What is wrong in this representation of relative reference systems?

Question

Up until now I've explained relative time to myself as looking at the 3D world from different four-dimensional perspectives, analogous to how looking at a 2D-ish object (eg. a sheet of paper) from different angles makes it appear to change shape.

Also, I've used this representation below to explain how the time at which an event occurs could change depending on the body's reference system. This representation uses two bodies that move relative to each other, $B1$ and $B2$, to represent the time of occurrence of events $E1$ and $E2$.

(Not very precise — blame MS Paint :P — but my intent should be clear.)

I am not sure, but as far as I know $t(E1) \le t(E2)$ should hold true for any reference system (order should be preserved), which means that the drawing is correct. But I just realized that in certain cases, it is possible for $E2$ to occur before $E1$ (according to my representation):

Here you can see that $B3$ and $B2$ have entirely different perceptions on the order of the two events, but that would break causality, so it cannot be true. What is wrong in my representation of relative systems of reference?

Answer 1

There is a mistake in your diagram, in that you drew the x-axis for body one in the rest frame of body 3 as perpendicular in the Euclidean sense to the t-axis of body one. The two lines are not Euclidean perpendicular, but Minkowski perpendicular.

The t axis is correct for body 1 in the second diagram, but the x-axis for body 1 slopes up, not down. It slopes up by the same amount as the t-axis for body 1 slopes to the right. This looks awkward in a Euclidean geometry diagram, but it is correct Minkowski geometry.

The direction of the slope is fixed by Einstein's argument about simultaneity at a distance, reproduced in this answer: Einstein's postulates $\leftrightarrow$ Minkowski space for a Layman . This argument also fixes the relative non-simultaneity of E1 and E2.

When you use the correct notion of perpendicularity, you will find that t(E2) is bigger than t(E1), not smaller. But they are indeed non-simultaneous, as you note.

Answer 2

The causality is not broken here. One event can cause (affect) another only if information from the first event can reach the second event. This is possible only if E2 is in the light cone of E1, i.e. light emitted from E1 can reach position of E2 before it happen.

Here is the picture of light cone:

The events inside the past light cone can affect the observer. The events inside the future light cone can be affected by observer. They are also called "absolute past" and "absolute future".

Let the event E1 happen at the origin point of the picture. If E2 is inside one of the light cones then one can select a reference frame where both events happen in the same place but at different time. There is no reference frame where the events happen simultaneously.

All events outside the light cones are "absolutely remote" events. They can not be neither the reason nor the consequence of the event at the origin.

Let the event E1 happen at the origin point of the picture. If E2 is outside the light cones then one can select a reference frame where both events happen simultaneously in different places. There is no reference frame where the events happen in the same place.

This is exactly the case you consider. Int the body 1 system of reference the events happen simultaneously. This means that they are absolutely remote and can not affect each other. Their order in time is not absolute and depends on the reference frame.

The relative position of two events is determined by the spacetime interval between them: $$ s_{1,2}^2 = \left|\mathbf{r}_1 - \mathbf{r}_2\right|^2 - c^2 (t_1 - t_2)^2 $$ This value is an invariant (i.e. does not depend on the reference frame). If $s_{1,2}^2$ is positive (space-like interval) then the events are absolutely remote. If $s_{1,2}^2$ is negative (time-like interval) then the events are in the light cones of each other and their order in time is determined.