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## Do photons have mass?

Do photons have mass?
The quick answer: NO.

However, this is where it gets a bit confusing for most people. This is because in physics, there are several ways to define and measure a quantity that we call "mass". Now, it doesn't create any confusion among physicists because we tend to know in what context such a quantity is defined. However, for the general public trying to decipher scientific papers and presentation, this is a trap that many fall into and can be the source of many confusion.

In physics, the most important definition of a bare mass (we are not going to deal with effective mass that is a part of solid state/condensed matter physics) is what is known as the invariant mass. Invariant mass ($m_0$) (aka rest mass, proper mass or intrinsic mass) is independent of reference frame. In other words, an object’s invariant mass has the same value no matter who is observing the object and no matter what their velocity is relative to the object. The invariant mass of a particle is defined as the total energy of the particle measured in the particle’s rest frame divided by the speed of light squared. More generally, the invariant mass is defined as

$$m_0 = \sqrt{\frac{E^2}{c^4} - \frac{p^2}{c^2}}$$

Now for a photon, this is zero since $E = pc$. In many aspect, this is all that we need to know. In physics, something that is invariant after some operation is very desirable.

But photons have energy. By $E = mc^2$, doesn't this mean that they have A mass?

The equation above was derived from this expression:

$$E^2 = (pc)^2 + (m_0 c^2)^2$$

A photon can still have zero invariant mass ($m_0$), and can still have energy. There's nothing inconsistent here. All of the photon's energy is in the term $pc$. Some people would say that this is the photon's "inertial mass", since it is similar to the inertia that one feels when trying to stop a moving mass. This may or may not be useful to consider. However, it certainly should not be confused with the concept of the ordinary mass that most people are familiar with.

There are, of course, other forms of mass. Most commonly used terminology is something called "relativistic mass". This mass is defined as

$$m = \gamma m_0$$

where

$$\gamma = \frac{1}{\sqrt{1-\beta^2}}$$

$$\beta = \frac{v}{c}$$

This relativistic mass is what most people attribute to the "gain in mass" of particle moving at relativistic speeds. However, one needs to be aware that in professional circles, such concept is very seldom used. One very seldom hears this when one attends a high energy physics seminar, for example, or read a particle collider experiment paper. This is because in citing a relativistic mass, one must also cite the speed of the particle with respect to what reference frame. This is cumbersome and unnecessary especially when the invariant mass would have been clearer (that's why we love invariant anything).