A frequent question in the context of special relativity (“SR”) is this: in the ordinary physical space, which is Euclidean, the null distance between two points is 0 and geometrically appears as an ideal point, which does not have any extension; however, in Minkowskian spacetime, the null interval between two spacetime points or events, which is also numerically 0, can be any lightlike interval, i.e. the interval between any two events connected by a light signal, which may have any extension when painted on a diagram (sometimes people call this extension the ruler length or Euclidean length; we will call it here "visual length"). How can that be?
A mystical answer is what you often find in pop science books and on the internet: things like "photons do not experience time", which in turn prompt all types of wild (but attractive to some people) speculations, about whether time is an illusion and stuff like that.
A more reasonable but incomplete answer is: you are blinded by your
Euclidean intuition; there is certainly an analogy between Euclidean and
Minkowskian spaces, but analogy is not identity and so things function
differently in the second context: just accept that the Minkowskian null interval
has 0 numerical value even if it has some visual length when painted in a diagram.
But the ambitious answer consists of delving into how the analogy plays and so see clearly how this has to be: why the numerical value of the lightlike interval must be 0, while the visual length is not.
For this purpose, the rule of thumb is as follows:
detach yourself from the specific situation that is problematic in the novel
area, elevate to a more abstract level and look for a general formulation
embracing both the old and the new area; then study how this generic spirit is
dealt with in each domain, given its specificities.
If you do this here and you adopt for this purpose (as one should always do) a positivistic or operational approach (that is to say, you define the intervals of a space based on what they measure and how it is measured), you will realize that non-null intervals are associated to the idea of "no measurement". At this stage, I have hesitated between two formulations. Initially, I focused on the object of measurement and thus stated that non-null intervals are what you can directly measure with the available physical instruments, whereas a null interval is what you cannot directly measure with the said instruments (only indirectly). Lately, I have thought of another version focusing on how you measure: any instrument contains an element or agent that participates in the measured property; thus you can obtain a non-null interval when you use an instrument having that feature and a null interval when your instrument does not have the measured property or you use it in a manner so that, in the end, the element in question has done nothing. In the end, I have thought that the two versions are complementary, as we will immediately see.
This definition fits ordinary space. Here what you measure is pure lengths of objects or spatial distances and you do it with instruments having such property, i.e. having physical extension, like rulers (X, Y and Z rulers). Hence non-null intervals are the values obtained with rulers, while null intervals are ideal points having no extension, which is what: in the first version, you don't directly get with rulers (only indirectly as an intersection between two lines); in the second version, you would get if you tried to measure with ideal points or you measure with rulers but you undo what you have done and return to the ideal point of departure.
But note that no visual length is the specific form or vest that a null interval takes in this particular space. If we now apply the above-mentioned general formulation to spacetime, then we must look at what you measure and how you do it in practice in this new space. What you measure is the distance between events and you do it by making use of an instrument that also mimics this reality, i.e. an agent traveling from one event to another, in particular an agent whose standard is light and which is active in both axes, i.e. what we usually call the X and the cT axis.
At this stage, we would probably need a lengthier explanation, which I will do elsewhere, but let me just make a telegraphic summary for our present purposes. I do not mean that all clocks and rulers must be constructed with light. Instead of light, we could talk about another agent traveling at the same speed, which is the universal speed limit (for example, gluons are also said to displace at the same speed). Also note that I do not mean that one of those best-in-class agents must forcefully appear in each instrument. Certainly, a clock can have inside its walls any other oscillating agent. But then you must convert its values into meters, which you do using the speed of light as a conversion factor. And how do you measure this speed? You do it by organizing a round-trip competition between any agent inside a clock and light oscillating inside another where it turns out that the latter traverses about 300 million meters while the former ticks 1 second. Then you fix once for all such ratio by redefining the meter as the length that light traverses in such time also in a one-way trip. But what if you need to measure distances between distant events? In that case, you need to synch the clocks at the two endpoints of the interval. And you do this through the so-called Einstein-Poincaré convention or radar convention, by virtue of which a clock is set to read the time for a light signal to cover a round-trip to its location, divided by 2. So there are massive walls at the start and end of the spacetime measurement operation in all cases.
Given this, the general formulation is adapted as follows: spacetime non-null intervals are the values obtained with spacetime clocks and rulers, having light oscillating between massive walls, while null intervals are the lightlike trajectories because they are: in the first version, what you don't directly get with spacetime instruments (only indirectly as a combination of distance traversed and time employed); in the second version, what you would get if you tried to measure with light alone because indeed light is the standard agent of the measurement instruments but it is not apt for measurement unless it is forced to make round trips within massive walls.
The geometry of spacetime confirms this. If you look at how spacetime is drawn in a Minkowski diagram, you will notice that the null interval must be placed where it cannot be overlapped by either the cT or the X axis of any frame. In such diagram, a frame takes a privileged position and has perpendicular axes, while the axes of the other frames get closer to each other as one increases the relative velocity with regard to the first frame, but they never overlap with what we could call the light axis, i.e. the null interval. Of course, all light intervals or so-called lightlike intervals can be described using a combination of a cT and an X interval (of the same size, so that their subtraction is 0). No problem, also in ordinary space a null interval (i.e. an ideal point) can be described using both an X and a Y axis (when they intersect, by the way). However, neither in ordinary space nor in spacetime do you get a single axis on top of a null interval, precisely for the indicated reasons: because the null interval is what cannot be directly measured (with the values that populate your axes) and because it is also what cannot measure (what does not furnish the values that populate those axes).
As a bonus, I would make a link with the concept of eigenvector.
In ordinary space, the null vector cannot be the eigenvector of the transformation matrix, i.e. it is not the vector that remains unaltered after a change of basis or transformation of values into the language of another frame. Here the eigenvector is the line crossing the origin of the system, which acts as a rotation axis, but it is not an ideal point without any extension. This makes sense because, in a generalized sense, the common thing is the element or agent with which you make measurements, which is the same no matter the orientation of the axis and the reference frame. Particularly, in this space, this is the extension of things, which as noted does not vary regardless the axis and regardless the perspective or frame, which corresponds to a rotation of the coordinate system.
But in spacetime things are different. Here the agent with which you make measurements is light or at least light's standard, so it is the eigenvector. But at the same time in this context the said eigenvector can be the null vector because, despite being a necessary condition for measuring, it is not sufficient: with it alone (i..e without walls where to bounce), you cannot measure.
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