Laws vs. initial conditions

Sometimes one finds inspiration in reading old articles, as it happened to me when I read Larry Sklar’s “Comments on Malament”, from the PSA 1984, Vol 2, 106-110. In this short paper, Sklar takes consistency constraints that arise in spacetimes with closed timelike curves (CTCs) to indicate that the principled distinction–a “distinction at the very root of the notion of physical possibility and necessity” (108)–between initial conditions and laws breaks down:

“If there is, relative to the proposition we now countenance as the laws of nature […], a lawlike constraint on ‘initial conditions’, if, that is, some specifications of the world ‘at a time’ are impossible since they are lawlike incompossible with themselves, what solid distinction is left between laws of nature and ‘mere matter of fact initial conditions’?” (108)

Well taken. According to Sklar, there have been, of course, previous reasons to doubt such a distinction: (i) the Humean view of laws as mere summaries of particular facts discourages such principled a gap, and (ii) the “existence of such more-than-mere-fact mere facts like the parallelism of likely entropic increase for branch systems” (ibid.) can be thought to have the same consequence. Translated into the terms of the recent debate, one might well debate whether the past hypothesis–the assumption that there was an early state of the universe with rather low entropy–is indeed an initial condition (as it is on the face of it), or more akin to laws of nature. But even if the past hypothesis were a law, it’s not a dynamical law. So perhaps there is still a principled distinction to be had between dynamical laws and their initial conditions?

Be this as it may, it (and other things Sklar says in this article) led me to ponder modality–a longtime hobby horse of mine. I would like to argue that even if the distinction between laws and initial conditions breaks down, there is still a clear notion of possibility and necessity relative to a theory (about the world). Consider the semantic view of theories. Here, we could say that a proposition is necessary just in case it is true in all models of the theory and possible (or contingent) just in case it is true in some models.

Thus, given a theory, modality is completely unproblematic. (Conversely, starting out from considerations of what is possible or necessary, one may arrive at a semantic characterization of a theory). Modal considerations become completely unfettered and loose beyond the framing confines of a theory. Thus, talk of whether the laws “could have been different” is meaningless unless it is embedded in a “higher-order” or more encompassing theory. If correct, this view entails that there is no such thing as possible or necessary “simpliciter”.

As an example, consider the structure of spacetime. Within special relativity (SR), the spacetime structure is necessarily Minkowskian, but relative to general relativity (GR), which contains the models of SR, but also many more, it is only contingently Minkowskian. Within SR, it simply makes no sense to ask whether the spacetime geometry could have been different, while of course in GR it does.

Of course, this view must offer a rigorous characterization of the models of a theory, or of the worlds that are possible according to it. For instance, more needs to be said what exactly the models of SR are; is it just Minkowski spacetime, or are there many models, i.e. all those physical situations compatible with SR such as the one where the birthday cake is pink and Obama is re-elected for a second term (assuming these scenarios are possible in a world without gravity, which they may well not be).

Any thoughts are welcome!



Filed under My half-baked ideas

6 responses to “Laws vs. initial conditions

  1. I was recently thinking about this too, so here’s a response: in physical theories, there are normally two levels at which we can characterize a laws and models, and Sklar is mixing them up.

    (1) Models as configurations of matter-energy. This is the picture you suggest above: the “laws” are the Einstein field equations, and a model is a spacetime that satisfies those equations.

    (2) Models as trajectories of test particles. In this picture, the “laws” are the geodesic equations of a given spacetime. These laws describe how an appropriate initial configuration of test-particles evolves forwards and backwards in time. (Notably, this picture of laws makes perfect sense in special relativity.)

    Both (1) and (2) are fair characterizations of “theory” and of “model.” But here’s what’s important: you have to specify (1) before we can specify (2), and Sklar (1984) seems to miss this. In Newtonian mechanics, we must specify a potential field before we can specify an initial configuration of particles and predict what they’ll do. In quantum theory, we specify a Hamiltonian before specifying an initial distribution predicting how it evolves. And in GR, we must specify a spacetime before we can specify the initial conditions of test particles (or Maxwell fields, or whatever).

    So, it’s true that “possible initial data configurations” are contingent upon a choice of a spacetime. In particular, the possible initial data in a time-travel spacetime will be contingent upon facts about that spacetime (specifically, the consistency constraints). And this is expresses nothing more than the fact that (2) is contingent on (1). It’s no threat to the laws/initial-data view, as long as we don’t conflate “laws and initial data in (1)” with “laws and initial data in (2).”

  2. That’s certainly a distinction that can be made. Not sure that the dichotomy between laws and initial conditions can be so easily salvaged though, for the following reason. In GR, of course, it’s not really the case that we specify a spacetime first, and only then specify initial conditions for test particles and fields evolving on the spacetime. Rather, if we think of the situation as being cast as a sort of initial-value problem, we specify the initial conditions for the spacetime geometry (e.g. a pair of canonical variables such as a three-metric and its conjugate momentum) and for all the particles and fields, and then plug all of this into the coupled Einstein-Maxwell-What-have-you equations that will then give us the dynamical evolution of the entire shebang.

  3. Sure — that’s just (1). When you think your initial data has enough mass-energy to perturb the background spacetime in a non-trivial way, you specify this data as “stress energy on a Cauchy surface.” Then your model is a maximal extension of this surface satisfying the field equations.

    But that’s not always how we specify as initial data. I mean, nobody couples a satellite to the Einstein equations in order to calculate its orbit around the earth — we just write down the Schwarzschild geodesic equations and calculate its evolution as a test-particle.

    So: when our initial data doesn’t non-trivially perturb the background spacetime, we just specify it as a distribution of this “test-matter” on a Cauchy surface. That’s (2). Then the geodesic equations for the (already fixed) maximally extended spacetime determine how they evolve, and this maximal evolution provides us with our model.

  4. You are certainly right as far as practical matters are concerned: Of course this is how the orbits of satellites are calculated. But this is only approximately valid. What this means is that somebody who would like to defend a principled distinction between laws and initial conditions would have to insist that this approximate way of computing evolutions is justified in the presence of CTCs. It may well be, but is it obviously so?

  5. Maybe not obviously. But there are theorems that justify it. For example, we can replace “test particle” with “Geroch particle” — Geroch showed that a world tube of matter-energy satisfying plausible conditions will contain a trajectory satisfying the geodesic equations. No approximation needed.

  6. OK, I see. But we agree then, it seems, that the defender of a principled distinction between laws and initial conditions would have to do some along the lines of what Geroch did.

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