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I have a question regarding a spacetime animation I am contructing, please. The animation is as follows. Einstein stands on the train station while Curie stands on board a train. At time 0, they are both at the same x position, and Curie sends a light signal to the right, as follows:

enter image description here

This represents the path over time traced from Einstein's point of view. To get Curie's point of view, we apply the Lorentz' transform xnew = gamma * (x - vt) and tnew = gamma * (t - v/c^2 * x) for every object at every point to get a new animated path:

enter image description here

This represents the path over time traced from Curie's point of view. We note that the light path stays at 45 degree to the right, as it should be, for this represents the exact line that the Lorentz transform aim to leave unchanged in all cases.

My question is, the time animated line obtained in this way is not "even" horizontally. Einstein is somehow "ahead" in time from Curie in this diagram, who is herself "ahead" in time from the light ray to the right of the diagram. What does this mean physically, if they are not at the same horizontal level on the diagram when time animated?

If we are to take out the y variance over time, this would result in a plot like so:

enter image description here

But how do we interpret the Lorentz transform result for Curie's point of reference, in which the three objects (Einstein, Curie and the Light) are not at the exact same horizontal time level? Can we draw a diagram when the "time" is not horizontally the same for all objects, or must we linear interpolate Einstein and the Light's position at the same time value Curie is currently at?

Thank you.

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    BTW, you can (and should) format equations here using MathJax. Here's a quick tutorial to get you started, which has a link to the full MathJax guide. https://math.meta.stackexchange.com/q/33179/207316 – PM 2Ring Oct 13 '21 at 20:50
  • Thank you, been wondering about how to do that... I have the page well-bookmarked now for future use :) –  Oct 13 '21 at 23:24

2 Answers2

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This is the relativity of simultaneity. Events that are simultaneous in Einstein’s frame are not simultaneous in Curie’s frame.

Dale
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  • @ Dave thank you, it's been of great help discussing with! –  Oct 13 '21 at 19:42
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    @James Wikipedia has a nice anim illustrating the relativity of simultaneity "Events A, B, and C occur in different order depending on the motion of the observer. The white line represents a plane of simultaneity being moved from the past to the future." – PM 2Ring Oct 13 '21 at 20:44
  • @ PM 2Ring Thank you for answering! Yes, it is clearer to me now, we always perceive time as the horizontal white line, and the non-even slope means that the events are not simultaneous to the frame of reference of the person in the middle. So, if I would like to draw an illustration of their instantaneous distance apart at t=tk, then I would need to take a ruler and linearly interpolate the right positions horizontally for Einstein and the Light at exactly tk, which are not the same as those two points "spitted out" by the Lorentz transform, is that right? Thanks. –  Oct 13 '21 at 23:20
  • Maybe. ;) I'm not totally clear on how you're doing that interpolation, so I don't want to say "yes". The point is that if you want to calculate the distance between events A & B you must do that in a frame where A & B are simultaneous (i.e., they have the same time component). There's a nice way to do this stuff using rotated graph paper, which robphy often uses in his answers, eg https://physics.stackexchange.com/a/383363/123208 & https://physics.stackexchange.com/a/545825/123208 – PM 2Ring Oct 16 '21 at 01:08
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Your first spacetime diagram shows Einstein, Curie and the light signal moving away from a common point for a while, and then suddenly vanishing from existence. If that's what you intended to show, then their vanishing is simultaneous with respect to the first coordinate frame and not simultaneous with respect to the second: that's the relativity of simultaneity, as Dale said.

But I suspect you didn't mean to show that. You mention "the time animated line" in your question, so I suspect you're animating worldlines that lengthen over time, and your images show how long they are at a certain frame of the animation.

That's not how special relativity works. Spacetime doesn't change over time; it just is. Worldlines don't just end in the middle of nowhere. The light may be absorbed, but the worldline of whatever absorbs it will intersect it at the absorption point. Einstein and Curie won't live forever (not to be morbid), but the matter that makes up their bodies will continue to exist after their death. All of those events are just points in spacetime. There's no such thing as a spacetime in which they "haven't happened yet".

Can we draw a diagram when the "time" is not horizontally the same for all objects

What you call "time" here is, I think, a sort of meta-time, the frame number in your animation. It has no physical significance. The only physically meaningful time is the vertical direction on the spacetime diagrams. Since the meta-time is your invention, you can make it behave however you want.

benrg
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  • @ benrg thanks for answering! I think I am understanding this better now, these lines are like "field lines" which exist for all the way to the end, so we can take any measurements of them we like horizontally and vertically and in any direction we like, as long as we can form an interpretation of what it actually "is" that we think we are really measuring, is this right? It seems that Lorentz transform is literally everything there is to special relativity, and everything else can be derived (length contraction, time dilation, proper length, twin paradox etc) by interpreting it right? –  Oct 14 '21 at 01:04
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    @James That sounds like a reasonable summary to me. Yes, the Lorentz transformation (or rather, requiring all physical laws to be invariant under the Lorentz transformation) is all there is to special relativity. – benrg Oct 14 '21 at 01:11
  • @ benrg thanks very much, you wouldn't know how relieving it is, and quite beautiful, to realize for once that everything in special relativity can be gotten by interpreting the Lorentz transformation the right way... Often as we first come across special relativity, it seems like a magician's bag of tricks... so many nonsensical predictions as audacious & cunning as the performing "physicist" can pull out of their hat, it is very nice to finally realize that everything actually comes out of a single consistent physical rule underneath. –  Oct 14 '21 at 01:19
  • Yep, GOOD: Lorentz Transform, Spacetime Interval; BAD: relativistic mass, time dilation, length contraction ("gamma slinging"). Now we just need to burn all those old books, and delete a million YT videos . . . ;) – m4r35n357 Oct 14 '21 at 11:29