A claim is made in Sean Carroll's GR book multiple times that goes along the following lines:
Given a problem in relativity, we can find a solution with ease if we choose a convenient reference frame (for instance, a frame co-moving with an object we're interested in, or if we're in curved spacetime, we could go to a frame using locally inertial coordinates). Then, we can find a tensorial expression that reduces to the answer we found in our specific frame. This tensorial expression will be the answer in any frame.
Examples of this can be found in pages 34-35, and 75-76 of Carroll's book. My problem with this method is that I'm not so sure there's a unique tensorial expression that can be extracted from an expression in a specific frame of reference. Is there something that guarantees this uniqueness? To me it seems intuitive that there could be multiple tensorial expressions that reduce to the same thing in the convenient frame we chose, but yet give different results in other frames. In this case, the method wouldn't actually work.
Answer
This can't happen, because if a tensor is zero in one frame it is zero in any frame. Suppose you have some expression which works in a specific frame, and you know that the tensor $T$ reduces to that in your frame. Now suppose that the tensor $S$ also reduces to your expression. Then the tensor $T-S$ is zero in our special frame, which means that it is identically zero and hence $T=S$.
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