[Centra (OP)]

I have a thought experiment type paradox, related to the Theory of Special Relativity, which I would like to see if anyone can resolve, the apparent paradox of light seemingly being sped up or slowed down to other than normal velocity in one inertial frame of reference when viewing light beams generated in another inertial frame in relative motion to it. This may have been addressed in other forms previously but I would like readers to try it again in my particular form, as sort of a challenge for amusement if nothing else. The paradox is as follows.

A rocket is in uniform motion and is of length, as measured in a stationary frame, of 300,000 km and equipped with lasers at each end, one on top and the other on the bottom, and targets on the other ends for each laser beam to hit after traveling the length of the rocket. When the back end of the rocket, which is moving left to right, as seen from the ground frame, reaches a point which is directly above a post on the ground both lasers are fired simultaneously as time is perceived in the rocket's frame. A second post is located 450,000 km to the right of the first post, from a ground based observer's perspective, and the rocket is moving toward it. There are distance markings in km between the two posts, with the zero mark at the position of the left post and running rightward to the right post.

There is a clock at the middle of the rocket and a clock on the ground in the middle of the two posts, both of which are visible to both observers in both frames. Both clocks read zero as observed from both frames at the time when the lasers are fired. The rocket is already up to speed when it passes over the posts and never slows down during the scenario. To the rocket based observers, the rocket is traveling at 150,000 km/s. I suppose it would be the same velocity to the ground based observers but I'll leave that unspecified in case a different velocity is indicated in someone's proposed solution.

At what distance markings do each observer see each target hit by its corresponding laser beam, and what are the readings on each clock when each happens, as perceived by each observer. The problem doesn't involve the actual times required for the light to reach the observers' eyes from the targets and clocks, it's assumed to be instantaneous for our purposes, the problem to be solved is the relativistic velocities of the beams in both frames. Can you perform any kind of relativity related manipulations which will result in both beams appearing to travel at 300,000 km/s to both observers? One beam in one direction would be easy, but can you do it with two beams in opposite directions at the same time? I'm sure it's been done before somewhere in some form but I'm not familiar with it so please humor me and explain how to do it one more time here. It's a discussion forum and this is my current subject for discussion.

[Centra (OP)]

Okay I'll give my own conclusions and others can see if they agree or if I went wrong somewhere.

Let's say the laser on top of the rocket is at the left, or back, end and fires to the right, in the direction of travel of the rocket. I conclude that it will hit its target at the 450,000 km mark, which is at the right post itself, and the bottom laser will hit its target at the 150,00 km mark and both will be at the time of 1 second on the rocket clock. It will be exactly the same results in the ground frame, no time dilation and no length contraction involved.

The rocket based observers will perceive the beams both to have traveled at normal velocity of 300,000 km/s. The ground based observers will perceive the rightward beam to have traveled 450,000 km in 1 second and the leftward beam to have traveled 150,000 km in 1 second. It will however have been an illusion, which they will discover if they consider the rocket to have been stationary and the ground to have been moving. They will then realize that both beams had in fact traveled at normal velocity, just as the rocket observers had perceived it, and the illusion of added and subtracted velocities was simply due to their changing perspective of the rocket.

[Eternal Student]

Hi and welcome to the forum.

   Basically the question describes an impossibe situation.  There's nothing you can do after that.

A rocket is in uniform motion and is of length, as measured in a stationary frame, of 300,000 km

   I'm going to assume the people on the ground are what you are considering as the stationary people.   The people on the rocket are the moving people.
   So this 300 000 Km is the proper length of the rocket just because the rocket was probably held stationary in the ground observer's rocket launcher at the start of the experiment.  Then someone ran up, lit a match and sent the rocket off accelerating which it did rapidly and by the time it reached the first post it was holding at a steady speed of 150 000 Km/s in the ground observers frame and will now continue at that velocity.   Ok, that's all fine, that can happen ( I mean it's a big rocket but this is theoretical physics not practical engineering).

When the back end of the rocket, which is moving left to right, as seen from the ground frame, reaches a point which is directly above a post on the ground both lasers are fired simultaneously as time is perceived in the rocket's frame.

   OK, that can happen but it's unusual to be jumping between frames of reference like this.   
1.  The location of the back of the rocket was precisely the location of the post in the ground observers frame.
2.  The lasers were fired simultaneously in the rocket's frame.

    One immediate consequence is that the lasers were NOT fired simultaneously in the ground observers frame.

There is a clock at the middle of the rocket and a clock on the ground in the middle of the two posts, both of which are visible to both observers in both frames. Both clocks read zero as observed from both frames at the time when the lasers are fired.

   Sorry, that is not possible.   The clock on the ground can read 0 when the first laser fired but it must read  +1 unit of time when the second laser fired.

  Anyway, there's nothing much you can do after that, the problem is an assumption of simultaneity in every frame.  The remainder of your reasoning might be good but it just doesn't matter.  The initial input was impossible or invalid.
   
Best Wishes.

[Halc]

There is a clock at the middle of the inertial rocket and a clock on the ground in the middle of the two posts, both of which are visible to both observers in both frames. Both clocks read zero as observed from both frames at the time when the lasers are fired.

This is the first problem. These clocks are not in the presence of the event of the laser being fired, and thus don't define a specific event at which they read zero.
Perhaps it would be better to say that there are clocks at the laser event, at both ends of the ship which read zero at the laser firing, and at the posts, synced in the ground frame, reading zero at the left laser firing event. So the ship clock is synced with the rear ship clock, etc. Ditto with a clock up front if there's on up there. If it reads X in one frame at the time of the laser firing, it is going to read something else in any other frame at the time of that same event. This is the essence of relativity of simultaneity.

The rocket is already up to speed when it passes over the posts and never slows down during the scenario. To the rocket based observers, the rocket is traveling at 150,000 km/s. I suppose it would be the same velocity to the ground based observers but I'll leave that unspecified in case a different velocity is indicated in someone's proposed solution.

At what distance markings do each observer see each target hit by its corresponding laser beam, and what are the readings on each clock when each happens, as perceived by each observer.

In ship time, the both lasers are going to hit 1 second later in ship frame since the ship is stationary in that frame and has a proper length of one light second. At a ship speed of 0.5c, the right-bound laser will hit a target at about the 519615 meter mark, not 450000. The left target is hit at about the 173190 meter mark if I did that correctly.

The problem doesn't involve the actual times required for the light to reach the observers' eyes from the targets and clocks, it's assumed to be instantaneous for our purposes

No such thing. The observer is irrelevant since all can be expressed in target X is hit when clock C (in the presence of the target at that event) reads time T. Everybody in any frame observes that, no matter how long it takes for the information to get to them. It's an objective event.

Can you perform any kind of relativity related manipulations which will result in both beams appearing to travel at 300,000 km/s to both observers?

Light moves at c relative to any frame, so no manipulation required. It is a premise of relativity.

One beam in one direction would be easy, but can you do it with two beams in opposite directions at the same time?

There is no 'at the same time' without a frame reference.  It doesn't matter when each laser is fired. It could be hours apart. The light will travel at c in any frame, and how long it takes depends only on how far it needs to go to reach the target, which is 1 light second in the ship frame.

Let's say the laser on top of the rocket is at the left, or back, end and fires to the right, in the direction of travel of the rocket. I conclude that it will hit its target at the 450,000 km mark, which is at the right post itself, and the bottom laser will hit its target at the 150,000 km mark and both will be at the time of 1 second on the rocket clock.

These numbers cannot be. It has light moving at different speeds.
Start with a specified relative speed between the two, and work from there. Don't just make up numbers.
If the light hits the right target at the 450,000 post, the ground is moving at about 0.384465c relative to the ship. At that speed, the left target will be hit near the 125,000 meter post, not the 150,000,

[Centra (OP)]

One immediate consequence is that the lasers were NOT fired simultaneously in the ground observers frame.

Quote from: Centra on Yesterday at 18:25:24
"There is a clock at the middle of the rocket and a clock on the ground in the middle of the two posts, both of which are visible to both observers in both frames. Both clocks read zero as observed from both frames at the time when the lasers are fired."

   Sorry, that is not possible.   The clock on the ground can read 0 when the first laser fired but it must read  +1 unit of time when the second laser fired.

  Anyway, there's nothing much you can do after that, the problem is an assumption of simultaneity in every frame.  The remainder of your reasoning might be good but it just doesn't matter.  The initial input was impossible or invalid.

Your statements are not logical, for the simple reason that if we choose to flip the two frames, so that the rocket is considered "stationary" and the ground is considered to be in motion relative to it, what would the ground based observer see? He would have to see the exact same thing regardless of which frame we arbitrarily choose to consider to be stationary or in motion. Would he see the lasers firing at different times? Like I said, he would have to, if that's what he saw when the rocket frame was considered in motion, if what you said is true. Nothing really changed, the relative motion is the same in both cases, there is no "true stationary" frame in the universe, so why would what he sees change? So by your logic he would see the two lasers firing at separate times even though we know that the lasers were fired using a single button in the rocket. I didn't write that but let's assume it now. Would that change your statements? No, you would still have said the same thing, right? By your logic, the rocket pilot would have had to have had two separate fire buttons, one for each laser, and pressed them at different times.

[Centra (OP)]

This is the first problem. These clocks are not in the presence of the event of the laser being fired, and thus don't define a specific event at which they read zero.

Like I said to Eternal Student, consider the rocket frame to be stationary and the ground frame to be moving at 150,000 km/s past it to the left, ignore the distance markers for now, what would the ground observer see as he moves past while keeping his eyes focused on the rocket? Like I said to Eternal Student, assume that both lasers are fired using a single button, not two separate ones. You answer that simple question and we'll proceed from there, same for Eternal Student. That's all I request either of you to post at the moment, though obviously you can post whatever you want. It would make things simpler, though, if you just describe what the ground observer would see and then I respond to that.

[Halc]

You seem to be unreceptive to corrections to your physics, so I've moved the topic.
To see the problem, it is best to run the numbers with your scenario, and you will see that either light is moving at a speed other than c, or that clocks with identical velocity must be running at different rates, either of which directly contradicts empirical physics.

So as far as I can tell, we have five distinct objective events (two emissions, two targets hit, and one crossing of signals), and these events need coordinates (location, time) in both frames.  The scenario seems to primarily take place in the rocket frame.  Assuming the rocket has a proper length of 300000 km, the coordinates of those five events in the rocket frame are:
LE (left emission): x=0 km, t=0 sec  beam emitted from left to right
RE (right emission) 300000, 0  beam emitted from right to left
LT (left target) 0, 1    RE beam detected on left end
RT (right target) 300000, 1    LE beam detected on right end
X  (beams cross)   150000, 0.5

Now you need to do it for the same five events in the coordinates of the ground frame, and you will see the contradictions.
Your description is unclear, so some of the numbers (mostly the times) are unknown. Feel free to correct numbers I guess wrong.
LE: x=0 km, t=0 sec
RE  ?, ?
LT  150000, ?
RT 450000, ?
X  225000?, ?

Notice that there are no observers in the above numbers. It is simply the coordinates of events, which, if it makes you happy, can be observed by observers present at those events. Any observer can also see the other events, but delayed due to the time needed for light to travel the distance from that event.  So for instance, each observer at the ends of the rocket will see the far clock be 1 second behind their own because it takes a second for light from the remote clock to reach them from 300000 km away.

Like I said to Eternal Student, consider the rocket frame to be stationary and the ground frame to be moving at 150,000 km/s past it to the left, ignore the distance markers for now, what would the ground observer see as he moves past while keeping his eyes focused on the rocket?

He sees a rocket that takes 1.732 seconds to pass by him. Fill out the chart above, and only then work out when this observer sees each of the five objective events.

Like I said to Eternal Student, assume that both lasers are fired using a single button, not two separate ones. You answer that simple question and we'll proceed from there, same for Eternal Student.

From your description, you said simultaneously in the rocket frame, which means the button needs to be mid-rocket and pushed at t=-0.5.  If I am wrong about this, please fill out the upper chart with corrections. If you want it to work this way, then the button being pushed is a 6th event with rocket coordinates of (150000, -0.5).

It would make things simpler, though, if you just describe what the ground observer would see and then I respond to that.

I posted numbers for the above scenario, but you rejected them. What a given ground observer sees depends on where you put him. The observers don't alter what the lasers do, so I ignored them. Those events occur at the coordinates shown, observed or not.

[Eternal Student]

Hi again.

    Firstly, it's an interesting situation that you were describing.  You've obviously spent some time thinking about this and you're obviously intersted in physics.   All of this is good.
    I'm not trying to run down what you've said or done at all.   The earlier comments shouldn't have seemed like an attack, they were intended to be just some discussion and I can only apologise if they seemed offensive.
    Everyone loves special relativity, it's just so peculiar.  The fundamental results from it suggest things like length contraction and time dilation which are just fascinating and extremely counter-intuitive.   We have a collection of bizarre examples of things that happen -  like the ability to fit a long pole into a short barn provided you run with the pole at great speed etc.
    I'm fairly sure Halc was also just trying to help and not criticise.  Actually you can see that Halc modified his answer and took a long time over it,  most probably because the question itself leads you astray.  It feeds you information that won't actually hold together consistently.  In the end Halc has tried to suggest modifications to the basic situation that was described by the question.

    Fundamentally two events that were simultaneous (happened at the same time) in one frame of reference do not have to be simultaneous in another frame of reference.  This is strange and it does take all of us quite a lot of time to accept and often this involves working through several examples.   Personally, I think it would actually be best if you looked at some examples of this or studied the effect directly.   After that you can come back to your own question and probably see where you have used an assumption of simulatenity across two different frames more easily.
    Where can you study the relativity of simultaneity:   1.  Just about any text book on relativity.
    2.  Most on-line texts on relativity.
    3.  You Tube, if you prefer videos.   Try this one: 

.    I've seen that video and it's factually sound, the presenter is unsual and his exmaples are based around human beings instead of inanimate objects like poles in barns.

   However, I know that a lot of people really want to see what happened in their example or their own working and it's just human nature to think that spending 30 minutes studying someone else's examples isn't going to be useful.   So, let's also just have a look at your own example and try to point out where things go wrong:
   

So by your logic he would see the two lasers firing at separate times even though we know that the lasers were fired using a single button in the rocket.

    You're presenting the idea that the lasers were operated by a single button and there is some electrical circuit between the button and the lasers.  Then you assume the lasers will fire simultaneously.   Unfortunately they don't and you need to have a carefull look at that electrical circuit to see why not.
Here's a diagram:

    End laser               piece of wire                     Button              piece of wire           Front laser
          *   ---------------------------------------------------  |-|  ----------------------------------------  *

You are assuming that when you press the button the power can flow through the wire and arrive at the lasers instantly.  In fact it can't and it doens't.   It travels at about the speed of light.   You'd barely notice the delay (when you and the circuit are both stationary relative to each other) but it is there.   The person in the rocket can press the switch but the lasers won't fire until a fraction of a second later.
    Now the people on the ground actually see much the same thing.  They can see the button and the wires (assuming the rocket is transparent or they have x-ray vision... the details aren't important the main thing is the circuit is still there).   However passing an electrical signal along these wires is exactly the same kind of thing as passing a laser from one place on the ship to another place.  The electrical signal is carried by electric and magnetic fields in and around the wire, you have a transfer of energy by electro-magnetic fields.  It really is much the same as an electromagnetic pulse just like a laser flash.
   Anyway, the people on the ground have the same physics applying for that electrical circuit.  The signal passes along the wires at light speed.   The only issue is that your rocket is travelling at 1/2 the speed of light from left to right in the observers frame.   So the the electrical signal from the button to the front laser actually has to go a lot further,  it has about 1.5 times the distance to cover that the people in the rocket would have thought it had to cover.  Similarly the people on the ground will see that the back end laser is actually being brought up to meet the electrical signal that was being sent down to it.  It will only need to travel about half the distance that the people on the rocket thought it had to travel.   Overall, the people on the ground will see the end laser has power and lights up BEFORE the front laser has power and lights up.
    I know this seems weird but it's just how things are.   The people on the rocket can observe the lasers firing simulatneously while the people on the ground do NOT see that, they see the lasers fired at different times.   Now you might think that perhaps the problem was using an electrical circuit, perhaps there is some other way you can make the lasers fire simultaneousy.   Well, that's why I suggested you just start by studying the general topic of the relativity of simultaneity.   It turns out that you cannot get any information from one place to another place faster than the speed of light.  It's not the electrical circuit that is acting just as some practical problem, there is an absolute or theoretical limit on what you can do.   There is nothing you can replace the electrical circuit with that will be any better to make those lasers fire simultaneously.   The best definition of simultaneity is that you have something that emits a light signal (or an elctrical signal) in the middle of the two objects or places you are trying to synchronise and you metaphorically hit the reset button on the stop-watch or consider the two places to be synchronised when the light signal reaches them.

Best Wishes.

[Origin]

I have a thought experiment type paradox, related to the Theory of Special Relativity, which I would like to see if anyone can resolve, the apparent paradox of light seemingly being sped up or slowed down to other than normal velocity in one inertial frame of reference when viewing light beams generated in another inertial frame in relative motion to it.

Your request to have the apparent paradox resolved has been answered by 2 members.  If you carefully read their replies you will see that scenario you presented is not a paradox and is in accordance with special relativity.  Your refusal to accept these answers makes me think you may just be another relativity denier, I hope not.