Answered 27 February 2020 and subsequently modified from time to time.

You might be mistaken about energy. According to the complete statement of Einstein’s most well known equation, energy content is a combination of a particle’s mass and its momentum. The equation you cite is abbreviated. It is a simplified version that is missing a term.

[math]E^{2}= \left ( pc \right )^{2}+\left ( mc^{2} \right )^{2}[/math] where m is rest mass.

Massless particles like photons have momentum that is correlated to their wavelengths (or frequencies). It’s their frequencies that give massless particles like photons their energy content. So without (rest) mass the equation becomes:

[math]E=pc[/math] where [math]p=\frac{hf}{c}[/math] for massless photons.

Of course, in classical physics ρ = mc. The mass term is critical.

On the other hand, in quantum mechanics the mass of a photon cannot be zero either—but photon mass in relativity theory is handled in an unusual way that has yet to be resolved by consensus as far as I know.

The Abraham-Minkowski controversy seems to touch the argument. The permittivity of “empty“ space qualifies as a dielectric, does it not? Is the argument related to a possible ambiguity concerning photon mass? Maybe an expert will comment to shed some light; I am out of my depth to go further.

Several physicists have claimed that the controversy is resolved by postulating an interaction inside dielectrics (like glass) of photons with electron-generated polaritons. But isn’t space itself—with its Maxwell assigned permeability and permittivity constants—a dielectric? Empty space isn’t necessarily empty, right?

Arthur Eddington wrote in chapter 6 of his book Space Time and Gravitation that the dielectrics of space around the Sun increase proportionally with the intensity of the gravitational field. Light waves closest to the sun slow down more, which pulls the wavefront that lies farther out to deflect still more to catch up. Like glass, gravity refracts light.

Light falls into the Sun like any solid rock, but refraction adds to light’s “Newtonian” deflection to give Einstein’s predicted result. Unlike slow rocks, light travels fast enough to avoid capture by the sun.

Eddington seems convincing, but he wrote his book about 100 years ago. In 1973 Thomas Pynchon published Gravity’s Rainbow. The idea of light refracting due to decreasing speed along gravitational gradients persists, even in literature.

It’s not clear to me how many physicists agree with Eddington, but then again, it’s not obvious whether humanoids are able to visualize reality. It’s one thing to write equations and symbolic algorithms that match well with observations. It’s quite another to acquire a natural intuition of what might be true.

All I think I know for sure is that the momentum and mass of quantum things have no meaning until brought into existence by measurements; in the math they look like nothing we know; sometimes they are used as operators that don't commute the way some might think they should.

I reviewed the math; I saw the term that makes the difference (it's really there) but did not understand enough at the time to tease out the reason why photons bend nearly twice more in a gravitational field than some acolytes of Newton conjectured.

According to Wikipedia, Einstein’s theory approximates the deflection to be [math]\frac{4GM}{(c^2)b}[/math]. b is the distance of a photon’s closest approach to a gravitational object like our Sun.

Here are some guesses:

Deeply-buried light in a gravity field near a star like the Sun experiences the flow of time more slowly—it’s an effect that is common to all objects in a gravity field; it affects every object in the same way and is unaffected by their mass or lack of it. It might have something to do with Schwarzchild geodesics. The geodesics of spacetime paths are longer and more curved in a gravity field than what anyone might expect from a simple application of Newton’s force law.

Schwarzchild metrics help explain “gravitational lensing” of faraway objects when their light approaches Earth from behind massive structures in the far reaches of space.

Here is another guess:

It might be that light spends more time in a gravitational field than it should due to special relativity induced time dilations so photons have more time to fall toward the star than they otherwise would. This guess is prolly wrong because the time differential would be governed by a Lorentz transformation.

Photons of light don’t undergo Lorentz transformations because, unlike massive objects that travel near the speed of light, they don’t have inertial frames of reference. Any line of reasoning that ties Lorentz transformations to photons leads folks into rabbit holes that contradict the current consensus about the nature of light.

Muons (which have rest masses 205 times that of electrons) are short-lived, but their relativistic speeds increase their lifetimes so that some of those that get their start in the upper atmosphere are able to reach the ground where they can be observed. Their increased life is described by a Lorentz transformation. It’s tempting to apply this principle to photons, but theorists say, no. It doesn’t work that way.

Time contractions and dilations are Special Relativity effects that apply to objects with masses that move at velocities close to the speed of light, yes, but never at the speed of light, right?

Nearly every expert will say that photons have no rest mass and that they travel at exactly the speed of light in every reference frame. Photons are never associated with inertial reference frames in the same way as fermions like muons and electrons.

Change in time and distance caused by a photon’s position in a gravity field is completely different; it’s described by the more complicated theory of Einstein, called General Relativity.

What makes General Relativity unique is it’s view that gravity and acceleration are equivalent. Acceleration is a change in the velocity and/or the direction of motion. Mass curves and elongates pathways that make the space around stars. Photons traveling on these longer spacetime paths are accelerated by their change in direction, but their velocity doesn’t change. Something has to give. What gives, what changes is the expected value of deflection. The light from distant stars bends more than it should.

No one who lived before 1900 could have known that the geodesics of space-time elongate (or curve) in the presence of mass and energy, which are equivalent, correct? No one in bygone eras could have known that time slows down for massive objects that are able to approach light-speed, either.

A man named Joann Georg Soldner did a calculation to show how much a Newtonian “corpuscle” of light would bend in the Sun’s gravity, which he published in 1804. He assumed that photons had mass and fell toward the Sun like any other object.

When Arthur Eddington’s observations showed that starlight deflected more than Soldner had calculated, Einstein’s theories of relativity got a boost in credibility that lives on into modern times.

I should add that Eddington knew of Einstein’s predictions when he made his experimental observations in 1919 because Einstein published his general theory

I would very much like to read a coherent, verbal (non-mathematical) explanation of exactly why and how Einstein’s general theory can lead to an accurate and reasonable prediction at odds with Newton about the angle of deflection of photons near a star.

Here is an explanation I heard from a working theorist: Newton’s view was used by Soldner to calculate deflection using only the time the photon spent in the gravitational field. Einstein did the same but then modified his calculation to account for the bending of space in the gravitational field. The space component nearly doubled the expected deflection. The theorist’s explanation satisfied me. It sounded right.

On the other hand, I believe (secretly and in agreement with Newton’s acolytes) that photons must have mass equivalence, which for some reason is overlooked, but no one I’ve read believes the idea makes sense beneath the shadow of relativity theory, which carries a reputation for being fundamental, flawless, and complete. Anyway, the mass of any object in a gravitational field is irrelevant to its trajectory because the math cancels it, right?

[math]F=ma=\frac{GMm}{r^2}[/math]

Little “m” appears on both sides of the equation so it can be divided away.

Nevertheless, mass-energy equivalence of photons might permit Lorentz transforms on light to help resolve certain problems in cosmology and the transmission of light through medias where gravity is not a factor. It might simplify understanding of annoying Shapiro effects, which slow down communications with explorer craft inside our solar system.

Since I haven’t yet found a good explanation—and with a promise to avoid nonsensical personal predispositions—here is my attempt to explain:

In GPS systems, dilations of time—in both the velocity of satellites in one frame and their acceleration in another frame (gravity)—must be added to provide accurate information to vehicles located in another frame.

This information can work at cross-purposes. It requires expensive infrastructure on the ground to coordinate the information so that drivers of vehicles don’t get lost.

A massless object moving at the speed of light is going to follow the geodesics of the gravity field. This field is a distortion of space and time induced by the presence of the mass of something like the Sun.

If massless energy does not obey the laws of special relativity (like GPS satellites do), then its velocity must necessarily have no influence whatever in the deflection of light near a star. It might seem like all the deflection comes from the distortion of spacetime, which is gravity.

Photons ride gravity geodesics like cars on a roller coaster. According to appendix III in Einstein's 3rd edition of his book Relativity, the Special and General Theory, published in English by Henry Holt & Company in 1921, it's only half the story. The other half of the measured deflection comes from the Newtonian gravitational field, which accelerates all objects in the same way. This field further deflects light across the spacetime geodesics toward the sun to double the expected angle.

I'm not entirely convinced that modern physicists believe it's quite that simple.

The theory of general relativity helps theorists to describe the distortion of metrics in spacetime near massive bodies to predict the deflection angle of passing photons of light. What we know is that predictions based on the theory never fail.

It’s like the theory of quantum mechanics. It never fails. It’s foundational. No one can explain why.

Somebody, please, tell me I’m wrong. Here is a link that addresses the math concerning the deflection disparity between Newton and Einstein.

Thanks to Tommy White and Harry McLaughlin who provided insights during the writing of this essay. I’m not sure either one would endorse it. All mistakes and foolish blunders are mine alone.