Special Relativity quick problem

[binbagsss]

The question is : Let s be the speed of light through water. If water is flowing at speed v in a given frame, find the speed of light in that frame, were the light propagates in the same direction as the water?

My question:

I am having a sign issue.
I can not see how the following is flawed, derived from the Lorentz boosts:

x' = γ(x-Vt)
t'=γ(t-Vx/c^2)

Where I define frame O' to be that of the water moving at v, and O to be the stationary water.

Then V=v.
x is the position vector in frame O. x = st.
=> dx'/dt'= $\frac{s-v}{1-\frac{sv}{c^{2}}}$

Which is not the correct answer : $\frac{s+v}{1+\frac{sv}{c^{2}}}$Many thanks to anyone who can help shed some light on this.

 

[TSny]

The question is : Let s be the speed of light through water. If water is flowing at speed v in a given frame, find the speed of light in that frame, ...

Which frame is being referred to here? O or O'?

Edit: OK. I see now. It seemed more natural to me to take frame O to be the Earth frame and O' to be the frame of the water. But you are using the opposite notation.

In which direction is the water (your O frame) moving relative to the Earth (your O' frame)? +x' or -x'?

 

[binbagsss]

Oh. I didnt include the Earth's frame initally. Instead I just went with two water frames - one stationary and the other moving at v wrt the stationary water.

If I include a water frame and a Earth frame, however, O' to be the frame of the water is definitely also the natural choice for me !

Working with this, I would follow the same argument as my first post. That V=+v (whilst I note that in the question it only specifies v as a speed). And that x=st.

 

[TSny]

OK. O is the Earth frame and O' is the water frame. Light is traveling through the water while the water is moving at speed v relative to the earth. From the perspective of the O frame, both the water and the light are traveling in the positive x direction.

Would you write x = st, or would you write x' = st' ?

 

[binbagsss]

I am still getting x=st.
x' is the position vector in frame O' = velocity of light in O' x t' , but velocity of light in O' is unknown.
Whereas we know that in frame O the speed of the light is s, so x=st.

 

[TSny]

So, we need to interpret the phrase "Let s be the speed of light through water". To me, that means the speed of light relative to the water. Since O' is the frame moving with the water, s would be the speed of light in the O' frame.

 

[binbagsss]

I interpret it as s is the speed of light with respect to the stationary water, and that the unknown speed is the speed of light wrt water moving at speed v.

 

[TSny]

I interpret it as s is the speed of light with respect to the stationary water, and that the unknown speed is the speed of light wrt water moving at speed v.

I agree that s is the speed of light wrt the stationary water. So, s is the speed of the light wrt to frame O'. (The water is at rest relative to frame O').

The unknown speed is the speed of the light wrt to Earth frame (the O frame). That's my interpretation anyway. And it leads to the same answer that you said was given for the problem.