How Big Is Time?
First of all, let me preface this by saying that I am not part of the physics community. I am a computer scientist, and work with high performance systems related to weather, but it is more of a “How can we make this go faster” perspective instead of developing new models or something cool like that. My math is good for my job, but nowhere near a true physicist level. So, in short if my thinking is way off I am very open to constructive criticism. I do have a very strong interest in physics in general and have noticed similar issues in what I deal with day to day to what is discussed in the physics world regarding time in general.
Time is by all accounts a physical dimension of sorts, excluding the constant movement we experience through it, things happen and objects exist at a point in three dimensional space and at a point or series in time, relative to the current observer. The definition of physical size and the passage of time are dependent on velocity and the interaction of nearby mass. However beyond that, the similarities break down. To calculate the distance from one point in space-time to another are dissimilar. The deltas of all three physical dimensions can be expressed in terms of Pythagorean’s theorem to arrive at a single number, but what single number represents the difference from point A to point B over a span of time? What benefit would a meter-second unit have in a general sense like either one alone?
Let us consider magnitude for a moment. For static objects, the greater the amount of space they take up, the bigger they can be said to be, or more to the point they represent a larger value. If said object exists for a microsecond, the value is low, but if it exists for a million years, its value is large given the same perspective. However, the relation to energy is the opposite. For an object to change over a shorter time-span represents a larger amount of energy involved, and vice versa, a static object that exists forever with no change represents no energy at all. And for energy, an amount of energy x in a small space represents a high energy density and therefore a larger magnitude. Likewise, for time, an amount of energy x released in a shorter time span represents a larger magnitude. So comparing with time, it is consistent with space with regards to energy, and in opposition with regards to matter. Not exactly the right angle geometry I was hoping for.
Now why am I on a quest to integrate time into interpolation? Those nice, pretty images on the evening weather forecast are not generated by a station at every point on the map. In fact, there may only be a handful of data points in a local forecast to go off of, including both real observations, and grid based forecast models. Every other pixel on the screen is usually interpolated using a weighted average method. However, these data points are also not all at the same time. The forecast model is at a specific hour, and the observations are spread out over an hours time more or less. Knowing this, there is not a realistic answer at an exact point in space time, it is merely a “good enough” guess, after all whether it is 10 degrees C or 11 at your house doesn’t really matter, does it? Even radar images, aren’t exactly what they say they are, it takes place over up to a minute for the dish to rotate. One side of the image is actually 15 to 20 seconds later than the other. Radar data is directly used in hazardous weather detection, and as such a lot of advancements are being made such as phased array radars which will give us a much clearer instantaneous picture. However, the fundamental problem remains, when comparing two elements in space time how do we relate a distance to a time interval?
Let’s start with a thought experiment (and a little math, I know we all have been longing for some equations here). A completely flat city sets all of their weather stations to sync with an atomic clock and report exactly on the hour. To get the interpolated value at any point in town, we use something like the following:
Unless D = 0 at any station, then just V for that station
Where V is the value at the station, and D is the distance from your current location to the station, for each station reporting.
Let’s examine the following scenario in Flatville:
From Stars house:
We can easily get the interpolated value of 9. In fact using this method, the function does tend towards the average of all the stations as distance increases to infinity, which is what we want.
And from Diamonds house:
Now, how do we integrate time into this? Since we have established that time has a dimension that is of fundamentally different units than space, we must treat it as its own axis, ie dimension. Lets look at a more realistic example, but move from Flatville to the Linetown, where we have one less physical dimension. There are only two weather stations in Linetown. Station A reported at 9:00 am, then went offline, and Station B reported at 9:00 am and 10:00 am. We want to find the temperature at Tri’s house at 9:30 am, who ironically, lives exactly midway between the two stations.
To solve this situation, we must find the values where the dashed line intersects the dotted line and then solve using our interpolation equation above. I made all of this easy, so the answer is 8.5. But as you can see, the addition of time adds another dimension to our equations, and what makes it hard is that it is not related to the physical ones. In fact, you can make the argument that the weather stations mentioned above are following a vector with components (v,t). That might explain why weather formulas rely heavily on linear algebra, and benefit greatly from OpenCL.
I am going to take a brief aside to mention that in mathematical terms, an “orbit” is a series of points in N-dimensional space, not to be confused with going around and around a mass in space, or a vector which can be described by 2 points.
Expanding on the vector idea, it could also be said that a weather station exists at a point in space, ie longitude, latitude, and elevation. It also exists at a series of points in time. If you are keeping count, we are up to four dimensions. It lastly exists at a certain value every time it reports. If we view it’s reporting as a truly unit-less component, we can call it “n” in mathematics, whereas the other values, all 5 of them have units, and can be represented as a dimension. In the example above, station A exists at (assuming it is at sea level and located at 0,0 lon/lat) a point of { 0,0,0,9,8 }, and station B (located at 1,1,0) represents a five dimensional vector of { 1,1,0,9,8 } → { 1,1,0,10,10 }. They both however, can be described as having orbits, one with one point, and one with two points. A hurricane hunter airplane flying through a storm, traverses an orbit that goes through a trajectory in 3 dimensional space, over time, with varying values recorded, each reading is a point along this orbit.
Now, why on earth would I take a simple matter of averages and turn it into 5 dimensional math? Am I sadistic? Maybe… But all jokes aside, that is a great way to tie space and time together. It is a quantum / cosmological idea that exists in every day life, and when perceived that way opens up a whole new realm. More than one physicist has come to the conclusion that things are a lot easier when adding another dimension, and better interpolation leads to better forecasts, especially near term.
I hope that this article was if not useful, at least thought provoking. Thank you for taking the time to read it. Special thanks to my friend Noah Brauer at OU for the detailed information about radar technology.
