Vector Addition in Curved Space: Invariance?

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SUMMARY

In curved space, vector addition is not translation invariant as it is in Euclidean space. In Einstein's framework, vectors can only be added locally where the curvature is negligible. This limitation affects the ability to directly sum momenta of objects in a curved manifold, leading to apparent violations of momentum and energy conservation laws. Instead, conservation laws are maintained along Killing Vectors, which are path-dependent and account for the curvature's influence on vector operations.

PREREQUISITES
  • Understanding of curved manifolds in general relativity
  • Familiarity with vector algebra in physics
  • Knowledge of Killing Vectors and their significance
  • Basic principles of momentum and energy conservation
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  • Study the implications of curvature on vector fields in general relativity
  • Explore the concept of local inertial frames in curved spacetime
  • Learn about the mathematical formulation of Killing Vectors
  • Investigate the relationship between curvature and conservation laws in physics
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The discussion is beneficial for physicists, mathematicians, and students studying general relativity, particularly those interested in the implications of curvature on vector operations and conservation laws.

amit_thakur
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in vector addition we assume that they are translation invariant .however in einsteins space
definition where we believe it to be curved unlike euclidean space ,is t not true that they
will no longer be translation invariant .in that case how could we add vectors?
 
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In that case, you don't add vectors unless they are "local". That is, the curvature over the amount of space you have to carry them is insignificant. So you can still do vector algebra with colliding particles, but you wouldn't be able to directly add momenta of all objects around a curved manifold to find the total momentum. This leads to apparent loss of momentum and energy conservation laws in curved manifolds. (Of course, the body causing the curvature picks up the slack, but that's usually ignored.) Instead, you have conservations along Killing Vectors, which are going to be path-dependent.
 

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