I [SR] light shot at an angle in a moving train

[roomie10]

SR explains how light travels longer for an observer outside the train when it's shot vertically upwards.
If the distance between the light source and the mirror on the ceiling is L and the train moves to the right at c/2,
the light travels 2L for the observer inside the train but sqrt(5)L for the observer outside the train for the roundtrip.
Since c is the same for both, it takes 2L/c for the observer inside and sqrt(5)L/c for the observer outside.
Time for the observer outside flows faster.

Now, when light is shot vertically toward the ceiling but at an angle to the left, say arctan(0.5), and the mirror on the ceiling is moved accordingly to the left to reflect the light correctly, the light travels sqrt(5)L for the observer inside the train but 2L for the observer outside for the roundtrip.
Since c is the same for both, it takes sqrt(5)L/c for the observer inside and 2L/c for the observer outside.
Time for the observer inside flows faster

What am I missing here?

 

[Ibix]

What am I missing here?

The relativity of simultaneity, as is almost always the case with SR "paradoxes".

Not only are clock rates different as measured using a frame where the clock is moving, the zeroing of clocks depends on their position. If I understand your setup correctly, the light pulses don't start and end their journey at the same place according to the train. According to the track frame, then, clocks in the train co-located with the start and end points of the light's flight path don't show the same time, and the difference between the clocks' zeroing will account for the discrepancy you see when you try to analyse only using time dilation.

Depending on your setup (I haven't worked through the details of the angles) you may find that the light pulses start and end their journey in the same place in the track frame. That would be a demonstration of the reciprocity of time dilation - both frames measure the other's clocks ticking slowly, and observers in both frames need to understand the relativity of simultaneity to understand how that is not paradoxical.

Generally, look up the Lorentz transforms and use them to transform coordinates between frames. My advice is to forget time dilation and length contraction, which are special cases of the Lorentz transforms, until you are familiar enough with the general case to know when you can use the special cases.

 

[Dale]

My advice is to forget time dilation and length contraction, which are special cases of the Lorentz transforms

I agree with this advice. Start with the Lorentz transforms. Time dilation and length contraction are too easy to misuse.

 

[roomie10]

Thank you all!

 

[Herman Trivilino]

Time for the observer outside flows faster.

No, that is an erroneous conclusion. The outside observer finds that clocks on the moving train run slower. But if the observer on the train were to perform the analogous experiment, he would find that clocks at rest relative to the outside observer run slower.

Paradoxically, each find that the other's clocks run slow. The Principle of Relativity requires that time dilation be symmetrical.

What am I missing here?

In the first scenario the light beam arrives back at its original location. The thought experiment is specifically engineered to contain this feature.

Sorting all this out requires an understanding of the relativity of simultaneity.

 

[robphy]

SR explains how light travels longer for an observer outside the train when it's shot vertically upwards.
...
Now, when light is shot vertically toward the ceiling but at an angle to the left, say arctan(0.5), and the mirror on the ceiling is moved accordingly to the left to reflect the light correctly, ...

The geometry involved in your scenario is encoded somewhere in this spacetime diagram

from my interactive visualization of circular light clocks: https://www.geogebra.org/m/pr63mk3j .

Some analysis is needed to identify the appropriate construction and set up the calculation.