Hypothetical: Can I determinate if my room is moving by measuring the time it takes for a photon to reach the oposite wall? Problem
I am locked in a room with no windows and I need to tell if the room is moving, I only have a lamp.
According to my professor I cannot tell if the room is moving because of the Special Relativity Theory.
My thought
What if the room moves to a speed very close to the speed of the light and I can see that the beam of light takes more time to reach the wall in the direction the room is moving. The speed of light is always the same no matter the initial frame of reference.
Can I tell the room is moving?
I guess my idea does not work because Galileo Theory of Relativity, but I still don't understand
 A: According to the special theory of relativity, the speed of light in your inertial frame of reference will be the same as in any other inertial frame, so the time it takes for the light pulse to reach the wall on the other side is just $d/c,$ where $d$ is the distance and $c$ is the speed of light.
For a person that you considers being in rest (staying on Earth?), the pulse will travel not only across your room, but also in the direction of your movement. The time it takes according to him is $(d/c)/\sqrt{1-v^2/c^2}$. During this time the pulse travels distance $d$ across the room, and distance $v(d/c)/\sqrt{1-v^2/c^2}$ in the direction of you movement. According to him your clock goes too slow (time dilation).
A: You assume that the room is moving close to the speed of light with respect to some absolute frame of reference (this would be the Galilean frame), and thus that the light would appear to you to move slower in one direction than the other.
This is the assumption that was proved false by the Michelson-Morley experiment.  That experiment was supposed to dot one of the last 'i's of physics, so that science could close the book on it before the close of the 19th century, and move on to more interesting things.  Instead, it absolutely broke the physics of the time.  It utterly disproved the existence of a Galilean frame of reference, as well as the luminiferous aether.
To you, light always travels at the same speed.  You are always at rest relative to yourself -- it's only relative to other things in the universe that you're moving.
So the whole entire point of Special Relativity is that if the speed of light is constant, then time and space must be variable.  So you you and your room are zipping by me at close to the speed of light, it looks to me like your room is foreshortened in the direction of our relative travel, and like your clock is slow -- and it looks to you like I am foreshortened, and that my clock is slow.
All the other stuff in Special Relativity is there to make the math work out, and make it consistent with physical measurement.
A: 
I am locked in a room with no windows and I need to tell if the room is moving ...

The actual problem is completely different than you think:
What exactly does "moving" mean?
According to special relativity theory, there is no "fixed" coordinate system in space, but you always have to specify some coordinate system relative to some object or similar when talking about motion.
Example:
You sit on your chair in your garden. How fast do you "move"?
If you are using a coordinate system which is relative to the earth, you are not "moving".
If you are using another coordinate system, you may be moving at the speed of about 1000 km/h (depending on where you live) due to the rotation of the earth around its own axis or even with 100000 km/h due to the rotation of the earth around the sun.
So the answer to your question is completely different than you think:

Can I determinate if my room is moving ...

If you use the coordinate system which is relative to your room, your room does not move. So you don't need to do any measurement to determine if your room moves: It does not move!
If you use a coordinate system which is relative to some other object (for example the earth), the question "Can I determinate if my room is moving?" is the same question as "Can I determine if the other object is moving relative to my room?"...
A: No, you cannot tell because the speed of light is the same for you, and in your own reference frame the room is not moving. Remember, there is no absolute motion, so for the observer inside room is  the platform that moves (which does not affect the measurements inside the room), not the room.
A: 
What if the room moves to a speed very close to the speed of the light and I can see that the beam of light takes more time to reach the wall in the direction the room is moving.

Well, how can you measure that?
You can start a timer and send a pulse of light towards the wall, and some of it will reflect off the wall and into a light sensor, which sends an electrical signal to stop the timer.
The photons will take a longer time to get to the wall, but they will also take a shorter time to get back to the sensor. Your measurement can only measure these things added together.
Also, the electronic timer ticks slower. All the electrons in the timer are going very fast in the direction of movement, and since they can't go faster than the speed of light, that means they're not going very fast sideways.
These kinds of effects cancel out so you get the same measurement you would if the room wasn't moving.
(everything in this answer is as measured by a "stationary" observer)
