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5 years ago in Mathematics , Modern Physics , Theoretical Physics By Nisha Ali

Can nonstandard space-curvature plots be described mathematically?

While analyzing numerical relativity data, I've generated some unusual embeddings and curvature plots that don't resemble the standard textbook examples. Before proposing novel geometry, I need to ground my work. Can these "nonstandard" shapes perhaps with intricate, fractal-like singularities or topology changes still be described rigorously by existing differential geometry and topology, or do they point to a need for new mathematics?

MathematicalPhysics DifferentialGeometry SpaceCurvature

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By Adi Answered 5 years ago

In principle, yes if it's a plot of a legitimate metric solution, it has a mathematical description. The power of differential geometry is its generality. I've worked with highly contorted embeddings from extreme spacetimes. The key is distinguishing coordinate artifacts from genuine geometric features. Your "nonstandard" plot likely represents a valid Riemannian or Lorentzian manifold, describable by metric tensors, curvature scalars, and topological invariants. However, if your plots suggest features like discontinuous metrics or non-Hausdorff topology, then you may be pushing at the boundaries of standard smooth manifold theory, which is where things get profoundly interesting.

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Can nonstandard space-curvature plots be described mathematically?

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