With a lot of help from ChatGPT I generated a little interactive animation to help visualize what is going on in the classic light-clock thought experiment. You are looking at a profile view of two experimenters, Linda and Ralph, each in their own frame of reference. By tradition this is a train car, but it can be a spacecraft or a shipping container or an RV, it really doesn't matter. The point is that Linda and Ralph have identical experimental apparatus, but they are moving relative to each other. In this animation, Lina, at the top wearing a purple shirt, is moving Left and Ralph, on the bottom wearing a green shirt, is moving Right.
(For a larger stand-alone version of the animation, or if this iframe doesn't fit in your screen, click here.)
Linda and Ralph each have an experimental appratus consisting of a light source (the yellow circles) and a clock on the floor of the train and a mirror on the ceiling (the white rectangles). The clock measures the time elapsed between when the light flashes and when the reflected light from the mirror returns to the same location. You can view the experiment from three different frames of reference: Linda's frame, where she is stationary and Ralph is moving, or Ralph's, where he is stationary and Linda is moving, or the ground where both Linda and Ralph are moving in opposite directions at equal speeds. The path of the light as seen by Linda and Ralph is traced out in colored lines. The purple lines are the paths that Linda sees and the green lines are the paths that Ralph sees. (Exercise: the light paths as seen by an observer on the ground are not shown. What would those look like?)
The important things to notice are:
- The physical situation is exactly the same no matter which frame you choose. The only thing that changes is what parts of the overall situation appear stationary.
- No matter frame you choose, each experimenter sees the other one's light traverse a longer path than their own.
- Both clocks end up with the same reading, namely, 2.00. The units that the clocks are measuring are not specified. On a realistically sized train car with a height of three meters the time units would be about ten nanoseconds.
- The path of Linda's light as seen by Ralph, and the path of Ralph's light as seen by Linda, are the hypotenuses of two right triangles. One of the legs of the triangle is the straight line distance from the light to the mirror, and the other leg is the distance moved by the trains in 2 time units.
- Linda and Ralph will agree on all of these distances. The only thing they will disagree on is which of them is moving and which of them is stationary, and hence which of their light beams traveled on the longer path.
- As the trains move faster, the distance between the trains after 2 time units will get longer and longer. But (and this is the really important bit) no matter how fast the trains are moving, that distance can never be as long as the apparent distance covered by the light in the other train car. This is a simple geometric fact: the hypotenuse of a right triangle is always longer than each of its legs.
Now think about what it would mean for the train cars to be moving "at the speed of light".
I'll give you the answer in the next episode, but you really should be able to figure it out.
(First sentence: "though" -> "thought")
ReplyDeleteThanks. Fixed.
DeleteCute visualization ... but one part that doesn't quite work is that you have your different light pulses going at different speeds. It's fine in the symmetric case (both light pulses go the same distance). But with either Linda or Ralph stationary, you have the angled light hitting the end clock at "the same moment" as the light pulse that just goes straight up and down. That's not what would happen. If light was going at a constant speed, the light on the angled path would need to take longer to reach the end point.
ReplyDeleteI know you know this already. But presumably this is intended to be an intuitive guide to those who don't (yet) know Relativity. Your visualization of the end clock reading at 2:00 appears to happen simultaneously for both clocks, and that isn't actually how it works (except for the symmetric case).
> That's not what would happen.
Delete> that isn't actually how it works
We'll see. :-) But note that if the Objective Reality Hypothesis is correct then what *actually* happens must be the same for everyone. It might *appear* different from different points of view, but what *actually happens* must be the same for everyone.
Note also that neither Ralph nor Linda can actually see the other one's clock nor the other's light beam while the experiment is underway. In fact, neither one of them can even see their own light beam while it is in transit. The only light that either one of them can see is light that is actually arriving at their eyes in the moment that they see it.
You clearly have some plan for presentation, so we’ll just see where you go with it.
DeleteYes, only one single consistent thing “actually happens”, but yes it can appear different for different observers.
But for the light “actually arriving at their eyes” … sure, you can add the complexity of propagation delays. That generally doesn’t make understanding easier. Einstein already “fixed” that long ago, by the thought experiment of filling all of space with inertial observers “at rest” with respect to each other. All those observers can (eventually) agree on all distances and times. And so all observations & measurements can always be assumed to be happening locally, without any propagation delays, by some collaborating observer in the same inertial reference frame.
I guess my issue is that your visualization suggests some kind of omniscient point of view. But in the two “stationary” cases, the infinite field of collaborating inertial observers don’t actually see what the visualization suggests. Perhaps you’ll talk later about what different observers see. But the visualization surely strongly suggests that there is at least SOME observer that could potentially see something like that. And there just isn’t. No observer sees that.
> All those observers can (eventually) agree on all distances and times.
DeleteOnly if they first agree on a synchronization convention.
Well, yeah, you say, of course. But Einstein took care of that too. Well, no, he didn't, not really. Einstein takes the one-way speed of light to be constant, and that works mathematically. It works epistemologically. But it doesn't work ontologically. Ontologically, the one-way speed of light is not merely indeterminate, it actually turns out to be *meaningless*. Here is a quick-and-dirty argument: for something traveling at the speed of light the Lorentz factor is zero, so from the "point of view" of a photon traveling from A to B, the time elapsed is also zero. But, crucially, the distance is also zero, so the "speed" is zero divided by zero. (I put "point of view" in scare quotes because the "point of view of a photon" also turns out to be equally meaningless.)
> I guess my issue is that your visualization suggests some kind of omniscient point of view.
Yes, that's by design. A single diagram can only show a single point of view. I chose the "omniscient" point of view specifically because it suggests that light travels faster in some directions than others, and our next project will be to reconcile that with the fact that all light appears to move at the same speed. (But note that the light which naively appears to move faster in some reference frames *do not complete round-trips* in those frames, at least not in this set-up.)
Ahh, thanks for doing this animation. Not too shabby. I think I figured something out here but I might have gotten something backwards.
ReplyDeleteTwo of the legs of each triangle both equal 1. And the hypotenuses equal about 1.414. This threw me off a bit because the light could not have travelled that far if 1 is the distance between the light source and the mirror. But it works if you think of it as a length of time instead of distance. When that extra .414 is how much *slower* Linda and Ralph see each others lights move. If the speed of their own lights are 1 and you divide that by 1.414, you get 0.707. So if it takes 1.414 t to travel the length between the mirrors, then you are moving at about 70% the speed of light?
I think I am seeing the mistake I made before when I said 50% so I am pretty sure I must have the right answer now.
> about 70% the speed of light?
DeleteYes!
Something else to ponder now that you have figured that out: let's call the height of the train car H. So from Ralph's point of view, his light travels a round-trip distance of 2H. But in that same time he also travels the same distance (2H) *sideways* (to the right in his case). How it possible for Ralph and the light to cover the same distance in the same time if Ralph is only traveling 70% of the speed of light? (Hint: remember how Zeno's paradox turned out to be a word puzzle, not a math puzzle.)
Delete> Yes!
DeleteYay!! :D
>Something else to ponder now that you have figured that out: let's call the height of the train car H. So from Ralph's point of view, his light travels a round-trip distance of 2H. But in that same time he also travels the same distance (2H) *sideways* (to the right in his case). How it possible for Ralph and the light to cover the same distance in the same time if Ralph is only traveling 70% of the speed of light? (Hint: remember how Zeno's paradox turned out to be a word puzzle, not a math puzzle.)
Ok, I'm on it!