The “relativistic mass” business is a misguided attempt to combine rest mass and kinetic energy together, but it really is a bad idea, since rest mass is an intrinsic property of an object, whereas kinetic energy depends on who is looking (an observer moving along with the object will see zero kinetic energy.)
Sometimes people like to say that the photon does have mass because a photon has energy E = hf where h is Planck's constant and f is the frequency of the photon. Energy, they say, is equivalent to mass according to Einstein's famous formula E = mc2. They also say that a photon has momentum, and momentum p is related to mass m by p = mv. What they are talking about is "relativistic mass", an old concept that can cause confusion (see the FAQ article Does mass change with speed?). Relativistic mass is a measure of the energy E of a particle, which changes with velocity. By convention, relativistic mass is not usually called the mass of a particle in contemporary physics so, at least semantically, it is wrong to say the photon has mass in this way. But you can say that the photon has relativistic mass if you really want to. In modern terminology the mass of an object is its invariant mass, which is zero for a photon.
https://www.desy.de/user/projects/Physics/Relativity/SR/light_mass.html
Thus, the mass of a box of light is more than the mass of the box and the sum of the masses of the photons (the latter being zero). Relativistic mass is equivalent to energy, which is why relativistic mass is not a commonly used term nowadays. In the modern view "mass" is not equivalent to energy; mass is just that part of the energy of a body which is not kinetic energy. Mass is independent of velocity whereas energy is not.
Another answer is that the light has energy and momentum which couples to gravity. The energy-momentum 4-vector of a particle, rather than its mass, is the gravitational analogue of electric charge.
force alters the momentum of the photon and the energy of the photon, but not the speed of the photon. See, force is not equal to , that's a derived relationship for particles that have mass. The fundamental definition of force that applies for all particles is
in words, force is the time rate of change of momentum.
As the OP mentioned, the energy of a photon E = hf. So as it goes into a gravitational well it increases its energy, and thus its frequent, it blue shifts. When it tries to get out it red shifts.
As a result, we claim that one of the basic axioms of General Theory of Relativity, the principle of equivalence, is incompatible with the existence of de Broglie’s wave‐lengths in Quantum Mechanics. So GTR and QM, as based on the wavelength postulate, are non‐unifiable. If we choose de Broglie’s phase harmony as fundamental, a new theory of gravity is needed.
https://www.sciencedirect.com/science/article/pii/S2211379719330943
some other experiment state that both properties do not co-exists simultaneously. According to electromagnetic theory, the rest mass of photon in free space is zero and also photon has non-zero rest mass, as well as wavelength-dependent. The very recent experiment revealed its non-zero value as . Even experimental results concluded that within matter (dispersive) the photon shows its imaginary rest mass. We have no exact answer as to why photon incarnates itself with versatile mass. Here we try to theoretically investigate about the rest mass of a photon. When it touches the surface of matter, it makes illusion and mathematically the rest mass is a complex number. Rest mass of photon depends upon scalar curvature of the surface of matter and wavelength of the photon. Photon itself reveals illusion posing with mass because of its dual nature. We have investigated the wave-particle duality of light, coexistence of wave and particle nature through morphing due to pliable character of light wave. Our theoretical work about the photon’s illusive mass will have to be experimentally verified and it might open plausible new applications in the secure communication of information.
https://sci-hubtw.hkvisa.net/10.1063/1.2216625
