Though this post is about heat waves, the allusion to the space-time of theoretical physics is not accidental. What happens across space matters if large areas are affected at the same time -- a bending of the fabric of the climate system, if you will. In this abstract realm, spatially large, long-lasting, and intense heat waves are the dominant players, and these are also what matter most from a societal-impacts point of view. A heat wave striking both Chicago and New York, or a hurricane lashing both Kolkata and Dhaka, or a drought afflicting both Ethiopia and Kenya may be of more concern than a more-concentrated event of greater magnitude. It's the full spatiotemporal picture, multiplied by other factors like population affected, that's in some sense the true measure of an event.
Space-time patterns, while powerful in a predictive sense, do fade. As you go farther apart, things are generally less connected. The first figure below illustrates that for heat waves -- the dates of LA heat waves have no connection whatsoever to those in NYC. But at the same time, this correlation is not entirely symmetric either; it's oriented preferentially in certain directions, in this case east-west, such that heat waves in Detroit resemble those in NYC Central Park more than heat waves in Virginia Beach, because it's more common for the same pressure system to cause a heat waves in Detroit and NYC than in Virginia Beach and NYC. Any major geographic feature, like the Central Valley, will have this asymmetric-correlational effect. Temperatures at beaches 100 miles apart will for the most part be more similar to each other than to those 10 miles inland from either location, because the common effect of sea and land breezes will dominate any effects resulting from differences in terrain or cloudiness. Besides geography, the other major source of climate 'action at a distance' is teleconnections -- both ones you'd expect from first principles and those that are somewhat more head-scratching (why the correlation off of Argentina??). While Einstein may have been wrong about quantum entanglement, he was right in the climatic sense: 'action at a distance' only appears that way from the ground, without the benefit of being able to observe the ocean and high atmosphere that are communicating information from one place to another. It may be a long game of Telephone, but some sort of message eventually gets through.
Space and time are correlated with other dimensions of weather events also-- for example, in the Northeast US (see second figure below) as well as in the Czech Republic (and likely throughout the mid-latitudes, if not the world), larger heat waves tend to be hotter. It's still uncertain as to whether this is a causation or merely a correlation, but nonetheless it implies that the stress on systems like the energy grid increases exponentially as heat waves get more severe. (But not necessarily that this exponential increase will occur in a warming climate, because most scientists' expectation is that dynamics will be relatively constant, i.e. a heat wave of the same size will increase temperatures by the same factor as now relative to climatology — that is, they won't somehow become 'more efficient' at transporting heat and moisture, just that there will be somewhat more of it available.)
One thing we know for sure, direct from observations, is that weather patterns of many sorts tend to 'pulse'. Rarely is there a monotonic change, or an abrupt onset with no subsequent slippage. This is as true for glacial onsets and terminations as it is for seasonal precipitation, and down to the timescale of minutes as shown in this hypnotic simulation. This pulsation is to be expected in a complicated system that never reaches true equilibrium because the forces that guide it are always themselves changing, even if that just means shifting slightly in position or strength. Such is the case for the infamous 2003 heat wave, nicely analyzed in many papers, but the Feudale and Shukla 2011 one is particularly relevant to the discussion here. Those authors found the focal point of the most-anomalous heat moved around Europe during that summer, and its movement could be summarized by two "empirical orthogonal functions" (see below) that represent prominent decompositions of the variability: the first in Central Europe with cooler air along the coasts, and the second over France and Benelux with relief from the heat in Russia and the Caucasus. These EOFs and the time series of their relative presence/absence are shown below. In this context, it is striking to note how, within what is popularly and even meteorologically considered a single extreme event, multiple different 'flavors' were observed, dramatically changing the experience of the summer depending on where exactly you were.
Another perspective on a similar event, this one occurring in 2010, is provided by a 2011 paper by David Barriopedro et al. They had the idea of representing the heat wave's combined spatial and temporal strength and persistence in one elegant figure that shows at a glance how large the heat wave was, on what time scale, when its peak occurred as measured by this metric, and overall how intense it was (by comparing to the same chart for the aforementioned 2003 European heat wave). This figure is included at the bottom of this post to marvel at the genius of.
With so many different dimensions and variables in the climate system, in some ways it's a wonder there aren't more than 26 heat-wave indices (Tables 2 and 3 of this paper) floating around. An analogy to economics seems appropriate -- an economic index could be tailored to perfectly match, say, your family's finances, including every DVD purchase and late-night run to KFC, but it would be quite unusable for anyone else. Similarly, making a very vague index is the more generalizable and also the least applicable to any one person or situation. A middle ground is necessary, to be able to usefully distill the great complexities of the climate system into something understandable -- because only once it's understandable, even with some degree of simplification (as with the potentially quantifiable autocorrelations given at the start of this post), can optimal decisions be made taking into account its many variations and apparent vagaries. When our species masters spacetime we may decide there's more-exciting things to do in the universe than investigate the details of terrestrial heat waves in old-fashioned nation-states, but who knows... even with competition from wormholes and time travel, they could well prove to be a subject of bottomless fascination.