Oxford Read And Imagine Starter

[Courtland Boland]

What is space? What is time? Do they exist independently of the thingsand processes in them? Or is their existence parasitic on those thingsand processes? Are they like a canvas onto which an artist paints;they exist whether or not the artist paints on them? Or are they akinto parenthood; there is no parenthood until there are parents andchildren? That is, is there no space and time until there are thingswith spatial properties and processes with temporal durations?

The hole argument was invented for slightly different purposes byAlbert Einstein late in 1913 as part of his quest for the generaltheory of relativity. It was revived and reformulated in the moderncontext by John3 = John Earman \(\times\) John Stachel\(\times\) John Norton.

See Stachel (2014) for a review that covers the historical aspects ofthe hole argument and its significance in philosophy and physics. Itis written at a technically more advanced level than this article. Foranother recent review of the hole argument, see Pooley (2021).

This one example illustrates the core content of the hole argument.With only a little further effort, the argument can be made moreprecise and general. This will be done concurrently in these notes,[1] intended for readers with some background in differential geometryand general relativity.

Manifold of Events. Consider our universe, whichrelativistic cosmologies attempt to model. Its spacetime is theentirety of all space through all time. The events of this spacetimeare generalizations of the dimensionless points of ordinary spatialgeometry. A geometric point in ordinary spatial geometry is just aparticular spot in the space and has no extension. Correspondingly, anevent in spacetime is a particular point in a cosmological space at aparticular moment of time.

Metrical Structure and Matter Fields. In specifyingthat events form a four dimensional manifold, there is still a lot wehave not said about the events. We have not specified whichevents lie in the future and past of which other events, how much timeelapses between these events, which events are simultaneous withothers so that they can form three dimensional spaces, what spatialdistances separates these events, and many more relatedproperties.

These additional properties are introduced by specifying the metricfield. To see how this field provides that information, imagine acurve connecting a given pairs of events in spacetime. The informationabout times elapsed and spatial distances is given by the timeselapsed and distances along such curves. See Figure 2:

These invariant properties are, loosely speaking, the ones that areintrinsic to the geometry and dynamics, such as distance along spatialcurves and time along worldlines of galaxies, the rest mass of thegalaxy, the number of particles in it, as well as a host of otherproperties, such as whether the spacetimes are metrically flat orcurved.

For example, if one makes a journey from one galaxy to another, allobservables pertinent to the trip will be invariants. These includethe time elapsed along the journey, whether the spaceship isaccelerating or not at any time in its journey, the age of the galaxyone leaves at the start of the trip and the age of the destinationgalaxy at the end and all operations that may involve signaling withparticles or light pulses.

Therefore, since the two spreadings or distributions of metric andmatter fields of a hole transformation agree on invariants, they alsoagree on all observables. They are observationallyindistinguishable.

One might wonder whether some of the further structures defined on themanifold represent further properties of spacetime rather than what iscontained within spacetime. In particular, the metric field containsimportant information on spatial distances and times elapsed. Oughtthat not also to be considered a part of the containing spacetime asopposed to what is contained within spacetime?

So, the metric field of general relativity seems to defy easycharacterization. We would like it to be exclusively part of spacetimethe container, or exclusively part of matter the contained. Yet itseems to be part of both. But in any case, the crucial point to notehere (contrary to some historical writing on the hole argument) isthat one need not settle this issue in order to get the hole argumentoff the ground! As long as one is dealing with a theory which isactively generally covariant in the sense articulated above, the holeargument will rear its head, as we will now see.

So far we have characterized the substantivalist doctrine as the viewthat spacetime has an existence independent of its contents. Thisformulation conjures up powerful if vague intuitive pictures, but itis not clear enough to be deployed in the context of theinterpretation of physical theories. If we represent spacetime by amanifold of events, how do we characterize the independence of itsexistence? Is it the counterfactual claim that were there no metric ormatter fields, there would still be a manifold of events? Thatcounterfactual is automatically denied by the standard formulationwhich posits that all spacetimes have at least metrical structure.That seems too cheap a refutation of manifold substantivalism. Surely,there must be an improved formulation. Fortunately, we do not need towrestle with finding it. For present purposes we need only consider aconsequence of the substantivalist view and can set aside the task ofgiving a precise formulation of that view.

Correspondingly, when we spread the metric and matter fieldsdifferently over a manifold of events, we are now assigning metricaland material properties in different ways to the events of themanifold. For example, imagine that a galaxy passes through some eventE in the hole. After the hole transformation, this galaxy might notpass through that event. For the manifold substantivalist, this mustbe a matter of objective physical fact: either the galaxy passesthrough E or not. The two distributions represent two physicallydistinct possibilities.

We can now assemble the pieces above to generate unhappy consequencesfor the manifold substantivalist. Consider the two distributions ofmetric and material fields related by a hole transformation. Since themanifold substantivalist denies Leibniz Equivalence, thesubstantivalist must hold that the two systems represent distinctphysical systems. But the properties that distinguish the two are veryelusive. They escape both (a) observational verification and (b) thedetermining power of cosmological theory.

(a) Observational verification. The substantivalist mustinsist that it makes a physical difference whether the galaxy passesthrough event \(E\) or not. But we have already noticed that thetwo distributions are observationally equivalent: no observation cantell us if we are in a world in which the galaxy passes through event\(E\) or misses event \(E\).

It might be a little hard to see from Figure 6 that the twodistributions are observationally equivalent. In the firstdistribution on the left, the middle galaxy moves in what looks like astraight line and stays exactly at the spatial midpoint between thegalaxies on either side. In the second distribution on the right, allthat seems to be undone. The galaxy looks like it accelerates intaking a swerve to the right, so that it moves closer to the galaxy onits right.

(b) Determinism. The physical theory of relativisticcosmology is unable to pick between the two cases. This is manifestedas an indeterminism of the theory. We can specify the distribution ofmetric and material fields throughout the manifold of events,excepting within the region designated as The Hole. Then the theory isunable to tell us how the fields will develop into The Hole. Both theoriginal and the transformed distribution are legitimate extensions ofthe metric and matter fields outside The Hole into The Hole, sinceeach satisfies all the laws of the theory of relativistic cosmology.The theory has no resources which allow us to insist that one only isadmissible.

So, the anti-substantivalist conclusion invited by the hole argumentis this. We can load up any physical theory with objects or properties(here: spacetime events) that cannot be fixed by observation. If theirinvisibility to observation is not already sufficient warning thatthese properties are illegitimate, then finding that they visitindeterminism onto a theory that is otherwise deterministic ought tobe warning enough. Therefore, such objects or properties (again, herespacetime events) should be discarded along with any doctrine thatrequires their retention.

The hole argument was created by Albert Einstein late in 1913 as anact of desperation when his quest for his general theory of relativityhad encountered what appeared to be insuperable obstacles. Over theprevious year, he had been determined to find a gravitation theorythat was generally covariant in the sense introduced above. He hadeven considered essentially the celebrated generally covariantequations he would settle upon in November 1915 and which now appearin all the textbooks.

There are at least as many responses to the hole argument as authorswho have written on it. In this section, we regiment the literature byconsidering five broad classes of response to the argument since itwas revitalised in the philosophical literature of the 1980s. In thecourse of scrutinizing the argument, by now virtually all its aspectshave been weighed and tested.

An alternative response to the hole argument is to accept thatgenerally covariant theories of space and time such as generalrelativity are indeterministic. Perhaps (the thought goes) thisindeterminism is not troubling, because it is an indeterminism onlyabout which objects instantiate which properties and not about whichpatterns of properties are instantiated. It is not obvious, however,that this is sufficient to defuse worries about indeterminism: at thevery least, if another response to the argument were available, theywould seem to be preferable.

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