# Defining simultaneity with a central light vs with clocks

So there's the classic example of the relativity of simultaneity involving two people on a train, with a light source exactly between them. Moments after the lights turn on, observers on the train will say the light struck the passengers simultaneously, while folks on the ground looking into the train will see the light hit one of them first.

Now let's say the passengers start out next to each other, synchronize their clocks, and then slowly proceed to their respective ends of a 2 lightsecond long table.

At 2:59:59 PM according to their clocks, the light between them turns on. Now there are 4 events:

1) The light hits Passenger A (the one closer to the front of the train).

2) The light hits Passenger B.

3) Passenger A's clock ticks 3 PM.

4) Passenger B's clock ticks 3 PM.

Passengers on the train should all agree the four events are simultaneous, but what will people outside the train see? Will the clocks stay simultaneous with each other, or will 2 and 4 happen simultaneously, followed by 1 and 3? And regardless of the answer, can you justify it in terms of the fixed speed of light?

The simple answer to this is in two parts:

(1) In all reference frames, A's clock reads 3pm when the light hits passenger A and B's clock reads 3pm when the light hits passenger B.

(2) Only in the trains' reference frame are the clocks synchronized. In any other uniformly moving reference frame, the clocks are not synchronized and, thus, the light hits one passenger first.

This is a profoundly important result from Special Relativity: the relativity of simultaneity.

I must address this comment below (at the time of this edit) from user12262:

-1 @Alfred Centauri: "Only in the trains' reference frame are the clocks synchronized. In any other uniformly moving reference frame, the clocks are not synchronized" -- By Einstein's definition the determination of whether two given clocks are synchronized, or not, is solely a matter of those clocks and (by transitivity) of other suitable members of "their frame"; not of any other participants (members of "any other frame"). – user12262

• So is the idea that the clocks fell out of sync (relative to the outside frame) while the passengers were walking to their respective seats? Even though they did so slowly? – hjfreyer Oct 22 '13 at 17:11
• @hjfreyer, I suppose that's correct. In the train's frame, the slow clock transport is symmetrical (clock's A & B can be swapped without changing the problem) but this isn't so from the perspective of a relatively moving frame. – Alfred Centauri Oct 22 '13 at 17:30
• Aha, right, if you think of the clocks as light clocks, the person moving towards the back of the train will have a slightly faster ticking clock from the outside frame, because he's moving slightly slower. Added up over such a long distance, it must make up for the difference. Thanks! – hjfreyer Oct 22 '13 at 18:05
• -1 @Alfred Centauri: "Only in the trains' reference frame are the clocks synchronized. In any other uniformly moving reference frame, the clocks are not synchronized" -- By Einstein's definition the determination of whether two given clocks are synchronized, or not, is solely a matter of those clocks and (by transitivity) of other suitable members of "their frame"; not of any other participants (members of "any other frame"). – user12262 Oct 22 '13 at 20:34
• @user12262, see edit to my answer. – Alfred Centauri Oct 22 '13 at 20:50

Since 1) and 3) happen at the same time and at the same place, every one in every frame will agree they are simultaneous. Likewise for 2) and 4).

Of course light doesn't know about watches so to fix the relationship between 1) and 2) we can appeal to the case where there were no watches.

The way to answer this, and indeed every other question (ESPECIALLY EXAM QUESTIONS¹) in special relativity is to choose a convenient pair of frames, write down the coordinates of the events you're interested in, then use the Lorentz transformations to transform the events into the other frame.

In this case we'll take the original frame to be the rest frame of the train and put the origin at the point the light is emitted. In that frame we have three events of interest:

• (0, 0) - the light is emitted
• (1, c) - the light is received by passenger A and their watch shows 03:00:00
• (1, -c) - the light is received by passenger B and their watch shows 03:00:00

Now take the primed frame to be the observer standing by the track, so the primed frame is moving at some velocity $v$. We'll take the origins of the two frames to coincide, so the light is emitted at (0, 0) in both frames. Now find the location of the events in the primed frame using:

$$x' = \gamma (x - vt)$$

$$t' = \gamma (t - \frac{vx}{c^2})$$

I won't do this because you might be interested to work it through for yourself. Ping me if you want me to update my answer to include the calculation.

¹ I wish I could highlight this point in 72 point flashing neon text!

So there's the classic example of the relativity of simultaneity involving two people on a train

... specifically Passenger A and Passenger B (who are separate from each other, and at rest to each other) ...

with a light source exactly between them.

The participant who is "middle between" (or "at the mid-point of") given participants A and B is typically called M; for instance in the classic example.

[...] observers on the train will say the light struck the passengers simultaneously

Yes, they as well as everyone involved can and will be able to say that Passenger A's indication of observing M's signal and Passenger B's indication of observing M's signal had been simultaneous;
at least once it has been conclusively and in mutual agreement established that Passenger A and Passenger B and Passenger M had indeed remained at rest to each other while exchanging these light signals.

while folks on the ground looking into the train

... it's useful to identify these participants by name as well; let's say specificly
- Bystander J who observed being passed by Passenger A while (in coincidence with) observing that M had stated the light signal and that A observed that M had stated the light signal, and
- Bystander K who observed being passed by Passenger B while (in coincidence with) observing that M had stated the light signal and that B observed that M had stated the light signal ...

will see the light hit one of them first.

Yes, everyone involved can and will be able to say that Bystander J's indication of observing M's signal and Bystander K's indication of observing M's signal had not been simultaneous;
at least once it has been conclusively and in mutual agreement established that Bystander J and Bystander K had indeed remained at rest to each other until observing the signals of M.

Indeed, since A is supposed to be the one closer to the front of the train and therefore A, B and M had been moving, approaching and passing "from K towards J", everyone can say that:
Bystander K's indication of observing M's signal (along with A passing) had been before Bystander J's indication of observing M's signal (along with B passing).

Now there are 4 events: 1) The light hits Passenger A (the one closer to the front of the train).

Strictly speaking that's merely the description of Passenger A's indication at some partcular event, but not the full description of an entire event; namely the event of Passenger A and Bystander J passing each other and, coincidently, both observing M's signal, (and, coincidently, 3) Passenger A's clock [indicating] 3 PM.).

2) The light hits Passenger B.

Strictly speaking that's merely the description of Passenger B's indication at some partcular event, but not the full description of an entire event; namely the event of Passenger B and Bystander K passing each other and, coincidently, both observing M's signal, (and, coincidently, 4) Passenger B's clock [indicating] 3 PM.).

Passengers on the train should all agree the four events are simultaneous

Everyone agrees that the four desribed indications (not entire events!) are simultaneous; and they are indications of passengers on the train (incl. their clocks).

will 2 and 4 happen simultaneously, followed by 1 and 3?

Indications (2) and (4) are coincident indications (of Passenger B and of B's clock). Likewise indications (1) and (3) are coincident indications (of Passenger A and of A's clock).

And:
Passenger B's indication of observing Bystander K's passing and Passenger M's stating of the signal was simultaneous to Passenger A's indication of observing Bystander J's passing and M's signal.

but what will people outside the train see?

Bystander K's indication of observing Passenger B's passing and passenger M's stating of the signal was not simultaneous to Bystander J's indication of observing Passenger A's passing and M's signal.
- Passenger A and Bystander J passing each other and
- Passenger B and Bystander K passing each other.
Will the clocks stay simultaneous with each other
As far as the simultaneous indications of these two clocks (of A and of B) were looking alike (e.g. both indicating "3 PM" simultaneously), the two clocks can be said having remained synchronous to each other. (Einstein's synchronism conditions are even stricter, however, and they involve assigning real numbers $t$ to indications.)