Time lines of observers meeting each other - doubts about their graphical representation

I am trying to make sense of Fig. 2.12, page 23 of Introducing Einstein's Relativity (D'Inverno, Oxford University Press). There it goes:

The book picture is in black. Scales are such that light rays are inclined by 45deg.

Observer A sees events P and Q happen simultaneously at equal opposite distance. According to the book, observer B (riding his own BLACK time line) meets A at the same moment as events P and Q happen according to A.

This does not convince me. In my view an observer that meets A the very moment A observes P and Q must be travelling on the RED line.

The book says that A sees P, Q and O happen at the same time. I'd say that A sees P, Q and O' happen at the same time.

Which is the correct time line for B? The black or the red one?

• If I undestand your question - B meets A at the same moment P and Q happen according to A - is true. The point is that it is only according to A P and Q are simultaneous (and meeting with B). Next step is to select a different observer ( B frame ), plot again and get picture different. – jaromrax Mar 22 '17 at 15:01
• If A and B meet, they are at the same place in the same time. If they meet in O, imho A hasn't yet seen P and Q, regardless of what B is experiencing. – Marco Faustinelli Mar 22 '17 at 15:05
• The book is correct when it says B meets A at the same moment as events P and Q happen according to A. Note that it doesn't say B meets A at the same moment as A sees events P and Q happening. It assumes that A knows how long the rays have taken to get to her location. – Holger Schmitz Mar 22 '17 at 15:10
• I do not think that "sees" refer to "when the light from those events reach A". I believe it means "what A considers that happened at t=0", and for that he has to take into account that the light he received is delayed. – user126422 Mar 22 '17 at 15:12
• I see what you mean. Can you please write me an answer, before the bots move all these comments to the chatbox? – Marco Faustinelli Mar 22 '17 at 15:14

The complete diagram from d'Inverno (p. 23) is shown below. It appears that observer-A has performed radar experiments on events P and Q. Observer-A assigns time-coordinates to events as the halfway time between emission and reception of the radar signal. So, Observer-A assigns P and Q the same time coordinate---to Observer-A, P and Q are simultaneous. Further, it appears that event O is the midpoint-event between emission and reception, and thus O is simultaneous with P and Q.

Granted, at the meeting event O, observer-A doesn't yet have the information needed to assign those time-coordinates to P and Q yet.

As others have pointed out, "sees" (or observes) is an imperfect term since one might not distinguish (1) a spacelike-relation of simultaneity with [i.e., assigning the same t-coordinates to] a distant event from (2) a past-lightlike-relation with [i.e., visually seeing] a distant event.

• "A sees P, Q and O happen at the same time." really means that A assigns the same time-coordinate to P, Q, and O.

• "A sees P, Q and O' happen at the same time." would be correct if it means that light-signals from P, Q, and O' reach A at the same time... according to A. If you said, with the meaning just given, "A sees P, Q and O' at the same event", then everyone would agree to that.

B's worldine passes through event O, as the book has drawn.

The purpose of the diagram is that Observer-B will assign distinct time-coordinates to events P, O, and Q.

• robphy: "The complete diagram [...] is shown below." -- Thanks; great to have this available here. And what a shame d'Inverno used the same notation style for participants (such as A) and events (such as P). "to Observer-A, P and Q are simultaneous." -- It's nonsense to say that entire events (which are spacelike separated) were simultaneous; by, or "to", anybody. "event O is the midpoint-event between emission and reception [events]" -- Correct. (As follows from $s^2$ values.) "thus O is simultaneous with P and Q." -- No: O is the midpoint-event between P and Q (by $s^2$ values). – user12262 Mar 23 '17 at 4:35
• I'm not sure what you don't like about the word "simultaneous"... so let me rephrase. Consider the radar-emission event and the radar-reception event on Observer-A's worldline. The intersection of the future light cone of the emission event and the past light cone of the reception event lie on a hyperplane that passes through the midpoint-event O, and this hyperplane is orthogonal to that observer's worldline. Thus, this observer will assign the same t-coordinate to events P and Q that are assigned to event O. Generally, another observer will assign different t-coordinates to those events. – robphy Mar 23 '17 at 5:35
• robphy: "what you don't like about the word "simultaneous"." -- I like that word; just not being applied to pairs of entire events, but to pairs of indications of certain participants (e.g. J and K from my answer here earlier, where A had been and remained the "middle between" J and K). "[Events P and Q are] intersection of the future light cone of the emission event and the past light cone of the reception event [of A]" -- Correct. There ought to be a word for that!: P and Q are "ping-symmetric" wrt. A!(?). "coordinates" -- Are secondary. – user12262 Mar 23 '17 at 5:55
• p.s. Since the originally posted question stipulated flat spacetime, we may indeed call event pair $(P \, Q)$ and event pair $(\text{emission} \, \text{reception})$ "orthogonal" to each other; i.e. in terms of spacetime intervals, $s^2$: $$s^2[ \, P, Q \, ] + s^2[ \, \text{emission}, \text{reception} \, ] = (4)^2~s^2[ \, P, \text{reception} \, ] = 0.$$ – user12262 Mar 23 '17 at 6:33
• I kinda understand that the fact that A has illuminated P and Q is crucial. So far I've left it aside, but I see that the book puts outgoing light rays all over the place. WHY is my outgoing light so important? Don't events just happen? Couldn't P and Q be explosions on their own that I see happen in that point of the sky? Lemme think some more about the meaning of "distance" :-) – Marco Faustinelli Mar 23 '17 at 8:49

Fig. 2.12, page 23 of Introducing Einstein's Relativity (D'Inverno, Oxford University Press). Observer A sees events P and Q happen simultaneously

This formulation constututes imprecise use of the notion of "simultaneity" according to Einstein's definition in two seprate senses. Instead:

(1): Observer A sees events P and Q "at the same time", i.e. in coincidence, at (only) one event (called O' in the sketch).

(2): The notion of simultaneity is not even applicable to entire events (but instead applicable to indications of individual observers, who took part in events; but who, unfortunately, are not explicitly drawn in the figure as reproduced above.

at equal opposite distance.

Distances are defined and to be determined between suitable pairs of observers (namely specificly: between observers who were and remained at rest between each other); not between an observer (such as A) and an event (such as P, or Q).

Again: Unfortunately, the figure doesn't show or name a specific observer who was and remained at rest wrt. A and who took part in event P (although such an observer might be easily considered in the setting of a thought-experment, let's call it J); nor another specific observer who was and remained at rest wrt. A and who took part in event Q (although such an observer might be easily imagined, too, let's call it K).

According to the book, observer B (riding his own BLACK time line) meets A at the same moment as events P and Q happen according to A.

Again: that's a misattribution of simultaneity to entire events. What can be said correctly and according to everyone involved, in mutual agreement, is instead:

Observer A's indication of having met observer B (namely at event O) was simultaneous to observer J's indication of having taken part in event P, as well as simultaneous to observer K's indication of having taken part in event Q.

In my view an observer that meets A the very moment A observes P and Q must be travelling on the RED line. [...] I'd say that A sees P, Q and O' happen at the same time.

Correct: Observer A observed all that in coincidence.

It is important to note the distinction:
The event in which A indicated perceiving events P and Q and being met (in passing) by the red-line-observer in coincidence (namely event Q) is not the same as
the event (O) in which observer A took part such that A's indication at this event (specificly: A's indication of having met, in passing, by observer B) was simutaneous to observer J's indication of having taken part in event P, as well as observer K's indication of having taken part in event Q.

• does the knowledge that A is actually making a radar experiment wrt P and Q (a fact that I blissfully omitted from my question but was pointed at by robphy in his answer) spare us from having to postulate observers J and K? – Marco Faustinelli Mar 23 '17 at 8:42
• @Muzietto: "does the knowledge that A is actually making a radar experiment wrt [events] P and Q spare [...] observers J and K?" -- Hardly: events are not observable "as such". Instead, observable are (indications of) participants/observers/"material points" who took part in events. Since therefore we must consider some participants in P and Q, then why not specificly J and K who were and remained at rest wrt. A; being jointly members of the same "inertial frame". – user12262 Mar 23 '17 at 20:32
• @Muzietto: "radar experiment [...] pointed at by robphy" -- With the specific outcome that for an "emission" indication of A subsequently A observed corresponding echo events P and Q in coincidence; while B's radar experiments obtained different results. However, in Einstein's coordinate-free definition of "simultaneity" (1917) there is apparently no (explicit) mentioning of "radar echos", "pings", "signal round trips" etc.; while "{determination of} mid-point" is explicitly required. – user12262 Mar 23 '17 at 20:34