In Hawking and Ellis, "The Large-Scale Structure of Spacetime" the following interesting remark appears:
"...the only relations defined by a manifold structure are tensor relations..."
Why is the above true?
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2$\begingroup$ Can you give us a page number so we have some context? $\endgroup$– user4552Commented Apr 22, 2018 at 22:03
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4$\begingroup$ "There will be variousfields on $\mathcal{M}$ [...]. These fields will obey equations which can be expressed as relations between tensors on $\mathcal{M}$ in which all derivatives with respect to position are covariant derivatives with respect to the symmetric connection defined by the metric $g$. This is so because the only relations defined by a manifold structure are tensor relations, and the only connection defined so far is that given by the metric." § 3.2, p. 59 of the 1994 ed. $\endgroup$– pglpmCommented Apr 22, 2018 at 22:42
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2$\begingroup$ The other the text on that page, quoted in part by @pglpm in the comment above, gives an explanation of why it's true. I don't see why it's helpful to ask the question as a fragment of a quoted sentence, without any mention of the explanation given there. $\endgroup$– user4552Commented Apr 22, 2018 at 23:08
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$\begingroup$ @BenCrowell If you mean the passage starting with "If there were another connection on $\mathcal{M}$, the difference between the two connections would...", that explains why considering an additional connection is equivalent to considering a tensor. It seems to me the question above is more generic; I personally find it interesting. Usually we see that there's nothing but "tensor relations" when we study what a manifold is, but I've never seen a proof of that – if such a proof made sense. Also, I wonder whether the OP has the full book or found that quotation only. $\endgroup$– pglpmCommented Apr 22, 2018 at 23:28
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2 Answers
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That is because spinor relations on a differentiable manifold are excluded in full generality, i.e. as long as one considers a differentiable manifold, one immediately has tensors (which come by considering tensor products of tangent and cotangent spaces to a particular point) and all algebraic operations on them, while for a manifold to have spinors, one needs a certain topological restriction (see page 365 of Wald 1984). I believe that this is meant by Hawking and Ellis.
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1$\begingroup$ Under that passage they add: "(The equations of the matter fields are sometimes expressed as relations between spinors on $\mathcal{M}$. We do not deal with such relations in this book, as they are not needed for the problems we wish to consider. In fact, all spinor equations can be replaced by rather more complicated tensor equations; see e.g. Ruse (1937).)" $\endgroup$– pglpmCommented Apr 22, 2018 at 22:53
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Suppose you want to build a physical theory on a manifold using no additional geometric structures like parallel transport or connections or some extra fibre bundles. Then the only objects you can use are tensors, which are naturally defined on a differential manifold. "Relations" – or better, operations on such tensors are: sum, tensor product, contraction, the Lie derivative of a tensor with respect to a vector field, the wedge product and exterior derivative of differential forms (totally antisymmetric tensors). (Am I forgetting any?)
If what you're asking is a proof that these operations are the only ones, to be honest I've never seen one.
You can do a great deal with this differential structure alone. For example, the two Maxwell equations for $E$ and $B$; the law of charge conservation, from which the two equations for $D$ and $H$ follow; and the relation between $(E,B)$ and $(D,H)$ can be written solely in terms of the objects and operations above (differential forms and exterior derivative). See e.g.:
A. Bossavit: Computational electromagnetism and geometry: Building a finite-dimensional "Maxwell's house" (2004), https://www.researchgate.net/publication/242462763_Computational_electromagnetism_and_geometry_Building_a_finite-dimensional_Maxwell\%27s_house (first published 1999–2000 in a journal).
F. W. Hehl, Y. N. Obukhov, G. F. Rubilar: Classical Electrodynamics: A Tutorial on its Foundations (1999) https://arxiv.org/abs/physics/9907046.
C. A. Truesdell, R. A. Toupin: The Classical Field Theories, in Flügge (ed.): Encyclopedia of Physics. Vol. III/1 (Springer 1960), Chap. F.
A brilliant book on geometric structures, with nice illustrations, is:
- J. A. Schouten: Tensor Analysis for Physicists (2nd ed., Dover 1989).
http://store.doverpublications.com/0486655822.html
https://archive.org/details/isbn_9780486655826
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$\begingroup$ what about jets and the like. as i understand jeta of local charts (diffeomorphisms) often dont transform like tensors. for example connections can be seen as sections of jet bundles but they arent tensors. and these arw natural bundles defined canonically $\endgroup$ Commented Nov 19 at 0:27