The first thing to say is that “at t0” means different things to different observers. Observers moving in different ways experience time differently and, e.g., count different sets of spacetime points as simultaneous.
There is a relativistic notion of “interval” which generalizes the conventional notions of distance and time-interval between two points of spacetime. It’s actually more convenient to work with the square of the interval. Let’s call this I.
If you pick two points that are spatially separated but “simultaneous” according to some observer, then I>0 and sqrt(I) is the shortest possible distance between those points for an observer who sees them as simultaneous. The separation between the points is said to be “spacelike”. Nothing that happens at one of these points can influence what happens at the other; they’re “too far away in space and too close in time” for anything to get between them.
If you pick two points that are “in the same place but at different times” for some observer, then I<0 and sqrt(-I) is the minimum time that such an observer can experience between visiting them. The separation between the points is said to be “timelike”. An influence can propagate, slower than the speed of light, from one to the other. They’re “too far away in time and too close in space” for any observer to see them as simultaneous.
And, finally, exactly on the edge between these you have the case where I=0. That means that light can travel from one of the spacetime points to the other. In this case, an observer travelling slower than light can get from one to the other, but can do so arbitrarily quickly (from their point of view) by travelling very fast; and while no observer can see the two points as simultaneous, you can get arbitrarily close to that by (again) travelling very fast.
Light, of course, only ever travels at the speed of light (you might have heard something different about light travelling through a medium such as glass, but ignore that), which means that it travels along paths where I=0 everywhere. To an (impossible) observer sitting on a photon, no time ever passes; every spacetime point the photon passes through is simultaneous.
So: does the distance as well as the time go to 0? Not quite. Neither distance nor time makes sense on its own in a relativistic universe. The thing that does make sense is kinda-sorta a bit like “distance minus time” (and more like sqrt(distance-squared minus time-squared)), and that is 0 for any two points in spacetime that are visited by the same photon.
(Pedantic notes: 1. There are two possible sign conventions for the square of the interval. You can say that I>0 for spacelike separations, or say that I>0 for timelike separations. I arbitrarily chose the first of these. 2. There may be multiple paths that light can take between two spacetime points. They need not actually have the same “length” (i.e., interval). Strictly, “interval” is defined only locally; then, for a particular path, you can integrate it up to get the overall interval. 3. In the case of light propagating through a medium other than vacuum, what actually happens involves electrons as well as photons and it isn’t just a matter of a photon going from A to B. Whenever a photon goes from A to B it does it, by whatever path it does, at the speed of light.)
Not quite either of those.
The first thing to say is that “at t0” means different things to different observers. Observers moving in different ways experience time differently and, e.g., count different sets of spacetime points as simultaneous.
There is a relativistic notion of “interval” which generalizes the conventional notions of distance and time-interval between two points of spacetime. It’s actually more convenient to work with the square of the interval. Let’s call this I.
If you pick two points that are spatially separated but “simultaneous” according to some observer, then I>0 and sqrt(I) is the shortest possible distance between those points for an observer who sees them as simultaneous. The separation between the points is said to be “spacelike”. Nothing that happens at one of these points can influence what happens at the other; they’re “too far away in space and too close in time” for anything to get between them.
If you pick two points that are “in the same place but at different times” for some observer, then I<0 and sqrt(-I) is the minimum time that such an observer can experience between visiting them. The separation between the points is said to be “timelike”. An influence can propagate, slower than the speed of light, from one to the other. They’re “too far away in time and too close in space” for any observer to see them as simultaneous.
And, finally, exactly on the edge between these you have the case where I=0. That means that light can travel from one of the spacetime points to the other. In this case, an observer travelling slower than light can get from one to the other, but can do so arbitrarily quickly (from their point of view) by travelling very fast; and while no observer can see the two points as simultaneous, you can get arbitrarily close to that by (again) travelling very fast.
Light, of course, only ever travels at the speed of light (you might have heard something different about light travelling through a medium such as glass, but ignore that), which means that it travels along paths where I=0 everywhere. To an (impossible) observer sitting on a photon, no time ever passes; every spacetime point the photon passes through is simultaneous.
So: does the distance as well as the time go to 0? Not quite. Neither distance nor time makes sense on its own in a relativistic universe. The thing that does make sense is kinda-sorta a bit like “distance minus time” (and more like sqrt(distance-squared minus time-squared)), and that is 0 for any two points in spacetime that are visited by the same photon.
(Pedantic notes: 1. There are two possible sign conventions for the square of the interval. You can say that I>0 for spacelike separations, or say that I>0 for timelike separations. I arbitrarily chose the first of these. 2. There may be multiple paths that light can take between two spacetime points. They need not actually have the same “length” (i.e., interval). Strictly, “interval” is defined only locally; then, for a particular path, you can integrate it up to get the overall interval. 3. In the case of light propagating through a medium other than vacuum, what actually happens involves electrons as well as photons and it isn’t just a matter of a photon going from A to B. Whenever a photon goes from A to B it does it, by whatever path it does, at the speed of light.)
Thanks, that was very helpful, especially the explanation of timelike and spacelike relations.