A Paradox in Special Relativity

Question

Two inertial frames K and k’ are considered. They are in relative uniform motion along the x-x’ direction with relative speed =v. In the frame K’ we have a cuboidal piece of dielectric [at rest wrt K’]with a flat face perpendicular to the x-x’ direction, that is, this particular face is parallel to the y-z direction.The dielectric is homogeneous and isotropic within itself. We now consider Maxwell’s equations [in a medium] wrt the dielectric in the rest frame of the dielectric,ie,K’. If these equations are transformed, they should retain their form in K[according to the first postulate of SR]. But the individual values of the variables may change

With this information we proceed into the paradox.

Speed of light in the dielectric as observed from k’=nc [n is a positive fraction less than 1]

Relative speed between the frames, v=cn’[n’ is also a positive fraction less than 1]

For normal incidence: Speed of light in the dielectric as observed from K[From Velocity-Addition Rule of SR]: $v=\frac{nc+n’c}{1+nn’}$ ------------------- (1)

For oblique ray inside the medium at $\theta$ degrees degrees with respect to the x’ axis in the K’ frame:

$v’_x=ncCos (\theta)$

$v’_y=nc Sin(\theta)$

$v’_z=0$

[$v'_z$ has been taken to be zero for the convenience of calculations]

Observations from K:

$v_x=\frac{nc Cos(\theta)+n’c}{1+nn’Cos (\theta)}$

$v_y=\frac{nc Sin(\theta)}{1+nn’Cos(\theta)}\sqrt{1-n’^2}$

$v_z=0$ Therefore,

$v=\sqrt{[\frac{nc Cos(\theta)+n’c}{1+nn’Cos (\theta)}]^2+[\frac{nc Sin(\theta)} {1+nn’Cos(\theta)}\sqrt{1-n’^2}]^2}$ ------------------- (2)

The results from (1) and (2) are not identical , though from the invariant Maxwell’s equations[in a medium] we understand that the speed of light should be the same in all directions inside the dielectric as observed from K. What would be your answer to this paradox. [My assessment:This paradoxical situation arises from the fact that we have applied SR in an incorrect context.It has been applied in an anisotropic and inhomogeneous configuration. You could of course have a different assessment] [The dielectric within itself is homogeneous and isotropic. But the overall space being considered is not homogeneous and isotropic]

If Maxwell's equations change their form wrt the Lorentz transformations,Gauss Law,Div B=0 etc may change if a piece of dielectric is loaded into a moving train!

Answer 1

The angle $\theta$ is used in both reference frames. As I recall from discussions of such SR paradoxi, the angle can change when you switch reference frames. Try to derive the values without explicit use of $\theta$.

Although, looking at the calculation, everything seems correct, with the angle used only in K'.

Answer 2

There's nothing wrong with the application of SR in this context. SR is where reference frames move at constant velocities, regardless to anisotropy or whatever.

I believe there's nothing wrong with the fact that the speed of light in a dielectric in motion is not the same in all the directions. In reference frame K the "cuboidal piece of dielectric" is shortened in X direction. Hence - it's not actually isotropic. One can say that electric susceptibility of such a moving dielectric is a tensor.