Maybe we can create spacetime

Suppose a substantivalist view of spacetime on which points of spacetime really exist.

Suppose I had taken a different path to my office today. Then the curvature of spacetime would have been slightly different according to General Relativity.

Question: Would spacetime have had the same points, but with different metric relations, or would spacetime have had different points with different metric relations?

If we go for the same points option, then we have to say that the distance between two points is not an essential property of the two points. Moreover it then turns out that spacetime has degrees of freedom that are completely unaccounted for in General Relativity, degrees of freedom that specify ``where’’ (with respect to the metric) our world’s points of spacetime would be in counterfactual situations. This makes for a much more complicated theory.

If we go for the different points option, then we have the cool capability of creating points of spacetime by waving our arms. While this is a little counterintuitive, it seems to me to be the best answer. Perhaps the best story here is that points of spacetime are individuated by the limiting metric properties of the patches of spacetime near them and by their causal history.

Substantivalism and locality

I find myself going back and forth between substantivalism and relationalism about spacetime. On a substantivalist theory, the points of spacetime are real.

But here is a problem. It seems essential to the concept of a point that geometric relations between points are essential to them. If two points are a certain distance apart, say, then they couldn’t be a different distance apart. But on General Relativity, where geometric properties are determined by the distribution of mass-energy in the universe, if geometric relations between points are essential to them, locality is violated. For imagine two events that are distantly spacelike separated. Then the geometric relation between the points at which the events are found depends on the distribution of mass-energy between the events. If the geometric properties are essential to the points, then influencing the mass-energy between the events will affect which points these events happen at. And that will be a non-local influence.

Perhaps we can say that only local geometric relations are essential to points. Perhaps the way to say this is that if a point x exists in worlds w₁ and w₂, then there is a set N of points such that every member of N exists in both worlds, and N is a neighborhood of x in both worlds, and the geometry on N is the same in both worlds.

Spacetime and Aristotelianism

For a long time I’ve been inclining towards relationalism about space (or more generally spacetime), but lately my intuitions have been shifting. And here is an argument that seems to move me pretty far from it.

Given general relativity, the most plausible relationalism is about spacetime, not about space.

Given Aristotelianism, relations must be grounded in substances.

Here is one possibility for this grounding:

1. All spatiotemporal relations are symmetrically grounded: if x and y are spatiotemporally related, then there is an x-to-y token relation inherent in x and a y-to-x token relation inherent in y.

But this has the implausible consequence that there is routine backwards causation, because if I walk a step to the right, then that causes different tokens of Napoleon-to-me spatiotemporal relations to be found in Napoleon than would have been found in him had I walked a step to the left.

So, we need to suppose:

2. Properly timelike spatiotemporal relations are grounded only in the later substance.

But what about spacelike spatiotemporal relations? Presumably, they are symmetrically or asymmetrically grounded.

If they are symmetrically grounded, then we have routine faster-than-light causation, because if I walk a step to the right, then that causes different tokens of x-to-me spatiotemporal relations to be found in distant objects throughout the universe.

Moreover, on the symmetric grounding, we get the odd consequence that it is only the goodness of God that guarantees that you are the same distance from me as I am from you.

If they are asymmetrically grounded, then we have arbitrariness as to which side they are grounded on, and it is a regulative ideal to avoid arbitrariness. And we still have routine faster-than-light causation. For presumably it often happens that I make a voluntary movement and someone on the other side of the earth makes a voluntary movement spacelike related to my movement (because there are so many people!), and now wherever the spatiotemporal relations is grounded, it will have to be affected by the other’s movement.

I suppose routine faster-than-light causation isn’t too terrible if it can’t be used to send signals, but it still does seem implausible. It seems to me to be another regulative ideal to avoid nonlocality in our theories.

What are the alternatives to relationalism? Substantivalism is one. We can think of spacetime as a substance with an accident corresponding to every point. And then we have relationships to these accidents. There is a lot of technical detail to work out here as to how the causal relationships between objects and spacetime points and the geometry of spacetime work out, and whether it fits with an Aristotelian view. I am mildly optimistic.

Another approach I like is a view on which spacetime position is a nonrelational position determinable accident. Determinable accidents have determinates which one can represent as values. These values may be numerical (e.g., mass or charge), but they may be more complex than that. It’s easiest in a flat spacetime: spacetime position is then a determinable whose determinates can be represented as quadruples of real numbers. In a non-flat spacetime, it’s more complicated. One option for the values of determinate positions is that they are “pointed spacetime manifold portions”, i.e., intersections of a Lorentzian manifold with a backwards lightcone (with the intended interpretation that the position of the object is at the tip of the lightcone). (What we don’t want is for the positions to be points in a single fixed manifold, because then we have backwards causation problems, since as I walk around, the shifting of my mass affects which spacetime manifold Napoleon lived in.)

Holes and substantivalism

Suppose substantivalism about space is correct. Imagine now that the following happens to a slice of swiss cheese: the space where the holes were suddenly disappears. I don’t mean that the holes close up. I mean that the space disappears: all the points and regions that used to be in the hole are no longer there (and any air that used to be there is annihilated). The surfaces of the cheese that faced the hole now are at an edge of space itself.

The puzzle now is that in this story we have an inconsistent triad:

1. There is no intrinsic change in the cheese.
2. The slice of cheese no longer has holes.
3. Changing with respect to whether you have holes is intrinsic.

Here are my arguments for the three claims. There is no intrinsic change in the slice of cheese as something outside the cheese has changed—space has been annihilated. The slice of cheese no longer has holes, as it makes sense to talk of the size or shape or volume of a hole, but there is no size or shape or volume where there is no space. And changing with respect to whether you have holes is change of shape, and changes of shape are intrinsic.
It seems that the above story forces you to reject one of the following:

4. Substantivalism about space
5. Intrinsicness of shape.

But there is another way out. Deny (3). Whether you have holes is not intrinsic. What is intrinsic is your topological genus with respect to your internal space and similar topological properties.

Note, also, a lesson relevant to the famous Lewis and Lewis paper on holes: the counting of holes should not involve the counting of regions, but the computation of a numerical invariant, namely the genus.

Substantivalism about space is a kind of relationalism

It’s just occurred to me that substantivalist views of space or spacetime are actually relationalist: they define location by relations between objects. It’s just that they introduce one or more additional objects—say, points or space or spacetime—to fill out the theory. An entity’s being located is then a matter of the entity standing in a certain relation to one or more of these additional objects.

Moreover, a substantivalist theory couched in terms of points may have to be even closer to relationalism, in that it may need to say that what makes the points be points of the kind of space or spacetime they are points of are their mutual spatial or spatiotemporal relations.

What has a hope of being a more radical alternative to relationalist theories are property theories, on which being in a location is a property very much like having a certain electric charge—the only difference being that the location properties have a three- or four-dimensional structure while the charge properties have a one-dimensional structure. Of course, having properties will be a matter of relation on heavy-weight Platonism and on trope theories, but these relations are not special spatial or spatiotemporal relations, but just general-purpose relations like instantiation or inherence.

Of course, maybe we don’t want an alternative to relationalism because we like relationalism.

Spacetime: Beyond substantivalism and relationalism

According to substantivalism, spacetime or its points or regions is a substance, and location is a relation between material things and spacetime or its points or regions. According to relationalism, location is constituted by relations between material things. Often, the two views are treated as an exhaustive division of the territory.

But they're not. Lately, I've found myself attracted to a tertium quid which I know is not original (it's a story other people, too, have come to by thinking about the analogy between location and physical qualities like charge or mass). On a simplified version of this view, being located is a determinable unary property. Locations are simply determinates of being located. This picture is natural for other physical qualities like charge. Having charge of 7 coulombs is not a matter of being related to some other substances--whether other charged substances or some kind of substantial "chargespace" or its points or regions. It's just a determinate of the determinable having charge.

This determinate-property view is more like the absolutism of substantivalism, but differs from substantivalism by not positing any "spacetime substance", or by making the locations into substances. Locations are determinates of a property, and hence are properties rather than substances. If nominalism is tenable for things like charge or mass, the theory won't even require realism about locations.