# Do Photons have Mass?

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Do photons have mass?

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 via the general relationship
$$E^2 = (m_0 c^2)^2 + (pc)^2$$
$$m_0 = \sqrt{\frac{E^2}{c^4} – \frac{p^2}{c^2}}$$
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?

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 definitions 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).

The following forum members have contributed to this FAQ:
Hootenanny
jtbell
ZapperZ

5 replies
1. mfb says:

One very seldom hears this when one attends a high energy physics seminar, for example, or read a particle collider experiment paper.

I never saw the relativistic mass used in a recent (not decades old) professional environment.

2. ZapperZ says:

Just as a clarification, Greg has been graciously reposting my old Blog entries, and a few other FAQs that I had made, to the Insight section. So unfortunately, many of these require a bit more "refinement", especially on the typos, grammatical errors, etc…etc, of which I'm too darn lazy to make right now.

So this is why I am not sure why this FAQ appears in the High Energy Physics section. It probably belongs in the General Physics section or even Relativity forum.

But I would also like to include a post that I've written on the issue of "relativistic mass". There is already a FAQ on this, but I want to include references on why the term "relativistic mass" should not be used anymore, and why many are starting to shy away from it.

Zz.

Edit: It was moved. Thanks!

3. anorlunda says:

Good job on the article. Maybe you could help clarify a point for me.

You said, "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."

In a recent PF thread, (https://www.physicsforums.com/threads/mass-of-an-electron.826015/#post-5187800) I learned that an atom with electrons in an excited state has (slightly) more rest mass than the same atom in the ground state. From that, I leap to the conclusion that a hot object has more mass than when it was cold, a spring gains mass as it is stretched, a molecule has different mass than its consituents, any chemical reaction must absorb or release energy and therefore does not conserve mass, and any solid structure has different mass than its constituents. The unifying principle is that the rest mass energy is any energy (regardless of type), that remains with the object when momentum is zero.

I recognize that the mass differences I'm talking about are tiny; almost too small to measure.

Thermal energy is tricky because it has to do with motion. But thinking of F=ma, if I accelerate a hot object i must accelerate its thermal energy with it.

I guess my question is this. Is there any difference between what you called "total energy of the particle measured in the particle’s rest frame" and what the others called "internal energy" in the other thread?

A second related question. For purposes of gravitation, all forms of energy gravitate equally, correct? That includes kinetic energy and the energy of massless particles. If the mass of a black hole was converted to massless energy in the singularity, we wouldn't be able to tell in terms of the external gravitational field, correct? F=mMG/R*R should be more properly written in terms of energies.

4. mfb says:
anorlunda

I recognize that the mass differences I'm talking about are tiny; almost too small to measure.

They are notable for nuclear reactions, they might become accessible for chemical reactions within the next decade or two.

anorlunda

For purposes of gravitation, all forms of energy gravitate equally, correct?

If they move in the same way, yes.

The composition of the interior of black holes is not known, and irrelevant in general relativity.