# Q & A: Massless Particles Traveling at the Speed of Light

Q:
Why do massless particles all travel at the speed of light? And is there some fundamental reason that c has the value of about 300,000k/s, rather than, say, 500,000k/s, or 50k/s?
- Matt Schur
A:
Hi Matt,

In order to answer your first question, we have to introduce the relationship between energy, mass, and momentum in special relativity. Most people are familiar with Einsteins E=mc**2, where c is the speed of light. This equation really only applies to a particle which is standing still, and it is misleading in the case of massless particles. What some people do (not those who routinely use special relativity in their daily work) is to redefine mass to mean E/c**2 to preserve this relationship; under this definition, the mass changes with speed, and we really only then have two names for the same concept. But there still is a valuable idea, called the "rest mass" of an object, which does not depend on the speed with which it travels. People sometimes refer to this mass as m0, and for people who use this stuff every day, thats what "mass" means under all circumstances. For a massless particle, m0 = 0.

The full relativisitic expression for energy in terms of the rest mass of a particle is

E**2 = (m0*c**2)**2 + (pc)**2

where p is the momentum of the particle. This relationship is true for all particles in all frames of reference and is very useful in practice. The momentum of a particle which is traveling at a speed v less than c is given by

p = m0*v*sqrt(1/(1-(v/c)**2))

This equation works for particles traveling at speeds less than c. For v=c, it involves dividing zero by zero. Suppose that a massless particle is traveling at a speed v less than c. Its momentum is zero in all frames of reference because m0=0. Its energy is therefore zero in all frames of reference. There are two problems with such hypothetical particles. 1) We cannot detect them even if they exist. Suppose that a particle with zero energy and zero momentum collides and bounces off of another particle with some energy and momentum. The particle it bounces from must satisfy the energy relationship above, with its rest mass m0, both before and after the collision. The particle bouncing off then bounces away with zero energy and zero momentum again. The normal particle then is undisturbed by the collision. So massless particles traveling at speeds less than the speed of light cannot bounce off of massive particles. The other problem is 2) there is no barrier to spontaneous creation of these energy-free particles. At any nonzero temperature, infinitely many of these particles will be created for free, but it wouldnt matter because we wouldnt be able to tell that they are there.

The process of science is to explain our observations using as simple a model as possible, and then to test our model with further observations. Introducing particles with no energy or momentum complicates our view of nature without giving us any ability to explain our observations, and gives us no ability to test it out, even in principle. So we ignore such a possibility.

The other possibility, that a massless particle travels faster than the speed of light, violates the principle of causality, if such a particle can interact with the particles we know about. Emission of a particle which travels faster than the speed of light, its absorption somewhere else, the readmission of another particle traveling faster than the speed of light back towards the original source implies that the returning particle would arrive before the first one was sent. Using interactions with these particles as triggers for observable events would mean that these events would have no cause. We physicists believe we can explain natural phenomena in terms of laws of physics. If we observe phenomena which violate the laws, we update our idea of what the laws are until they match our observations. By denying that events have any cause whatsoever by allowing particles to travel faster than light and interact with us, we would admit that these events can never be explained or be governed by laws. The good thing so far is that we have no evidence that this is happening (random processes do happen, but even these processes are governed by well-known laws which do not violate the principle of causality). Since the particle may travel neither faster than c nor slower than c in order to interact with us, we find that it travels at the speed of light c.

As for the numerical value of c, it really depends as much on how we defined the meter and the second as on the properties of light. The scale for these was set by the French (10,000,000 meters from the North Pole to the equator on a meridian through Paris), and the second has to do with dividing up the day into things divisible by 60 and 24 (the Babylonians were very fond of base-60 numeration of things). On a more important level, however, the meter and the second are "people-sized" units -- most people are between 1 and 2 meters tall, and we react to things in times which are measured easily in seconds. Then the question becomes, "why is c so huge when expressed in these units?" And the answer probably is that we are complicated and many electro/chemical/mechanical processes must happen within us for us to be aware of something and respond to them, making us unable to be sensitive to times much smaller than 1 second without the help of equipment. If we were of a different size or reacted faster or slower, wed pick different units and c would seem either faster or slower.

The definition of the meter and the second are arbitrary and are chosen to make precise physical measurements easier (using standards are easily available in the laboratory). In 1983, the General conference on Weights and Measures defined the meter as the distance that light travels in 1/299,792,458 of a second. The definition of the second comes from a frequency of oscillation of a particular excited atomic state. A committee therefore has defined the speed of light to be 288,792,458 m/s, with no uncertainty on its value; they could have defined it to be any other number they pleased, and only the meter would change in size. This standard was chosen because the speed of light is universally available to anyone who would like to measure it in a lab, and the atoms used to measure time are also easy to come by. One does not have to go to Paris, for example, and measure a rod to some accuracy to get a precise length standard for calibrating meter sticks.

Some of us professional physicists use units in which c=1 and distance and time are measured in the same units.

Tom

(published on 10/22/2007)

## Follow-Up #1: More on the speed of light

Q:
Greetings. I found this to be informative. However, I interpreted the second part of the person's question differently. I think they were asking (if not, then I am) why is the speed of light the value that it is? I think that the answer is something along the lines of: that is the speed at which you obtain maximum momentum. Another question: as the universe has expanded and contracted, has the speed of light always been the same value?
- Douglas Bishop (age 44)
Fort Wayne, IN
A:
The speed of light is what it is.   There is no rhyme nor reason.  What you sees is what you gets. All evidence we know of shows that it has been the same for billions of years.  Other constants of nature, for example the charge of the electron and Newton's gravitational constant have also been shown to be the same a billion years ago as they are now, within experimental error.  These measurements involve subtle comparisons of light-emitting atomic transitions that we can observe with sensitive telescopes.

The experimentally determined  relationship between the speed of light and momentum of a photon is p = hf/c where h is Plank's constant and f is the frequency of the photon.  There is no particular maximization involved.   Plank's constant is another one of those givens.  No particular reason is known for it's being what it is.

LeeH

(published on 04/21/2008)

## Follow-Up #2: why massless particles travel at speed of light

Q:
Thank you for the well-phrased answer, but I find is missing.You give p = m0*v*sqrt(1/(1-(v/c)**2)). But since for massless particles this gives a nonsensical answer, the momentum of massless particles has to be determined by other means, namely from quantum theory by p=h/Î».Since something can have a frequency (Î») at any non-zero speed this brings us full circle and doe snot answer why a massless particle can no manifest at a velocity other the c.
- Alik (age 52)
San Diego, CA
A:

First, there's a little confusion in your question.  λ in p=h/λ is the wavelength, not the frequency.

At any rate, let's see what the problem would be for a particle with rest mass m=0 if it were traveling at speed  less than c.

The full (E,p) relation is E2=p2c2+m2c4. Now for anything moving at less than c, you have p = m*v*sqrt(1/(1-(v/c)2) which comes out to be 0 when m=0. So massless particles traveling at less than c would always have p=0 and E=0. They wouldn't be able to trade energy with other particles or trade momentum (i.e. exert forces). It's hard to see how their existence would be detectable.

Mike W.

(published on 05/05/2016)

## Follow-Up #3: E=pc if massless

Q:
I'm not sure if this completes the answer. Based on your formula for momentum p, even if a massless particle travels at speed C, p would still be zero (since mass is 0) and energy zero, therefore even massless particles travelling at speed C shouldn't exist either. Either their mass cannot be zero, and there is a minimum mass limit in the universe, or their potential must come from something other than mass.
- Jack (age 34)