Existential inertia and spacetime

According to the principle of existential inertia:

1. If x exists at t₁ and t₂ > t₁ and there is no cause of x’s not existing at t₂, then x exists at t₂.

This sounds weird, and one way to get at the weirdness for me is to put it in terms of relativity theory. Times are spacelike hypersurfaces. So, then:

2. If x exists somewhere on a spacelike hypersurface H₁ and H₂ is a later spacelike hypersurface and there is no cause of x’s not existing on H₂, then x exists on H₂.

This seems weird to me. Why should being in one specific area of spacetime metaphysically push one to exist in another specific area of spacetime? I can see how existing in one area of spacetime could physically push one to exist in another. But metaphysically? That seems odd.

Metaphysical semiholism

For a while I’ve speculated that making ontological sense of quantum mechanics requires introducing a global entity into our ontology to ground the value of the wavefunction throughout the universe.

One alternative is to divide up the grounding task among the local entities (particles and/or Aristotelian substances). For instance, on a Bohmian story, one could divide up 3N-dimensional configuration space into N cells, one cell for each of the N particles, with each particle grounding the values of the wavefunction in its own cell. But it seems impossible to find a non-arbitrary way to divide up configuration space into such cells without massive overdetermination. (Perhaps the easiest way to think about the problem is to ask which particle gets to determine the value of the wavefunction in a small neighborhood of the current position in configuration space. They all intuitively have “equal rights” to it.)

It just seems neater to suppose a global entity to do the job.

A similar issue comes up in theories that require a global field, like an electromagnetic field or a gravitational field (even if these is to be identified with spacetime).

Here is another, rather different task for a global entity in an Aristotelian context. At many times in evolutionary history, new types of organisms have arisen, with new forms. For instance, from a dinosaur whose form did not require feathers, we got a dinosaur whose form did require feathers. Where did the new form come from? Or suppose that one day in the lab we synthesize something molecularily indistinguishable from a duck embryo. It is plausible to suppose that once it grows up, it will not only walk and quack like a duck, but it will be a duck. But where did it get its duck form from?

We could suppose that particles have a much more complex nature than the one that physics assigns to them, including the power to generate the forms of all possible organisms (or at least all possible non-personal organisms—there is at least theological reason to make that distinction). But it does not seem plausible to suppose that encoded in all the particles we have the forms of ducks, elephants, oak trees, and presumably a vast array of non-actual organisms. Also, it is somewhat difficult to see how the vast number of particles involved in the production of a duck embryo would “divide up” the task of producing a duck form. This is reminiscent of the problem of dividing up the wavefunction grounding among Bohmian particles.

I am now finding somewhat attractive the idea that a global entity carries the powers of producing a vast array of forms, so that if we synthesize something just like a duck embryo in the lab, the global entity makes it into a duck.

Of course, we could suppose the global entity to be God. But that may be too occasionalistic, and too much of a God-of-the-gaps solution. Moreover, we may want to be able to say that there is some kind of natural necessity in these productions of organisms.

We could suppose several global entities: a wavefunction, a spacetime, and a form-generator.

But we could also suppose them to be one entity that plays several roles. There are two main ways of doing this:

1. The global entity is the Universe, and all the local entities, like ducks and people and particles (if there are any), are parts of it or otherwise grounded in it. (This is Jonathan Schaffer’s holism.)

2. Local entities are ontologically independent of the global entity.

I rather like option (2). We might call this semi-holism.

But I don’t know if there is anything to be gained by supposing there to be one global entity rather than several.

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.

Theism and the absolute present

Some people believe in an absolute present. An absolute present would define a privileged absolute reference frame. Suppose that there is an absolute present. Would we have any reason to think that the privileged absolute reference frame is anywhere close to our reference frame? If not, then for all we know, the things around us have an absolute geometry quite different from the one we think they have: that clock on the wall isn’t absolutely a circle, but an oval, say.

If the reason for accepting an absolute present is doing justice to common sense, then we not only need an absolute presnet, but an absolute present that defines a frame close to our frame. And that would be almost literally a version of anthropocentrism.

Of course, if we are in the image and likeness of God, the anthropocentrism may be defensible. And maybe only then.

If this is right, then the A-theory of time (which seems to require an absolute present) makes a lot more sense on theism. (Anecdotally, there is a correlation between being a theist and accepting the A-theory of time.) But on the other hand, the A-theory of time requires God’s beliefs to be changing.

Saving a Newtonian intuition

Here is a Newtonian intuition:

1. Space and time themselves are unaffected by the activities of spatiotemporal beings.

General Relativity seems to upend (1). If I move my hand, that changes the geometry of spacetime in the vicinity of my hand, since gravity is explained by the geometry of spacetime and my hand has gravity.

It’s occurred to me this morning that a branching spacetime framework can restore the Newtonian intuition of the invariance of space. Suppose we think of ourselves as inhabiting a branching spacetime, with the laws of nature being such as to require all the substances to travel together (cf. the traveling forms interpretation of quantum mechanics). Then we can take this branching spacetime to have a fixed geometry, but when I move my hand, I bring it about that we all (i.e., all spatiotemporal substances now existing) move up to a branch with one geometry rather than up to a branch with a different geometry.

On this picture, the branching spacetime we inhabit is largely empty, but one lonely red line is filled with substances. Instead of us shaping spacetime, we travel in it.

I don’t know if (1) is worth saving, though.

Does general relativity lead to non-locality all on its own?

A five kilogram object has the determinable mass with the determinate mass of 5 kg. The determinate mass of 5 kg is a property that is one among many determinate properties that together have a mathematical structure isomorphic to a subset of real numbers from 0 to infinity (both inclusive, I expect). Something similar is true for electric charge, except now we can have negative values. Human-visible color, on the other hand, lies in a three-dimensional space.

I think one can have a Platonic version of this theory, on which all the possible determinate properties exist, and an Aristotelian one on which there are no unexemplified properties. There will be important differences, but that is not what I am interested in in this post.

I find it an attractive idea that spatial location works the same way. In a Newtonian setting the idea would be that for a point particle (for simplicity) to occupy a location is just to have a determinate position property, and the determinate position properties have the mathematical structure of a subset of three-dimensional Euclidean space.

But there is an interesting challenge when one tries to extend this to the setting of general relativity. The obvious extension of the story is that determinate instantaneous particle position properties have the mathematical structure of a subset of a four-dimensional pseudo-Riemannian manifold. But which manifold? Here is the problem: The nature of the manifold—i.e., its metric—is affected by the movements of the particles. If I step forward rather than back, the difference in gravitational fields affects which mathematical manifold our spacetime is isomorphic to. If determinate position properties are tied to a particular manifold, it means that the position of any massive object affects which manifold all objects are in and have always been in. In other words, the account seems to yield a story that is massively non-local.

(Indeed, the story may even involve backwards causation. Since the manifold is four-dimensional, by stepping forward rather than backwards I affect which four-dimensional manifold is exemplified, and hence which manifold particles were in. )

This is interesting: it suggests that, on a certain picture of the metaphysics of location, general relativity by itself yields non-locality.

Existential inertia and relativity

According to the doctrine of existential inertia, objects have a metaphysical tendency to continue existing absent interference. Existential inertia differs from ordinary physical inertia, in that existential inertia is supposed to come from the metaphysics of time and existence, while physical inertia comes from the laws of nature.

Let’s now imagine that Bob is a physical object that pops into existence at some point z₀ in spacetime, in a universe with nothing that would annihilate Bob (and with Bob lacking any means of self-annihilation). Then according to existential inertia, Bob will continue to exist. But what does that mean in a relativistic setting? Times correspond to spacelike hypersurfaces. A time is future with respect to z₀ provided that it corresponds to a spacelike hypersurface intersecting the future light-cone centered on z₀. Thus, what we get is:

1. If Bob exists at z₀, then for every spacelike hypersurface H that intersects the future light-cone of z₀, Bob exists at some location or other on H.

This is strange on two counts. First, it seems odd that for every spacelike hypersurface that intersects the future light-cone of z₀, we have some sort of a metaphysical guarantee that Bob is somewhere on it—not at any particular place, mind you, but somewhere or other on it. Maybe it doesn’t seem as odd to you as it does to me, though.

Perhaps more seriously, however, (1) describes a metaphysical tendency that makes crucial reference to the speed of light.

In fact, (1) seems to somehow make speed-of-light limits have some sort of metaphysical force. For suppose that Bob faster-than-light travels from z₀ to some point z₁ outside the light-cone of z₀. Then after that episode of faster-than-light travel, Bob could continue living a normal slower-than-light life, while violating (1) with respect to a number of hypersurfaces (that intersect the future cone of z₀). Thus, while violations of existential inertia are supposed to come from destructive powers, a violation of (1) can come from simple faster-than-light travel.

If we suppose that the metaphysics of time requires a privileged reference frame, the above problems disappear. They also disappear if we think that objects come along with a privileged internal time sequence, and that existential inertia is defined with respect to that internal time sequence.

A simple causal theory of the arrow of time

Typical causal theories of time say that the order of time is determined by the order of causal relationships between events in time. This tends to be a difficult theory to develop, if only because of the possibilities of simultaneous causation and time travel. But suppose with substantivalists that there really are moments (or intervals) of time. Then it is possible to have a very simple and elegant theory of the order of time:

1. Time u is prior to time v if and only if time u causes time v.

This theory is, I suppose, inspired by Tooley's idea that earlier points of spacetime cause later ones. It has no worries about the possibility of simultaneous causation. For even if there is simultaneous causation, there is no simultaneous causation between times, since distinct times can never be simultaneous (if u and v are times and they are simultaneous, they are the same time--and of course nothing can cause itself). Likewise, while there is some cost in denying time travel and backwards causation, one can accept time travel and backwards causation while denying that a later time can cause an earlier one.

There is also an interesting and slightly more complex variant:

2. Time u is prior to time v if and only if some event at u at least partially causes v.

This variant, too, is compatible with simultaneous and backwards causation between events--it just rules out simultaneous or backwards causation between an event and a time. It has the advantage that perhaps the dynamic evolution of times isn't just causally driven by earlier times, but by the events at those earlier times. For instance, if a time is a maximal spacelike hypersurface, it is very natural to think on the grounds of General Relativity that the distribution of matter at earlier times causes that hypersurface. We can, further, deem each time's existence to be an event that happens at that time. If we do that, then (2) becomes a generalization of (1).

General Relativity and the dimension of reality

When people first hear of General Relativity and the idea that spacetime is curved, they naturally imagine a higher-dimensional uncurved space in which our four-dimensional spacetime curvily sits. They may even be shown pictures of a two-dimensional sheet warping into three dimensions. But it's usually then explained to them that that's a misleading way to look at things: reality is just four-dimensional, but has a metric that makes it behave like it's curving in a higher-dimensional space.

What if the natural way of thinking about this is right? What if, say, reality is an 89-dimensional Euclidean space with signature (2,87), but physical objects are constrained to live on a 4-dimensional subset of it? The constraint could be effected, for instance, by a global discontinuous scalar field on the 89-dimensional space that takes two values: 1=allowed and 0=forbidden.

I suppose the main reason not to go for an ontology like this for General Relativity is that it's messier.