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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1$\begingroup$ 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. $\endgroup$– Zo the RelativistMay 24, 2013 at 3:41
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5$\begingroup$ 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. $\endgroup$– VibertMay 24, 2013 at 6:42
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2$\begingroup$ 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. $\endgroup$– TrimokMay 24, 2013 at 8:43
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$\begingroup$ 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. $\endgroup$– MBNMay 25, 2013 at 11:13
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2 Answers
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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?
Actually this isn't quite true. General relativity doesn't have frames of reference (except locally, which is trivially true because GR is the same as SR locally). A better way of saying this would be:
The purpose of tensors is to make equations coordinate-independent.
The idea is that when we assign coordinates to something, that's just a name. The laws of nature should be expressible in a manner such that the names don't matter.
I'm looking for a mathematical argument/proof about this fact.
A tensor is defined as something that transforms in a certain way under a change of coordinates. Since the transformation of tensors is well-defined, it follows that a tensorial equation retains the same form under a change of coordinates.
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2$\begingroup$ Indeed. What replaces a global coordinate system in GR is an atlas, which is a patchwork of coordinate systems with are glued to each other via homeomorphic biyections in a subset of their overlap regions $\endgroup$– lurscherSep 29, 2014 at 2:56
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1$\begingroup$ @lurscher: Right, I think we're in agreement. I was trying to be nontechnical in my answer, but basically everywhere that I say "coordinates," technically it should be "atlas." $\endgroup$– user4552Sep 29, 2014 at 2:58
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What we want of a law of nature is that is has the same form for every equivalent observer.
Therefore, these laws should be construct with geometrical objects which transform into themselves up to multiplicative factors. This is also known as an homogeneous transformation under certain group (typically Lorentz or diffeomorphism).
The geometrical object which satisfy this homogeneous transformation rule are tensors (there are also spinors). Thus physical theories are described (so far) successfully by these objects (or fields).
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$\begingroup$ Tensorial relationships are form-invariant under a change of coordinates. In SR, a choice of coordinates can be identified with a choice of observer, but that's not the case in SR. $\endgroup$– user4552Sep 29, 2014 at 0:48