Introduce Newtonian Metric for SR & Lorentz

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SUMMARY

The discussion centers on the introduction of a Newtonian metric defined as ##ds^2 = dx^2 + dy^2 + dz^2##, which parallels the Lorentz metric used in special relativity. While this metric is valid for Euclidean three-dimensional space, it is inaccurately labeled as a "Newtonian metric" since it does not directly relate to Newtonian physics. The conversation emphasizes that tensor methods are essential for expressing fundamental laws in Newtonian mechanics, rather than being an overcomplicated approach for simple manifolds.

PREREQUISITES
  • Understanding of special relativity and Lorentz metrics
  • Familiarity with Euclidean geometry
  • Basic knowledge of tensor calculus
  • Concepts of manifolds in general relativity
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  • Explore the application of tensor methods in Newtonian mechanics
  • Study the differences between Newtonian and relativistic metrics
  • Investigate the role of coordinate transformations in general relativity
  • Learn about non-trivial manifolds and their significance in physics
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Physicists, mathematicians, and students interested in the foundations of relativity and the mathematical frameworks used in theoretical physics.

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Is it meaningful to introduce a Newtonian metric ##ds^2 = dx^2 + dy^2 + dz^2## in analogy with special relavity and the Lorentz metric?
 
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That's the metric for Euclidean three-dimensional space, and as such it's meaningful. Tensor methods are generally a bit of overkill for this trivial manifold, but it's educational to rewrite the metric in some truly bizarre coordinate system with off-diagonal elements as a demonstration of how the formalism works before taking on the non-trivial manifolds of GR.

However, it's not a "Newtonian metric" and it doesn't have much to do with Newtonian physics.
 
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Tensor methods are no overkill but the only way to express the fundamental laws of physics already in Newtonian mechanics ;-)).
 
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