That is helpful, and interesting, though I think I remain a bit confused about the idea of ‘moving through time’ and especially ‘moving through time quickly/slowly’. Does this imply some sort of meta-time, in which we can measure the speed at which one travels through time?
And I think I still have my original question: if a photon travels through space at c, and therefore doesn’t travel through time at all, is the photon at its starting and its final position at the same moment? If so, in what sense did it travel through space at all?
[Is] the photon at its starting and its final position at the same moment?
At the same moment with respect to whom? That is the question one must always ask in relativity.
The answer is: no, emission and arrival do not occur at the same moment with respect to any actual reference frame. However, as we consider an abstract sequence of reference frames that move faster and faster approaching speed c in the same direction as the photon, we find that the time between the emission and the reception is shorter and shorter.
Does this imply some sort of meta-time, in which we can measure the speed at which one travels through time?
No it doesn’t. Remember, in relativity, time is relative to a frame of reference. So when I talk about a moving object traveling slowly through time, I’m not relativizing its time to some meta-time, I’m relativizing time as measured by that object (say by a clock carried by the object) to time as measured by me (someone who is stationary in the relevant frame of reference). So an object moving slowly through time (relative to my frame of reference) is simply an object whose clock ticks appear to me to be more widely spaced than my clock ticks. In the limit, if a photon could carry a clock, there would appear to me to be an infinite amount of time between its ticks.
I will admit that I was using a bit of expository license when I talked about all objects “moving through space-time” at the constant rate c. While one can make sense of moving through space and moving through time, moving through space-time doesn’t exactly make sense. You can replace it with this slightly less attractive paraphrase, if you like: “If you add up a non-accelerating object’s velocity through space and its (appropriately defined) rate of motion through time, for any inertial frame of reference, you will get a constant.”
And I think I still have my original question: if a photon travels through space at c, and therefore doesn’t travel through time at all, is the photon at its starting and its final position at the same moment? If so, in what sense did it travel through space at all?
Again, it’s important to realize there are many different “time” parameters in relativity, one for each differently moving object. Also, whether two events are simultaneous is relative to a frame of reference.
Relative to my time parameter (the parameter for the frame in which I am at rest), the photon is moving through space, and it takes some amount of (my) time to get from point A to point B. Relative to its own time parameter, though, the photon is at point A and point B (and every other point on its path) simultaneously. Since I’ll never travel as fast as a photon, it’s kind of pointless for me to use its frame of reference. I should use a frame adapted to my state of motion, according to which the photon does indeed travel in non-zero time from place to place.
Again, this is all pretty non-technical and not entirely precise, but I think it’s good enough to get an intuitive sense of what’s going on. If you’re interested in developing a more technical understanding without having to trudge through a mathy textbook, I recommend John Norton’s Einstein for Everyone, especially chapters 10-12. One significant simplification I have been employing is talking about a photon’s frame of reference. There is actually no such thing. One can’t construct an ordinary frame of reference adapted to a photon’s motion (partly because there is no meaningful distinction between space and time for a photon).
That is helpful, and interesting, though I think I remain a bit confused about the idea of ‘moving through time’ and especially ‘moving through time quickly/slowly’. Does this imply some sort of meta-time, in which we can measure the speed at which one travels through time?
And I think I still have my original question: if a photon travels through space at c, and therefore doesn’t travel through time at all, is the photon at its starting and its final position at the same moment? If so, in what sense did it travel through space at all?
At the same moment with respect to whom? That is the question one must always ask in relativity.
The answer is: no, emission and arrival do not occur at the same moment with respect to any actual reference frame. However, as we consider an abstract sequence of reference frames that move faster and faster approaching speed c in the same direction as the photon, we find that the time between the emission and the reception is shorter and shorter.
No it doesn’t. Remember, in relativity, time is relative to a frame of reference. So when I talk about a moving object traveling slowly through time, I’m not relativizing its time to some meta-time, I’m relativizing time as measured by that object (say by a clock carried by the object) to time as measured by me (someone who is stationary in the relevant frame of reference). So an object moving slowly through time (relative to my frame of reference) is simply an object whose clock ticks appear to me to be more widely spaced than my clock ticks. In the limit, if a photon could carry a clock, there would appear to me to be an infinite amount of time between its ticks.
I will admit that I was using a bit of expository license when I talked about all objects “moving through space-time” at the constant rate c. While one can make sense of moving through space and moving through time, moving through space-time doesn’t exactly make sense. You can replace it with this slightly less attractive paraphrase, if you like: “If you add up a non-accelerating object’s velocity through space and its (appropriately defined) rate of motion through time, for any inertial frame of reference, you will get a constant.”
Again, it’s important to realize there are many different “time” parameters in relativity, one for each differently moving object. Also, whether two events are simultaneous is relative to a frame of reference.
Relative to my time parameter (the parameter for the frame in which I am at rest), the photon is moving through space, and it takes some amount of (my) time to get from point A to point B. Relative to its own time parameter, though, the photon is at point A and point B (and every other point on its path) simultaneously. Since I’ll never travel as fast as a photon, it’s kind of pointless for me to use its frame of reference. I should use a frame adapted to my state of motion, according to which the photon does indeed travel in non-zero time from place to place.
Again, this is all pretty non-technical and not entirely precise, but I think it’s good enough to get an intuitive sense of what’s going on. If you’re interested in developing a more technical understanding without having to trudge through a mathy textbook, I recommend John Norton’s Einstein for Everyone, especially chapters 10-12. One significant simplification I have been employing is talking about a photon’s frame of reference. There is actually no such thing. One can’t construct an ordinary frame of reference adapted to a photon’s motion (partly because there is no meaningful distinction between space and time for a photon).