Philosophy of Space and Time: A Primer for Physicists

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......Philosophy of Space and Time: A Primer for Physicists

Antonio [email protected]

Section de PhilosophieUniversité de Lausanne

October, 2nd 2015

Antonio Vassallo (UNIL) Philosophy of Space and Time October, 2nd 2015 1 / 20

Philosophical Questions about Space and Time

Does space exist?Does time exist?If yes, in what sense do they exist?Are they two sides of the same coin?Does the flow of time have a privileged direction?What are past, present, and future?Are space and time fundamental categories?...

! If we want to address these question, we need to look not only at ourbest worked-out scientific theories, but also at the process of scientificdevelopment that led to them.


Philosophical Questions about Space and Time

“Everything real is somewhere, sometimes”. This is a plausible claim.Space and time seem to be two “pillars” of reality.First appearance of the problem: motion relative to what?Substantivalism: Space and time exist independently of materialobjects & they are not reducible to relations among them.Relationalism: Spatiotemporal facts supervene on facts about thespatial and temporal relations between bodies.Ontic Structuralism: Spacetime is neither a substance, nor a set ofrelations between material bodies, but a structure in its own right.


Newtonian Mechanics

Newton presents the classic statement of spatial and temporal realismin the famous Scholium and, more explicitly, in “De Gravitatione”.Space is a necessary being. Extension is “exceptionally” clearlyconceivable even without bodies.It is not (strictly) a substance, since it affects nothing, but is closer tosubstance than to accident.Newton’s space and time are immutable for metaphysical reasons.Parts of space and time have no “hint of individuality apart from orderand position” so it makes no sense to think they could swap placesand keep heir identity in doing so.


Newtonian Mechanics

Leibniz challenges Newton’s views on space and time.For him, these are metaphysically suspicious structures (“red herring”).Space is just the order of coexisting things.Time is the order of succession of things, i.e. time has to beabstracted from change.His reasoning is based mainly on two metaphysical principles:

...1 The Principle of the Identity of Indiscernibles.

...2 The Principle of Sufficient Reason.

Leibniz mounts two arguments against Newtonian space and time:Argument from static shifts;Argument from kinematic shifts.

Cast in modern terms, the conclusion of both arguments is the same:commitment to space and time entails commitment to ontologicallydifferent yet physically indistinguishable universal states of affairs.


Newtonian Mechanics

Newton’s reply: we cannot give up the commitment to absolute spaceand time because they have physically observable consequences notaccountable just in terms of instantaneous spatial relations.Bucket argument:

Conclusion: relational motion is neither necessary nor sufficient toestablish inertial effects.


Newtonian Mechanics

Rotating globes argument:

Conclusion: Inertial effects are not reducible to instantaneous relations.


Newtonian Mechanics

Later replies:Enriched relationalism.

Relationalism plus Humeanism.Mach/Barbour.


General Relativity

Einstein’s field equations:

G[g] = κT[g,Φ].

G[g] is the Einstein’s tensor. It encodes information about the4-dimensional spacetime geometry.T[g,Φ] is the stress-energy tensor. It encodes information about thedistribution of a matter field Φ in spacetime.A model of GR is a solution of Einstein’s field equation.It is a triple < M, g,T >.


General Relativity

Einstein’s field equations:

G[g] = κT[g,Φ].

Space acts on matter, telling it how to move. In turn, matterreacts back on space, telling it how to curve.(C.W. Misner, K.S. Thorne, J.A. Wheeler - Gravitation. Freeman & Co., 1973, p.5)


General Relativity

It is commonly said that GR unifies gravity and physical geometry.GR reduces gravitational forces to curvature effects of spacetimegeometry.In this sense, GR succeeds in “geometrizing away” gravity.Prima facie, it seems that GR gives us a picture of spacetime asbearing genuine physical properties.This might hint at the fact that, in GR, spacetime (or, better,spacetime points) is something that exists over and above materialfields.


General Relativity

Main arguments for substantivalism in GR:GR admits cosmological models of the form < M, g, 0 >, which meansthat it makes physical sense in GR to think about a universe totallydeprived of matter, where still there is spacetime.There are several models of the form < M, g, 0 >, each of which isphysically distinguishable from the others (e.g. different curvatures),this means that the properties of spacetime are not ontologicallyparasitic on matter fields.Matter fields require spacetime in order to be defined. Thestress-energy tensor is basically a specification of material propertiesinstantiated at spacetime points.


General Relativity

Replies from relationalists:Agreed, GR admits empty solutions. But once the theory is properlyinterpreted, such solutions turn out to be mere mathematicalpossibilities.Agreed, we need points to which attach field values but, again, this isjust a mathematical construction. There are no prima facie obstaclesto interpret physically this picture as a web of relations amongmaterial relata. (In particular, no relata → no relations).

All of the above are “defensive” arguments. Let us now turn torelationalists’ counter-attack.


General Relativity

.Diffeomorphism..

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A diffeomorphism over a (neighborhood of a) manifold M is a functionf : M → M such that:

It is bijective (i.e. one-to-one and onto).It is differentiable together with its inverse f −1.

Just to have a rough idea, you can visualize a diffeomorphism as acontinuous deformation of the manifold..Gauge theorem for GR (Substantive General Covariance)..

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If < M, g,T > is a model and f : M → M is a diffeomorphism, then if we“deform” < M, g,T > with f , we obtain another model < M, f ∗g, f ∗T >that is physically indistinguishable from the starting one.


The Hole Argument in General Relativity

Let us start with a nice and well-behaved picture compatible with thedynamics of GR:

Now let us consider a spacetime point E and “carve a hole” around it wheresome diffeomorphism acts:



Question: Do the above pictures differ in some respect?Substantivalist reply: Yes, they differ in the central galaxy’s trajectory.But the two pictures agree on all physically observable aspects! (Sowhether the trajectory passes or not through E is not a matter ofobservable fact).Hence, the substantivalist is committed to the existence of states ofaffairs which are ontologically different yet physically indistinguishable.This leads to indeterminism: the two pictures fully agree on the initialdata but disagree on the subsequent dynamical development.



Possible reactions:This is an instance of metaphysical indeterminism, so the argumenthas no disruptive consequences for the physical picture.Modify the notion of determinism involved in order to avoid the threat.Recognize that the argument involves a straw man, i.e. it is directedagainst a very naive and untenable version of substantivalism, namely,bare manifold substantivalism:

What we reify is not just the manifold M, but the manifold plus themetrical field g!



Two main lines of attack:Spacetime essentialism: Spacetime points bear their metricalproperties and relations essentially.

There is at most one metaphysically possible model among the twoconsidered.

(Moderate) Spacetime structuralism: Spacetime is a web of relationswhose relata (spacetime points) have no intrinsic identity.Furthermore, both relata and relations are ontologically on a par.

Diffeomorphisms do not alter the structure of spacetime, so the twomodels represent one and the same scenario.

Is spacetime structuralism really a substantivalistic stance? Or is itrelationalism in disguise? Or, perhaps, is it a tertium quid?


References

Earman, J. (1989). World enough and space-time. Absolute versusrelational theories of spacetime. The MIT Press.

Friedman, M. (1983). Foundations of Space-Time Theories. RelativisticPhysics and Philosophy of Science. Princeton University Press.

Maudlin, T. (2012). Philosophy of physics: space and time. PrincetonUniversity Press.

Nerlich, G. (2003). Space-time substantivalism. In D. Loux,M.J. Zimmerman (Ed.), The Oxford Hanbook of Metaphysics,Chapter 10, pp. 281–314. Oxford University Press.

Norton, J. (2015). The hole argument. The Stanford Encyclopedia ofPhilosophy . http://plato.stanford.edu/archives/fall2015/entries/spacetime-holearg/.

Sklar, L. (1974). Space, Time and Spacetime. University of CaliforniaPress.


http://plato.stanford.edu/archives/fall2015/entries/spacetime-holearg/

http://plato.stanford.edu/archives/fall2015/entries/spacetime-holearg/

.Image Credits..

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Image on slide 6 c⃝Hrvoje Crvelin.

Image on slide 7 c⃝Soshichi Uchii.

Image on slide 8 c⃝Michael Friedman.

Image on slide 10 was generated with Mathematica’s notebook “Gravitation versuscurved spacetime” contributed by B.L. Blinder:http://demonstrations.wolfram.com/GravitationVersusCurvedSpacetime/.

Images on slides 15, 16 c⃝John Norton.


http://demonstrations.wolfram.com/GravitationVersusCurvedSpacetime/

Philosophy of Space and Time: A Primer for Physicists

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