# Q: How can photons have energy and momentum, but no mass?

Physicist: Classically (according to Newton) kinetic energy is given by $E=\frac{1}{2}mv^2$ and the momentum is given by $P=mv$, where m is the mass and v is the velocity.  But if you plug in the mass and velocity for light you get $E=\frac{1}{2}0c^2=0$.  But that’s no good.  If light didn’t carry energy, it wouldn’t be able to heat stuff up.

The difficulty comes from the fact that Newton’s laws paint an incomplete (and ultimately incorrect) picture.  When relativity came along it was revealed that there’s a fundamental difference in the physics of the massive and the massless.  Relativity makes the (experimentally backed) assumptions that: #1) it doesn’t matter whether, or how fast, you’re moving (all physical laws stay the same) and #2) the speed of light is invariant (always the same to everyone).

Any object with mass travels slower than light and so may as well be stationary (#1).

Anything with zero mass always travels at the speed of light.  But since the speed-of-light is always the speed-of-light to everyone (#2) there’s no way for these objects to ever be stationary (unlike massive stuff).  Vive la différence des lois!  It’s not important here, but things (like light) that travel at the speed of light never experience the passage of time.  Isn’t that awesome?

The point is: light and ordinary matter are very different, and the laws that govern them are just as different.

Light and Matter: different

That being said, in 1905 Einstein managed to write a law that works whenever: $E^2=P^2c^2+m^2c^4$.  The same year (the same freaking year) he figured out that light is both a particle and a wave and that the energy of a photon isn’t governed by it’s mass or it’s velocity (like matter), but instead is governed entirely by f, it’s frequency: E=hf, where h is Planck’s constant.

For light m=0, so E=Pc (energy and momentum are proportional).  Notice that you can never have zero momentum, since something with zero mass and zero energy isn’t something, it’s nothing.  This is just another way of saying that light can never be stationary.

Also!  Say you have an object with mass m, that isn’t moving (P=0).  Then you get: E=mc2 (awesome)!

Unrelated tangent: It took a little while, but the laws governing the massive and the massless are even more inter-related than the ‘Stein originally thought.  He figured out that the energy of a photon is related to it’s frequency (E=hf), but why are photons so special?  Why do they get to have frequencies?  They’re not special.  Years later (1924) de Broglie drew the most natural line from Einstein’s various equations from light to matter.  $mc^2=E=hf$  So for a given amount of matter you can find it’s frequency.  Holy crap!  Everything has a frequency!

On the off chance that anyone out there got unduly excited about that last statement: the frequencies never go out of wack, you can’t tune them, more importantly they are utterly unimportant on the Human scale, or even the single-cell scale, and don’t ever buy a bracelet or anything else with “quantum” in the name.

No, no, no, no, no, no, no, no, no.

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### 120 Responses to Q: How can photons have energy and momentum, but no mass?

1. Ben Offertaler says:

I apologize if this is a bit unrelated to the article. Based on what I understand, the electromagnetic force is mediated by photons – light. If two electrons were approaching each other with the same speed, they should end up moving in opposite directions, again with the same speeds, and in the intervening time period had to have exchanged a photon. If one of the electrons (while they are still approaching each other) emits a photon that the other eventually absorbs, then clearly that photon must carry momentum to be able to turn both electrons around, but since the KE of the electron does not change, it cannot carry energy, or so it would seem. That would disagree with Einstein’s equation which says that the energy and the momentum of a photon are directly related. Where is my thinking flawed?

2. David Smith says:

If mc2=E=hf isn’t the reverse true (hf=E=mc2)? If not, why not? And if so, then light’s energy can be converted into mass (i.e., has mass).

Okay, next. Do electrons have mass? They certainly have energy. But Maxwell showed that electricity moves at the speed of light, correct? Again, we have an instance of energy but no mass even though mass and energy are supposedly interchangeable.

Please clarify in simple terms if possible. Thank you.

3. lawal oluwaseun says:

Pls can you be more explanatory on why photon does not have mass and electric charge?

4. David Smith says:

What they are saying is that e=mc^2 only applies to rest mass and the equation when you include “relativistic” or “inertial” mass is e=mc^2+p(hc)^2 or something like that but the upshot is that if m=0 as in the case with light, you can still have energy expressed, in a kinetic way. But it’s just a tool for explanation to save the simpler equation in cases of light. They don’t understand light, and the wave-particle duality, double slit Feynman sum of all routes etc drives them nuts so this is their agreed upon cop out.

5. David Smith says:

Gravity controls it because it is not having mass that affects a photons path through a gravitational field, it is the fact that the space itself that the photon travels through in that gravitational field has been warped (distorted).

6. Xerenarcy says:

@David Smith

“If mc2=E=hf isn’t the reverse true (hf=E=mc2)? If not, why not? And if so, then light’s energy can be converted into mass (i.e., has mass).”

it is true. the problem is the energy associated with mass is far far in excess of what your typical photon will carry. pions, being some of the lightest particles, are likely candidates here – the neutral pion decays to two gamma rays so in theory it should work in reverse under specific conditions.

“Gravity controls it because it is not having mass that affects a photons path through a gravitational field, it is the fact that the space itself that the photon travels through in that gravitational field has been warped (distorted).”

be that as it may, because of the energy-mass equivalence, i could argue that the miniscule amounts of energy each photon has would be equivalent to a mass of the same energy, which would experience and exert gravitational forces, however small. probably too small to factor into gravitational light deflection for every-day energies.

7. David Smith says:

Assigning ANY mass, however negligible, to anything traveling at constant “c” (light speed here) would violate Einstein’s special relativity theory though. My understanding is that since mass increases at it approaches c, and would become infinite at c, requiring infinite energy to propel it.

8. furkan says:

If the light is made up of photons, and photons are particles illuminated According to Law Article (everything from running into the universe has mass) of that conclude that the mass of photons, what is the mass of light and how it calculated?Is the photon sphere?
If Be a sphere, what size and diameter?
And how to penetrate the human body?
Thank you

9. I gotta be honest. That sounds like complete crap to a layman like myself. All this effort to explain how light can have energy but no mass or no rest mass or no mass when it is convenient smacks of convenience to me. I thought I could come here and understand something. What I am coming away with is the realization that the scientific community does not understand something. You guys sound like people in 1904 trying to describe the aether. We need a new Einstein.

10. huma says:

i want to know the precise explanation about the momentum of QUANTUM PARTICLES like photon,neutrino, anti neutrino.just explain how they posses momentum with zero mass,no charge.moreover please add one thing more to my knowledge what is the basic difference between the classical idea of momentum and quantum idea of momentum.

11. John says:

The way I understand it is photons travel at the speed of light only in a vacuum and at slower speeds through different things. Electricity or electrons to be exact have mass and travel slower (in wires they drift around a few centimeters per second). The electromagnetic field that electricity generates travels near the speed of light moving all of the electrons along the wire. For example the electrons in an antenna only vibrate back and forth but the EM field generated travels near the speed of light in our atmosphere.

12. Gg says:

Photons always travel at light speed.
No matter what the medium is.

13. David Smith says:

Photons travel at light speed in a vacuum. Physics 101. Gravity slows photons otherwise how would a black hole be black? The electrons in a prism deflect photons (ok absorb and re-emit) but this process does not occur at constant C.

14. A.Marson says:

Photons don’t have rest mass but they do have relativistic mass. If you put alot of photons in a box with reflective walls, the box would be measurably heavier.

The way I see it, Light is , in simple terms, supreme and probably the only thing which is absolute. Everything else changes and is manipulated in order to maintain this ‘absolute’ quality of light. All rules, be it Quantum or Classical , accommodate to this exclusive property of light. Like Einstein said, c will always remain constant No Matter What! SO … your length can contract , your time can dilate and loads of other stuff can happen, but the speed of light will remain the same. There are a lot of unexplained phenomenon out there . All left for scientists is to discover yet newer ways in which the space time and the properties of matter are compromised to adjust to Light and it’s manipulating ways, because Light won’t ever deter. No sir, because Light is Supreme…

16. David Smith says:

No light is not supreme. The speed at which things travel is supreme. It’s not just space that is important it’s spaceTIME. We too travel at the constant c but through the time dimension. This is the interesting thing. Everything travels at c through spacetime and it’s simply v+t=c. Light being massless has no t. Neutrinos are no different.

17. LearningThings says:

I understand why a photon has to move to be something, but why does it have to move at the speed of life

18. michael says:

why (if protons already travel at light speed) cant we cause protons to travel at a speed they can o on their own?

19. The Physicist says:

@michael
Protons
Photons

20. David Smith says:

I’ll handle this one. There is one speed in the universe, call it C (the constant). Since space and time ante one thing, “spacetime” EVERYTHING moved at C through spacetime. A body with intrinsic mass is slowed down through space so it travels forward at the speed of C, mostly through time. That’s why we age. Light doesn’t. From the perspective of light, it’s still the Big Bang since it has always used it’s “C” in a maximum space speed, zero time. Hope this blows your mind.