# Need help with time dilation and length contraction.

## Main Question or Discussion Point

Can anyone help me with this thought experiment.

I have a spaceship moving at very close to the speed of light. Inside are the captain and two crew members. The captain is positioned at the front of the spaceship with a clock and a light. One crew member is positioned at the back of the spaceship, a certain distance away from the light, with a mirror and a clock, and the other crew member is positioned at the top of the spaceship, directly above the light, the same distance away from the light as the other crew member, and also with a mirror and a clock. All the clocks are synchronized. The captain notes the time on his clock and then flashes the light. When the crew members at the back and top of the ship see the flash they note the time on their respective clocks. The light then reflects off of the mirrors and returns to the captain, and he notes the time/times.

When the three men compare notes, what do they find?

Did the crew members at the back and top of the spaceship see the flash of light at the same time?
Did the reflection from both mirrors reach the captain at the same time?

Now let's consider the viewpoint of a stationary observer, standing on a nearby planet.

To the observer, did the light reach the two crew members at the same time?
To the observer, did the light return to the captain at the same time?

Thanks in advance for any insight that you can offer.

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ghwellsjr
Gold Member
Can anyone help me with this thought experiment.

I have a spaceship moving at very close to the speed of light. Inside are the captain and two crew members. The captain is positioned at the front of the spaceship with a clock and a light. One crew member is positioned at the back of the spaceship, a certain distance away from the light, with a mirror and a clock, and the other crew member is positioned at the top of the spaceship, directly above the light, the same distance away from the light as the other crew member, and also with a mirror and a clock. All the clocks are synchronized. The captain notes the time on his clock and then flashes the light. When the crew members at the back and top of the ship see the flash they note the time on their respective clocks. The light then reflects off of the mirrors and returns to the captain, and he notes the time/times.

When the three men compare notes, what do they find?

Did the crew members at the back and top of the spaceship see the flash of light at the same time?
Yes.

Did the reflection from both mirrors reach the captain at the same time?
Yes.

Now let's consider the viewpoint of a stationary observer, standing on a nearby planet.

To the observer, did the light reach the two crew members at the same time?
No.

To the observer, did the light return to the captain at the same time?
Yes.

Thanks in advance for any insight that you can offer.

Nugatory
Mentor
I have a spaceship moving at very close to the speed of light.......
You've been sloppy in the way that you posed the problem because you didn't say what that speed was relative to. If you had been more careful about the wording, it would have been clear that the ship is moving relative to the observer, and that as far as the ship and the men in the ship are concerned, it's the ship that is rest while the observer and the planet are moving away from them "at very close to the speed of light".

This isn't just a quibble, because with the more precise phrasing your first question is trivial:
When the three men compare notes, what do they find?
Did the crew members at the back and top of the spaceship see the flash of light at the same time?
Did the reflection from both mirrors reach the captain at the same time?
They're just sitting at rest exchanging flashes of light across the same distance. They both receive and reflect the flash at the same time according to their clocks, and both reflected flashes make it back to the source at the same time.

Now let's consider the viewpoint of a stationary observer, standing on a nearby planet.

To the observer, did the light reach the two crew members at the same time?
To the observer, did the light return to the captain at the same time?
You mean a moving observer who, along with his planet, is receding away from the stationary spaceship, right? OK, so you really do mean a stationary observer watching the spaceship move? They're both the exact same thing.

The planet observer will see both light signals return to the captain at the same time. They have to, because we can replace the captain with a device that explodes if it is struck by two light signals at the same time. All observers have to agree that either the device explodes or it doesn't.

The planet observer will also agree that both observers' clocks read the same at the moment of reflection (suppose the clocks stop when the reflection happens - all observers have to agree about the physical position of the hands of the stopped clock).

The planet observer's own clock will say that the two reflections did not happen at the same time. This is relativity of simultaneity at work.

The planet observer will see both light signals return to the captain at the same time. They have to, because we can replace the captain with a device that explodes if it is struck by two light signals at the same time. All observers have to agree that either the device explodes or it doesn't.

The planet observer will also agree that both observers' clocks read the same at the moment of reflection (suppose the clocks stop when the reflection happens - all observers have to agree about the physical position of the hands of the stopped clock).

The planet observer's own clock will say that the two reflections did not happen at the same time. This is relativity of simultaneity at work.
First, let me thank both of you for your replies, I really appreciate it, and sorry for the sloppiness of the original question. (The spaceship and the observer are both in inertial frames)

I can't help but notice however, that there is some disagreement as to whether the stationary observer will see the light flash reach both crew members at the same time according to the crew members' clocks. Nugatory, you pointed out something that was indeed one of my concerns. It seems logical that the stationary observer should see the two crew members' clocks as reading the same time at the moment of reflection. But it seems just as logical that there is no way that this can happen. If the spaceship is significantly contracted along the direction of motion then the distance up to the mirror at the top should be longer than the distance to the mirror in the back. Not only that, but the mirror at the top is moving away from the point of origin of the flash, meaning that from the stationary observers perspective the light has to follow an even longer diagonal path to the mirror. Meanwhile from the stationary observer's perspective, the mirror in the back is moving toward the origin of the flash. Thus the light traveling to the mirror in the back has to follow not only a contracted distance, but one that is further shortened by the fact that the mirror is moving toward the point of origin.

So that's one of the reasons that this thought experiment confuses me. Logically, the stationary observer should see the crew members' two clocks as reading the same time at the moment of reflection. But unless I'm overlooking something (which I quite often do) there is no way that this could happen.

I also noticed that you both agreed that the light reflected off of both mirrors would reach the captain at the same time. But again, in my mind at least, from the stationary observer's perspective, this just isn't possible. Even disregarding the fact that the light should reach the rear mirror first, giving it a head start on the way back, the length of the return trip is longer for the light from the upper mirror, than it is for the light from the rear mirror. To illustrate this let's say that from the stationary observer's perspective the spaceship is now fifty feet tall, but only one inch long. At the moment of reflection, from the stationary observer's perspective, the captain is moving away from the origin of both reflections at the same speed, thus the light from the rear only has to travel one inch, plus whatever distance the ship travels during the time it takes for the light to catch up to the captain. But the light from the upper mirror has to cover fifty feet, plus whatever distance the ship travels during the time that it takes to catch up. Again they're both chasing the captain, who is moving away from both points of origin of the reflections at the same speed, but the rear reflection not only starts closer, it takes a straight path, while the light from the upper mirror must traverse a longer, diagonal path.

So as you can see, I'm confused, because in my mind, from the stationary observer's perspective, the light should reach the rear mirror first, and it should get back to the captain first. But this leads to all sorts of problems, so it must be wrong. I just don't know how.

If someone could point out my error I would probably feel like an idiot, but I would also appreciate it. So, anybody, where's my mistake?

Thanks

Last edited:
Nugatory
Mentor
It seems logical that the stationary observer should see the two crew members' clocks as reading the same time at the moment of reflection....Logically, the stationary observer should see the crew members' two clocks as reading the same time at the moment of reflection. But unless I'm overlooking something (which I quite often do) there is no way that this could happen.
In this case, you're overlooking the relativity of simultaneity. The two events "light reflected from rear mirror when rear clock read T" and "light reflected from side mirror when side clock read T" happen at the same time T for the shipboard observers. However, they're not simultaneous for the planetary observer. One of them happens at time T1 according to his clock, the other happens at time T2 according to his clock, and neither T1 nor T2 is equal to T.

in my mind, from the stationary observer's perspective, the light should reach the rear mirror first, and it should get back to the captain first.
Don't just handwave the distance traveled on the two paths, calculate it. It will come out the same. However, that's very much the hard way of doing this problem, because it's so much easier to solve the problem in the ship frame and then transform into the planet frame:

Assume the ship is moving at .6c and is one light-second long (according to the crew). Now we have four relevant events. Using the coordinate system in which the ship is at rest and the captain is at the origin, we have four events:
(t=,0,x=0,y=0): The flash is emitted
(t=1,x=0,y=1): Light signal reaches side mirror and is reflected
(t=1,x=-1,y=0): Light signal reaches rear mirror and is reflected
(t=2,x=0,y=0): Reflected light signals reach captain

Hit each of these with the Lorentz transforms and you'll get the coordinates of these events using a coordinate system in which the planet is at rest.

ghwellsjr
Gold Member
First, let me thank both of you for your replies, I really appreciate it, and sorry for the sloppiness of the original question. (The spaceship and the observer are both in inertial frames)

I can't help but notice however, that there is some disagreement as to whether the stationary observer will see the light flash reach both crew members at the same time according to the crew members' clocks.
Now let's consider the viewpoint of a stationary observer, standing on a nearby planet.

To the observer, did the light reach the two crew members at the same time?
The planet observer's own clock will say that the two reflections did not happen at the same time.
Nugatory also addressed the question that you did not ask and which I did not address, so again, there's no disagreement.

Nugatory, you pointed out something that was indeed one of my concerns. It seems logical that the stationary observer should see the two crew members' clocks as reading the same time at the moment of reflection.
Yes, and that is what Nugatory said. How could it be otherwise?

But it seems just as logical that there is no way that this can happen. If the spaceship is significantly contracted along the direction of motion then the distance up to the mirror at the top should be longer than the distance to the mirror in the back. Not only that, but the mirror at the top is moving away from the point of origin of the flash, meaning that from the stationary observers perspective the light has to follow an even longer diagonal path to the mirror. Meanwhile from the stationary observer's perspective, the mirror in the back is moving toward the origin of the flash. Thus the light traveling to the mirror in the back has to follow not only a contracted distance, but one that is further shortened by the fact that the mirror is moving toward the point of origin.

So that's one of the reasons that this thought experiment confuses me. Logically, the stationary observer should see the crew members' two clocks as reading the same time at the moment of reflection. But unless I'm overlooking something (which I quite often do) there is no way that this could happen.
You seem to have forgotten that you stated that "All the clocks are synchronized." That means that before the "real" experiment is performed, an identical experiment was repeated and the two crew members' clocks are adjusted until they read the same so that when the experiment is performed "for real" they again read the same.

I also noticed that you both agreed that the light reflected off of both mirrors would reach the captain at the same time. But again, in my mind at least, from the stationary observer's perspective, this just isn't possible. Even disregarding the fact that the light should reach the rear mirror first, giving it a head start on the way back, the length of the return trip is longer for the light from the upper mirror, than it is for the light from the rear mirror. To illustrate this let's say that from the stationary observer's perspective the spaceship is now fifty feet tall, but only one inch long. At the moment of reflection, from the stationary observer's perspective, the captain is moving away from the origin of both reflections at the same speed, thus the light from the rear only has to travel one inch, plus whatever distance the ship travels during the time it takes for the light to catch up to the captain. But the light from the upper mirror has to cover fifty feet, plus whatever distance the ship travels during the time that it takes to catch up. Again they're both chasing the captain, who is moving away from both points of origin of the reflections at the same speed, but the rear reflection not only starts closer, it takes a straight path, while the light from the upper mirror must traverse a longer, diagonal path.

So as you can see, I'm confused, because in my mind, from the stationary observer's perspective, the light should reach the rear mirror first, and it should get back to the captain first. But this leads to all sorts of problems, so it must be wrong. I just don't know how.

If someone could point out my error I would probably feel like an idiot, but I would also appreciate it. So, anybody, where's my mistake?

Thanks
I made an animation that shows the captain with a circle of mirrors all around him, not just one behind him and one above him but hopefully it will make sense to you:

This is from a thread called "A graphical explanation of Special Relativity".

Dale
Mentor
I can't help but notice however, that there is some disagreement as to whether the stationary observer will see the light flash reach both crew members at the same time according to the crew members' clocks.
There was no disagreement. You just misread.

I agree very strongly with Nugatory's recommendation to actually work through the math. Even if you don't "get it" that way you will at least become confident that there are no contradictions.

In this case, you're overlooking the relativity of simultaneity. The two events "light reflected from rear mirror when rear clock read T" and "light reflected from side mirror when side clock read T" happen at the same time T for the shipboard observers. However, they're not simultaneous for the planetary observer. One of them happens at time T1 according to his clock, the other happens at time T2 according to his clock, and neither T1 nor T2 is equal to T.
Nutgatory, thanks for responding again, and I'm sorry, but I'm still not getting this. When I dig a logical hole, I dig it deep. But as they say, when you're stuck in a hole the first thing to do is stop digging, so that's what I'm trying to do here.

I thought that I had a grasp on simultaneity, Lorentz transformations, and what each observer sees from their own inertial frame, but I'm still stuck. I understand that due to simultaneity the stationary observer sees the events happen in a different order than the crew on the ship does, I also realize that the stationary observer's clock is running at a different rate than the ones on the ship. But what I don't understand is that from the stationary observer's perspective the clocks on the ship must be running at different rates relative to each other. The captain's clock, and the crew member's clock in the rear of the ship, must appear to be running faster than the crew member's clock at the top of the ship. This has to be true if the stationary observer is to see their clocks as reading the same time when the light reaches them. He should see these things happen in a different order than the crew on the ship, but he should still see the shipboard clocks read the same time when the light reaches them. Which I can't find a way to account for. I know this sounds confusing, so let me try to simplify this, and not get sloppy again.

Let's say that the stationary observer has a powerful camera, such that he can take a picture of the ship as it passes, and let's say that he takes a picture at the exact time that the captain flashes the light, and also that this picture shows that all of the shipboard clocks show exactly the same time. We also know that from the crew members' perspectives' the light will reach their mirrors at exactly the same time. Let's call this time T1. This is the time that all the shipboard clocks will say that the light hit the mirrors.

Now let's say that our stationary observer takes another picture the instant that he sees the light reach the rear mirror. In this picture the captain's clock and the crew member in the rear's clock should read the same time, the aforementioned T1, because that's the time they all agree that the light hit the mirrors, but from the stationary observer's perspective the light hasn't reached the crew member's mirror at the top yet, so his clock hasn't reached T1. This must be true if the crew member's are correct when they say that the light reached each of their mirror's at T1. In the second picture, the clocks at the front and back have reached T1, but the crew member's at the top hasn't. We know that this is true, because we know that from the stationary observer's perspective, the light hasn't reached him yet. So from the stationary observer's perspective the clock at the top is running slower than the other two. The same thing happens on the reflection back from the mirrors.

I hope that in trying to clarify the problem, that I haven't totally confused anyone. The problem is simply this, from the stationary observer's perspective, the clocks on the ship are running at different rates. This is something that runs contrary to my understanding of relativity. Since the clocks on the ship are all moving at the same speed, relative to the stationary observer, they should all be running at the same rate from his perspective. The stationary observer should see all the clocks running at the same rate, but he doesn't. Since he sees the events happen in a different order, he must see their clocks as having different times.

Like I say, I'm confused. Lorentz transformations aside, the stationary observer doesn't see what relativity says he should see, all the shipboard clocks running at the same rate.

Sorry for being so verbose and confusing, but I really am trying to make this as clear as possible.

(ghwellsjr, thank you for the clarification, and for the animation. AWESOME!!! But I wrote this response before I had the chance to read it, and I didn't want to go back and rewrite it. So I will respond to it separately)

...
such that he can take a picture of the ship as it passes, and let's say that he takes a picture at the exact time that the captain flashes the light,
...
...
the instant that he sees the light reach the rear mirror.
These phrases need to be explained. How does the photographer know 'the exact tiime' or 'the instant that ...' ?

All the clocks on the ship will tell the same time if they have been synchronised and are relatively at rest wrt each other.

A better experiment is for the clocks to broadcast their times when they receive the light signal. Any observer receiving these signals will see that each clock broadcast the same time.

I made an animation that shows the captain with a circle of mirrors all around him, not just one behind him and one above him but hopefully it will make sense to you:

ghwellsjr, thank you very much for the animation, it really is helpful. But unfortunately it didn't clear up my confusion.

We know that the crew members will all agree about the time at which the light reaches their mirror. (The blue circle in your animation representing the flash of light) Let's call the time at which the light reaches the mirrors, T1. But we see in the animation that the light reaches the rear mirror first. (At 0:02) At this point his clock must say T1, but the light hasn't reached the mirror at the top yet, (the top of the circle) so his clock must not read T1 yet, his clock must be running slower from the perspective of the stationary observer. (Us watching the animation) The light won't reach the mirror at the top until 0:04, at which time that clock must say T1. Thus the clock at the top appears to be running slower.

This is just a quick reply. But where am I getting confused?

ghwellsjr
Gold Member
(ghwellsjr, thank you for the clarification, and for the animation. AWESOME!!! But I wrote this response before I had the chance to read it, and I didn't want to go back and rewrite it. So I will respond to it separately)
In that case, I don't want to spend too much time responding to the rest of your previous post in detail but I do want to make these general points:

You have to also consider how long it takes for the light to traverse from several different events to the camera. Unless the camera is equidistant (as defined by the rest frame of the camera) from all the events it is taking a picture of, you won't see the events as happening simultaneously. So if the objects are moving in relation to the camera, then there is no way that the camera can take two pictures and remain equidistant from all the events for each picture (unless the objects are moving in a circle around the camera).

A stationary observer will not actually see moving clocks at different positions running at the same rate unless they are coming directly in line towards or away from him. But if he takes additional steps to essentially calculate what his rest frame will establish what their rates are, he will then determine that they are running at the same rate. For example, if an observer is watching a single clock approaching him, he will see it running faster than his own and after it passes him, he will see it running slower even though it continues to run at the same speed.

I would suggest that you tackle inline scenarios before you move on to 2D scenarios. That way, we can draw spacetime diagrams to explain what's going on in the different frames. It is very difficult to draw a spacetime diagram for a 2D scenario which is why an animation is helpful in these situations. Even if you do the Lorentz Transformations on specific events, you're still not going to see simultaneous issues unless you specifically apply them to additional events. If you just do the coordinate transformations in the 1D inline case, it's going to be very confusing unless you also make a drawing because the drawing for a scenario will automatically recast it so that you see simultaneity issues easily.

A better experiment is for the clocks to broadcast their times when they receive the light signal. Any observer receiving these signals will see that each clock broadcast the same time.
This is actually quite helpful, thanks.

You have to also consider how long it takes for the light to traverse from several different events to the camera. Unless the camera is equidistant (as defined by the rest frame of the camera) from all the events it is taking a picture of, you won't see the events as happening simultaneously. So if the objects are moving in relation to the camera, then there is no way that the camera can take two pictures and remain equidistant from all the events for each picture (unless the objects are moving in a circle around the camera).
You and Mentz114, have really helped to clear this up.

Thanks to both of you

ghwellsjr