the geodesic equation. Key steps include expanding the equation using the product rule and recognizing the symmetry properties of the Christoffel symbols. The user encounters confusion regarding the cancellation of terms and the proper handling of indices, indicating a need for clarity in tensor notation and operations.

PREREQUISITES
  • Understanding of General Relativity concepts, specifically geodesics
  • Familiarity with tensor calculus and the properties of Christoffel symbols
  • Knowledge of the metric tensor and its derivatives
  • Proficiency in applying the product rule in the context of differential equations
NEXT STEPS
  • Study the properties of Christoffel symbols, focusing on their symmetry and role in geodesic equations
  • Learn about the metric tensor and its derivatives in General Relativity
  • Explore tensor notation and index manipulation to avoid common pitfalls
  • Review the product rule in the context of vector calculus and its applications in GR
USEFUL FOR

This discussion is beneficial for students and researchers in theoretical physics, particularly those studying General Relativity and tensor calculus. It is also useful for anyone seeking to deepen their understanding of geodesics and the mathematical framework of GR.

Sparkyboy
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Hey guys I'm a bit new to GR and stuck on this question? :/. So we are given that:

d2xi/dλ2+\Gammaijk dxi/dλ dxj/dλ = 0

and asked to show that d/dλ(gijdxi/dλdxj/dλ) = 0

So I expanded using the product rule to get:

\Gammaijkd2xi/dλ2 dxj/dλ +\Gammaijk dxi/dλd2 xj/dλ2

Then rearranged the first equation to get:

d2xi/dλ2 = - \Gammaijkdxi/dλ dxj/dλ

and substituted for the second order differential equations. That's where I get stuck as I don't know how to get rid of the christoffel symbols. I read somewhere that they have a high degree of symmetry - so maybe I can change the dummy indices to get a symmetric form and they cancel? Very confused - any help would be appreciated.
 
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I believe you are missing a term d(gij)/dλ*...?

I guess I don't follow exactly your "expand" term. If you skipped a lot of steps, I can't do them in my head haha.

There's something wrong with the indices though, you have two of the same indices "upstairs" (implied sum or not?), which almost never happens.
 

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