# Relativity of simultaneity doubt

• I
Summary:
My question is about on of the "famous" Einstein's mental experiment: the train and the lightnings, i don't understand why the result is what it is, I thought for day but I don't understand this particular case, If you can help me to understand I'll very grateful.

(excuse me for my english, but i'm studying physics and I am not a native English speaker)
One observer OE, is on the ground, we take him as the fixed frame of reference.
The other OT is on the train that is moving relatively to the OE at a costant velocity ( they are both inertial frame of reference).
So, two lighning strike the two points A and B when the train is with back and end in the same points ( A and B) and the two observer are aligned.

The observer OE is at equal distance from A and B, so the light, which travel at constant speed c, arrive at the same time at his eyes, the two events are simultaneous.
The observer OE anyway see OT traveling to B and move away from A, the speed of light is always c, so he conclude that OT observers first the light from B and then the light from A.
The result: OT see first ligh from B and then from A in his frame of reference.( like what think OE)

I agree with that, but now there is the problem( more than one):
1- If the speed of light is the same in EVERY inertial frame of reference, in the reference of OT that speed is c, so i think when the two lightning strike in A and B, they are equal distance from the center where there is OT 'cause the first assumption (when A and B are ALIGNED) so i think light will arrive at OT at the same time also in his frame of reference, if i don't observe that i think light speed is greater than c from B to OT and lower from A to OT, but this is a contradiction, and I don't understand how to escape from that my surely wrong interpretation.

2-If A and B are fixed with OE, and they arrive at OE in the same time, when I am OT, I see OE traveling in the opposite direction so I see light from A arrive first to OE and the B, ( because always c is the speed of light), but in this case OT see the "same" event but from different perspective and this for me is clear, also with Minkowskii diagrams, i see two events simultaneous for OE and for the OT perspective they're not.

3-If A and B are fixed with OT, they are simultaneous for OT but OE see what the original experiment say: OT see first A and then B, this time I am totally agree with that.

So what is the difference between 1 and 3, and why if the speed of light is the same, c, OT see first B and then A, if they are aligned and equal distance from OT.

PeroK
Homework Helper
Gold Member
2020 Award
Would you consider learning about the relativity of simultaneity from a different experiment?

I find this Einstein experiment confusing and unsatisfactory. There are easier ways!

lukka98
Dale
Mentor
2020 Award
1- If the speed of light is the same in EVERY inertial frame of reference, in the reference of OT that speed is c, so i think when the two lightning strike in A and B, they are equal distance from the center where there is OT 'cause the first assumption (when A and B are ALIGNED) so i think light will arrive at OT at the same time also in his frame of reference, if i don't observe that i think light speed is greater than c from B to OT and lower from A to OT, but this is a contradiction, and I don't understand how to escape from that my surely wrong interpretation.
You are making an incorrect assumption here. The two light pulses arrive to OT at the same time if all three of the following conditions are met:

IF
1) OT is equidistant from where the flashes occurred
AND
2) The speed of both flashes are the same
AND
3) The flashes occurred at the same time
THEN
The flashes arrive at OT at the same time

We know that 1) and 2) are true, but we do not know 3). In fact, since we know that the flashes do not arrive at OT at the same time, we know that at least one of 1), 2), or 3) must be false. And since we know that 1) and 2) are true, that implies that 3) must be false.

cianfa72
PeroK
Homework Helper
Gold Member
2020 Award
So what is the difference between 1 and 3, and why if the speed of light is the same, c, OT see first B and then A, if they are aligned and equal distance from OT.

The time that light signals take to travel from an event to reach an observer is irrelevant. Signal delay has nothing to do with relativity. This is one of the things I don't like about this experiment. It suggests that signal delay is part of the theory of SR.

The time of an event in a inertial frame of reference is determined by a local observer; or, by a distant observer who takes the signal delay into account.

sysprog
Would you consider learning about the relativity of simultaneity from a different experiment?

I find this Einstein experiment confusing and unsatisfactory. There are easier

You are making an incorrect assumption here. The two light pulses arrive to OT at the same time if all three of the following conditions are met:

IF
1) OT is equidistant from where the flashes occurred
AND
2) The speed of both flashes are the same
AND
3) The flashes occurred at the same time
THEN
The flashes arrive at OT at the same time

We know that 1) and 2) are true, but we do not know 3). In fact, since we know that the flashes do not arrive at OT at the same time, we know that at least one of 1), 2), or 3) must be false. And since we know that 1) and 2) are true, that implies that 3) must be false.
Thank you for the answer, but i don't understand one thing, we know that 3) must be false, but from observation made by OE which is in motion,
so for example if there are two flash light at the end of the train instead of lightnings, in this case the observation of OE is the same, they don't arrivere at same time but for OT there are simultaneous....
Is possible to replace lightning with two flash light? Because this will be more clear but result will not the same...
And last question: if i had two flash light at the same distance from me and I go to one of that and away from the other, if i think of light as a ball, I say " first I met one and then the other, but because light has a constant velocity of c, if I met one before the other i think it go faster than the other.....

PeroK
Homework Helper
Gold Member
2020 Award
Thank you for the answer, but i don't understand one thing, we know that 3) must be false, but from observation made by OE which is in motion,
so for example if there are two flash light at the end of the train instead of lightnings, in this case the observation of OE is the same, they don't arrivere at same time but for OT there are simultaneous....
Is possible to replace lightning with two flash light? Because this will be more clear but result will not the same...
And last question: if i had two flash light at the same distance from me and I go to one of that and away from the other, if i think of light as a ball, I say " first I met one and then the other, but because light has a constant velocity of c, if I met one before the other i think it go faster than the other.....

I'm not sure I really understand your question. One of the things I think is most important in learning SR is to be able to transform an experiment (set of events) from one reference frame to another. In this case, let's first describe the experiment in the platform frame:

1) Platform Frame of reference:

There are two markers on the platform at points A and B. These are a distance ##L## apart, say.

The train has length ##L##, so that the front of the train is at ##B## when the rear of the train is at ##A##. We can call this time ##t = 0##.

At ##t = 0## two lightning strikes hit ##A## and ##B## (and the front and rear of the train).

NOTE: the time of both events in the platform frame is ##t = 0##.

At time ##t = L/2c## the light from the lightning strikes reaches the midpoint between A and B, where OE is standing. NOTE: this is largely irrelevant.

2) Train frame of reference.

The train is longer than ##L## (as it is length contracted in the platform frame).

The distance between A and B is shorter than ##L## (as the platform is length contracted in the train frame).

When the front of the train is at ##B##, the rear of the train is still not at ##A##. The point ##A## is somewhere near the middle of the train. Because the train is (much) longer than the platform.

At this time, let's call this time ##t' = 0##, a lightning strike hits B (and the front of the train).

Later, the rear or the train reaches point A. At this time a lightning strike hits point A and the rear of the train.

So, we can see that the lighteing strikes are not simultaneous in the train frame.

NOTE: the whole experiment looks very different in the train frame from the platform frame.

MikeeMiracle, sysprog and Dale
PeroK
Homework Helper
Gold Member
2020 Award
PS part of the problem here is when you rely on a diagram drawn in the platform frame. You are trying to think in the train frame, but the diagram is always the platform frame. This is confusing.

The first thing I would do is draw a new diagram for the train frame.

LBoy and Dale
Dale
Mentor
2020 Award
we know that 3) must be false, but from observation made by OE which is in motion
No, 3) is false regardless of who is making the observation. It is merely easiest to see in OE's frame.

For clarity, suppose that OT carries an accurate clock, and that clock has optical sensors that detect exactly when a flash arrives at OT and prints on a piece of paper a permanent record of that time. All frames must agree on what is printed on that piece of paper. In this case, all frames agree that the two times are different.

So in any frame OE receives both flashes at some point in time. At that point in time OT is to OE's right. That means that the flash from the right has already passed OT and thus been received by OT. That also means that the flash from the left has not yet reached OT and therefore has not been received by OT. Therefore in every frame there exists some time when OT has received the flash from the right but has not received the flash from the left.

Therefore the flashes are not received at the same time and all frames must agree that the two times printed on the paper are different. The analysis holds in any frame. They may disagree that the times are correct, but they agree what is printed and that it is different.

Last edited:
cianfa72 and sysprog
PeroK
Homework Helper
Gold Member
2020 Award
PPS If we accept length contraction, then we can simplify this experiment:

We don't need the lightning strikes. We don't need observers OE and OT. We don't need accurate clocks or optical sensors.

In the platform frame the events: "front of train at B" and "rear of train at A" are simultaneous.

In the train frame the event ""front of train at B" occurs before "rear of train at A".

Therefore: simultaneity is relative.

MikeeMiracle, sysprog and Dale
Janus
Staff Emeritus
Gold Member
Thank you for the answer, but i don't understand one thing, we know that 3) must be false, but from observation made by OE which is in motion,
so for example if there are two flash light at the end of the train instead of lightnings, in this case the observation of OE is the same, they don't arrivere at same time but for OT there are simultaneous....
Is possible to replace lightning with two flash light? Because this will be more clear but result will not the same...
And last question: if i had two flash light at the same distance from me and I go to one of that and away from the other, if i think of light as a ball, I say " first I met one and then the other, but because light has a constant velocity of c, if I met one before the other i think it go faster than the other.....

From the ground observer view, the train is moving away from strike A and towards strike B.
By the time he sees the light from both strikes, the train observer has moved to a point closer to strike B and further from strike A, in other words, the light from strike B will have already passed the train observer on its way to our Ground observer, but the light from Strike A will not have yet reached the train observer, and won't reach him until the train observer has moved even closer to where strike B occurred.
This means that the train observer had to be next to two different points of the tracks when each light flash reached him.
Now, here's the important part. The train observer must agree to this fact. You can't have the ground observer say that that light flash B reached the observer when he was next to point C of the tracks and Light flash A reached him while next to point D while the train observer says that both flashes reached him when he was next to point E. That would involve a physical contradiction.
So we are forced to accept that the train observer must see the flashes at different times, And, being at the midpoint of the train, with the strikes hitting the ends, and light traveling at the same speed from each flash, he must conclude that the lightning strikes that produced those flashes could not have occurred simultaneously in the rest frame of the train.

cianfa72 and sysprog
So we are forced to accept that the train observer must see the flashes at different times, And, being at the midpoint of the train, with the strikes hitting the ends, and light traveling at the same speed from each flash, he must conclude that the lightning strikes that produced those flashes could not have occurred simultaneously in the rest frame of the train.
I think an important point is that when the lightning strikes occur each of them leaves actually two co-located 'marks'. A mark on the tracks and a mark on one end of the train (strike A at the rear end and strike B at the front end of the train).

The observer in the midpoint of the train knows the strikes are originated from the rear end and the front end hence he must conclude lightning strikes that produced those flashes could not have occurred simultaneously in its rest frame.

Dale and PeroK
Nugatory
Mentor
, in this case the observation of OE is the same, they don't arrivere at same time but for OT there are simultaneous....
Is possible to replace lightning with two flash light? Because this will be more clear but result will not the same...
In that case the train observer will find the two flashes to be simultaneous but the platform observer will find that they are not. It will be easier to analyze this variation if we imagine that the train is standing still while the platform is moving backwards (and if you are not satisfied that that's an equivalent way of describing the situation you'll want to back up and learn about Galilean relativity, discovered a few centuries before Einstein).

Okay I have two last doubts,
1)The speed of light is c, for all inertial frame of reference, so what I'll see if I run to a flash light? And there is difference from a stationary observer at the same distance when the light will turn on?
I think no because the speed is the same, but intuitively i think if i move towards the light i see it first, but first than who.....?

2)When I did SR, I found always trasformation from event occurs in a frame of reference to another frame of reference to see what different observes see, but I think my miss-understanding of the "Einstein train lightnings" became from the fact that I don't understand if the two lightning "occurs" in the "rest" or "stationary" observer or in the train frame.

Maybe the main problem that I don't understand Is that:
If I think they occur in the "stationary" frame of reference, they met the observer simultaneous but when I change frame of reference and go in the train frame, usually what I do is to see the same event from that perspective, and i will see that the observer at the rest see one first and then the other and I think the two event are not simultaneous, but In the explanation I found is different I think, because is what the train observer see about "his event"

PeroK
Homework Helper
Gold Member
2020 Award
Okay I have two last doubts,
1)The speed of light is c, for all inertial frame of reference, so what I'll see if I run to a flash light? And there is difference from a stationary observer at the same distance when the light will turn on?
I think no because the speed is the same, but intuitively i think if i move towards the light i see it first, but first than who.....?

2)When I did SR, I found always trasformation from event occurs in a frame of reference to another frame of reference to see what different observes see, but I think my miss-understanding of the "Einstein train lightnings" became from the fact that I don't understand if the two lightning "occurs" in the "rest" or "stationary" observer or in the train frame.

Maybe the main problem that I don't understand Is that:
If I think they occur in the "stationary" frame of reference, they met the observer simultaneous but when I change frame of reference and go in the train frame, usually what I do is to see the same event from that perspective, and i will see that the observer at the rest see one first and then the other and I think the two event are not simultaneous, but In the explanation I found is different I think, because is what the train observer see about "his event"
I think you are very confused here. I think the advice I gave in post #2 is valid. The Einstein lightning experiment is so complicated and confusing that it is a potential disaster for many students.

Note that events do not occur in one reference frame or another. Events are events - the things that happen. They occur in all reference frames. Each reference frame assigns spacetime coordinates to each event.

Your problems are also caused by having an observer at rest in each reference frame and having light from every event be transmitted to that observer. What happens if you do an experiment in a dark room? Is the physics different if you switch the lights off?

In general, the time that a single observer sees an event (in terms of receiving a light signal directly from an event) is not relevant. That's another confusion that you need to overcome - and that's a confusion that the Einstein experiment has directly caused.

I would rewind, rethink and learn SR without this crazy, complicated, confusing experiment.

Dale
Mentor
2020 Award
I don't understand if the two lightning "occurs" in the "rest" or "stationary" observer or in the train frame.
All events occur in every frame. Different frames merely disagree about their coordinates, but not their existence.

It always bothers me when you read things like “Bob is in an inertial frame”. What they mean is “Bob is at rest in an inertial frame”. Everything and every event is in all frames. They may be moving in some and at rest in others, but they are all in every frame.

cianfa72 and PeterDonis
Hans de Vries
Gold Member
One observer OE, is on the ground, we take him as the fixed frame of reference.
The other OT is on the train that is moving relatively to the OE at a constant velocity
So, two lightning strike the two points A and B when the train is with back and end in the same points ( A and B) and the two observer are aligned.
View attachment 289031
The observer OE is at equal distance from A and B, so the light, which travel at constant speed c, arrive at the same time at his eyes, the two events are simultaneous.
The observer OE anyway see OT traveling to B and move away from A, the speed of light is always c, so he conclude that OT observers first the light from B and then the light from A.
The result: OT see first ligh from B and then from A in his frame of reference.( like what think OE)

Once you understand non-simultaneity it is really not that complicated. It is also very natural: Just what you would expect.

You can also already see why non-simultaneity occurs without any reference to relativistic effects like Lorentz contraction and Time dilation. This because at low speeds non-simultaneity is proportional to the velocity ##v## and therefor it is already relevant while the other effects can still be neglected.

If the two propagated light flashes meet each other at one point then any observer at that point will see them (receive the light flashes in his eyes) at the same time. This is totally independent of the velocity of the observer. However, they draw different conclusions about the simultaneity of the light flashes.

So both observers see two flashes at the same time. Nevertheless, your observer at rest concludes that they happened at the same time, while your moving observer concludes that the flash in front of him occurred earlier then the one behind them.

Why?

Say you see two flashes at house b and c at the same time. Your conclusion is that the flash at house b occurred earlier because, from the size, you conclude that house b is further away.

This is the whole point. The moving observer sees a bigger flash in front of him and a smaller flash behind him, so he concludes that the flash in front of him occurred later.

The explanation for this effect is (without any need to involve special relativity) just a simple vector addition in the rest frame. You need to add the speed vector of the observer to the light ray vectors coming from the light source.

The angle of the light rays change depending on the relative velocity and the apparent size changes.
The image above illustrates this effect. Now lets go into more detail using your image.

What does the OT observer see if he moves at 1% of the speed of light? We take the formula for ##t'## and for low speeds we can set ##\gamma\approx 1## so we have for the non-simultaneity:

$$t' = t - vx/c^2$$
Now the light rays coming of an object are wave-fronts build up from emitted light over a period of time. So a wavefront from a moving object is different than from an object at rest.

- OT will see the house B blue-shifted by 1% and larger by 1%.
- OT will see the house A red-shifted by 1% and smaller by 1%
- OT sees no red or blue shifting in the wavefront from both ends of the train.
- OT neither sees an apparent change in size of both ends of the train.

Because OT sees the objects in the frame at rest changed in size by 1% he concludes that there is a 1% shift in simultaneity as expressed by the formula above. This is all still without Special Relativity.

In more detail: House A and B have clocks. The light from clocks A and B reaches the eyes of OT (and OE) and will show the exact same time since both observers are in the middle. However, OT knows about propagation speed so he assumes that the clocks must have ticked further during the propagation. Now, OT assumes that clock B is closer as clock A so he assumes that clock B has progressed less than clock A and concludes their actual times are different. He assumes they are different as given by the non-simultaneity formula given above.

-------​

If you want to understand Special Relativity and the beautiful way in which this non-simultaneity effect perfectly reverses the relativistic effects of Lorentz contraction and Time dilation I can recommend to this chapter of my book:
http://www.physics-quest.org/Book_Chapter_Non_Simultaneity.pdf

It involves the ellipsoids of simultaneity. The ellipsoids on the top are defined as simultaneous by our moving observer by the effect described above. The ellipsoids in the middle reverses the vector addition in a non-relativistic way. The lower spheres are corrected for Lorentz contraction: Because we end up with spheres the moving observer concludes that the speed of light is still C in any direction, even after changing to a new reference frame.

Other chapters:
Lorentz contraction: http://www.physics-quest.org/Book_Chapter_EM_LorentzContr.pdf
Time dilation: http://www.physics-quest.org/Book_Chapter_Time_Dilation.pdf

The first one may be too technical since it involves Liénard Wiechert potentials. The start of the second is very readable though.

Last edited:
If the two propagated light flashes meet each other at one point then any observer at that point will see them (receive the light flashes in his eyes) at the same time. This is totally independent of the velocity of the observer. However, they draw different conclusions about the simultaneity of the light flashes.
I would say they (the two propagated light flashes) meet each other at the same spacetime point (event). This is frame independent.

Last edited:
Hans de Vries
Summary:: My question is about on of the "famous" Einstein's mental experiment: the train and the lightnings, i don't understand why the result is what it is, I thought for day but I don't understand this particular case, If you can help me to understand I'll very grateful.

View attachment 289031

1- If the speed of light is the same in EVERY inertial frame of reference, in the reference of OT that speed is c, so i think when the two lightning strike in A and B, they are equal distance from the center where there is OT 'cause the first assumption (when A and B are ALIGNED) so i think light will arrive at OT at the same time also in his frame of reference, if i don't observe that i think light speed is greater than c from B to OT and lower from A to OT, but this is a contradiction, and I don't understand how to escape from that my surely wrong interpretation.
Lukka98, Perhaps this may help resolve your problem #1: OT observes the light from each lightning strike at different times because OT was no longer at a location equidistant from both lightning strikes at the time the light from both strikes reached OT.

Let's stop time when the lightning strikes occur so we can establish what is occurring at that time. When the two lightning strikes occur, the light has not traveled toward OT yet because light takes time to travel. When the lightning strikes occur, OT is at a certain location we will call LOCATION 1. OT has not seen the light yet because light takes time to travel.

Now, let's start time moving very slowly so we can observe events in order. Let's first consider OT's relation to only the lightning strike at the front of the train. The light from the lightning strike is traveling toward OT, and OT is traveling away from LOCATION 1 (where OT was when the lightning strike occurred), and OT is traveling toward the light. The light reaches OT sooner than it reaches LOCATION 1 because OT has been moving toward the light while the light was traveling toward OT. The light traveled a shorter distance, which took less time to reach OT than it would have if OT was back at LOCATION 1, so OT observes the light sooner than if OT was back at LOCATION 1 when the lighting struck.

If we look at what happens with the lightning strike at the back of the train in the same manner, we notice that OT was moving away from that lightning strike while the light from it was moving toward OT, and OT was moving away from LOCATION 1, where he was when the lightnings struck. By the time the light from the back of the train reached OT, OT had traveled farther away from the light and farther away from LOCATION 1, so the light took longer to travel the longer distance to reach OT. So, OT observed the light from the back of the train at a later time than if OT were back at LOCATION 1.

Therefore, OT observes the light from each lightning strike at different times because OT was no longer at a location (LOCATION 1) equidistant from both lightning strikes at the time the light from both strikes reached OT.

PeroK
PeroK
Homework Helper
Gold Member
2020 Award
Therefore, OT observes the light from each lightning strike at different times because OT was no longer at a location (LOCATION 1) equidistant from both lightning strikes at the time the light from both strikes reached OT.
This whole analysis simply makes the platform frame a universal frame of reference. As fas as OT is concerned, the lightning strikes hit either end of the train, from which OT is permanently equidistant. As fas as OT is concerned it is OE and the platform that are moving relative to the locations that the lightning struck.

- If the speed of light is the same in EVERY inertial frame of reference, in the reference of OT that speed is c, so i think when the two lightning strike in A and B, they are equal distance from the center (LBoy: but they are niot simultaneous) where there is OT 'cause the first assumption (when A and B are ALIGNED) so i think light will arrive at OT at the same time also in his frame of reference

Bold: here you have made the implicit assumption that both events are simultaneous for the OT on the train. And they aren't. That's where the misunderstanding comes from.

In the OE frame A and B are simultaneous (obvious I suppose). The OT has the same distance to the beginning and end of the train, where the lightning strikes, but by the time he sees the light from B (the beginning) a certain amount of time will have passed in which he approaches B and moves away from A.

The point or time at which it receives the light is crucial - that point is moving toward B and away from A.
Therefore, if the speed of light is constant for both observers - OE will see the light from B earlier and the light from A later, because distance from B to him is shorter due to movement, and bigger from him to A.

For him A and B are not simultaneous.

OE will see the light from B earlier and the light from A later, because distance from B to him is shorter due to movement, and bigger from him to A.
Why would OE see the light from B earlier and the light from A later? Doesn't OE see both flashes at the same time?

LBoy and Hans de Vries
Why would OE see the light from B earlier and the light from A later? Doesn't OE see both flashes at the same time?

Thank you, my bad! There should be OT (observer in the train) instead of OE of course, I will change it immediately.

Last edited:
David Lewis
The point or time at which it receives the light is crucial - that point is moving toward B and away from A.
Therefore, if the speed of light is constant for both observers - OT (in my previous post there was an error, it was OE instead of OT) will see the light from B earlier and the light from A later, because distance from B to him is shorter due to movement, and bigger from him to A.

For him A and B are not simultaneous.

As I see I cannot edit my previous post,OK, so I'm doing it here.

PeroK
Homework Helper
Gold Member
2020 Award
This particular thought experiment is good and bad at the same time.
This thought experiment, IMO, is simply bad. It confuses everyone who is new to SR. It over-emphasises a single observer over a reference frame, and creates misunderstanding about the need for light signals. It also over-emphasises the platform frame over the train frame and requires so many physical coincidences that it is genuinely quite complicated. It gets SR off on the wrong foot altogether.

Relativity of simultaneity can be shown much more simply.

I would advise students to steer clear of it, but obviously they are wowed by the fact that it was Einstein's original thought experiment and can't resist.

LBoy
Relativity of simultaneity can be shown much more simply.

I would advise students to steer clear of it, but obviously they are wowed by the fact that it was Einstein's original thought experiment and can't resist.

I would like to see it if you wouldn't mind, train experiment works for me, however I see the whole environment, a galaxy of possibilities of making an error due to a lot of irrelevant effects present. I can filter it out I suppose however I see that Hans might have a problem with this, he focuses on the less relevant things.

Maybe I can use something new, simpler to explain the essence of simultaneity.