How to interpret Lorentz transform in Special Relativity when the time axis position for different objects are not horizontally even?

Question

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:

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:

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:

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.

Answer 1

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

Answer 2

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.