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# Bend It Like Newton

## Brian Koberlein

1 August 2013

Yesterday’s post on testing the assumption that photons are massless raised a few questions for readers. One of the most common was the idea that the gravitational lensing of light must mean that photons have mass. After all, if a star or galaxy can deflect light gravitationally, doesn’t that mean the light is gravitationally attracted to it? If that is the case, doesn’t that mean that light has mass?

Before we delve into the question, we first need to be clear about what we mean by “mass.” There are actually several different types of mass. The type that best corresponds to our intuitive understanding is known as inertial mass. Inertial mass is determined by its resistance to acceleration. If you push on objects with a force, an object with less inertial mass will accelerate more than one with more inertial mass.

Another type of mass is known as gravitational mass. Gravitational mass is what (in Newton’s gravity) causes the gravitational attraction between objects. When you step on a scale in the morning, you are measuring your gravitational mass. While technically gravitational mass and inertial mass are not the same thing, we generally treat them as the same thing because of the “principle of equivalence.”

If you release a ball from the leaning tower of Pisa, the gravitational force on the ball causes it to fall. The strength of that force depends on the gravitational mass of the ball, but the rate at which it falls depends on the inertial mass of the ball. But experiments have shown that masses all fall at the same rate in a gravitational field, so that means the gravitational and inertial masses must have the same value. This equality between gravitational and inertial mass is called the principle of equivalence. While this was known since at least Galileo’s time, it was Einstein who made the idea central to our understanding of gravity.

The third type of mass is known as relativistic mass. This stems from Einstein’s theory of special relativity and the equivalence of mass and energy (the famous E equals m c squared). In that famous equation, E is the energy of a particle, and c is the speed of light. So if you divide the energy of a particle by the speed of light squared, you get a “mass”, known as the relativistic mass of the particle.

Now if an object is at rest (relative to you) then the relativistic mass has the same value as the inertial mass. This is sometimes called the “rest mass” of an object. But in general, relativistic mass is *not* the same thing as inertial or gravitational mass. Unfortunately this point isn’t often made clear, so it leads to a great deal of confusion. When someone says “the mass of an object increases as it approaches the speed of light”, that’s really the relativistic mass. A fast moving object has not only energy due to its rest/inertial mass, but also a kinetic energy due to its motion. The relativistic mass due to its total energy is what increases. Its inertial (and gravitational) mass is unchanged.

This is the key difference. Relativistic mass is an apparent mass that depends on how the object is moving relative to you. Inertial and gravitational mass are inherent properties of an object, and don’t depend on your point of view.

So what does this have to do with whether photons have mass? Photons have energy, so we can define the relativistic mass of a photon by taking its energy and dividing by the speed of light squared. The energy of a photon depends upon its wavelength. Long wavelength (reddish light) photons have less energy than short wavelength (bluish light) photons. This means photons have different relativistic masses.

Photons don’t have “rest mass” or inertial mass. Despite popular news articles about “stopping light”, you can’t hold a photon in place. The “light stopping” experiments are effects of light waves, which is a whole other rabbit hole. You also can’t accelerate light with a force. The speed of a photon is constant, so again, no inertial mass. By the equivalence principle, that also means they have no gravitational mass.

At least that is the accepted answer. Maybe for photons, their relativistic mass *is* their inertial/gravitational mass. How do we know it’s not? Actually, we have an experiment that proves it, and Arthur Eddington first did it in 1919.

In 1919 Eddington photographed the positions of stars near the Sun during a total eclipse. He compared those positions to their positions when the Sun wasn’t there, and found that they had appeared to shift away from the sun. This is because the Sun gravitationally deflected the starlight slightly. This bending of light made the stars appear to be in a different direction. Einstein predicted this light bending due to the curvature of space in his theory of general relativity. Thus, Eddington proved that Einstein’s theory was correct.

When this story is presented, it’s often said that since photons have no mass Newton’s model predicts light shouldn’t bend. Einstein’s theory predicts light bending, so this proved Einstein right. But actually that isn’t entirely the case. If the relativistic mass of a photon is equated to its inertial and gravitational mass, then Newton’s gravity *does* predict light bending.

The catch is that the amount of bending predicted by Newton’s model is half what Einstein’s model predicted. Eddington actually demonstrated not only that light was gravitationally deflected, but that the amount matched Einstein, and not Newton. You can see this in the figure above, which shows three possible outcomes for light pending: Newton is right, and photons are massless (no deflection), Newton is right and photons have mass (some deflection), or Einstein is right, photons are massless and space is curved (more deflection).

So the gravitational lensing we see from stars and galaxies actually demonstrates that photons aren’t being gravitationally attracted in the Newtonian sense. Instead, space is warped by mass of stars and galaxies, and the path of light is warped accordingly. Light really is massless. You can’t bend it like Newton, but you can bend it like Einstein.