1. Has anyone come across the following argument somewhere?
Can anyone suggest a relativistic solution to the paradox?
I've done my best for the diagrams.
Regards.

The standard imaginary model of special relativity (without the mirror) is that of a beam of light (AB) seen as perpendicular to the motion inside the reference frame. While the observer outside the moving reference frame sees the beam of light (AC) pointing ahead of the perpendicular.
____A .

v->

____B .---. C

If c is the speed of light and v is the speed of the moving reference frame and if t is the time inside the reference frame and t' the time outside, then :
AB = ct, AC = ct' and BC = vt'
If ABC is a right angle triangle, then :
(ct)Â² = (ct')Â² - (vt')Â²
t = t sqrt(1 - vÂ²/cÂ²) ... (1)

Now imagine that the beam of light, as seen inside the reference frame (AB), is pointing slightly ahead of the perpendicular.

____A .

v->

____O .---B .---. C

If c is the speed of light and v is the speed of the moving reference frame and if t is the time inside the reference frame and t' the time outside, then :
AB = ct, AC = ct' and BC = vt'
If AOB and AOC are right angle triangles and if OA = y and OB = x, then :
(ct)Â² = xÂ² + yÂ² and (ct')Â² = (x + vt')Â² + yÂ²
Therefore:
(ct)Â² - xÂ² = (ct')Â² - xÂ² - 2xvt' - (vt')Â²
(ct)Â² = t'Â²(cÂ² - vÂ² - 2xv/t')
t = t' sqrt(1 - vÂ²/cÂ² - 2xv/t'cÂ²) ... (2)

Then imagine that the beam of light, as seen in the reference frame (AB), is pointing slightly behind the perpendicular.

______A .

v->

____B .---. C

If c is the speed of light and v is the speed of the moving reference frame and if t is the time inside the reference frame and t' the time outside, then :
AB = ct, AC = ct' and BC = vt'
If ACB is a right angle triangle, then :
(ct)Â² = (ct')Â² + (vt')Â²
t = t' sqrt(1 + vÂ²/cÂ²) ... (3)

This means that the proportion t : t' varies according to the direction of the beam of light, with respect to that of the moving reference frame, and that the relativity of time depends on what is being shown.

2.

3. I have tried to offer the idea you seem to presenting, not though with equations, but putting known pieces together in presenting the idea that the "light", for instance (you refere to), the different references of light, would cause, for the equations to be upheld, as physics understands them, to be bent, for the light to undergo a change in direction.

I am arguing, well, where I am coming from, is that light would bend as an a-priori to satisfy the relativity conditions, as opposed to being secondary to the so-called "curvature of space".

4. Two things:

(ct)Â² = xÂ² + yÂ² and (ct')Â² = (x + vt')Â² + yÂ²
Where x=OB

In the second equation, you neglect the length contraction of the distance OB as measured from the "outside" frame. From the "outside" frame, OB does not equal x, but x', where x' = x sqrt(1-vÂ²/cÂ²)

Second, by removing the mirror and only considering the light path of A-C, rather than the round trip of A-C-D (where D is the point the light ends up on the return trip), You have to incorporate the 'Relativity of Simultaneity' into the scenerio.

Thus, if we were to put clocks at A and B, such that they are sychronized(read the same time) in the AB reference frame, then:
In the second scenerio, where B is slighty ahead of A in the direction of motion, then from the "outside" frame, the clock at B will run ahead(time wise) of the clock at A.

In the last example, the clock at B will lag behind(timewise) the clock at A as measured from the "outside" frame.

This assures that when the clock at B and the light meet at C, the clock will read the same time in both reference frames.

If you allow the light to complete the round trip, you find that the ratio of t':t for the clock that starts at A, will always follow the standard time dilation equation no matter what direction the light is aimed.

5. In addition, here's a post I made in answer to a simular claim that stated that time dilation in the light clock experiment depended on the angle (alpha) at which the light was emitted. The argument was that light emitted fromt he center of a moving sphere would return to the center of the sphere at different times according to Relativity.

This is my mathematical refutation of that claim.

I apologise ahead of time to linking to a post on another forum, but the post contains quite a bit of math in LaTex, and converting the LaTex formatted equations to text is a task I just don't feel like doing (Plus the fact that I doubt that I could do it without transcription errors creeping in.)

http://www.scienceforums.net/forum/s...0&postcount=68

6. Of course! The Lorentz transformations. But then what about BC? How long is BC, if x=x'? In this case BC is shorter than it appears. It equals BC' = BC sqrt(1-vÂ²/cÂ²). What about AB and AC and what does Lorentz have to say about that?
It is my opinion that introducing Lorentz makes things even more incoherent. So it seems that my questions remain unanswered. Thanks for trying.

7. Originally Posted by Ken
Of course! The Lorentz transformations. But then what about BC? How long is BC, if x=x'? In this case BC is shorter than it appears. It equals BC' = BC sqrt(1-vÂ²/cÂ²). What about AB and AC and what does Lorentz have to say about that?
Since BC is the distance traveled by the whole arrangement as measured from the outside frame it does not contract because the points B and C are at rest from this frame. Part of your confusion lay in the the fact that you interchangeably use point "B" in two ways. You use it for both the Starting point for the apparatus as measured in the outside frame, and the reference point from which the distance to O is measured as measured in the reference frame.

It is my opinion that introducing Lorentz makes things even more incoherent. So it seems that my questions remain unanswered. Thanks for trying.
???
The Lorentz transformations are a natural result of the situation and the postulates of SR, they are not "introduced".

As to answering your question, I will repeat what I already said; The answer lies in understanding the concept of "The Relativity of Simultaneity" This is a keystone concept of Relativity and not understanding it will prove to be a stumbling block on the road to grasping Relativity.

8. It's very kind of you to spread the "grasp" of relativity. But learning rules does not help solve my problem. Why does the rule apply to one case (1) only. I asked if any one had come accross this argument elsewhere and you are saying that there is no argument. In that case, why does t/t' give three different values. And, in fact, as many values as there are possible orientations of the light beam.
As for the Lorentz transformation, it depends on the exact position of the "stationary" observer, with respect to the moving "event". Opposite C, it is moving towards him and will be longer. Opposite B, it is moving away and will be shorter, but OB will be longer.
As I said, as soon as one tries to apply the SR rule to other cases, it goes haywire.

9. You didn't follow the proof that Janus posted did you?

10. Originally Posted by Ken
Of course! The Lorentz transformations.
No these are not the lorentz transformations. But it would be a good idea to go back to these instead of looking at the derived equations of lorentz contraction and time dilation, which you have to be very careful about using for a situation in which they do not apply.

The lorentz transformations are these when the velocity v is in the x direction and x,y,z,t are the coordinates of the frame that is moving with velocity v with respect to the frame in which we use x',y',z',t'.

t'=(t+(v/c^2)x)/sqrt(1-v^2/c^2)
x'=(x+vt)/sqrt(1-v^2/c^2)
y'=y and z'=z.

and the inverse transformations

t=(t'-(v/c^2)x')/sqrt(1-v^2/c^2)
x=(x'-vt')/sqrt(1-v^2/c^2)
y=y' and z=z'.

These are all coordinate positions in time and space and NOT the length of special intervals in space and time as is the case in length contraction and time dilation.

You have taken overly simplistic derivation of time dilation and treated it as foundational which it is not while the lorentz transformations ARE foundational and are what physicists use to check the logic of arguements like one you use to derive time dilation in your first example.

Originally Posted by Ken
But then what about BC? How long is BC, if x=x'?
In this case BC is shorter than it appears. It equals BC' = BC sqrt(1-vÂ²/cÂ²).
If x = x' and x is in the direction of velocity v then v = 0, and what you call BC is therefore also equal to zero.

BC does not "appear" anything, BC is a diference between measures in two different coordinate systems and it certainly is not something to which you can apply length contraction to. BC' has no meaning for BC is not the distance between two events. B and C are actually the same event marked in two different coordinate systems. That event is the arrival of the light at the destination.

Originally Posted by Ken
What about AB and AC and what does Lorentz have to say about that?
AB and AC are the same interval of both space and time in two different coordinate systems. AC in your first example and both AB and AC in your second and third examples are not lined up in the same direction of the velocity and so these simple length contraction and time dilation formulas would not apply even if they were the proper type of intervals. For these you must assign space time coordinate in the two different reference frames in which you can make A the origin and therefore the same in both coordinate systems if you like. Then leaving z out as uneccessary you can x,y,t for the coordinates at B and x',y',t' for the coordinates for what you call C (the same event in the other frame) and then the lorents transformations do indeed apply

Now in your first example x=0 so the inverse transformations
t=(t'-(v/c^2)x')/sqrt(1-v^2/c^2)
x=(x'-vt')/sqrt(1-v^2/c^2)
y=y' and z=z'.

give you

t = t'/sqrt(1-v^2/c^2) or t' = t sqrt(1-v^2/c^2)
0 = (x'-vt')/sqrt(1-v^2/c^2) which gives us x' = vt'

And so your first example works out. But in your second example x is not zero and so we should check your assumption that what you call BC = vt'. Well what you call BC is given by x'-x according to what I call x and x' here, and so we see that,

BC = x' - (x'-vt')/sqrt(1-v^2/c^2)

which is nothing like vt' at all and so we see that this assumption in your second and third examples that BC = vt' is invalid.

Sometimes it helps to look at extreme cases to understand things like this. Consider the case when y is small and x is extremely large so that the light is going practically directly away from you in the x direction. In that case you can see that your assumption that BC = vt' is basically claiming that you can reduce the speed of light by your velocity v, which clearly wrong. It is only in your first example, where light is going completely pependicular to your direction of motion where the claim that BC = vt' does not imply any reduction of the speed of light.

Originally Posted by Ken
It is my opinion that introducing Lorentz makes things even more incoherent.
And what kind of opinion is that? Do you have some kind of training or expertise in the field of special relativity?

Originally Posted by Ken
So it seems that my questions remain unanswered. Thanks for trying.
Your question was flawed and Janus was pointing out some of the flaws, which is the only proper answer to a flawed question.

11. Well, I donâ€™t need to ask who and what you are. I recognised you straight away. I dropped out of Kingâ€™s College (London) forty years ago, partly because of people like you, full of certainties and no sense of humour.
It happens that Iâ€™ve read Einsteinâ€™s little book. I donâ€™t have it on hand, but you seem to have followed the non Galilean part all right. However, what about PoincarÃ©â€™s non Euclidean geometry? If I remember rightly, t is in fact it.
Non Euclidean geometrical space, irrational numbers and non Galilean relative time, plus a dose of non Newtonian gravity, Charles Dodgson would have loved this, had he still been around.
At least the quantum people manage to poke fun at themselves (you quark!). But I do realise that they are under less pressure than your side is. They get all the funding. While the relativity people and the big bang ones for that matter, as these are often shared opinions, are kept on a shoe string, never having managed to demonstrate anything.
So, with every new telescope, Creation has to be pushed ever farther back in time. While, to my knowledge (I would welcome information to the contrary), the isotropy of light has never been tested for direction (with a laser beam) or for colour shift (with a spectrograph). The M&M experiment for speed is supposed to have settled all these aspects. Though, at the time (1887), neither test was possible. This being quite contrary to scientific method (and to Peirceâ€™s pragmatic maxim), one may ask the question: Is this science or ideology?
The whole structure is rooted in the 19th century, as though time slowed down when it was declared relative, and came to a standstill when its finitude was announced. We are told to believe that the passage of time is in the eye of the beholder, that the four-dimensional space/time continuum is curved by the presence of a three-dimensional object and that light travels along these curves. We must also believe that the nature of space is to expand, that distance varies with time, that the universe was once the size of a pin head and yet was already infinitely big, as it was all that was.
Of course, nobody really cares, myself included. So many other ideas from the distant past clutter up the present and help to make nonsense of all attempts at change. There Is No Alternative, as Truth is a Â© for all eternity.
I apologise for wasting your time.

12. Hmmm... such hostility.

Well so much for the idea that this person was actually interested in the science or the answer to his proposed paradox.

It seems that his real agenda is as a spokesman for the anti-relativity pseudo-science subculture that has arisen recently, who spends (wastes?) his time looking for a way to shoot down a scientific theory he doesn't approve of. The Creationists do it, so why not the Trekkies.

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