Doppler effect changes the intensity of the blackbody radiation as well (see 'Relativistic Doppler Effect' on wikipedia and further reference). This effect will be rather small for the plate transverse to the direction of observer's motion and very large for the plate parallel to the motion.

No paradox. It's just that the statements about the sizes of the plates don't make sense, that's all, since the sizes aren't invariant under Lorentz transformations. (Not to mention the fact that temperature defines an energy, that does transform under Lorentz transformations.)

If different observers measure the same values for Lorentz invariant quantities, they're equivalent; if they don't, they aren't. That's how they can be told apart.

If they measure different (or the same) values for quantities that aren't invariant under Lorentz transformations, this doesn't mean anything.

Stam Nicolis Hi Stam, thanks for your input. However, temperature is considered a Lorentz scalar in modern literature, so it is in fact invariant under Lorentz transformations. It is the identical time-dependent temperature readings that, I claim, create a 'peculiar' or even 'paradoxical' situation here. Let me stress the peculiarity by going into more detail about the geometry of the plates: for dramatic effect lets say that the thickness, a, is much smaller than the sides, L; this means we are dealing essentially with 'square sheets' of side L. When v-->c we have that the total effective surface area of sheet A approaches (LxL)+(LxL), which is barely different from what it is in the rest frame, while the effective area of sheet B becomes infinitesimal (!!).......So, for the observer in the rest frame S there is nothing strange going on: he heated two identical objects to the same temperature and sees them cool at exactly the same rate in an isotropic & homogeneous medium. The observer in S' must necessarily have to perceive some very extreme physical processes in order to account for the identical time-dependent temperature readings between an object of infinitesimal surface area and an object with essentially the same surface area as in its rest frame!

Once more: The statement that temperature is a Lorentz scalar means that only those Lorentz transformations make sense that leave it invariant-not the full group. And any other quantity that matters must be invariant under this subgroup, too.

So observers that measure the same value for the invariants are equivalent and those that measure different values for the invariants aren't equivalent. No problem.

Quantities that aren't invariant simply can't be compared, that's all-they refer to different physical systems, not the same system.

The size of the plates, on the other hand, isn't invariant under Lorentz transformations, that's why it doesn't make sense using it to make any statement that can be compared by two observers if they're assumed to be equivalent; and it doesn't make sense discussing it, if they're not equivalent.

Stam Nicolis "The statement that temperature is a Lorentz scalar means that only those Lorentz transformations make sense that leave it invariant-not the full group."

That looks like a very strange statement to me. The statement that temperature is a Lorentz scalar would mean, IMHO, that it must be invariant under Lorentz transformations, i.e. the full Lorentz group. That is the meaning of "scalar" in the combination of the two words.

On the other hand, temperature clearly is not a Lorentz scalar. The temperature of the cosmic microwave background has a dipole anisotropy due the motion of Earth in that background. i.e. the temperature of the radiation is observer dependent, hence it cannot be a Lorentz scalar. In fact, it should behave like the time component of a four vector under Lorentz transformations, since a Planck spectrum remains Planckian under these, but with varying temperature.

I guess the solution to the evoked "paradox" lies somewhere in the direction of Lerner's answer. The thermal radiation cooling the plates will change drastically in the forward and backward directions due to the motion (let's suppose they are cooled by radiation cooling only).The sideways radiation will also change due to the transverse Doppler effect. The directional characteristics of the radiation will change, it will not be isotropic in the moving frame and its temperature will be different in different directions of observation. A moving observer may not even assign a single temperature to each plate. Whether there is some sense in which he will "see" the temperatures of the two plates to be the same, seems doubtful. Well, I guess there is one sense: He will see the same kind of thermal radiation whether the two plates are in contact with each other or whether they are not. So he may conclude that they were in some kind of equilibrium, thermal or other, if he does both experiments.

Invoking the Doppler effect amounts to defining a preferred frame, therefore of breaking Lorentz invariance.

Indeed, temperature, also, implies a preferred frame, that one, where the bath, with which the object under study is in equilibrium.