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I quite often hear that GR is formulated in terms of tensors because laws of physics expressed in terms of tensor equations are indepedent of choice coordinates because they `transform nicely'.

I thought the motivation for tensors was that since spacetime is curved, we locally linearize it by introducing the tangent space to a point. Then since all physics happens within the tangent space, all quantities of interest can be expressed as multilinear mappings from the tangent/cotangent space to the real numbers. Coordinate invariance then follows because of the same reason that vectors have a coordinate independent existence in the tangent space, tensors have a coordinate independent existence in tensor product space.

I don't see the connection between coordinate invariance and ``transforming nicely'' under a change of coordinates. Ie, what would be the supposed `bad' implication of not transforming according to the tensor transformation law??

I thought the motivation for tensors was that since spacetime is curved, we locally linearize it by introducing the tangent space to a point. Then since all physics happens within the tangent space, all quantities of interest can be expressed as multilinear mappings from the tangent/cotangent space to the real numbers. Coordinate invariance then follows because of the same reason that vectors have a coordinate independent existence in the tangent space, tensors have a coordinate independent existence in tensor product space.

I don't see the connection between coordinate invariance and ``transforming nicely'' under a change of coordinates. Ie, what would be the supposed `bad' implication of not transforming according to the tensor transformation law??

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