What Are the Differences Between Riemannian and Lorentzian Spin Networks?

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SUMMARY

The discussion clarifies the differences between Riemannian and Lorentzian spin networks, emphasizing their dependence on the underlying manifold. Riemannian spin networks utilize a Euclidean tangent space, while Lorentzian spin networks operate on a Minkowskian tangent space, reflecting the nature of spacetime in General Relativity (GR). Lorentzian networks present additional complexity due to the necessity of managing links in timelike directions, often leading to research being conducted on a spacelike slice within a three-dimensional Riemannian manifold embedded in a higher-dimensional Lorentzian framework.

PREREQUISITES
  • Understanding of Riemannian geometry
  • Familiarity with Lorentzian geometry
  • Knowledge of spin networks
  • Basic principles of General Relativity (GR)
NEXT STEPS
  • Research Riemannian geometry applications in quantum gravity
  • Explore Lorentzian geometry and its implications in theoretical physics
  • Study the construction and properties of spin networks
  • Investigate the role of spacelike slices in quantum gravity theories
USEFUL FOR

The discussion is beneficial for theoretical physicists, mathematicians specializing in geometry, and researchers focused on quantum gravity and spin networks.

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Can someone tell me what's the difference between a Riemannian spin network and a Lorentzian spin network?
 
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It's the underlying manifold they are implemented on. Although spin networks look forward to replacing metric geometry, they must as yet be built on an underlying manifold. That manifold can be either Riemannian, with a Euclidean tangent space, or Lorentzian, with a Minkowskian tangent space. The spacetime of GR is Lorentzian.

Generally speaking a Lorentzian network is harder to work with because you have to deal with links in timelike directions. For this reason you will see basic work on networks being carried out in a spacelike slice - a three dimensional Riemannian manifold embedded in the higher dimensional Lorentzian one.
 

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