# Problem Perceiving Simultaneity (Special Relativity)

1. May 21, 2015

### Heatherfield

Hi,

This is not a question aimed at answering a homework problem, but since it is part of a course on Special Relativity, I believe I'm supposed to put my question here. The course itself, however, is a Dutch High School syllabus on SR, so I won't be able to link to source material. The question arose fairly early on, while discussing simultaneity.

The textbook has already explained how two particles at rest can synchronise their clocks by emitting light beams and reflecting these. Subsequently, the textbook used the famous train analogy (a lamp is in the middle of a high-speed train, yet the light will reach both sides of the train at the same time) to explain how simultaneity is relative (thus, the interpretation of time within the train must differ from the interpretation outside the train, right?).

Furthermore, the textbook developed the theory on spacetime diagrams to include more than one set of axes (for example, a black set of axes parellel to the line paper, and a red set of axes drawn inside the black set to represent the train).

To me, this theory of simultaneity is very understandable, as long as the analogies are kept inside the train. In an example question, however, the textbook introduces something that goes fully against my intuition.

The textbook takes a set of 'regular' black axes: w for time, and x for space. Inside this set of axes, a red set of axes is drawn with w' and x', to represent a moving object. Relative to the black set of axes, two events A and B occur. Let's say they are at coordinates (7,3) and (8,3), thus sharing their value for w.

The textbook then asks the reader why events A and B are not simultaneous in reference frame w'. The answer to this question is not very strange: they are both, in the red frame of axes, on different time coordinates. To me, however, this seems very peculiar.

I copied the assignment onto graph paper of myself and drew reference frames w and w' and events A and B (simultaneous in reference frame w). I then drew two photons being emitted from the events, towards the origin. As events A and B were only a light second apart, there was a second in between their arrivals. This is very much expected and proves their simultaneity.

As expected, due to the two events having different coordinates for w', the photons hit the w' axis only about three quarters of a second apart. On my own graph paper, events A and B were also not simultaneous as seen from w'. In that perspective, I have solved this homework problem all on my own. I can't, however, create a mental explanation for these events.

I tried explaining it to myself in terms of this analogy:
1) there is a train riding on insanely long tracks, at a very high speed
2) in front of this train stand two people (A and B). Their clocks are synchronised
3) at the exact same moment (from their perspective), they flash a flashlight in the direction of the train

According to the diagrams in my notebook, if these men were standing a lightsecond apart, these photons will hit the train with only about three seconds apart. To me (and I'm absolutely sure I'm wrong, but I can't disprove my own logic so far), this seems to violate the constant speed of light: according to the train, the light still approaches it with 300,000 km/s, thus the photons should hit the train at a speed proportional to the distance between A and B. If this doesn't hold, wouldn't that mean that one photon travelled faster (seen from the train)?

I know that I'm wrong: I've proven my logical mistake with the diagram described above. However, in my mind, the analogy still holds. Could someone please correct my analogy (preferably with the example used above) so that it explains while events A and B are simultaneous in one frame of reference, but not in the other?

Your help would be greatly appreciated, but I'd already be amazed at someone actually reading through all of this. I understand it's a lot of text, but since this question is aimed at fixing my thinking process, I felt like I had to describe my thinking process in as much detail and clarity as possible.

With regards,
Heatherfield

P.S. I uploaded the diagram I drew myself at www.g2f.nl/08eynfo, but I felt like it was too sloppy (and on a weird type of grid paper) to make my story clearer.

EDIT:
I spent the previous hour drawing more diagrams and watching videos and I'm beginning to think I'm seeing where my logic is wrong. To do so, I used another train.

Inside a new, black set of axes (labelled W), I drew a worldline for my train and made a red set of axes to go along with it (labelled W'). I then had two people, standing on either side of the train, fire a photon/lightbeam/flashlight towards the train. As the theory predicted, but as I previously did not understand, the photons didn't arrive at the train simultaneously, despite travelling at the exact same speed (using this approach, it was easier to check this speed), thus the only CONCLUSION is that the light beams were fired at different moments.

Come to think about it, this is a very obvious conclusion to the novel I posted above, and it's a little shameful to come up with he correct answer so soon after posting such a question (but I swear, I spent two days in existential crisis after not being able to solve this problem), so I must apologise for that (especially seeing the fact that I posted such a problem in a section meant for advanced problems).

I have, however, included a link to my new diagram at the bottom of the page, and would be very happy if someone could check:
1) the diagram I drew for errors
2) my above reasoning for errors

Last edited: May 21, 2015
2. May 22, 2015

### BvU

Hello heather, welcome to PF !

I had some remarks on your first treatise you might want to consider:

Your diagram suggest smaller time steps in W' than the formulas indicate. See e.g. here
(your steps go to $\sqrt 2$ for $v\rightarrow c$. They should go to $\infty$).

In your edit you confuse yourself by writing "standing on either side of the train". B is in front of the train, A is way behind the thing.
The light pulses were fired simultaneously in W. But not in W'.

This kind of exercises is very good to develop some transformation skills. Keep it up.
One I remember is someone halfway inside the train flashing a light. He sees the flashes reaching the ends of the train simultaneously. Someone outside the train does not !
Someone next to the tracks can flash simultaneous with the guy inside. What do both think of the times the flashes arrive at the two ends ?

--

3. May 22, 2015

### Heatherfield

Thanks for the kind words and the helpful link. In the end, I think I managed to find my logical error when I started thinking in terms of the analogy where the train is hit by two flashes of lightning on either side. At first, I thought that there were no convincing arguments to say that
- the observer in the train perceives the flashes to happen not simultaneously. They disagree on simultaneity.