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

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Second, the dimensionality diﬀerence between time and space doesn’t seem deep enough to be the fundamental diﬀerence between the two. I take it that the fundamental diﬀerence between space and time will illuminate the other, less funIn Minkowski spacetime the maximum dimension of a subspace containing only null vectors (lightlike vectors and the zero vector) is one. But Dimension does not entail that these vectors are timelike. In section 3 I argued that null vectors are not timelike on geometrical grounds alone; Dimension, like the other principles I will later advance, is only meant to determine which of the remaining vectors are timelike. (Recall I said in section 3 that talk of two vectors being of the same kind is reserved for vectors that (unlike null vectors) are either spatial or temporal.) Judith Thompson tries to describe such a world in her (1965). (This example is discussed in more detail in (MacBeath 1993).) It’s a world in which two people disagree about the temporal order of certain pairs of events. Each person thinks he outlives the other. The argument that time is two-dimensional in this world goes, in outline, like this. (1) There is no way to temporally order events on a line so that both people are right. (2) But both are equally well-placed to determine the temporal order of events, so we do not want to say that one is right and the other wrong about the temporal order. So (3) the events cannot be temporally ordered on a line; time has more than one dimension.

But this conclusion does not follow. Time can be one-dimensional even if there is no uniquely correct way to temporally order events on a line. In Minkowski spacetime, for example, two observers can disagree about the temporal order of events. But time is still one-dimensional in that spacetime.

damental, diﬀerences between them, and help us explain those other diﬀerences.

But the diﬀerence in dimensionality doesn’t do this.

Third, and most important, it seems possible that both time and space be onedimensional. But in a possible world with two-dimensional Minkowski spacetime, all the (non-null) vectors meet the condition in Dimension. So Dimension entails that all (non-null) directions in that world are temporal, and so (since no direction is both spatial and temporal) that that world contains no spatial directions at all. But that can’t be right: surely it’s possible that special relativity be true and that time and space both be one-dimensional.

The problem is that while vectors of one kind satisfy the condition in Dimension, vectors of the other kind do as well; while we have already established that all timelike vectors are of the same kind. Someone who maintains that dimensionality is the only factor that does any work to distinguish timelike from spacelike directions, then, must admit that the condition in Dimension is necessary but not suﬃcient for a direction to be timelike; and that no condition is suﬃcient in every

world. He might then revise his view as follows:

–  –  –

In worlds where just one kind of vector satisﬁes the condition in Dimension, then, Dimension and Dimension∗ agree that vectors of that kind are timelike. But in worlds with two-dimensional spacetimes Dimension∗ entails that it is indeterminate which kind of vector is timelike.

In the end, though, this move to indeterminacy fails. It fails not because I insist that it must be perfectly determinate in every world which directions are timelike. But surely in some two-dimensional worlds, complicated worlds in which plenty is happening, there is a fact of the matter about which directions are temporal.

So we should reject Dimension∗ as well as Dimension.

I have looked at two ways to distinguish temporal from spatial directions in geometrical terms, and found reasons for rejecting both. Might some other attempt to distinguish them in geometrical terms succeed where these have failed? The answer is ‘no.’ The two-dimensional spacetimes that make trouble for Dimension also make trouble for any attempt to use geometry alone to distinguish temporal from spatial directions. For the roles that timelike and spacelike directions play in the geometry of two-dimensional Minkowski spacetime (and two-dimensional Newtonian spacetime) are symmetric. Since timelike and spacelike directions play symmetric roles in the two-dimensional spacetime geometries, any attempt to distinguish temporal from spatial directions by isolating a geometrical role that one but not the other plays is bound to fail.

So what else other than or in addition to the geometry of spacetime makes the diﬀerence between spacelike and timelike directions?

5 Laws of Nature

Timelike and spacelike directions play diﬀerent roles in the laws of physics that we have taken seriously as the fundamental laws governing our world. Those laws govern the evolution of the world in timelike directions, but not in spacelike directions.

This claim might look analytic (‘of course evolution happens in time’), but I’m using ‘govern the evolution of the world’ in a stipulated sense that does not build time into its meaning. The laws govern the evolution of the world in some direction just in case the laws, together with information about what is going on in some region of spacetime, yield information about what is going on in regions of spacetime that lie in that direction from the initial region.

My meaning can be made more precise using an example. Earlier I said that the spacetimes of Newtonian mechanics and special relativity, as well as some of the spacetimes of general relativity, can be partitioned into a sequence of times— a sequence of three-dimensional submanifolds. Now in Newtonian gravitational theory, given information about what is going on on one time, the laws determine what is going on on the rest of the times.8 These laws govern the evolution of the I am pretending here (for purposes of illustration) that Newtonian gravitational world from one time to the others. And a similar claim is true for other laws we’ve considered fundamental.

Now, timelike vectors are not tangent to any time, on any way of partitioning any given spacetime into times. Rather, no matter which partitioning of spacetime into times you use, timelike vectors point from one time toward others. So timelike vectors point in the directions in which the laws govern the evolution of the world.9 The same is not true of points of space. If I know what is going on right here (at this location in space) for all time, the laws tell me nothing about what is going on anywhere else at any time. The laws do not govern the evolution of the world in spacelike directions.

I used the laws of Newtonian gravitational theory as an example, and these laws are deterministic. If some laws are deterministic, then given information about the state of the world on some time, those laws yield complete information about the state of the world on other times. But not all possible laws are deterministic;

on some interpretations the laws of quantum mechanics are an example of indeterministic laws. But even here there is a diﬀerence between the roles timelike and spacelike directions play in the laws: given information about the state of the world at a time, these laws assign probabilities to possible states at other times; but they do not do so for points of space.

(There is another role that timelike directions play in some familiar laws that spacelike directions do not: quantities like mass, charge, and energy are conserved in timelike directions, but not in spacelike directions. But when there are conservation laws like this, they are usually derived from the dynamical laws (as, for example, the law of conservation of charge follows from Maxwell’s equations—the dynamical laws for electromagnetism). So I need not explicitly mention this as a theory is deterministic, even though there are good arguments that it is not. Earman discusses these arguments in his (1986).

As I’ve said, not every general relativistic spacetime can be partitioned into times. But the laws of general relativity still govern the evolution of worlds with those spacetimes in timelike directions at a local level: some four-dimensional regions of those spacetimes can be partitioned into times, and (if the region is the right shape) the laws determine what is happening at all times given information about what is happening at one time.

second role in the laws that distinguishes timelike from spacelike directions.) It is not controversial that timelike directions play these roles in the laws with which we are most familiar. I propose that we take these roles as constitutive: what it is to be a timelike direction is to play these roles in the laws. To be explicit, the

proposal is this:

Laws: Any direction (that is either spatial or temporal according to the geometry) in which the laws govern the evolution of the world is a timelike direction.

First, I do not claim that it is necessary that the laws of nature govern the evolution of the world in some direction(s) in spacetime.10 I do not claim, that is, that it is necessary that there be some timelike role in the laws to be ﬁlled. Perhaps there are possible worlds with strange laws of nature in which there is no such role.

But I do claim that in such worlds no direction is a timelike direction.

Second, my proposal presupposes that the laws of nature are more fundamental than the distinction between timelike and spacelike directions. It presupposes, roughly, that it is possible to state the laws of nature without using the words ‘time’ and ‘space.’ For if in order to state the laws we had to presuppose that time and space had already been distinguished, then it would be going in a circle to then appeal to the laws to distinguish time from space. This is not a problem, though. We standardly state laws without appealing to the distinction between time and space.

In formal presentations of, say, Newtonian gravitational theory, the distinction between timelike and spacelike directions is always made in some informal remarks after the author has described the geometrical structure of spacetime and before he writes down the equations of the theory. But the equations can be understood perfectly well with the informal remarks removed.

Or, to take a more concrete case, consider Newton’s ﬁrst law: a body not acted on by any forces moves with a constant velocity. ‘Velocity’ means the same as ‘change in spatial location with respect to change in temporal location.’ I claim that we can re-write this law to remove the reference to space and time. We can do this by using quantiﬁers: ﬁrst, replace ‘velocity’ with ‘change in location along the I’m using ‘the laws of nature’ non-rigidly.

Figure 4.5:

–  –  –

x direction with respect to change in location along the y direction’; then, preface the laws with the quantiﬁcational phrase, ‘there are two (distinct) directions, x and y, such that x and y play such-and-such geometrical roles, and....’ 6 Testing Our Intuitions Let’s take a look at a particular world with a two-dimensional spacetime and see if we agree that the laws distinguish space from time even though the geometry of spacetime does not. Consider the spacetime diagram in ﬁgure 4.5.

This diagram depicts the distribution of matter in spacetime in some possible world. (Suppose that spacetime has the familiar Newtonian structure, so that talk of space and time makes sense.) Normally we read the vertical axis of spacetime diagrams as the time axis. I ask you to forget about that convention for now and suppose you do not know which axis represents time. I contend that if you do not know what the laws are, you are unable to tell which axis is time. And that is evidence that it is the laws that are doing the work to make one axis time.

It certainly seems that there are possible worlds correctly represented by this diagram in which the vertical axis represents time; and also possible worlds correctly represented by this diagram in which the horizontal axis represents time. To make this plausible, let me describe for you one world of each kind. Call the world in which the vertical axis represents time ‘the vertical world,’ and the other, ‘the

horizontal world.’ First, a description of the vertical world:

Vertical: there are two particles that accelerate toward each other, until they meet in an elastic collision and rebound back the way they came;

then they slow down, turn around, and accelerate back toward each other, repeating this cycle for all time.

And here’s a description of the horizontal world:

Horizontal: for a long time nothing happens. Then an inﬁnite number of pairs of particles come into existence; one member of each pair moves oﬀ to the right, the other to the left. Each particle bounces oﬀ a particle coming from the other direction, then is annihilated in a collision with the particle with which it was created. Then nothing happens for the rest of time.

By itself the diagram doesn’t favor one of these descriptions over the other.

Things are diﬀerent, I think, when I tell you what laws of nature govern the world the diagram depicts. Let the laws be Newton’s three laws of mechanics and a slightly amended version of Newton’s law of universal gravitation.11 Of course Newton’s laws contain terms like ‘velocity’ and ‘acceleration’ which are deﬁned in According to Newton’s law the force between two bodies is inversely proportional to the square of the distance between them. As the distance between two bodies goes to zero, the force gets inﬁnitely large. A natural way to extend Newton’s laws to deal with this case is to have the particles bounce oﬀ each other in a perfectly elastic collision.

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