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I'm challenging anyone who can answer the following question objectively:

As usual, imagine a railway station and trains which are equipped with single photon sources (one each in the platform and inside the trains) and a narrow strip of detector which is positioned right above the photon sources. The photons are fired vertically upwards, perpendicular to the train’s direction of travel. In addition, let us assume that apart from the photo detectors on the train and the platform, the roof of the train is open, and that the ceiling of the station above the train also has photo detectors, corresponding to the position of the photo detectors on the platform. Now, the photon sources in the station and the trains are set to trigger at the same moment – i.e. when the train’s photon source passes in front of Alice. Now, Bob is sitting in front of the photon source in the train which is travelling at approximately half the speed of light, towards the station.

When the train in which Bob travels crosses Alice, both photon sources emit a photon (on platform and on the train). Hence, one photon is emitted towards the roof of the railway station by the photon source stationed in the platform; another is emitted inside the train above which is the photo detector on the train (moving w.r.t. Alice) and the photo detector on the station’s roof (stationary w.r.t. Alice) above the train. Which means, Alice will (? might) record two photons hitting the photo detectors on the ceiling of the station (the photon from the platform and the photon from the train). The photon emanating from inside the train should hit the station’s ceiling since the train’s detector would have moved away by the time the photon travels upwards. Now from Bob’s view point, the photon emitted inside the train will be detected by the sensor IN the train; the photon emitted by the emitter on the platform will miss the detector on top of it (since the station is moving relative to Bob’s train). Either way, SR would not work for both observers at the same time.

In case you are thinking that there is some trickery with which this question can be answered in a manner similar to the barn door paradox, let Charlie travel in the track parallel to Bob, in the opposite direction (both Bob and Charlie are equidistant from Alice’s view point and are approaching with the same speed). Now we will have three different observers with different predictions; but due to law of conservation of energy, the photons can be detected by only one of the sensors, which should settle the confusion (or if they do separate, we would be able to tell because of the lesser energy liberated by the photon). In general, the experiment suggests that depending on the relative speed of the train to the station, there are an infinite number of positions in which the photon can be found, which could either support the many worlds theory, or is at least at odds with the Special relativity.

I have been digging around this for almost 13 months; still no solution. Any ideas? Would theory of General relativity help?

Should I rephrase my question -or- add details?

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I want to add that this seems like a trivial question - and probably it is, since this is similar to the Michelson-Morley experiment, only with a slight modification. But for the heck of it, I cannot figure what is wrong in my PFP gedanken! – Vignesh Mar 12 '14 at 19:17
The suspense is killing me - can some body tell me if I'm right or wrong - I REALLY need an answer... – Vignesh Mar 12 '14 at 20:19
up vote 1 down vote accepted

can some body tell me if I'm right or wrong - I REALLY need an answer

Your premise that Alice and Bob observe both photon trajectories as vertical is false.

The locus of events that form the worldline of either photon are absolute but the coordinates Alice and Bob assign to these events are not.

If Alice observes the station emitted photon to have a constant $x$ coordinate $x = 0$ then, assuming the photon is emitted at $t = t' = 0$, the Lorentz transformation to Bob's coordinate is

$$x' = -\gamma vt $$

In other words, Bob does not observe the photon to have a vertical trajectory but, rather, to have a horizontal component too.

Similarly, Alice observes the train emitted photon to have x coordinate

$$x = \gamma vt $$

Are you not familiar with the well-known light clock thought experiment?

enter image description here

If the Galilean transformation were true, Alice would observe Bob's photon to have a vertical speed of $v_z = c$ and a horizontal speed of $v_x = v$ and, thus, the photon would have a total speed of $u = \sqrt{c^2 + v^2}$

According to the Lorentz transformation, Alice observes Bob's photon to have a total speed $u = c$ and a horizontal speed of $v_x = v$ thus, the vertical speed of the photon is reduced, according to Alice, to $v_z = \sqrt{c^2 - v^2}$.

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I think I'm missing elementary things - I will read throug the answers thoroughly (I think all answers point in the same direction, that one of my assumptions is wrong - will correct myself). Thanks. – Vignesh Mar 14 '14 at 9:48
Many thanks for your detailed answer - this clears it! – Vignesh Mar 17 '14 at 18:09

No issue with SR and no many worlds - when Bob is firing his photon upwards towards the train's detector, Alice as well sees the photon of Bob moving to the train's detector, but for her the photon's direction is not upwards but diagonal. For her the photon is taking more time than it takes for Bob because it is going diagonally - this phenomenon is called time dilation.

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Hmmm.. That doesn't answer my question - Why would stationary Alice see a diagonal path for the photon? – Vignesh Mar 12 '14 at 21:06
Why would stationary Alice see a diagonal path for the photon? For the exact same reason Bob observes a diagonal path for the station emitted photon. This is elementary. The events that constitute the worldline of the photon are absolute - the coordinates Alice and Bob assign to those events are not. If Alice observes that the station emitted photon has constant x coordinate of zero, then by the Lorentz transformation, Bob observes the photon to have non-constant x' coordinate: $x' = -\gamma vt$ – Alfred Centauri Mar 12 '14 at 22:35
Sorry, did not get it the first time around. I stand corected. – Vignesh Mar 17 '14 at 18:03

The point you are confused about is the very fundamental point that the laws of physics are the same in all inertial frames. Let's look at what Bob sees. When the photon is emitted, bob sees the photon go straight up and hit the detector on the train, becuase photons move in straight lines in inertial frames, and the train is an intertial frame. Since the photon is going straight up in Bob's frame, and Bob's frame is moving with respect to Alice's frame, Alice sees the photon going up diagonally. Since the photon was emitted directly underneath the detector on the station, and the photon travels diagonally, Alice will not observe the photon from the train hitting the detector in the station. Thus both people will observe the station photon hitting the station detector, and the train photon hitting the train detector.

Notice that the only thing used here was the laws of physics being the same in all reference frames. Thus we can do the same thought experiment in newtonian dynamics with a ball being thrown into the air (ignore gravity). Then you would have said that the ball thrown upward from inside the train "should hit the station’s ceiling since the train’s detector would have moved away by the time the ball travels upwards." But this is not true because we know that the train is an inertial frame and so in the frame of the train the ball must move straight up and so Alice will see the ball moving diagonally, so the ball will hit the train's detector and miss the station's detector.

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Thanks for the answer, I think I got it this time. I have one more question regarding this - 'Do photons have a preferential frame?' - I thinks that is another question, so I will frame a new query... – Vignesh Mar 17 '14 at 18:06

When the photons are emmited, Alice will see that Bob's photon has a finite horizontal momentum component. That is, it's not travelling vertically towards the roof, but diagonally towards the place where Bob's sensor will be some time later (how much time depends on Bob's speed relative to Alice, which determines the horizontal component of Bob's photon momentum). Both Alice and Bob will agree that Bob detects his own photon. Think of it as if Bob threw a ball upwards inside the train (an ultrarelativistic, massless ball traveling at exactly the speed of light).

By the way, you may want to reformulate your question. Saying that the photon sources trigger "at the same moment" does not make sense: the same moment respect to which reference frame? Simultaneity notions change when you go from one frame to another in SR.

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Got it! But instead of reformulating the silly question, I will ask another question which might be more relevant! Thanks for your answer. – Vignesh Mar 17 '14 at 18:08

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