Past exam of the physics 1 paper of the University of Oxford / 2014 / 5

Use the inverse Lorentz transformation, with ${\gamma }_u=(1-u^2/c^2{)}^{-1/2}$:Taking differentials along the object's trajectory givesThis is the collinear relativistic velocity addition law. It concerns the velocity of one object measured in two frames. By contrast the rate of change of the distance between two objects in one laboratory frame can be as large asattained by oppositely directed light signals; for massive objects it is approached but not attained. This is not an object's speed in the other's rest frame and cannot transmit a signal faster than $c$.

Take the light to travel along positive x, while the water moves with signed velocity $-v$. In the water's rest frame its light phase velocity is $c/n$. The relativistic velocity addition law from a gives the laboratory phase velocityExpanding to first order in $v/c$, with $nv/c\ll 1$, yieldsEquivalently $w\simeq c/n-v(1-1/n^2)$: the counterflow reduces the forward light speed. This is the nondispersive Fresnel drag coefficient. The refractive index is taken as constant over the relevant frequencies; dispersion and its frequency-shift corrections are outside this approximation.