# How to make appropriate statements concerning simultaneity or sequence of pitches?

This answer The example of relativity of simultaneity given by Einstein to a recent question related to Einstein's thought-experimental definition of (how to determine) simultaneity contained the following statement:

Suppose two people, $C$ and $D$, stand equal distances from you and are known to pitch balls at exactly the same speed. With everyone standing at rest, $C$ and $D$ each toss you a ball. You get the ball from $C$ before the one from $D$. This is not a logical inconsistency. It simply means $C$ threw a ball before $D$ in your reference frame [emphasis added].

I believe that I understand the described setup and the conclusion ("It simply means $C$ threw a ball before $D$") as such. But I question whether it is necessary to add the qualification "in your reference frame".

Carefully applying Einstein's definition of (how to determine) simultaneity, as referenced above, which for the given setup involves a suitable observer "at the midpoint between $C$ and $D$", is there even any reference frame at all (necessarily other than "your reference frame") "in which" $C$ threw a ball *simultaneous to* $D$ throwing a ball ?

Or is there even any reference frame at all (again necessarily other than "your reference frame") "in which" $C$ threw a ball *after* $D$ ?

(If there are no such reference frames, then the qualification "in your reference frame" is apparently not necessary; and, indeed, it would seem inappropriate and misleading to add such a qualification as if it were necessary.)

Try thinking in terms of events and spacetime. Event C is the toss of the ball from pitcher C and Event D is the toss of the ball from pitcher D. In your frame of reference, C occurs before D.

Now, if events C and D have a spacelike interval (meaning they are farther apart in space than in time), then there is a frame of reference in which they are simultaneous and there are frames of reference in which D occurs before C.

But, if events C and D have a timelike interval (meaning they are farther apart in time than in space), or a light-like interval (meaning they are as far apart in time as in space), then there is no frame of reference in which C does not occur before D.

Here is an interactive spacetime diagram that you can "play" with to help make these results visually clear.

• Alfred Centauri: "Try thinking in terms of events and spacetime." -- Fine, as long as the notion "event" is carefully distinguished from other notions appearing in the quoted statement. "Event C is the toss of the ball from pitcher C [...]" -- Already your notation starts to conflate "event" and "one particular participant of an event (and perhaps also of other events)": "pitcher $C$". May I therefore suggest (also trying to be consistent with notation used in the reference): ... (to be continued.) – user12262 Oct 17 '13 at 6:14
• (contd.) ... May I suggest: call "$A$" the ball thrown by $C$; call "$\mathcal{E}_{AC}$" the event in which both $A$ and $C$ took part ($C$ tossing $A$, and $A$ being tossed by $C$; and perhaps even other participants, such as a passing butterfly clapping its wings, or a passing muon decaying); call "$C_A$" pitcher $C$'s indication of tossing $A$; call "$A_C$" ball $A$'s indication of being tossed by $C$. Likewise call "$B$" the ball thrown by pitcher $D$, introduce event "$\mathcal{E}_{BD}$" and so on. ... (to be continued.) – user12262 Oct 17 '13 at 6:23
• (contd.): Finally, note that the quoted statement and my question is specificly about the pitchers ("people") $C$ and $D$ and their indications (pitches, or "acts or throwing", $C_A$ and $D_B$); and not necessarily about indications of any other participants in events $\mathcal{E}_{AC}$ or $\mathcal{E}_{BD}$). – user12262 Oct 17 '13 at 6:23
• p.s. user12262 wrote: "May I suggest: call "$A$" the ball thrown by $C$ ... Likewise call "$B$" the ball thrown by pitcher $D$". -- That's unfortunately a poor choice of names for the balls under consideration because name pair "$A$, $B$" is typically given to (and suggestive of) two participants who are at rest to each other; which these balls are not. So the ball thrown by $D$ might be better called "$E$", for instance; and the event in which $D$ and $E$ took part correspondingly $\mathcal{E}_{DE}$. – user12262 Oct 17 '13 at 10:19
• @user12262 In relativity the term "event" has a specific meaning: it's a point in spacetime. And being a point it's instantaneous, and it has no spatial extent. – PM 2Ring Jan 18 '18 at 5:17