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In Weinberg's 1964 paper "Photons and Gravitons in S-Matrix Theory: Derivation of Charge Conservation and Equality of Gravitational and Inertial Mass" where he proved the leading soft photon and graviton theorems, he discussed several implications of these theorems.

For example, the soft theorems together with on-shell gauge invariance (Lorentz invariance at its core) imply that

  • electric charges (defined as the low-energy coupling of matter to photons) are conserved; and
  • gravitons couple to matter in a universal way.

Moreover, by considering the soft graviton exchange of particles, he was able to show that the gravitational mass $\tilde m$ equals the inertial mass $m$ for massive particles, if the universal coupling is chosen to be unity.

However, for massless particles, the soft theorem implies $\tilde m=2E$, twice the energy of the massless particle. I wonder if there is a classical setting (e.g., gravitational light-bending vs. scattering of non-relativistic particles) where one can intuitively understand this relation.

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    $\begingroup$ Light does, in fact, bend twice as much as would be predicted by Newtonian gravity. $\endgroup$
    – Brian Bi
    Commented Mar 9, 2023 at 13:20
  • $\begingroup$ Linked. $\endgroup$
    – Cosmas Zachos
    Commented Mar 9, 2023 at 15:49
  • $\begingroup$ @BrianBi Yes, it does seem like a reasonable explanation. I am aware of the results quoted by user203234, however, before, I was not sure whether it is sensible to treat photons simply as particles moving at light speed, with kinetic energy $E=mc^2/2$. But I think user203234's answer cleared my doubts; one should consider photons as having an infinitesimal mass in Newtonian relativity and gravity, because in Newton's mind, there is no such thing as a massless particle. So his and Einstein's predictions indeed differ by a factor of two. $\endgroup$
    – chaostang
    Commented Mar 10, 2023 at 13:55

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