Tuesday, May 1, 2018

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.


M. J. Glaeser said...

Every massive particle is a clock; so your problem only affects massless particles... except it doesn't either, since it is well known that the proper time is zero for massless particles :)

Alexander R Pruss said...

How is it a clock? What are its "hands"?

M. J. Glaeser said...

I do not think Einstein had common clocks (with hands and so forth) in mind when he defined time as that which clocks measure. Such an interpretation strikes me as quite uncharitable. Here a clock is any physical system in which certain event occurs periodically. For example, a photon bouncing between two static walls. Every "bounce" marks a "tick". Since we know that the photon's speed is constant, and that the distance traveled by it remains always the same, it follows that the time between two "bounces" is also always the same, so they satisfy the periodicity condition. Ok, all particles are clocks in this generalized sense which presumably Einstein had in mind. I really recommend the following post by German theoretical physicist Sabine Hossenfelder on her blog, it is very nicely explained:

Alexander R Pruss said...

The photon bouncing back and forth requires a photon and two walls.
The method described in the post requires a particle and a laser and other measuring apparatus.
There is also a historical question: Did Einstein think that every physical system had a periodic event?