 # md65536

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1. ## Another go at my spacetime interval question from a week or so ago.

Here's a bit of a meander through how I understand this topic. There are other ways to look at it that you might prefer. First, if you move a light clock across a timelike interval in a particular frame, you can derive the time dilation factor for the moving clock using Pythagoras theorem. It looks quite similar to the scene you described in the initial post. I'd take a look at this if you've never seen it, I could post a video. If you repeat this for a bunch of different frames, you'll find you're looking at a bunch of different triangles that all have one of their sides in common: the side representing the proper time measured by the clock. Or working in the opposite direction: If you look at the time dilation factor with these equations: t/tau = 1/sqrt(1-v^2/c^2), and r = vt, you get (c tau)^2 = (ct)^2 -r^2, the spacetime interval. You can also consider space-like intervals, and either add a ruler (a proper length) instead of a light clock, and/or replace the proper lengths with times measured by light passing over those lengths, effectively swapping time and distance to get the same situation as above. From this you have a simple geometric picture of the spacetime interval components, in a triangle that gets stretched for different frames of reference, but has one side remaining invariant, and you can see the equation of the spacetime interval in the Pythagoras theorem. I get the sense that you're trying to say something like, "The spacetime interval is some natural measure of separation, and the fact that it's invariant must say something fundamental about relativity." The way I see it, the definition of the interval was chosen because it's something that is invariant---proper time---and as seen above, simply relates time and distance in different frames. Rather than starting by defining it and then assuming it is invariant, I think we start by defining it as something that is already assumed to be invariant. Rather than deriving relativity from it, I'd say the opposite is true; since it represents the measurements (eg. of time) in a particular frame, it shows that Newtonian time in a "rest frame" can be derived from special relativity. Like Markus's video suggests, it shows that not everything becomes relative when going from a Newtonian model to SR. The fact that proper time is invariant is not saying anything more (to me at least) than that 1) there's a single measure of time between a pair of events in a single frame (same as with Newtonian time). Or, different clocks sharing a rest frame don't measure time differently, and 2) while different observers measure time differently than each other, they all agree on the (proper) measurements that each other is making.
2. ## Another go at my spacetime interval question from a week or so ago.

Thanks for that. Is the repeated use of the word "true" in the video not a standard scientific term (even misleading)? If someone said that if a measure of something being length contracted isn't a true measure of length, I'd argue that's wrong, whereas saying it's not its proper length is using a scientific term. About exact specification of intervals: If you have 2 events and you don't care about their locations, only their relative separation, then you're talking about their spacetime interval. Further if you don't care about how they're measured in one particular frame of reference, then all you need to completely specify the interval is the one value, s^2. So in OP's example, suppose that s^2 is about 39999.9999999999889111 light seconds squared. This describes two events that, in one frame, are separated by say 1000 meters and 200 seconds. But it also describes the same two events that, in another frame, are in the same place and separated by 199.99999999999997227774 seconds. The latter is a measure of the proper time between the two events. (Sorry for those numbers, but if I don't use that many digits, the 1000 meters gets completely lost to rounding.) But also... that same s^2 describes the same 2 events separated by billions of light years and billions of years, in yet another frame. It also describes 2 completely different events a long time ago in a galaxy far far away that had the same separation relative to each other. This seems to imply that all light-like intervals are "the same." They all have s^2 = 0. What this means physically, is that if you have a light signal from A to B, no matter how near or far they are in your frame of reference, you can find other frames of reference where that light signal is arbitrarily short, and others where it is arbitrarily long. Those all describe the same thing, and there is no one frame in which the distance or timing of the light signal is "proper". An exception is the interval between an event and itself??? That seems to produce a valid interval where s^2 = 0, yet there are no frames of reference that can separate the events in time or space. I've never seen any mention of this. Is "spacetime interval" only defined for two different points?
3. ## Another go at my spacetime interval question from a week or so ago.

I think you should! If you think it's a bad job and don't know how to make it better, then we haven't done a good job in explaining it. There are aspects of this topic that I'm going to keep getting wrong until I see it in the right way. I know for myself it'll take repetition, to keep looking at it. Besides, I don't think that you did a bad job. Originally you didn't specify the two events of the interval precisely, but 1) it was good enough to understand what you implied, and 2) the imprecision only changes the 200 seconds value by +/- 3.3 microseconds, so imprecision is not a problem there. I'm not concerned with the exactness of the example interval, but I'm concerned about the meaning of it especially with respect to multiple frames of reference.
4. ## Another go at my spacetime interval question from a week or so ago.

No, I think you're right. When OP wrote, you're basically saying that the 200 LS is imprecise or an assumption about where the reflection point is. I was treating it as though it was supplying the previously missing information, but that's not explicit.
5. ## Another go at my spacetime interval question from a week or so ago.

No, I didn't see that as a problem. The original spec is, "So we have 2 events ; the emission and the recapture of the signal," and "200 light.seconds taken by the signal to make the round trip." It's not specified where the reflection point is, but I don't think that matters because it's only used to establish the time between the two events, and that's given. For me the light reflection path is irrelevant. It's only used here as a clock, and any stationary clock would do. Yes, there are light-like intervals between the reflection point and each of the 2 events, but I wasn't thinking of those. I think I've completely confused the meaning of hyperbolic angle, which I tried to relate to the derivative dr/dt. With a given invariant spacetime interval, a change in the t and r parameters doesn't involve moving along a world line between the two events. It involves rotating the fixed interval through different frames of reference, to vary the t and r components that make up the same fixed interval. If you do consider a particle moving along such a world line, it's moving through different points along that line, ie. different events, each of which makes a different spacetime interval between it and the initial event. (Though, in the case of light-like paths, all of those intervals are 0! But they can still be rotated so that different observers measure light between the two events traveling a different distance during a different time. Lol I'm sure there's a simpler way to look at this.)
6. ## Another go at my spacetime interval question from a week or so ago.

You're in over my head! Hopefully someone else can help? That's the derivative of a hyperbola. I don't see it saying anything about switching places. When y (or r) is small, it changes quickly. As y gets bigger, it approaches x (or ct), and the rate of change approaches constant; a unit hyperbola asymptotically approaches the line y=x. If you take the spacetime interval and make r a function of t, I think what that means physically is... It describes how the spatial distance of the interval changes as a function of the time component of the interval, as you go through different frames of reference. The infinitesimal changes in t for example would mean, if you change inertial frames by just a little (ie. with infinitesimal change of speed), the time and spatial distance components of the interval change like a hyperbolic function does. Or, an infinitesimal change in speed corresponds with an infinitesimal hyperbolic rotation of the spacetime interval. Edit: I'll leave that there but it's wrong! When both x and y are very large, an infinitesimal rotation (I think) can still mean a huge change in x and y. So to correct that: A small change in inertial frame involves an infinitesimal change in t and an infinitesimal change in r. However, as speed approaches c, a small change in speed (but huge change in rapidity) can involve a huge change in t and r. That makes sense with respect to velocity composition, right? Maybe it's correct to say "an infinitesimal change in rapidity corresponds with an infinitesimal hyperbolic rotation of the spacetime interval", but I might change my mind again after learning more...
7. ## michel123456's relativity thread (from Time dilation dependence on direction)

"If it's not moving, it's not Lorentz contracted" seems like a good rule to me. "If it's moving, it's length-contracted" could be made into a rule of thumb, but it's problematic (point particles, c, distances between relatively moving points, I think don't easily fit). "If B is moving, then (something else) is length contracted" is not a rule as you seem to think. I don't remember stating a rule though, so you might not find it. I did ask you which of certain objects (not distances) were moving and which were length-contracted. Misinterpreting things like that, and misinterpreting what SR says, and assuming it says something that you've invented, is a recurring problem here. I wrote: Not that it'll matter for you, but it would have been helpful if I'd instead said something like that the "rest or proper distance between the dice is contracted in the frame where they're moving." If you're talking about a length being contracted, it's only relative to the length measured in another frame. Here the frames of reference are implied, but if you want to think about rules, it'd be better to be explicit about frames. If you want to think of a distance being contracted, think of a ruler that is measuring the distance. If that ruler is moving (in frame F) then distances being measured using that ruler are contracted (in frame F). If you're talking about measuring a distance to B, as measured by X, using X's ruler (that is not moving relative to X), then distances as measured using that (relatively) stationary ruler are not contracted according to X. I expect you to either ignore or twist this idea. Also, this is just my attempt to explain things as far as they make sense to me. It's not an "official rule of SR", and if it disagrees with SR or can be so easily misinterpreted, it's not a good rule. Certainly there are clearer and/or more precise ways to explain it.
8. ## michel123456's relativity thread (from Time dilation dependence on direction)

What rule is that? That's not my rule.
9. ## michel123456's relativity thread (from Time dilation dependence on direction)

You don't know how to tell if something is moving?
10. ## michel123456's relativity thread (from Time dilation dependence on direction)

What are you trying to achieve here? I'd say the question of whether or not you will learn relativity is answered, your refusal to do so is just too strong. I think you've convinced others that you're interested in relativity, even though you've stated that you're not. I don't see what's in it for you, to waste time on this. Do you hope to have your mind changed? Do you hope to change anyone's mind? I'm fairly certain, no one's changing their minds here. This will go on to page 140+. Would you persist, knowing you'll never change the mind of someone who understands relativity? For others, how long is it worth persisting if the result is what we currently have? (For myself, it's only worth it to write about relativity in this thread if I'm doing it for myself, not to try to inform michel123456.) When you say "hyperbolic rotation", michel123456 reads "pirouette". You can't force-feed understanding to someone willing to put in the effort to avoid it.
11. ## Another go at my spacetime interval question from a week or so ago.

It's meters or lightseconds or any distance units, squared. The use of distance units is apparently a convention, see https://physics.stackexchange.com/questions/519707/is-the-unit-for-spacetime-intervals-time-or-space-distance Yes, for a time-like interval, the time component will be greater than the spatial. For time-like (or light-like for that matter) intervals, the ratio of r/t is the constant speed of a particle that moves between the two events. ct/r would be the ratio of the distance that light travels between the two events (along any path that gets it there, like your 200 lightsecond example) to the straight-line spatial distance between the two events, in the given frame. This ratio is frame-dependent, and undefined in frames where r=0. (ct)^2/r^2... I'm not sure of any meaning to that. As squares, the equation of the interval s^2 (a constant) =(ct)^2-r^2 is that of a hyperbola, and relates to the pythagorean theorem.
12. ## Another go at my spacetime interval question from a week or so ago.

Conventionally in SR "observer" refers to a frame of reference. If you used that convention (not that you have to, just that it can be helpful to think in these terms), then A and B are the same observer when they're not moving relative to each other. They measure times and distances the same. Local measurements differ, like the relative timing of perceived light from distant events, that each can see (ie. locally measure) in different orders, but that doesn't matter in your example. A and B measure the same as each other, the time between the two events, and the distance between the two events. They measure an interval with a length of negligibly less than two seconds. A moving observer (another reference frame) would measure a generally different time between the two events, and a different distance between the two events, but end up with the exact same interval length. The interval you're describing is a timelike interval (meaning a clock could travel between the events, and record its length as a proper time). The time component that A measures is simply c multiplied by the time on A's clock measured between the events. But of course that's the distance that A measures light traveling in that time, so you're right that it is 200 light seconds. The reflection point being 100 light seconds away doesn't really matter either, for the interval you're describing. Basically you're measuring the time between the events using a very big light clock that ticks just once between the events. A smaller light clock that reflects a light signal multiple times, can measure the same thing.
13. ## michel123456's relativity thread (from Time dilation dependence on direction)

I don't care whether you ever learn relativity. It's still interesting to find errors in what seems like paradoxes, but you're just adding complication on top of previous errors. Why not go simpler instead of more complicated? You don't have a solid foundation to build on, but you're building anyway. I think that's wrong. How do you get that X takes 45 minutes? If B starts at E, and the length to X is length-contracted to 0.6 LH (in B's frame), then X is already at that location (in B's frame) at B's time 0. A problem when introducing rods like this is that you can't just compare both ends of a rod at a single time that applies in multiple frames. You have to consider relativity of simultaneity (the real one, not "what I'm calling RoS" etc). You could always label the events that you're describing, in the frames you're describing (so it's not just an x-coordinate like Xb, but an x and a time coordinate, and they're different in different frames). But I still think you're wasting your time. I think you would do better trying to learn Galilean relativity.
14. ## michel123456's relativity thread (from Time dilation dependence on direction)

Yes, I agree. Even without the derivations, just much simpler examples, starting with the basics and without already deciding the answers before looking at the examples. One of the many problems here is that we're all looking at a relatively complicated example and trying to explain/understand step 10 of it, and Michel is effectively saying "I replaced step 3 with my own ideas, but can you keep explaining step 10 over and over? You're doing it wrong because I'm getting different results." Though, I still think giving up and not misusing the language of SR is a good option for him.
15. ## michel123456's relativity thread (from Time dilation dependence on direction)

Correct! Their numbers made sense and I could repeat the calculations of SR to get them, and when they referred to "relativity of simultaneity" they were using the established meaning of the term. Your numbers are based only on a denial of time dilation (your "?=30" is based only on having B's clock match X's, nothing else), and you use your own personal redefinition of RoS that seems to mean some combination of "light is delayed, and I've modified Galilean relativity so that it is not symmetric". Anyway, I'm not interested in discussing your alternative model, so... good day, sir.
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