Time doesn't stop for a beam of light. It ceases to exist.
What would it be like to catch up to a beam of light? That was the question that the 16 year old Einstein asked himself. Ten years later, he would publish his answer: you cannot catch it at all.
That doesn’t stop physicists from trying to use Einstein’s theory to explain what a beam of light might “experience”. Unfortunately, they almost always get it wrong because they try to transfer our concept of time to light inappropriately.
There are different ways to think about time in physics. A dimension is “timelike” if it has a metric signature that is the opposite of your space dimensions. We define a dimension using a coordinate and moving in a particular dimension means making a small displacement such that all but that coordinate stay the same. A metric, meanwhile, is a mathematical object that allows us to measure distances in multidimensional spaces with arbitrary coordinate systems.
You can see right away that it is possible to create coordinate systems where our concepts of time and space might be very complicated over long distances. For example, if I were to take a spacetime and define a hyperspherical coordinate system on it somewhat arbitrarily, there is no guarantee that any particular coordinate direction will align with my familiar concept of time. In such a system I would have three angles and a radial direction. If I were to define it such that time is in an angular dimension rather than the radial one, that might work for a small amount of time but eventually time which I perceive would deviate from that dimension. You could say this is just a bad choice of coordinates and you’d be right, but nevertheless at any given point there will be a particular direction that is “timelike” in the sense that if I apply my metric to it, it will be negative (assuming we choose our signs so that timelike is negative and spacelike is positive which is often written -+++).
In the familiar four dimensions we normally work with there is only one dimension of time. That means that, for a given observer, there is a timelike Killing vector which defines time for them. A Killing vector is another useful mathematical object and all it really says is that if I displace a small amount in the direction of this Killing vector then all the other points get displaced such that they stay the same in relation to one another.
This is an example I copied from Wikipedia just for illustration purposes. In this case the Killing vectors are the white arrows. If you move along these vectors, the circle rotates, so nothing actually changes.
If I have a particle moving through spacetime, it traces out a line called a world line. In this case, its Killing vector is simply in the direction of its motion in spacetime. If we are in the particle’s reference frame, meaning it is not moving at all from our perspective, then that vector would simply be in time alone.
Another perspective on time is a little different and that has to do with the perception of time. Things don’t just move in time, they experience change in time. As far as we know, we never experience more than one moment of time simultaneously. That sense of experience of time is sometimes called a temporal dimension. It is not present in the theory of general relativity at all because in that theory time is treated like an ordinary dimension. Time is only special because it is timelike so it has a causal structure. That causal structure, however, doesn’t tell us anything about how time is experienced.
Our experience of time, the fact that time has an arrow and things flow from past to future, is only contained in one physical theory: thermodynamics (as well as its quantum equivalent). Yet, these theories are not fundamental but only look at how particles (or more generally what are called microstates) behave in large numbers. Fundamentally, there is no real explanation for the arrow of time.
Our dimension of time, the one we are familiar with, has both of these properties: it is timelike, and it is temporal.
I came across an idea for how to explain what time is like for light while playing around with a coordinate system called the Infalling Eddington-Finklestein coordinate system. This coordinate system was first written down by Sir Roger Penrose but he credited papers by Arthur Eddington and David Finklestein. The classic textbook on general relativity, Gravitation by Misner, Thorne, and Wheeler, affectionately known by those in the biz as “MTW” also used this name. It has stuck ever since.
This coordinate system was developed to look at black holes from the perspective of beams of light falling into the black hole. Thus, imagine you are riding a beam of light as if falls in.
In general relativity, a beam of light follows a null trajectory, meaning that if you calculate the length of its velocity vector in four dimensions, its length is zero. That may seem counter-intuitive but in spacetime, when computing distances, the square of the time component of the velocity has the opposite sign of the space components. If the length of the time component is equal to the spatial speed, you have a null trajectory. The only objects that have spatial and temporal speeds the correct sizes for this to happen are those going the speed of light. All such objects also, by definition, have no “rest mass”, meaning if we could ride those beams of light we would measure no mass.
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