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, regardless of the reference frame from which you look at it.
In turn, 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").
This is due to the spacetime metric: to calculate a spacetime interval you don't add the squares of the components and take the square root of that (like you do in ordinary space with -imagine a 2D world- the X and Y components), as ruled by the Pythagorean Theorem; here you have a negative sign between the squares of the space and time components); since the so called time component is not really so, but it is "spatialized" by multiplying it by the speed of light (c), this means that a lightlike interval always contains same number of X units as cT units, so any spacetime interval travelled by light is forcefully zero.
So we have a null spacetime interval that is painted in a Minkowski diagram, from the perspective of any reference frame, as the line bisecting the X and cT axes and hence having a visual length, as well as visual lengths for its identical X and cT components.
This can more or less puzzle you. But things get more complicated when you consider the hypothesis of a reference frame that is the photon itself. From this perspective, it is said that the proper time of the photon would be zero.
To understand why, let us consider an example. We have two events 1 and 2 that happen at two planets that are co-moving: one is when a projectile is fired, the other when it reaches its destination. From the frame of the projectile itself, the two events happen at the same place, so there is no space component, only time, ie what is called the proper time. As you make the projectile faster with regard to the planets, the proper time of the interval becomes shorter. Well, certainly in the frame of the projectile, the latter is still and what becomes faster is the planets, but that doesn't change the issue: as the planets become faster, also the proper time of the destination planet from (in this case) event 1 bis (happening at such planet when the projectile was fired) and 2 becomes shorter. Then, ideally, as the relative velocity between projectile and planets became the speed of light, the proper time of the projectile (as judged by the planets) or of the planet (as judged by the projectile) should be 0. T
A mystical reaction to this that you may find in pop science books and on the internet is declaring that "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; as to the issue of the proper time for a photon being zero, note that a photon is not a valid reference frame, so the question is simply unreal and inapplicable.
This is fine but I would like to be deeper in two senses:
A) First, a more ambitious answer is 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.
B) Second, explain why the photon is not a valid reference frame.
I have made several versions of this post in an attempt at finding those answers. In the end (Sept. 2025) I think that I have a simple answer.
The guiding point is taking a positivistic or operational approach: what you feed the equations and the geometry with are data obtained through measurements made with physical instruments. In the context of the Minkowskian space, those instruments can be visualized as follows: a clock which is a box where a light pulse oscillates after reflecting against its walls; to sync distant clocks, following the Einstein-Poincaré convention, you send a light signal to each clock and note the time taken by the round-trip, so you assume that the time of the go-trip was half of that. Which conditions are always present here? The light must oscillate between massive walls. Without massive walls there is no operational spacetime instrument.
Therefore, the answer to the above-mentioned questions is that:
A) The general formulation is that the null interval means "no measurement" because the interval in question is not measurable in itself with the available instruments.
The specific formulation is:
- In ordinary space, the null interval is an extensionless point, which can only be measured as an intersection between the X and Y axes, but not directly with X and Y rulers, because a thing with extension cannot measure a thing without extension.
- In Minkowskian space, the null interval is a trajectory that no clock (endowed with massive walls) can follow; certainly with clocks (endowed with massive walls) one can measure the coordinate time (difference between the readings of distant clocks) required by the photon to cover a distance and compose those things into a spacetime zero interval through the relevant equation but not directly.
B) The photon is not a valid reference frame because reference frames play (do math and geometry with data fed by measurements and from a photon frame you could make no measurements, in the absence of massive walls.
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 shared thing across frames 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, it 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.
In spacetime, the agent with which you make measurements is light or at least light's standard, so it is the eigenvector. But, unlike what happens with ordinary space, where the extension of things (eigenvector) by definition cannot be extensionless (null vector), in spacetime 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.
