Graduate Revisiting the Light Clock

[DmitryS]

A good one to everyone.



My previous post on this subject here on the forum was a fiasco. I’d like to apologize to everyone who did their best to comment and got ignored by me. In defence, I could tell you I had really little time to spend on discussion, and just overlooked the explanations that seemed irrelevant (why they seemed irrelevant, I will tell you at the end of this).



Before we get to the point, I will kindly ask you to comment having considered this text carefully, because every word here has been contemplated on and well-weighed. I have come up with a better explication of how I see this problem, namely why I don’t think the model of the light clock works.



I will consider the ‘stationary’ and ‘moving’ frames sliding relatively to one another along x/x’ lines called respectively O and O’, and so are the points of origin of the coordinate systems.



1. My first point is again the photon count argument that we didn’t have the opportunity to discuss properly. If the angle arccos(v/c) embraces N photons in O, the same quantity falls within the angle π/2 in O’. The photon sensor grid located to the ‘right’ of the sphere of light will catch fewer photons for O’ than for O.



2. Another point is something I came up with after the previous thread got locked. Let’s consider how we recalculate angles within the Lorentzian context.



For O which we consider stationary the photon slides along the line of inclination arccos(v/c). For O’, the fixed angle of the trajectory α is transformed into α' according to this equation:



sin^2(α') = (sin^2(α))/(1-(v^2/c^2)cos(α)),



and it is not the same as the transform for the angle of trajectory resulting from a composition of motions. The trajectory angle α'' will be transformed as



sin(α'') = (sin(α))/γ(1-(v/c)cos(α)).



Now, the question is, is there a non-trivial situation when α'' = α'?



It all comes down to the quadratic equation with respect to cos(α):



((v^4/c^4) – 2(v^2/c^2))cos^2(α) + (2v/c) cos(α) – (v^2/c^2)=0.



Unfortunately, this equation has a solution, and it’s not α = 0. For this angle, the ray of light originating from x=0, t = 0 and ending at x=x_1, t=t_1, in O’ will originate from x’=0, t’ = 0 and end at x’= -x _1 (its trajectory will be inclined in O’ to the opposite side), t’=t_1.



Within the context of the thought experiment which affirms the relativity of simultaneity, we can set up a different thought experiment which testifies to the opposite.



3. Yet another objection to the light clock scenario is even more interesting – I only recently came up with it. In the light clock scenario, we are considering a singular photon emitting from O at an angle arccsos(v/c). Instead, we can consider a flat wave of light whose infinitely wide front is inclined to x at (π/2 – arccos(v/c)). We can see that the ‘top’ leg of this front will cross the horizontal line y = ct/γ at the point where t’= t/γ, while its ‘bottom’ leg crosses the x axis at the point where t’ = 0 – and forever so. No matter how much of t time passes, the crossings of the wave front with the horizontal planes will have the same t’ coordinates, each one individual at its own level of y.



Thus, the front which will never fully cross the x axis will cross x’ momentarily, and that corresponds to the transform of arccos(α) to π/2 or arcsin(π/2 - α) to zero: the front of the light wave for an O’ observer should be seen rising vertically.



That is to say, what takes an infinite amount of time in O happens in O' momentarily. Thus, we come to this seeming contradiction:

1. Time t elapsing in O corresponds to time t/γ elapsing in O'; and

2. Any however large amount of t elapsing in O corresponds to Δt’=0 in O'.

The contradiction as has been said is only a seeming one, since Item 1 speaks of time t' measured for a specific location, while Item 2 speaks of the situation in O' at large. In 1, we state that there's a 'time wave' of t' in O' as observed by O, and for any specific location the resulting time t' elapsed follows from the composition of motion of an O' observer and said 'time wave', while in 2 we're dealing with the problem of an ultimate composition of the 'time wave' and O' motion which should result in a set of synchronized clocks for O' (universally synchronized time).

Marrying these two statements, that is, aspiring to the above ultimate composition of the motion of O’ and the ‘time wave’ represents the real problem, since there's no mathematics describing this transition: the Lorentz transforms describe the result but not the transition itself, as for the transition we will have to operate with infinites and bump into singularities (the asymptotes of the velocity composition formula).



Thus, the problem I'm describing cannot be dismissed as a case of aberration, as it is the definition of aberration which is problematic within this context that we are talking about.

 

[Dale]

Hi Everyone, this thread is related to @DmitryS previous thread, but after that thread he and I discussed the issue that led to closure. He understands the need to be responsive to explanations and not simply repeat his previous posts.

This thread will be actively monitored to ensure that it progresses, and hopefully it will be productive.

 

[PeterDonis]

Before we get to the point, I will kindly ask you to comment having considered this text carefully, because every word here has been contemplated on and well-weighed.

Sorry, but as it stands, I see no point in doing this, because you have left out crucial things.

Here is what I think you need to do to analyze a light clock scenario:

Start in one chosen frame (the obvious one is the rest frame of the light clock). Assign coordinates to key events in that frame (the obvious ones are the light pulse starting from one mirror, the light pulse reflecting off the other mirror, and the light pulse arriving back at the first mirror).

Then Lorentz transform the coordinates of those key events to another frame (an obvious one is a frame in which the light clock is moving with speed $v$ in a direction parallel to the mirrors, i.e., perpendicular, as seen in the light clock rest frame, to the movement of the light pulse). As you have set things up, that would mean the relative motion is in the $x$ direction, and the light pulse moves in the $y$ direction, as seen in the light clock rest frame. (An important exercise for you is to figure out why I have been careful to specify "as seen in the light clock rest frame" when I give these directions.)

Then compute whatever other quantities of interest you want in each frame, from the coordinates of the events in each frame.

When you do the above things properly, you will find that everything else that is talked about in relativity textbooks--length contraction, time dilation, relativity of simultaneity, aberration of light, relativistic Doppler shift--pops out automatically. But if you don't do the above, but instead try to wave your hands about those other things--which is what you're doing--you're going to get yourself confused and end up with wrong conclusions. That was the root cause of your problems in the previous thread, and you haven't fixed it, and unless and until you fix it, you're going to continue to have the same problems, continue to make the same wrong claims, and this thread will end up suffering the same fate as the previous one.

I strongly suggest that you take whatever time you need to consider the above and do as I suggest.

 

[Dale]

I have come up with a better explication of how I see this problem, namely why I don’t think the model of the light clock works.

Before we start, let's be very clear. The light clock works. The purpose of this thread is not for you to explain why it doesn't work (it does), nor for you to repeat your arguments (you have presented them). The purpose of this thread is to teach you why it works. Your specific arguments may not be important/useful for that purpose, and may not all be addressed by people posting here.

To see why and how the light clock works, it is important to actually use the Lorentz transform. Often, the light clock is presented in introductory "derivations", but it is important to understand that relativity was not actually derived that way. The light clock is a pedagogical tool used by people already familiar with the Lorentz transform to explain time dilation to laymen. It is not the foundation of relativity.

With that, we start as follows. In the unprimed frame, we have a pulse of light that goes vertically ($y$ direction) starting at the origin. So its position is $x=0$, $y=ct$, $z=0$. This can be written more concisely with four-vector notation where the position in spacetime is written as a 4-D vector $R=(ct,x,y,z)$. In spacetime notation that is therefore $R=(ct,0,ct,0)$, which is called the worldline of the pulse.

The Lorentz transform is the transform $$t'=\gamma \left(t-\frac{vx}{c^2} \right)$$$$x'=\gamma (x-vt)$$$$y'=y$$$$z'=z$$ where $$\gamma=\frac{1}{\sqrt{1-v^2/c^2}}$$

So, when we Lorentz transform $R$ into the primed frame we get $$R' =(c t',x',y',z')= (c t', v t', c t'/\gamma,0)$$

The fact that $x'=vt'$ follows directly from Lorentz transforming the spacetime path of the vertical light pulse. Any explanation or reasoning that does not conform to that is not consistent with the Lorentz transform, so it is not a correct application of relativity.

I will kindly ask you to comment having considered this text carefully, because every word here has been contemplated on and well-weighed

I suspect that my post will not be what you wanted because you want to focus on your arguments.

I read your arguments, but I wanted to first address your conclusion. Once you are clear that your conclusion is incorrect, then I think you will be more receptive to understanding where the arguments fail. If we go the other way then I think your inclination will be to defend the arguments, which will be less productive overall.

 

[Ibix]

1. My first point is again the photon count argument that we didn’t have the opportunity to discuss properly. If the angle arccos(v/c) embraces N photons in O, the same quantity falls within the angle π/2 in O’. The photon sensor grid located to the ‘right’ of the sphere of light will catch fewer photons for O’ than for O.

This is easy to address. Photon counting arguments are risky because not all states preserve photon number. A better approach is to consider a source that emits rays - perhaps a lot of lasers mounted on a sphere and pointing radially outwards. You can pulse the lasers to produce discrete classical light pulses, the number of which is conserved.

The number of such pulses absorbed by a screen is an invariant. This is obvious from the Lorentz transforms. Qualitatively, you will find that different frames describe the screen as being at different distances from the emission location when it absorbs a pulse, and thus subtending a different angle. That difference cancels with the different angular distribution of the pulses to give you the same number of pulses absorbed.

Edit: note that, as discussed in your previous thread, the amount of energy absorbed, both in absolute terms and as a fraction of energy emitted, will depend on the frame in general. The number of pulses will not.

 

[PeterDonis]

In the unprimed frame, we have a pulse of light that goes vertically ( direction) starting at the origin.

Note that in a light clock scenario, this pulse reflects off the second mirror and reverses direction. I don't know that accounting for that part is actually required to address the OP's confusion (which seems to be more about things like aberration), but to fully model the light clock (since a "tick" of the clock is supposed to be the light pulse returning to its starting point on the original mirror) it needs to be taken into account.

 

[Dale]

Note that in a light clock scenario, this pulse reflects off the second mirror and reverses direction. I don't know that accounting for that part is actually required to address the OP's confusion (which seems to be more about things like aberration), but to fully model the light clock (since a "tick" of the clock is supposed to be the light pulse returning to its starting point on the original mirror) it needs to be taken into account.

Yes, that is completely correct. I think his confusion is about this part, which is why I focused on it, but I completely agree that this is only half of a tick of the light clock.

 

[A.T.]

every word here has been contemplated on and well-weighed

That's nice.

But what is missing, is a systematic approach:
1) Define the full scenario in one frame.
2) Use the full Lorentz Transformation to all aspects of the scenario.
3) Show that a quantity, which really must be frame invariant (not just should be according to your intuition), isn't invariant under the full Lorentz Transformation.

No amount contemplating and weighing of words can replace that.