Weyl Vacua Solutions to GR: Derivation from Riemann Tensor

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SUMMARY

The discussion centers on the derivation of vacuum solutions in General Relativity (GR) using the Riemann tensor, specifically the Weyl tensor, rather than the traditional Schwarzschild solution. Participants clarify that the Einstein Field Equations (EFE) do not explicitly involve the Weyl tensor; instead, they emphasize that solving the vacuum EFE requires setting the Einstein tensor to zero and finding the corresponding metric. Once a metric is established, the Weyl tensor can be computed, but it cannot be used as a basis for deriving the solution itself.

PREREQUISITES
  • Understanding of General Relativity and the Einstein Field Equations (EFE)
  • Familiarity with Riemann and Weyl tensors
  • Knowledge of differential equations in the context of physics
  • Experience with metric tensors and their properties
NEXT STEPS
  • Study the derivation of the Schwarzschild solution in General Relativity
  • Learn about the properties and implications of the Weyl tensor
  • Explore advanced topics in differential geometry related to GR
  • Investigate alternative vacuum solutions in GR beyond the Schwarzschild metric
USEFUL FOR

This discussion is beneficial for physicists, mathematicians, and students specializing in General Relativity, particularly those interested in advanced tensor calculus and vacuum solutions in gravitational theories.

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Where can I find a derivation of the vacuum solution for GR directly from the Riemann tensor of zero trace, i.e., Weyl tensor, instead of the more traditional Schwarzschild derivation?
 
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I'm not sure what you mean. The Einstein Field Equation, which is what gets solved, does not say anything about the Weyl tensor. Solving the vacuum EFE means solving the differential equations you get when you set the Einstein tensor equal to zero. After you find a metric that solves the vacuum EFE, you can of course compute its Weyl tensor; but I don't see how you could obtain a solution via the Weyl tensor.
 

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