# A question about space-time diagram

Examine the image carefully, this is a space time diagram plotted by Brian Greene. The event co-ordinates are shown according to both reference frames.

My question is, did the event, $$E$$, happen simultaneously for both the blue and red frames (the relativity of simultaneity would not be applicable here as I am not talking about whether two events happened simultaneously or not)? In other words, is the event in the "present" moment for both the frames or not?

• I'm not sure if one can say that an event is simultaneous for two reference frames since there is no absolute time, but maybe someone with more experience can provide an in-depth answer. Commented Feb 10, 2021 at 12:18
• How do you define simultaneous? That they have the same time label?
– JGBM
Commented Feb 10, 2021 at 12:33
• I personally define, two events are simultaneous if they belong to same timeline i.e. two event happened "now" then it's simultaneously happened (for me). Commented Feb 10, 2021 at 13:05
• "In other words, is the event "present" moment for both the frames or not?" - the "present" moment for each frame is its corresponding x-axis (all the events on the x axis, for each frame). So at t=0 the event is in the future of both frames. As time progresses, this "simultaneity line" moves parallel to the x-axis along the t direction (for each frame - so they are angled). By your definition, the only point (event) that is "now" for both frames is the one where these "simultaneity lines" of each frame intersect. Commented Feb 10, 2021 at 13:32
• "did the event, $E$, happen simultaneously for both the blue and red frames " --- asking whether an event is "simultaneous" is like asking whether an argument is "relevant" --- every argument is relevant to some issues and not others, and every event is simultaneous (in a given frame) with some events and not others. Commented Feb 10, 2021 at 15:34

"Simultaneously" is a comparison which requires two events and one reference frame. If event $$A$$ is simultaneous with event $$B$$ according to reference frame $$S$$ then that means that the time coordinate of $$A$$ is the same as the time coordinate of $$B$$ both according to $$S$$.

In principle, you could set $$B=A$$ and then have one event and one frame, but there is no way to express the concept of simultaneity for one event and two reference frames. It just doesn't fit.

In the diagram event $$E$$ is simultaneous with any event on the white dotted line $$t_0$$ according to the unprimed reference frame and event $$E$$ is simultaneous with any event on the red dotted line $$t'_0$$ according to the primed reference frame. That is really all that can be said.

• is it allowed to draw a trajectory of faster than light object on a space time diagram. it would be a slope of less than 45 degrees. however, does this make sense? Commented Apr 14 at 11:44
• @AlexanderCska certainly you can draw such a line. You can even parameterize such a line with an affine parameter. As far as we know such a line does not correspond to the path of any actual object
– Dale
Commented Apr 14 at 17:42
• Actually, I asked because space time diagram with such a line is used to demonstrate causality vialoation. Say one trows a superluminar magic dart. In this case it could be seen from another moving reference frame as going backwards in time or even launching the dart and it arriving on target get simultaneous. How credible is such graphical analysis? Commented Apr 16 at 14:23
• @AlexanderCska the graphical analysis is correct. It gives a good theoretical reason why we do not expect that such darts actually exist.
– Dale
Commented Apr 16 at 15:44

In the light diagram there is only one point such that an event that happens in that point happens in the "present" moment for both the frames (in the sense that I interpreted from your question), and it's the origin. And this means that if we define as the event $$A$$ the one in which "one hand of one observer from the blue frame touches one hand of one observer of the red one", then a statement of the type "events $$A$$ and $$B$$ happened simultaneously" can be true only if both their coordinates happen in the origin. Indeed, if we consider events $$A$$ and $$B$$ simultaneous for the blue frame, it means that they have to be aligned horizontally in the graphic, and because of to be simultaneous also for the red frame they should be aligned not horizontally but in a way parallel to the red $$x$$ axis, we can conclude that it's not possible that they happen simultaneously both in the red and in the blue axis, unless they stay both in the same point, and it can be only the origin because event $$A$$ has to be in the $$t=0$$ for both the frames.