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In the context of general relativity it is often stated that one of the main purposes of tensors is that of making equations frame-independent.

Question: why is this true?

I'm looking for a mathematical argument/proof about this fact.

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I would probably just refer you to the first chapter of Carlo Rovelli's book. Or any book on general relativity that starts with a tetrad formalism. –  Jerry Schirmer May 24 '13 at 3:41
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A flavour of why this is true: a vector, say a normal one in 3d, is really a physical object $\mathbf{v}.$ You can of course project unto a certain basis $\{\mathbf{e}_i\}_i$, so $\mathbf{v} = \sum v_i \mathbf{e}_i$, and then the components $v_i$ are frame-dependent numbers, but as long as you don't choose a particular basis you're safe. You can then manipulate vectors and tensors (take sums, multiply with scalars, contract them etc.) in a way that doesn't depend on any basis chosen. –  Vibert May 24 '13 at 6:42
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The answer of @Vibert is perfectly correct. In fact, the real equations are between physical objects, because intrinsic physical reality, and intrinsic relations between physical objects, do not depend on observers (frames). But, practically, you need to express them into a particular frame, so you need to use components of vectors, or components of tensors. –  Trimok May 24 '13 at 8:43
    
If a tensor is zero in one frame it is zero in any frame or if a tensor equation holds in one frame it holds in any. –  MBN May 25 '13 at 11:13
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