Vector Addition in Curved Space: Invariance?

[amit_thakur]

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?

 

[K^2]

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.