Philosophy of Space and Time: a Primer for Physicists

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

Section de Philosophie Université 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 our best worked-out scientific theories, but also at the process of scientific development that led to them.

Antonio Vassallo (UNIL) Philosophy of Space and Time October, 2nd 2015 2 / 20 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 material objects & they are not reducible to relations among them. Relationalism: Spatiotemporal facts supervene on facts about the spatial and temporal relations between bodies. Ontic Structuralism: Spacetime is neither a substance, nor a set of relations between material bodies, but a structure in its own right.

Antonio Vassallo (UNIL) Philosophy of Space and Time October, 2nd 2015 3 / 20 Newtonian Mechanics

Newton presents the classic statement of spatial and temporal realism in the famous Scholium and, more explicitly, in “De Gravitatione”. Space is a necessary being. Extension is “exceptionally” clearly conceivable even without bodies. It is not (strictly) a substance, since it aﬀects nothing, but is closer to substance 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 order and position” so it makes no sense to think they could swap places and keep heir identity in doing so.

Antonio Vassallo (UNIL) Philosophy of Space and Time October, 2nd 2015 4 / 20 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 be abstracted 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 ontologically different yet physically indistinguishable universal states of affairs.

Antonio Vassallo (UNIL) Philosophy of Space and Time October, 2nd 2015 5 / 20 Newtonian Mechanics

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

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

Antonio Vassallo (UNIL) Philosophy of Space and Time October, 2nd 2015 6 / 20 Newtonian Mechanics

Rotating globes argument:

Conclusion: Inertial eﬀects are not reducible to instantaneous relations.

Antonio Vassallo (UNIL) Philosophy of Space and Time October, 2nd 2015 7 / 20 Newtonian Mechanics

Later replies: Enriched relationalism.

Relationalism plus Humeanism. Mach/Barbour.

Antonio Vassallo (UNIL) Philosophy of Space and Time October, 2nd 2015 8 / 20 General Relativity

Einstein’s ﬁeld equations:

G[g] = κT[g, Φ].

G[g] is the Einstein’s tensor. It encodes information about the 4-dimensional spacetime geometry. T[g, Φ] is the stress-energy tensor. It encodes information about the distribution 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 >.

Antonio Vassallo (UNIL) Philosophy of Space and Time October, 2nd 2015 9 / 20 General Relativity

Einstein’s ﬁeld equations:

G[g] = κT[g, Φ].

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

Antonio Vassallo (UNIL) Philosophy of Space and Time October, 2nd 2015 10 / 20 General Relativity

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

Antonio Vassallo (UNIL) Philosophy of Space and Time October, 2nd 2015 11 / 20 General Relativity

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

Antonio Vassallo (UNIL) Philosophy of Space and Time October, 2nd 2015 12 / 20 General Relativity

Replies from relationalists: Agreed, GR admits empty solutions. But once the theory is properly interpreted, such solutions turn out to be mere mathematical possibilities. Agreed, we need points to which attach ﬁeld values but, again, this is just a mathematical construction. There are no prima facie obstacles to interpret physically this picture as a web of relations among material relata. (In particular, no relata → no relations). All of the above are “defensive” arguments. Let us now turn to relationalists’ counter-attack.

Antonio Vassallo (UNIL) Philosophy of Space and Time October, 2nd 2015 13 / 20 General Relativity

. Diffeomorphism . A diffeomorphism over a (neighborhood of a) manifold M is a function f : 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 a continuous deformation of the manifold. . Gauge theorem for GR (Substantive General Covariance) . 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.

Antonio Vassallo (UNIL) Philosophy of Space and Time October, 2nd 2015 14 / 20 The Hole Argument in General Relativity

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

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

Antonio Vassallo (UNIL) Philosophy of Space and Time October, 2nd 2015 15 / 20 The Hole Argument in General Relativity

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! (So whether the trajectory passes or not through E is not a matter of observable fact). Hence, the substantivalist is committed to the existence of states of affairs which are ontologically different yet physically indistinguishable. This leads to indeterminism: the two pictures fully agree on the initial data but disagree on the subsequent dynamical development.

Antonio Vassallo (UNIL) Philosophy of Space and Time October, 2nd 2015 16 / 20 The Hole Argument in General Relativity

Possible reactions: This is an instance of metaphysical indeterminism, so the argument has 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 directed against 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 the metrical ﬁeld g!

Antonio Vassallo (UNIL) Philosophy of Space and Time October, 2nd 2015 17 / 20 The Hole Argument in General Relativity

Two main lines of attack: Spacetime essentialism: Spacetime points bear their metrical properties and relations essentially. There is at most one metaphysically possible model among the two considered. (Moderate) Spacetime structuralism: Spacetime is a web of relations whose 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 two models represent one and the same scenario. Is spacetime structuralism really a substantivalistic stance? Or is it relationalism in disguise? Or, perhaps, is it a tertium quid?

Antonio Vassallo (UNIL) Philosophy of Space and Time October, 2nd 2015 18 / 20 References

Earman, J. (1989). World enough and space-time. Absolute versus relational theories of spacetime. The MIT Press. Friedman, M. (1983). Foundations of Space-Time Theories. Relativistic Physics and Philosophy of Science. Princeton University Press. Maudlin, T. (2012). Philosophy of physics: space and time. Princeton University 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 of Philosophy. http://plato.stanford.edu/archives/fall2015/ entries/spacetime-holearg/. Sklar, L. (1974). Space, Time and Spacetime. University of California Press.

Antonio Vassallo (UNIL) Philosophy of Space and Time October, 2nd 2015 19 / 20 . Image Credits . Image on slide 6 ⃝c Hrvoje Crvelin. Image on slide 7 ⃝c Soshichi Uchii. Image on slide 8 ⃝c Michael Friedman. Image on slide 10 was generated with Mathematica’s notebook “Gravitation versus curved spacetime” contributed by B.L. Blinder: http://demonstrations.wolfram.com/GravitationVersusCurvedSpacetime/. Images on slides 15, 16 ⃝c John Norton. .

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