Very interesting discussion, e2m2a.
Concerning your original question:...
Does the magnitude of the Poynting vector depend on the frequency of the wave hitting the surface?
and your follow up question...
e2m2a said:
... But I have also learned, according to Einstein's special theory, that the intensity of the light will be affected too. That is, the intensity of the blue-shifted light will increase and the intensity of the red-shifted light will decrease. Since the Poynting vector is an intensity entity, joules per second per meter squared, then the magnitude of the Poynting vector relative to this moving observer will be greater on the blue-shifted side and less on the red-shifted side. So, it appears that frequency and the Poynting vector(intensity) are linked, at least in this relativistic sense. Am I misunderstanding something here?
Of course, you are missing something and that is the experimental verification.
The Nichols and Hull Experiment verified the Maxwell (Poynting) formula quantitatively. It is not commonly realized, however, that the experiment was done in three differeent wavelengths and the conclusion was that
"the radiation pressure depends only upon the intensity of the radiation and
IS INDEPENDENT OF THE WAVE-LENGTH". (Astrophysics Journal, Vol.17,pp. 351).
The thing you apparently have forgetten in your analysis is that (in wave mechanics) "intensity" is NOT the Poyting Vector...it is the
TIME AVERAGED Poynting Vector, <S> , and, as "Born2bwire' mentioned, the TIME AVERAGED Poynting vector formula removes the frequency dependence...meaning that the momentum delivered is frequency independent.
Written in terms of the electric fiield only, the total Time Averaged Poynting Vector (Intensity) is simply...
<S> = (ec/2)(E^2)... (e is the vacuum permittivity...and E is the MAXIMUM E field)
Thus the radiation pressure,
P = <S>/c , is not predicted by Maxwell (Poynting) to be frequency dependent, (at least not in the same inertial frame), and that is exactly what was empirically verified by Nichols and Hull.
Now, not knowing the formula you are using for a relativistic "intensity", a comparison seems somewhat ambiguous.
Having said all that, I can imagine how, in
different (boosted) frames, the intensity (even based on time averaging)
may change with doppler...(based on the more general Poynting Theorem involving time rate of change of energy density).
Creator