Nijenhuis on the Formal Definition of Natural Bundles

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"The Hole Argument and Some Physical and Philosophical Implications"

John Stachel

	Abstract
1	Introduction
	1.1	Why should we care?
	1.2	Summary: Where we are headed
	1.3	Outline of the article
2	Early History
	2.1	From the special theory to the search for a theory of gravity
	2.2	From the equivalence principle to the metric tensor
	2.3	From the metric tensor to the hole argument
	2.4	From the hole argument back to general covariance
	2.5	The point coincidence argument
	2.6	From general covariance to Kretschmann’s critique
	2.7	The Cauchy problem for the Einstein equations: from Hilbert to Lichnerowicz
3	Modern Revival of the Argument
	3.1	Did Einstein misunderstand coordinate transformations?
	3.2	Einstein’s vision and fiber bundles
4	The Hole Argument and Some Extensions
	4.1	Structures, algebraic and geometric, permutability and general permutability
	4.2	Differentiable manifolds and diffeomorphisms, covariance and general covariance
	4.3	Fiber bundles: principal bundles, associated bundles, frame bundles, natural and gauge-natural bundles
	4.4	Covariance and general covariance for natural and gauge-natural bundles
5	Current Discussions: Philosophical Issues
	5.1	Relationalism versus substantivalism: Is that all there is?
	5.2	Evolution of Earman’s relationalism
	5.3	Pooley’s position: sophisticated substantivalism
	5.4	Stachel and dynamic structural realism
	5.5	Relations, internal and external, quiddity and haecceity
	5.6	Structures, algebraic and geometric
	5.7	Does “general relativity” extend the principle of relativity?
6	Current Discussions: Physical Issues
	6.1	Space-time symmetry groups and partially background-independent space-times
	6.2	General relativity as a gauge theory
	6.3	The hole argument for elementary particles
	6.4	The problem of quantum gravity
7	Conclusion: The Hole Argument Redivivus
A	Nijenhuis on the Formal Definition of Natural Bundles
B	Appendix to Section 4: Some Philosophical Concepts
	B.1	Things: Synchonic states and diachronic processes
	B.2	Open versus closed systems: Determinism versus causality
	B.3	Properties: Intrinsic and extrinsic
	B.4	Relations: Internal and external
	B.5	Quiddity and haecceity
	References
	Footnotes

A Nijenhuis on the Formal Definition of Natural Bundles

This appendix contains excerpts from Nijenhuis (1994 , pp. 109–110).

Natural bundles …are defined through functors that, for each type of geometric object, associate a fiber bundle with each manifold. To formalize this, we need two categories.

First, consider the category $ℳ fm$ of $m$ -dimensional (smooth) manifolds. Its morphisms are diffeomorphisms (into). Every open set of an $m$ -manifold belongs to $ℳ fm$ : the theory is fundamentally a local one.

Second, consider the category $ℱ ℳ$ of fibered manifolds $(N, p,M ),$ where $M, N$ are manifolds and $p : N → M$ is a surjective submersion. The inverse images $− 1 p (x )$ , for $x ∈ M$ , are the fibers, $N$ is the total space, and $M$ the base space. The morphisms of $ℱ ℳ$ are the fiber-preserving (smooth) maps. The base functor $B : ℱ ℳ → ℳ f$ assigns to each fibered manifold $(N, p,M )$ its base manifold $M$ and to each morphism in $ℱ ℳ$ the induced map on the base spaces.

With these definitions, a bundle functor on $ℳ fm$ , or a natural bundle over $m$ -manifolds, is a covariant functor $F : ℳ fm → ℱ ℳ$ with these simple properties:

(1) (Prolongation) The base space of the fibered manifold $F M$ is $M$ itself.

(2) (Locality) If $U$ is an open subset of $M$ , then the total space of $FU$ is $p− 1(U )$ , the part of $N$ above $U$ .

…

The concept of natural bundle was first formalized by the reviewer. It was little more than a definition, however, until some real theorems were proved. Work by Epstein and Thurston shows that natural bundles are of finite order (i.e., the structure group is a homomorphic image of finite-order jets of diffeomorphisms of $ℝm$ fixing the origin). In addition, their work shows that natural bundles have a natural smooth structure that automatically satisfies a regularity condition (not stated here) in the original definition of natural bundle. Basic to all of this is Peetre’s Theorem, with a number of refinements. Other fundamental work, initiated by Palais and Terng, deals with the classification of natural vector bundles.

…

The pursuit of D[ifferntial]G[eometry] consists to a large extent of performing operations on sections of natural bundles. Connections are constructed from Riemann metrics, covariant derivatives are taken, Lie brackets of vector fields are formed, etc. These operations are natural; they commute with point transformations, yield smooth sections in natural bundles from the same, and have non-increasing supports. All such natural operations are, as implied by Peetre-like theorems, of finite order and so induce natural transformations between corresponding jet bundles.