Brian,
Yes, I agree that physics classes should have different weight.
wrt your question about axiom II and MWI, in a word, no.
Thank you for raising this issue. I think that conservation of momentum presents the toughest challenge to my idea.
Newton's third law generally does not hold in classical Electrodynamics. If you consider, for instance, two charges q1 and q2 approaching a common point, the magnetic force of q1 on q2 is generally not equal and opposite to that of q2 on q1. But momentum is still conserved because we consider the fields themselves to carry and "store" momentum.
I would respond to your question then as follows: Yes, I agree that if the gravitational field of photons is zero then it is true that N3 is violated. But I would like to raise the possibility that overall momentum may still be conserved if one takes into account the overall consequences of the interaction.
Let me give a concrete scenario: Suppose a photon comes in from a distant star, and is deflected by the sun's gravitational field in such a way that it is absorbed by an electron on the surface of the earth (call it m_e), whereas in the absence of the gravitational field it would have been absorbed by an electron on the surface of mars (call it m_m) (say). Here the classical language of trajectories is once again shorthand for saying that the gravitational field changed the wave function of the photon so that it collapsed at the location of m_e instead of m_m.
As a result of the absorption, the electron goes in an excited state, which will be marked by *. So we assume we have with the gravitational field the configuration (m_e*, m_m) and without the field the configuration (m_e, m_m*). The binding energy in the excited state is higher, and since it is associated with an entity that ages, it also increases the gravitational field of the electron.
Of course, these are ridiculously small effects, but the point is that the action of the Sun's gravitational field on the photon, while by my idea not being met with an equal and opposite reaction, nevertheless resulted in a slight reconfiguration of gravitational fields in its vicinity.
Let me be the first to point out that this is a rather unsatisfactory argument.
The main problem I see is, how do we know that the reconfiguration occurs in exactly such a way as to conserve momentum overall? It seems to me highly non-trivial to prove that it does or does not. Also, what if the photon is not absorbed by anything within the solar system but just keeps on traveling? Even if the eventual reconfiguration does conserve momentum, we would have to wait until the photon is absorbed by something (unless we consider the alteration in the photon's "direction" (classically) or wave function as a result of the gravitational field itself as a sort of "momentum storage").
So I don't have a good answer. I can neither show that my idea does result in momentum conservation nor does it seem to me that it can be clearly shown that it violates it. In my view the best way to settle this is to just attempt to measure the gravitational field of electromagnetic radiation, but of course that is not easy either.
There is another major challenge that one could raise against my idea, and fortunately (for me) I have a much better response to that. One may ask whether my idea does not violate the principle of equivalence.
The argument, as I understand it, goes as follows:
From SR we know that inertial mass is equivalent to Energy. The principle of equivalence says that inertial mass is equal to gravitational mass. Therefore, Energy is equivalent to gravitational mass and photons should produce gravitational fields (i.e. spacetime curvature).
Now I confess that I have never seen this argument stated exactly in this way, because every GR text I have consulted simply treated the energy density of massless particles and that of massive particles as it pertains to their inclusion in the Energy momentum tensor exactly the same without ever explicitly providing a justification for doing so (as if to imply that this should be obvious).
So because of that I am not 100% certain that the argument above is the actual relativistic argument for considering electromagnetic radiation to produce gravitational fields. If it is something else, I would be most grateful to anyone who would point it out to me.
But, given my caveat, let us suppose that it really is the reason why one may believe that because of the principle of equivalence, photons should produce
gravitational fields.
This argument contains a subtle but elementary logical error, and to highlight it, let me express the argument in the form of a syllogism:
premise 1: m_i c^2=E_i for all m_i
premise 2: m_i=m_g for all m_i and all m_g
Conclusion: m_g c^2=E_g
m_i is inertial mass
c^2 is the speed of light squared
E_i is the energy equivalent to inertial mass
m_g is gravitational mass
E_g is the energy equivalent to gravitational mass (in the sense of producing spacetime curvature)
the conclusion does not follow because it makes an extra assumption not contained in the premises, namely that E_i=E_g.
If photons do not produce gravitational fields, then the logically correct form of this syllogism is
premise 1: m_i c^2=E_i for all m_i
premise 2: m_i=m_g for all m_i and all m_g
Conclusion: m_g c^2=E_i
If photons do produce gravitational fields, then the logically correct form of this syllogism is
premise 1: m_i c^2=E for all E_i and E_g
premise 2: m_i=m_g for all m_i and all m_g
Conclusion: m_g c^2=E
The bottom line is that the equivalence principle in the standard form m_i=m_g has nothing to say about whether photons produce gravitational fields or not. If one wishes to extend the empirically established domain of the equivalence principle to E_i=E_g, then one must do the experiment.
Sorry for the length, any feedback is appreciated.
Armin