Measuring rods

In his popular book on relativity theory, Einstein says that distance is just what measuring rods measure. I am having a hard time making sense of this in Einstein’s operationalist setting.

Either Einstein is talking of real measuring rods or idealized ones. If real ones, then it’s false. If I move a measuring rod from one location to another, its length changes, not for relativistic reasons, but simply because the acceleration causes some shock to it, resulting in a distortion in its shape and dimensions, or because of chemical changes as the rod ages. But if he’s talking about idealized rods, then I think we cannot specify the relevant kind of idealization without making circular use of dimensions—relevantly idealized rods are ones that don’t change their dimensions in the relevant circumstances.

If one drops Einstein’s operationalism, one can make perfect sense of what he says. We can say that distance is the most natural of the quantities that are reliably and to a high degree of approximation measured by measuring rods. But this depends on a metaphysics of naturalness: it’s not a purely operational definition.

Time and clocks

Einstein said that time is what clocks measure.

Consider an object x that travels over some path P in spacetime. How long did the travels of x take? Well, if in fact x had a clock traveling with it, we can say that the travels of x took the amount of time indicated on the clock.

But what if x had no clock with it? Surely, time still passed for x.

A natural answer:

• the travels of x took an amount of time t if and only if a clock would have measured t had it been co-traveling with x.

That can’t be quite right. After all, perhaps x would have traveled for a different amount of time if x had a clock with it. Imagine, for instance, that x went for a one-hour morning jog, but x forgot her clock. Having forgot her clock, she ended up jogging 64 minutes. But had she had a clock with her, she would have jogged exactly 60 minutes.

That seems, though, a really uncharitable interpretation of the counterfactual. Obviously, we need to fix the spacetime path P that x takes. Thus:

• the travels of x over path P took an amount of time t if and only if a clock would have measured t had it been co-traveling with x over the same path P.

But this is a very strange counterfactual if we think about it. Clocks have mass. Like any other massive object, they distort spacetime. The spacetime manifold would thus have been slightly different if x had a clock co-moving with it. In fact, it is quite unclear whether one can make any sense of “the same path P” in the counterfactual manifold.

We can try to control for the mass of the clock. Perhaps in the counterfactual scenario, we need to require that x lose some weight—that x plus the clock have the same mass in the counterfactual scenario as x alone had in the actual scenario. Or, more simply, perhaps we can drop x altogether from the counterfactual scenario, and suppose that P is being traveled by a clock of the same mass as x.

But we won’t be able to control for the mass of the clock if x is lighter than any clock could be. Perhaps no clock can be as light as a single electron, say.

I doubt one can fix these counterfactuals.

Perhaps, though, I was too quick to say that if x had no clock with it, time still passed for x. Ordinary material substances do have clocks in them. These clocks may not move perfectly uniformly, but they still provide a measure of length of time. Alice’s jog took 396,400 heartbeats. Bob’s education took up 3/4 of his childhood. Maybe the relevant clocks, then, are internal changes in substances. And where the substances lack such internal changes, time does not pass for them.

Relativity of simultaneity

I've been thinking about Einstein's nice argument for the relativity of simultaneity in his popular book. The argument starts with the assumption that the speed of light is the same in every inertial reference frame, and uses this to construct a method for determining whether two events are simultaneous. Basically, this method involves having an inertial observer spatially equidistant between the two events checking whether light reaches her simultaneously from the two events. Given the constancy of the speed of light and the equidistance assumption, it seems to follow that the two events are simultaneous if and only if light reaches the observer simultaneously from the two events. And then Einstein gives a very nice argument that applying this method gives different answers depending on the osberve, and concludes that simultaneity is relative to the reference frame.

But there is something that has been worrying me conceptually about Einstein's account of simultaneity. That account takes for granted that we know what it means for the observer to observe two events simultaneously. But isn't the task to define simultaneity?

I guess not. Einstein seems to presupposing that we already have the notion of simultaneity of events at the location of an observer. Moreover, the details of Einstein's argument assume this principle which I think he doesn't discuss:

• Two events befalling the same observer occur simultaneously in the reference frame of the observer if and only if they occur at the same point in spacetime.

(Einstein also tacitly makes the simplifying assumption that observers are point-sized. I won't worry about that assumption in this post.) I am a little troubled by this principle. It's not clear that it's conceptually necessary (might we not think it's violated in cases of time travel?). Still, maybe the best way to take Einstein's account of simultaneity in the book is this. First, we define simultaneity for events befalling the observer who defines a reference frame by requiring sameness of spacetime location. Second, we use this and the equidistant-observer thought experiment to define simultaneity for events not both located at the observer. Third, we show that by this two-part definition of simultaneity, simultaneity is frame-relative.