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When we formulate transformation laws for vectors and tensors, the transformation rule for $x^\mu$ is calculated via arguments from total derivatives considering $x^\mu=x^\mu(x^{'\nu})$ that in turn gives us the jacobians and stuff. This conserves the infinitesimal distance $ds^2$ between two points. This seems both mathematically appropriate and physically makes sense as we don't want distances to depend on coordinate systems. Coming to special relativity, we can follow similar arguments to define the transformation laws with the added condition that the transformation should be Lorentz.

Now if I want to write the transformation law for a four-vector $A^\mu$ that is not $x^\mu$, what argument do we use?

  1. It is a vector so it should transform like a vector. This is true but not helpful.
  2. We can use the fact that $A_\mu x^\mu$ should be invariant under transformations. But what is the motivation behind taking $A_\mu x^\mu$ is invariant under a transformation? I mean, why should the inner product between vectors of two different vector spaces be invariant under transformations made to one of the vector space (assuming $A^\mu$ isn't known to be a function of $x^\mu$)?
  3. If $A^\mu=A^\mu(x^\mu)$, then once we know the functional form, we can calculate the transformation law.
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