[hamdani yusuf (OP)]

Here is a picture of two trains moving at + and - 0.5c near a station. Train A moves to the right while train B moves to the left. The length of the trains as well as the station is 1 light second.

at t=0, the tail of train A coincides with head of train B and a lamp on the left side of the station, which starts to lit.
at t=0.67s, tail of train B is 0.67 light second away from the lamp, which coincides with the light front.
at t=1s, the light front arrives at right side of the station.
at t=2s, the head of train A is 2 light seconds away from the lamp, which coincides with the light front.

[Halc]

Here is a picture of two trains moving at + and - 0.5c near a station. Train A moves to the right while train B moves to the left. The length of the trains as well as the station is 1 light second.

You're drawing the two moving trains the same length as the station. So your trains, if the same length as the station when moving, are actually a proper length of 1.1547 ls, but length contracted to 1 ls in the station frame. This is not immediately relevant to your station-frame observations, but you need to know that if you're to make any observations from the PoV of either train.

at t=0, the tail of train A coincides with head of train B and a lamp on the left side of the station, which starts to lit.
at t=0.67s, tail of train B is 0.67 light second away from the lamp, which coincides with the light front.
at t=1s, the light front arrives at right side of the station.
at t=2s, the head of train A is 2 light seconds away from the lamp, which coincides with the light front.

All correct, and all expressed in the frame of the station, so only true in the frame of the station.

You've not asked any questions, so not sure where to go from here. You've identified no locations of any observers.

The train proper lengths are 1.15 ls each.  Each train, relative to the other, is moving at 0.8c and is length contracted to 0.6928 ls.
In either train frame, you can make t=0 coincide with the event of the light flash, in which case it arrives at the other end of the stationary train at t=1.1547s. In either train frame, the station length is 0.866 ls.

[hamdani yusuf (OP)]

The question is in the title. To state it more clearly, how to demonstrate that speed of light is still c when observed by train A as well as train B?

[hamdani yusuf (OP)]

Relativity doesn't demonstrate constant speed of light. It assumes it as one of its postulates. The assumption is made due to the empirical observation  that any measurement taken of light speed seems completely independent of the frame in which the experiment is performed.

Every once in a while we need to check and recheck whether or not our assumptions are justifiable and consistent between one another and compatible with observations.

[Halc]

Every once in a while we need to check and recheck whether or not our assumptions are justifiable and consistent between one another and compatible with observations.

You're not going to do that with a thought experiment. You go out and measure the speed of light using a better method than last time. Last we checked, it hadn't suddenly become frame dependent. This became evident back in the Michelson–Morley days, before Einstein, which is what prompted the need for the theory of relativity in the first place.

Anyway, if you have a specific inconsistency you see in your trains scenario, you need to point it out so your error can be identified. Nobody seemed to be measuring light speed in the scenario you described. I saw no errors in your descriptions of 4 points in time in the OP except for the lack of frame references, which I took to be the station frame.

[hamdani yusuf (OP)]

Here is a picture of two trains moving at + and - 0.5c near a station. Train A moves to the right while train B moves to the left. The length of the trains as well as the station is 1 light second.

at t=0, the tail of train A coincides with head of train B and a lamp on the left side of the station, which starts to lit.
at t=0.67s, tail of train B is 0.67 light second away from the lamp, which coincides with the light front.
at t=1s, the light front arrives at right side of the station.
at t=2s, the head of train A is 2 light seconds away from the lamp, which coincides with the light front.

Since the length of the trains are 1 light second in station's frame, their proper length should be 1.1547 light second.

Here is the spacetime diagram in station frame of reference.

How should it be transformed for the trains' frame of references?

[hamdani yusuf (OP)]

here is the same diagram but viewed in train A's frame

In this frame, velocity of the station is -0.5c. Velocity of train B is -0.8c as per relativistic velocity addition.
In this frame, length of train A is 1.1547 ls. Length of the station is 0.866 ls. Length of train A is 0.693 ls

[hamdani yusuf (OP)]

here is the same diagram but viewed in train B's frame.

The lengths and speeds are similar to train A's frame, but in reverse direction.
The light hit the tail of train B after 1.15 seconds.
The light hit the head of train A after 3.45 seconds.

[hamdani yusuf (OP)]

Here is a picture of two trains moving at + and - 0.5c near a station. Train A moves to the right while train B moves to the left. The length of the trains as well as the station is 1 light second.

at t=0, the tail of train A coincides with head of train B and a lamp on the left side of the station, which starts to lit.
at t=0.67s, tail of train B is 0.67 light second away from the lamp, which coincides with the light front.
at t=1s, the light front arrives at right side of the station.
at t=2s, the head of train A is 2 light seconds away from the lamp, which coincides with the light front.

Let's add additional lamps in each trains, they are placed on the tail of train A and head of train B. Those lamps start to lit at t=0. The second postulate of special relativity requires that the light of those lamps propagate at the same speed c. Here is how the sequence would look like from station observer.

[hamdani yusuf (OP)]

Let's add additional lamps in each trains, they are placed on the tail of train A and head of train B. Those lamps start to lit at t=0. The second postulate of special relativity requires that the light of those lamps propagate at the same speed c. Here is how the sequence would look like from station observer.

From the diagram, we can conclude that station observer measures the speed of light as constant if it is defined as change of distance between light front and the observer per unit time. But if speed of light is defined as change of distance between light front and the lightsource per unit time, it depends on the velocity of the light source relative to the observer.

[hamdani yusuf (OP)]

How do you determine a location in space without a physical object to refer to?

[hamdani yusuf (OP)]

How acceleration affect the speed of light? Let's say an observer is accelerating at 1 m/s² in the same direction as the light propagation. What is the measured speed of light according to the observer?

[Halc]

How acceleration affect the speed of light? Let's say an observer is accelerating at 1 m/s² in the same direction as the light propagation. What is the measured speed of light according to the observer?

The observer doesn't really observe the light after it leaves him, so hard to say.
He can note the time on some remote clock when the pulse gets to it, but lacking that clock being in sync with said observer, not sure what can be concluded from that.
I can say that if the light comes from below (in the same direction as acceleration) from about 10 light year distance, it will never reach the observer at 1 m/s² acceleration.

[Janus]

How acceleration affect the speed of light? Let's say an observer is accelerating at 1 m/s² in the same direction as the light propagation. What is the measured speed of light according to the observer?

It depends on whether he his measuring the local "proper" speed of light or the coordinate speed of light.
Locally, he will measure light moving at c.   The coordinate speed for light at points "ahead of him increases, the further ahead, the greater the coordinate speed.  "Behind" him, the coordinate speed decreases with distance.  At a far enough distance behind him you get a "Rindler horizon",  from beyond which, light can never reach him, as Halc alluded to.

[Bill S]

An observer is accelerating at 1 m/s², and a beam of light is travelling in the same direction at c. If the observer measures the speed of the light, would he measure the proper speed, or the coordinate speed?

[Bill S]

It's not like there's a device that you can just stick into a light beam and it tells you the speed.

I couldn't think of one, so I was fishing to see if see if there was something among the many things I don't know.

[hamdani yusuf (OP)]

Here is another problem:

A light source shine a laser beam to the right, λ=300 nm, c=3*10^8 m/s, f=c/λ =10^15 Hz.
Observer O doesn't move relative to the light source.
Observer A move to the left, observer B move to the right, with velocity vA and vB, respectively. Let's say +/- 3000 m/s
What is the frequency and wavelength of the laser beam as measured by each observer?

[alancalverd]

It's not like there's a device that you can just stick into a light beam and it tells you the speed.

The undergraduate laboratory syllabus used to include diffraction of microwaves of known frequency, then c = f λ, both being easily measurable. Nowadays you can use a LED to produce photons of known energy and measure λ with a diffraction grating made from a DVD in a school laboratory. Or use x-ray diffraction from crystals at higher energies. In each case you are not measuring speed at a point directly, but if c ≠ fλ  there is something wrong with our definition of speed!

[Bored chemist]

The local speed of light in a vacuum is C; if that's what you want to measure then a piece of cardboard with "C" (or 299792458 metres per second if you prefer) written on it is perfectly accurate.

Otherwise, the thing that measures the speed of light is a refractometer.

[alancalverd]

If the frequency and wavelength are 'known', what's the point of measuring them?

The original question is implicitly querying the constancy of c. We know the frequency and wavelength of the source because it is written on the source or we can measure them directly. The point is that we can carry out exactly the same measurements on the moving train and discover that whilst f and λ are different (thanks to Doppler) the product is the same. So there is indeed a device, or at least an assembly of devices, that can measure c at a point anywhere.

This is quite different from asserting that it is constant.