Someone argues: " is a set of four numbers attached to each point of spacetime

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Relativity & Gravitation · Relativistic dynamics and fields in flat spacetime

The Geometry of Gravity, Ch Tensors in flat spacetime: index notation and covariant field equations

Problem

Someone argues: "AμA^\mu is a set of four numbers attached to each point of spacetime. I take an arbitrary four-vector BμB^\mu and form S=∑μAμBμS = \sum_\mu A^\mu B^\mu. Whenever I try this in different inertial frames, boosting both AμA^\mu and BμB^\mu with the same Lorentz transformation and recomputing SS, I keep getting the same number. By the quotient rule, since SS is a scalar for every choice of BμB^\mu, AμA^\mu must be a tensor of type (1,0)(1,0)." ((a)) Find the error in this reasoning. ((b)) Explain, using the idea of index balance and the transformation law, why the sum ∑μAμBμ\sum_\mu A^\mu B^\mu (both indices upper) cannot in general be relied upon to come out the same in every frame, even though a genuine contraction AμBμA^\mu B_\mu always does.

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