# «Once Upon a Spacetime by Bradford Skow A dissertation submitted in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy ...»

Once Upon a Spacetime

by

Bradford Skow

A dissertation submitted in partial fulﬁllment

of the requirements for the degree of

Doctor of Philosophy

Department of Philosophy

New York University

September, 2005

Hartry Field

c Bradford Skow

All Rights Reserved, 2005

Acknowledgements

Thanks to my advisors Gordon Belot, Cian Dorr, and Hartry Field, for their comments, suggestions, and criticisms, and for providing models of what good philosophy can and should be. Thanks to my parents—all of them—for supporting me through graduate school. And special thanks to Peter Graham for the frequent meetings and for the discussion that led to the writing of chapter 4.

iv Abstract This dissertation concerns the nature of spacetime. It is divided into two parts.

**The ﬁrst part, which comprises chapters 1, 2, and 3, addresses ontological questions:**

does spacetime exist? And if so, are there any other spatiotemporal things? In chapter 1 I argue that spacetime does exist, and in chapter 2 I respond to modal arguments against this view. In chapter 3 I examine and defend supersubstantivalism— the claim that all concrete physical objects (tables, chairs, electrons and quarks) are regions of spacetime.

Four-dimensional spacetime, we are often told, ‘uniﬁes’ space and time; if we believe in spacetime, then we do not believe that space and time are separately existing things. But that does not mean that there is no distinction between space and time: we still distinguish between the spatial aspects and the temporal aspects of spacetime. The second part of this dissertation, comprising chapter 4, looks at this distinction. How is it made? In virtue of what are the temporal aspects of spacetime temporal, rather than spatial? The standard view is that the temporal aspects of spacetime are temporal because they play a distinctive role in the geometry of spacetime. I argue that this view is false, and that the temporal aspects are temporal because they play a distinctive role in the geometry of spacetime and in the laws of nature.

v Table of Contents Acknowledgements iv

**Abstract**

v Table of Contents vi List of Figures ix 1 An Argument for Substantivalism 1 1 The Standard Argument........................ 1 2 Characteriz

An Argument for Substantivalism 1 The Standard Argument Distinguish relationalism about motion from relationalism about ontology. Relationalists about ontology deny that spacetime exists. In doing so they oppose substantivalists, who aﬃrm that spacetime exists.1 (When I use it without qualiﬁcation, ‘relationalism’ means the same as ‘relationalism about ontology.’) Relationalists about motion assert that all motion is the relative motion of bodies. In doing so they oppose absolutists, who aﬃrm that there are some states of motion that are not states of motion relative to this or that material object.

I will work in the context of pre-relativistic physics. The standard argument

**for substantivalism in this context, going back to Newton, has two premises:**

Some philosophers claim that relationalists do not deny the existence of spacetime, but merely assert that spacetime is a logical construction from the spatiotemporal relations among material objects. (Forbes (1987) is an example.) I’ve always thought that to say that x’s are logical constructions is just another way to say that x’s don’t exist. Speaking in terms of ‘logical constructions’ obscures the debate.

(1) Relationalism about motion is false.

(2) If relationalism about motion is false, then spacetime exists.

To argue for (1), substantivalists claim that no adequate physical theory is consistent with relationalism about motion. Newtonian gravitational theory has a well-posed initial value problem: there is a unique future evolution from any complete state of the world at a time.2 But no competing theory of gravitation that is consistent with relationalism about motion has a well-posed initial value problem. And so, given the availability of Newton’s theory, no relationalist theory can be adequate.

The argument that no theory that is consistent with relationalism about motion has a well-posed initial value problem has premises. First we suppose (i) that our world, and all physically possible worlds, are worlds of massive point particles.

Then if relationalism about motion is true, the history of the world is a history of inter-particle distances. We also suppose (ii) that Newtonian gravitational theory is empirically adequate and a good guide to physical possibility. That is, we suppose that one of the solutions to the equations of Newtonian gravitational theory captures the actual history of inter-particle distances; and that each solution captures a physically possible history of inter-particle distances. Finally we suppose (iii) that if relationalism about motion is true, then to give the complete state of the world at a time is to specify the distances between each pair of particles and the velocity of each particle relative to each other.

Given what we have supposed, there can be no alternative to Newtonian gravitational theory that both is consistent with relationalism about motion and has a well-posed initial value problem. For any complete state of the world at a time— for any speciﬁcation of the distances between and relative velocities of each pair of particles—there is more than one physically possible future evolution. Here is an example (from Newtonian gravitational theory supplemented with Hooke’s law).

Consider a (physically) possible world that contains just two lead balls connected by a spring. At some given time the distance between them is one meter and the rate at which this distance is changing is zero. According to Newtonian mechanics, Setting aside problems with space invaders (Earman 1986) and collision singularities.

there are at least two physically possible futures: either the two balls are rotating around their center of mass at such a rate that the ‘centrifugal force’ exactly balances the force from the spring, and so they remain at relative rest for all time; or they are not rotating, and so the spring collapses and then expands, and the initial time was at the moment when the balls turn from moving apart to moving back together.3 (Absolutists diagnose this failure as follows: what relationalists about motion call ‘the complete state of the world at a time’ is not complete. There are distinct instantaneous states of the world that the relationalist cannot distinguish. In the example, there are at least two ways to ‘complete’ the relationalist description of the ball-and-spring system, completions that diﬀer over the value of each ball’s absolute velocity.) That is the argument for (1). The argument for (2) is shorter. If relationalism about motion is false and there are some states of motion that are not states of relative motion, then spacetime exists. For the only way to deﬁne these absolute motions is by reference to spacetime. (A particle is undergoing absolute acceleration at a time, for example, just in case its worldline is not tangent to a geodesic at that time.) Relationalists like Leibniz and Mach claimed that the ﬁrst part of this argument failed. But no relationalist produced a theory with a well-posed initial value problem. Until recently. Julian Barbour has developed such a theory.4 (So where does the argument that there can be no such theory go wrong?

There are diﬀerent ways to develop Barbour’s theory. Diﬀerent developments reject diﬀerent parts of the argument. On one way, relationalists reject (iii), and say instead that the complete state of the universe at a time is given by the distances between and relative velocities of all particles and, in addition, a holistic property of the entire universe at that time (holistic because not reducible to the properties of its parts), its angular momentum. (Relationalists who take this approach say someI ﬁrst heard this example from David Albert.

This theory is discussed in detail in (Barbour 1999), (Belot 2000), (Butterﬁeld 2002), (Pooley and Brown 2002), among other places. Belot notes that theories like this were discovered as early as 1924.

thing about why we do not need to believe in spacetime in order to make sense of angular momentum.) On another way, relationalists reject (ii), that Newton’s theory is a good guide to physical possibility. They deny that worlds in which the entire universe is rotating, including one where two lead balls connected by a spring are rotating around their center of mass, are physically possible.) With theories like Barbour’s available, substantivalists are in a weaker position. No longer able to argue for (1) by pointing out that relationalists about motion have no physical theory, they must now argue that absolutist theories like Newton’s are better than relationalist theories—and not better because more adequate to the empirical phenomena, for both theories do just as well on that score, but better for

**some more subtle reason. But it is not clear that the absolutist theory is better:**

Barbour (1999) and Pooley and Brown (2002) argue that the relationalist theory is better than Newton’s because it predicts a phenomenon that in Newton’s must be taken as a brute fact: namely, the fact that the universe is not rotating.

To argue for substantivalism I take a diﬀerent approach. In the debate over the standard argument, it is widely assumed that we need spacetime to make sense of absolute acceleration, but that we do not need spacetime to make sense of the distances between and relative velocities of material objects.5 I think this is wrong.

Even if relationalism about motion is true, and we can do physics while recognizing only distances between and relative velocities of material objects, we still need spacetime (I will argue) to make sense of the distances between material objects.

Relationalism about motion demands substantivalism just as much as its denial.

Any adequate characterization of the spatiotemporal structure of the world—even one consistent with relationalism about motion—must make reference to spacetime.

Before outlining my argument in any detail, I will explain this terminology.

2 Characterizing the Spatiotemporal Structure of the World

**Relationalism about motion is an answer to the question:**

(3) What is the spatiotemporal structure of the world?

Sklar (1974) disputes the ﬁrst half of this claim.

Both substantivalists and relationalists must answer this question, though they will not answer it the same way. It is easiest to see what a complete, detailed answer to this question will look like if we look at some substantivalist answers ﬁrst.

Substantivalists believe in spacetime. Just as there are many possible geometries that space may have (it may be Euclidean, or hyperbolic, for example), there are many possible geometries that spacetime may have: it may have a neoNewtonian structure, for example, or a full Newtonian structure. When substantivalists disagree about the answer to (3), then, they are disagreeing about which of these geometries the spacetime in our world has.

A substantivalist might answer (3) by saying that spacetime has a Newtonian geometry. How might a relationalist answer (3)? She denies that spacetime exists, so she cannot give the same answer. But she does not think that (3) is an empty question, and she might even think that, except for their ontological disagreement, the substantivalist’s answer is right.

To see the general form a relationalist answer may take, look more closely at a substantivalist answer. Suppose some substantivalist says that spacetime has a Newtonian geometry. We may ask: in virtue of what does it have that geometry?

And the answer is: it has that geometry because the points of spacetime instantiate certain spatiotemporal relations in a certain pattern. Which spatiotemporal relations? There are many possible answers to this question. Here is one: spacetime has a Newtonian geometry because the points of spacetime instantiate the spatial distance between x and y is r and the temporal interval between x and y is r in a certain pattern. (That is, there are geometrical laws that these relations satisfy.) A relationalist who says that the world has a Newtonian spatiotemporal structure might say, then, that it has this structure (at least in part) because the material objects, rather than the points of spacetime, instantiate certain relations in a certain pattern. (Maybe the relationalist uses the same relations the substantivalist does;

maybe not.) So both substantivalists and relationalists produce lists of spatiotemporal relations and laws governing those relations in order to characterize the spatiotemporal structure of the world. Now, for any given spatiotemporal structure the world might have there are many diﬀerent sets of relations that characterize it. Euclidean space, for example, may be characterized using the relation the distance from x to y is r; or using the relations y is between x and z and x and y are as far apart as z and w; and there are other choices as well. But I am interested in which relations are, according to substantivalists and to relationalists, the fundamental relations that characterize it. Fundamental relations, I assume, have the following feature: facts about the instantiation of non-fundamental relations obtain in virtue of the facts about the instantiation of the fundamental relations.6 Facts about the instantiation of the fundamental relations, on the other hand, are brute, ‘bottom-level’ facts. (The non-fundamental relations, then, supervene on the fundamental relations. Since it is also true that the fundamental relations supervene on themselves, every relation supervenes on the fundamental relations.) Now I can outline my argument in more detail. I will argue that even if the correct answer to (3) entails relationalism about motion, relationalist accounts of which spatiotemporal relations are fundamental are objectionable. So we should prefer substantivalism to relationalism.7 In my argument I impose a constraint on which spatiotemporal relations may be fundamental. To state this constraint I need to introduce some terminology. Say that relations which relate abstract objects to concrete objects (x is n years old is an example) are mixed relations; other relations—relations that relate only abstract