# Space-time diagram from the point of view of someone who accelerates in a $\delta(t)$ way

Suppose the situation shown in the following space-time diagram: This diagram was drawn by the inertial observer standing in "the blue system", called A from azul in Spanish. In it we see the trajectory of "the green system" called V (verde) which, as can be seen, is also inertial. Finally the trajectory of an observer standing at "the red system" (called R for red $\overset{\cdot . \cdot}{\smile}$). This last system R is inertial everywhere except at $t=0$ where it experiences a short and strong enough acceleration that can be seen as a Dirac delta.

I wonder how the person in R would draw a space-time diagram for this?

I have tried to answer this question as follows: first I have added some "identified events" (events 1, 2, $A_1^+$ and $A_1^-$) in order to see how things change from the perspective of the blue person and the green one. Doing some straightforward math I arrived at the following S-T diagrams: At the left there is the same S-T diagram as the original one but with the events marked down. At right the S-T diagram representing the point of view of the green person.

As can be seen each blue and green person is co-moving with the red one before and after the acceleration, respectively. The red person would have two space-time diagrams one for each tram of its trajectory, like in the following drawing? Would it have only one space-time diagram consisting of a mixing of this two? The coordinates of some event will change before and after acceleration? Would some event appear more than one time in a "unified space-time diagram"?

• No, it changes! I didn't make the plots very accurate, but if you see in the second image,the event "2" in the plot of the right is not only at $t<0$ but also farther in $x$, i.e. with $x>x_0$. – user171780 Apr 4 '18 at 11:46
• You may be interested in the following material in my special relativity book, lightandmatter.com/sr : example 17 in section 1.4; section 3.9.3; section 7.1. Although my original comment was not quite correct, it seems to me that there is still something goofy about the graphical conventions you're using in drawing these diagrams. There are two ways people would normally do this: (1) keep the events fixed on the page while distorting the axes, or (2) draw the (t',x') axes right on top of the (t,x) axes, but move the points. You seem to be doing some combination of those things. – user4552 Apr 4 '18 at 15:30

Call the two frames of reference $R_{\mathrm{old}}$ and $R_{\mathrm{new}}$. The coordinates of events will certainly be different between $R_{\mathrm{old}}$ and $R_{\mathrm{new}}$. Some events which have not yet happened at the instant of the acceleration according to $R_{\mathrm{old}}$ will have already happened at the instant of acceleration according to $R_{\mathrm{new}}$. And some events which have already happened at the instant of the acceleration according to $R_{\mathrm{old}}$ will have not yet happened at the instant of acceleration according to $R_{\mathrm{new}}$. $R_{\mathrm{old}}$ and $R_{\mathrm{new}}$ would only agree on the spacetime coordinates of one event, which would presumably be chosen to be the location of the Red person at the moment of the acceleration.
It really wouldn't work well to try to create a spacetime diagram that combines the use of $R_{\mathrm{old}}$ and $R_{\mathrm{new}}$ in one diagram, unless perhaps if you did something like plot two dots for each event, one for each of the two coordinate systems, in two different shades of red. It may or may not be useful to draw a unified diagram like that, in which almost all events are drawn twice.