Lorentz Transformations and time of event

Question

Consider two inertial frames, $F$ and $F'$, such that $F'$ moves at $\mathbf{v} = (v,0,0)$ with respect to $F$ (assume $v > 0$). Suppose tat $x = x\prime = 0$, when $t = t' = 0$, where $x,t$ refer to $x$ axis and time coordinates in $F$, and $x',t'$ refer to those quantities in $F'$.

Consider the event that occurs at $x = a, t = 0, a > 0$. By the Lorentz transformations, $$ t' = \gamma t - \beta \gamma\ \ \text{ and } \ \ x' / c = -\beta \gamma a/c, $$ where $\beta = v/c$ and $\gamma = 1/\sqrt{1 - \beta^2}$. But, in $F'$, this is even before its origin strikes the origin of $F$, that is even before $t' = t = 0$. So $F'$ experiences the event even before its actually done? How is this possible, or am I slipping up somewhere?

Answer 1

This effect is called relativity of simultaneity. It means that two observers need not agree on the simultaneity of two events, or on their temporal order.

This effect depends critically on whether the events are spacelike separated (i.e. $\Delta s^2=-c^2\Delta t^2+\Delta x^2>0$) or timelike separated (i.e. $\Delta s^2=-c^2\Delta t^2+\Delta x^2<0$).

• If events $A$ and $B$ are timelike separated, and one observer $O$ observes $A$ to happen before $B$, then all observers agree that $A$ happened before $B$. The only way for events to be timelike-separated and simultaneous is for them to happen at the same point in space (in all frames), which means that they are the same event.

• If events $A$ and $B$ are spacelike separated then there will always exist

  (i) an observer $O_1$ for whom $A$ and $B$ are simultaneous,

  (ii) an observer $O_2$ for whom $A$ happens before $B$, and

  (iii) and observer $O_3$ for whom $A$ happens after $B$.

Because of this, it is meaningless to talk about the temporal order of spacelike-separated events. Such events are causally disconnected from each other: the only events which $A$ can influence (or be influenced by) are within its future (or past) light cone. A spacelike-separated event $B$ is outside the light cone and can do neither.

So $F′$ experiences the event even before its actually done?

This is an incorrect interpretation. The events are placed by $F$ and $F'$ after all the observations have been made; specifically, events are not equivalent with observations of the events. $F'$ disagrees with $F$'s observation that $A$ and $B$ were simultaneous, but there is nothing going causally back through time.

Answer 2

Consider two inertial frames [...]
[...] its origin strikes the origin of [the other]

Let's give these two distinct participants (which are both called "origin" of their respective inertial frames) some more distinctive short names, for reference below; say $\mathsf J$ and $\mathsf P$.

The event of the two "origins", participants $\mathsf J$ and $\mathsf P$, "striking (and passing)" each other is then readily denoted as $\varepsilon_{\mathsf{JP}}$.

Participant $\mathsf J$ (as well as all other members of the inertial frame to which $\mathsf J$ belongs) and participant $\mathsf P$ (as well as all other members of the inertial frame to which $\mathsf P$ belongs) are moving with respect to each other uniformly (straight and with constant speed, $\beta~c := v$).

Consider the event that occurs [... without participation of either $\mathsf J$ and $\mathsf P$]

Let's explicitly identify some of the relevant participants in this event, too: say $\mathsf K$ and $\mathsf Q$,
where $\mathsf K$ and $\mathsf J$ were and remained at rest to each other (thus both being members of the same inertial frame),
and likewise $\mathsf Q$ and $\mathsf P$ were and remained at rest to each other (thus both being members of the same inertial frame).

The event of these two participants, $\mathsf K$ and $\mathsf Q$, "striking (and passing)" each other is then readily denoted as $\varepsilon_{\mathsf{KQ}}$.

$x$ [...] $t$ [...] coordinates [...] quantities [...]

Presumably these variables don't refer just to (distinctive, but otherwise arbitrary) coordinate labels, but instead to numbers which are proportional to geometric quantities: namely to distances and durations measured by and among members of one or the other inertial frame. Correspondingly, the assignment of the same value (such as "$t = 0$" in several cases) is based on the determination of simultaneity, and
the assignment of different values (such as "$t' = 0$" vs. "$t' = \frac{-\beta~a}{c~\sqrt{1 - \beta^2}} \lt 0$") is based on the determination of dis-simultaneity.

Therefore, in the given setup, participants $\mathsf J$ and $\mathsf K$ (along with other suitable members of their inertial frame, in particular another suitable participant identified as "the middle between" $\mathsf J$ and $\mathsf K$) determined that $\mathsf K$'s indication of being struck and passed by $\mathsf Q$ was simultaneous to $\mathsf J$'s indication of being struck and passed by $\mathsf P$;

while participants $\mathsf P$ and $\mathsf Q$ (along with other suitable members of their inertial frame, in particular another suitable participant identified as "the middle between" $\mathsf P$ and $\mathsf Q$) determined that $\mathsf Q$'s indication of being struck and passed by $\mathsf K$ was not simultaneous to (but, indeed, before) $\mathsf P$'s indication of being struck and passed by $\mathsf J$.

(That's provided if, according to the setup prescription, the "sequence of encounters" had been such that participant $\mathsf Q$ had first been struck and passed by $\mathsf J$, and only subsequently by $\mathsf K$.)

So [... $\mathsf Q$] experiences the event $\varepsilon_{\mathsf{KQ}}$ even before it's actually done?

No ... The experience and indication of $\mathsf Q$ being struck and passed by $\mathsf K$ simply and obviously belongs to (the observational contents of) event $\varepsilon_{\mathsf{KQ}}$, i.e. the event of participants $\mathsf K$ and $\mathsf Q$, "striking and passing" each other.

And that's of course perfectly consistent with the determinations of simultaneity, or of dis-simultaneity, mentioned above; of course in application of Einstein's coordinate-free defintion of (how to determine) "simultaneity".

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p.s.
Further comment on the notion of simultaneity; especially in response to this answer by @Emilio Pisanty (which had been submitted earlier):

The setup as given in the OP question and interpreted in my answer above, especially the joint, mutually consistent and agreeable conclusions that

• participants $\mathsf J$ and $\mathsf K$ determined that $\mathsf K$'s indication of being struck and passed by $\mathsf Q$ was simultaneous to $\mathsf J$'s indication of being struck and passed by $\mathsf P$;
  i.e. for short: "the indications $\mathsf {K_Q}$ and $\mathsf {J_P}$ were simultaneous to each other", and

• participants $\mathsf P$ and $\mathsf Q$ determined that $\mathsf Q$'s indication of being struck and passed by $\mathsf K$ was not simultaneous to $\mathsf P$'s indication of being struck and passed by $\mathsf J$;
  i.e. for short: "the indications $\mathsf {Q_K}$ and $\mathsf {P_J}$ were not simultaneous to each other"

provide an example of relativity of simultaneity. It means that simultaneity (or just as well dis-simultaneity, or temporal sequence), as determined by Einstein's operational definition, is not determined of, and cannot be ascribed to, pairs of entire events; but instead refers to a relation between indications of a pair of suitable particpants in separate events.

This is not a concern for pairs of (distinct, separate) events which have an unambiguous causal order relation between each other; i.e.

• either if at least one participant (and generally several participants) may be imagined, or even be found, who took part in both events under consideration;
  such as participant $\mathsf Q$ having been struck and passed by $\mathsf J$, and, separately, by $\mathsf K$, thus having taken part both in events $\varepsilon_{\mathsf{JQ}}$ and $\varepsilon_{\mathsf{KQ}}$,
  and being able to consistently determine the (temporal) sequence of own experiences/indications and which causal effects (memories etc.) had been carried from one event to the other, instead of vice versa;

• or if participants of one event thereby (in coincidence with striking and passing each other) could have first observed (seen, learned, been causally affected by) what had happened/struck/passed at the other event under consideration.

So Einstein's method is not required (nor usable) for such pairs of events which are "time-like" or "light-(signal-)like" related to each other.

It remains to consider pairs which are otherwise (namely "space-like") related to each other; such as events $\varepsilon_{\mathsf{JP}}$ and $\varepsilon_{\mathsf{KQ}}$. If such a "space-like" related pair of events were contained in a flat region, i.e. such that there could be an inertial frame found at all of which one of its members had taken part in one event and another of its members had taken part in the other event, and yet another member could be identified as the "middle between" the former two, then other inertial frames can be found, too, which move wrt. the first uniformly, at any speed $0 \lt v \lt c$, in any "direction of motion (from the origin)". Among all of them

• there are several inertial frames (e.g. in the described setup, the inertial frame to which both participants $\mathsf J$ and $\mathsf K$ belonged) whose members determine that the indication of one member at one event (such as $\mathsf K$'s indication $\mathsf {K_Q}$) was simultaneous to the indication of the other member at the other event (correspondingly $\mathsf J$'s indication $\mathsf {J_P}$), and

• there are several inertial frames (such as the one to which both participants $\mathsf P$ and $\mathsf Q$ belonged) whose members determine that the indication of one member at one event (such as $\mathsf Q$'s indication $\mathsf {Q_K}$) was not simultaneous to the indication of the other member at the other event (correspondingly $\mathsf P$'s indication $\mathsf {P_J}$), but instead before, or after, depending on "direction of motion" (or to put it more directly: depending on the "sequence of encounters", as sketched above).

Accordingly, it is incorrect to attribute "simultaneity" (or just as well "dis-simultaneity" and any "temporal sequence") to pairs of entire events which are not related by an unambiguous causal relation.