First, imagine that we ignore the time data. Now we just have a bunch of temperature data points [T0, T1, ...] and strain data points [e0, e1, ...]. In fact, in order to truly ignore time data, we cannot even order the points according to time! But that means that we no longer have any way to line up the points T0 with e0, T1 with e1, etc. Without any way to match up temperature points to corresponding strain points, the temperature and strain data are randomly ordered, and the correlation disappears!
That is not how d-separation works. If you control for time, the temperature and the strain on the tuning fork should be uncorrelated. We would expect the temperature to roughly follow a 24-hour cycle, plus some random noise. If the tuning fork has a 24-hour period, then we would expect the same thing to be true of the strain. But it would be very strange if the random noise in the temperature were correlated with the random noise in the mechanical strain (e.g. if, once we already knew it was 3 pm, reading the thermometer on the window tells you something about the strain on the tuning fork). That could plausibly happen if the material that the tuning fork is made of has different properties at different temperatures, changing the strain, but I think you meant to imply that neither one of these causes the other, so I'll ignore that for now.
In a Bayes net, not conditioning on a variable doesn't mean that you stop lining up the data into samples with a value for each variable, and declare that everything is uncorrelated with everything else; it just means that keep the data lined up in samples and look for correlations without paying attention to the variable you are not conditioning on. In this case, the temperature and mechanical strain should be very highly correlated if you do not condition on time, because time will still be there as a lurking variable.
Yup, you're right. That's the right way to handle it, and it yields time as the common cause of temperature and strain, as we'd expect.
Now that I'm knee-deep in it, I do think this crazy concept of separating sets of values of variables from the mappings between the sets has something to it. It isn't necessary for the example in the main post, but I think the example I gave with mice in one of the other comments still applies. The mapping between points is legitimately a variable unto itself, so it seems like it should be possible to handle it like other variables. It might even be useful to do so, since the mapping is nonparametric.
Anyway, thanks for giving a proper analysis of the problem.
In a recent comment, I suggested that correlations between seemingly unrelated periodic time series share a common cause: time. However, the math disagrees... and suggests a surprising alternative.
Imagine that we took measurements from a thermometer on my window and a ridiculously large tuning fork over several years. The first set of data is temperature T over time t, so it looks like a list of data points [(t0, T0), (t1, T1), ...]. The second set of data is mechanical strain e in the tuning fork over time, so it looks like a list of data points [(t0, e0), (t1, e1), ...]. We line up the temperature and strain data according to time, yielding [(T0, e0), (T1, e1), ...] and find a significant correlation between the two, since they happen to have similar periodicity.
Recalling Judea Pearl, we suggest that there is almost certainly some causal relationship between the temperature outside the window and the strain in the ridiculously large tuning fork. Common sense suggests that neither causes the other, so perhaps they have some common cause? The only other variable in the problem is time, so perhaps time is the common cause. This sort of makes sense, since changes in time intuitively seem to cause the changes in temperature and strain.
Let's check that intuition with some math. First, imagine that we ignore the time data. Now we just have a bunch of temperature data points [T0, T1, ...] and strain data points [e0, e1, ...]. In fact, in order to truly ignore time data, we cannot even order the points according to time! But that means that we no longer have any way to line up the points T0 with e0, T1 with e1, etc. Without any way to match up temperature points to corresponding strain points, the temperature and strain data are randomly ordered, and the correlation disappears!
We have just performed a d-separation. When time t was known (i.e., controlled for), the variables T and e were correlated. But when t was unknown, the variables were uncorrelated. Now, let's wave our hands a little and equate correlation with dependence. If time were a common cause of temperature and strain, then we should see that T and e are correlated without knowledge of time, but the correlation disappears when controlling for time. However, we see exactly the opposite structure: controlling for t induces the correlation. This pattern is called a "collider", and it implies that time is a common effect of temperature and strain. Rather than time causing the oscillations in our time series, the oscillations in our time series cause time.
Whoa. Now that the math has given us the answer, let's step back and try to make sense of it. Imagine that everything in the universe stopped moving for some time, and then went back to moving exactly as before. How could we measure how much time passed while the universe was stopped? We couldn't. For all practical purposes, if nothing changes, then time has stopped. Time, then, is an effect of motion, not vice versa. This is an old idea from philosophy/physics (I think I originally read it in one of Stephen Hawking's books). We've just rederived it.
But we may still wonder: what caused the correlation between temperature and strain? A common effect cannot cause a correlation, so where did it come from? The answer is that there was never any correlation between temperature and strain to begin with. Given just the temperature and strain data, with no information about time (e.g. no ordering or correspondence between points), there was no correlation. The correlation was induced by controlling for time. So the correlation is only logical; there is no physical cause relating the two, at least within our model.