Conclusion: The Hole Argument Redivivus

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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

7 Conclusion: The Hole Argument Redivivus

The problem becomes even more delicate for quantum systems, in which the existence of the quantum of action $h$ is taken into account. The finite value of $h$ precludes the measurement of a complete set of classical observables by a single compound procedure. It becomes important to show that a complete set of quantum observables, as defined by the theory, can indeed be so measured in principle. Non-relativistic quantum mechanics and quantum electrodynamics, have been show to meet this criterion; and it has been employed as a test of proposals for what should be the fundamental physical quantities defined in quantum gravity (see Bergmann and Smith, 1982 ; Amelino-Camelia and Stachel, 2009 ). Rovelli (2004

) and Oeckl (2008 , 2013 ) have shown how to define such measurements on the hypersurface bounding a four-dimensional region of space-time, even in a background-independent theory.

In field theory, the analog of the data set $(x,t,x′,t′)$ is the couple $[Σ,φ ]$ , where $Σ$ is a 3d surface bounding a finite spacetime region, and $φ$ is a field configuration on $Σ$ . …The data from a local experiment (measurements, preparation, or just assumptions) must in fact refer to the state of the system on the entire boundary of a finite spacetime region. The field theoretical space $𝒢$ is therefore the space of surfaces $Σ$ and field configurations $φ$ on $Σ$ . Quantum dynamics can be expressed in terms of an [probability] amplitude $W [Σ, φ]$ . Following Feynman’s intuition, we can formally define $W [Σ, φ]$ in terms of a sum over bulk field configurations that take the value $φ$ on the boundary $Σ$ . …Notice that the dependence of $W [Σ,φ ]$ on the geometry of $Σ$ codes the spacetime position of the measuring apparatus. In fact, the relative position of the components of the apparatus is determined by their physical distance and the physical time elapsed between measurements, and these data are contained in the metric of $Σ$ . …What is happening is that in background-dependent QFT we have two kinds of measurements: those that determine the distances of the parts of the apparatus and the time elapsed between measurements, and the actual measurements of the fields’ dynamical variables. In quantum gravity, instead, distances and time separations are on an equal footing with the dynamical fields. This is the core of the general relativistic revolution, and the key for background-independent QFT (Rovelli, 2004 , p. 23).

In this sense, Einstein’s hole, as a symbol of process, has reasserted its physical primacy over Hilbert’s Cauchy surface, as a symbol of instantaneous state (see Section 2.7).