Speed of Light in Special Relativity: Explained & Questioned

[Sameeran N Rao]

While trying to understand special relativity, i came across the following explanation several times:

Imagine a train moving at c/4 from left to right. If a person A, inside the train standing near the back of the train, flashes a light, the light is supposed to travel in the direction of the train's motion (to the right) with a speed of c with respect to person A. But in this case, a person B looking at it from outside should observe light to move at a speed of c+c/4=5c/4 with respect to his frame. This cannot happen since all observers should measure the speed of light to be c. Therefore, the time inside the train must be slower than the time of person B. This will ensure that if person A observes light to move at c towards the front of the train, then person B will observe time going slower in the train and will observe light moving at 3c/4 with respect to the train. So person B will observe light to move at 3c/4+c/4=c with respect to his frame of reference. Thus both of the observers will observe the same speed of light.

The explanation is fine but I started wondering about the following case:

If you take into account a person C standing at the front of the train and flashing a light towards the back of the train, he too must observe the light to travel at c. If this should be the case then person B should observe it as c-c/4=3c/4 with respect ot his frame. But since time has slowed down in the moving train, B will observe the light from person C's flashlight moving at 3c/4 with respect to train. Therefore, person B should observe light to move at 3c/4-c/4=c/2 with respect to his own frame. Now again, there is a change in speed of light observed.

The explanation mentioned above satisfies one case (light moving in direction of motion) but fails to explain how time dilation accounts for same speed of light in opposite direction. The same happens with length contraction also. How is this possible? Time dilation and length contraction are supposed to ensure that light moves at c in whichever direction meassured. Please help me out with this dilemma.

 

[PeroK]

While trying to understand special relativity, i came across the following explanation several times:

Imagine a train moving at c/4 from left to right. If a person A, inside the train standing near the back of the train, flashes a light, the light is supposed to travel in the direction of the train's motion (to the right) with a speed of c with respect to person A. But in this case, a person B looking at it from outside should observe light to move at a speed of c+c/4=5c/4 with respect to his frame. This cannot happen since all observers should measure the speed of light to be c. Therefore, the time inside the train must be slower than the time of person B. This will ensure that if person A observes light to move at c towards the front of the train, then person B will observe time going slower in the train and will observe light moving at 3c/4 with respect to the train. So person B will observe light to move at 3c/4+c/4=c with respect to his frame of reference. Thus both of the observers will observe the same speed of light.

The explanation is fine but I started wondering about the following case:

I'm not sure this is a very good explanation. For light in the same direction of motion, you have a) time dilation, b) length contraction and c) the relative motion to take into account. The full explanation of why they both measure the same speed of light needs to consider all these factors.

Normally, you need to consider transverse light motion first (as there is no transverse length contraction) to establish time dilation. Then, considering light in the direction of travel implies length contraction (using your already calculated time dilation).

If you take into account a person C standing at the front of the train and flashing a light towards the back of the train, he too must observe the light to travel at c. If this should be the case then person B should observe it as c-c/4=3c/4 with respect ot his frame. But since time has slowed down in the moving train, B will observe the light from person C's flashlight moving at 3c/4 with respect to train. Therefore, person B should observe light to move at 3c/4-c/4=c/2 with respect to his own frame. Now again, there is a change in speed of light observed.

The explanation mentioned above satisfies one case (light moving in direction of motion) but fails to explain how time dilation accounts for same speed of light in opposite direction. The same happens with length contraction also. How is this possible? Time dilation and length contraction are supposed to ensure that light moves at c in whichever direction meassured. Please help me out with this dilemma.

Again, you need to take all factors into account. Let's consider light traveling the length of the train:

B sees the light travel a shorter distance because of length contraction and also because the rear of the train is moving towards the light.

I don't think there is a neat explanation of why they both measure c. I think you have to work through the equations of motion (carefully) to check that there is no contradiction.