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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139 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.

21. sami allah says:

I am a P.hD scholar of Entomology, dept of
Zoology, University of Kashmir. I want to
know that if photons doesn’t have mass, how
could they have energy because Einstein’s
equation E-mc2 tells us that mass and
Thank you.

22. David Smith says:

Photons have no rest mass but do have kinetic energy in the form of angular momentum. The full equation is E=mc2(hf)

23. al clemence says:

Does a photon simultaneously exhibit amplitude and frequency in multiple planes about a common axis? If so, could it be described as a vibration? Also, for this discussion are we defining a vacuum as an area of space devoid of the influence of mass/gravity?

24. al clemence says:

Can electricity be generated on planets without an iron core? Do protons and neutrons exchange or share an electron? If a neutron looses an electron, does the mass/energy relationship in the nucleus of the atom chance? Is electrical current the result of displaced electrons or transferred electrons?

25. Laszlo G Meszaros says:

The above explanation is very, very weak and makes some “historical” mistakes, too. Although I have studied physics at the university, I am lately getting the feeling that modern physics is now more like a religion, than science. (But it might be my fault, as I am a Greek, who just can give up on logic and being rational.)

26. David Martin says:

You make one statement that to me is seriously mistaken. On the one hand you say that the laws for light and matter are very different. I agree absolutely. You then try to apply special relativity to light! This is ridiculous – you say that “things (like light) that travel at the speed of light never experience the passage of time. Isn’t that awesome?”. No, it’s not, because it’s not true. Light is capable of internal change as it travels, we can in principle observe changes to wavelength, polarisation etc. If matter could travel at c, it would be observed without any internal change, and would look frozen to an observer. Light doesn’t do that, so you can’t apply the rules about time in SR to light.

27. kmguru says:

I have difficulty understanding that Light (photons) do not have rest mass and yet does not go out of Black Holes. Does that mean BH does absorbs the photons to something else? Or the photons are still there hanging around with so much gravity? How does gravity work on photons? Thank you.

28. Elaine Puricelli says:

I’ve been thinking about the picture that’s been displayed with the beams of light versus a plain red brick. Same idea, I’m told photons are massless and the beams of light picture doesn’t at all resemble a brick. And one can’t feel X-ray photons going through them when they get an X-ray taken. But what about gamma ray photons?
Doesn’t that pack a punch? I’m also told that “resting mass” is just a mathematical
value assigned to make the math work. ??? I’m glad I am amongst the legions of
folks who agree that in some way, whether black hole dwelling (trapped) photons or
radioactive gamma ray photons there is a measure of mass. I have also been assured that photons emitted from the Sun are massless. Also recall that what we see in a bolt of lightning is just pure energy. But if we can see the lightening…..then why isn’t that a photon light display as well? Sorry to seem as if I have flight of ideas.

29. frank gibbs says:

If light is not “pulled in” but distorted I’m just guessing not trying to get too deep but we know black holes are so powerful light can not escape if that’s the case then could that mean A) light is stuck in a infinite loop around a black hole because it is so strong even tho technically light is not effected or could that imply B) a black hole is so dense that it creates a hole in space time and light “falls” in to a black hole which could mean there may be a opening on the other side “maybe” but we can not survive the journey to investigate

30. David Smith says:

For Elaine:
A bolt of lightning does not move at the speed of light, it moves at the speed of electricity (much slower). Think of a light bulb. When you flip the power switch on a lamp, the light is produced by a current of electricity flowing through a filament (old-style light bulbs). The filament doesn’t travel at the speed of light nor does the electricity traveling through it. The “bolt” in lightning bolt is simply an electrical current like that in a filament. The light itself travels from the current to your retina at the speed of light.

For Frank:
The laws of physics break down inside a black hole so we don’t know what is going on inside. But it is safe to say that light is not “pulled in” from anywhere since if the warping of spacetime at the event horizon is sufficient to prevent light from “escaping” then the warping of spacetime (or “gravity”) at all points inside the black hole is just as limiting. So, nothing could emit photons, in theory, nor could they travel in any direction beyond the Planck length. But again, laws of physics don’t apply so it could be Alice and the mad Hatter in there.

31. Louis Gerard says:

If anything with mass was to be accelerated to C , then it would approach infinite mass , and would need infinite energy to be propelled to C .Yet a photon travels at C and has no mass . So a photon traveling at C , having no mass , would not experience [metaphorically ] Time . So , if a photon is not affected by time , has no mass , then it can exist forever , and since we do not know what lies beyond the leading edge of the Universe ,which we determine is expanding , and know not what it is expanding into , that photon is [ or should ] be traveling at atleast the same speed as the Expansion . This would seem to be a Constant whether in Classical or Quantum physics , so as we look and observe the distant galaxies , we measure, that the farther they are away , the farther back in time we are seeing , the faster they seem to be travelling , we determine this by the shift in the spectrums of the sources of light from these distant objects .
I am not a physicist , nor a scientist , but simply one who has been studying this stuff since childhood , 55 years have passed since my interest began , and what I have stated is by no means scientific beyond that, it is what my understanding of it all at present encompasses . There is no explanation , which is definitive , of why or how light behaves as both a particle and wave , we have observed it , and have proven that in experiments , but still don’t know why or how . To date , it remains a mystery , as does the Universe and it’s origin . I remain in awe , as I was in first understanding the immensity of what the Universe was . Maybe one day we will have an answer , however IMHO I think those two things , as well as Time itself will remain inscrutable .

32. David Jackson says:

Photons want to travel in straight lines but are constantly colliding with bosons (tiny particles that exist everywhere, even in the so-called vacuum of space). If there were no bosons, a photon would travel not only in a straight line but instantaneously. With bosons around, photons zig-zag randomly – light-speed (186k m/s) being their average rate of progress. All moving sub-atomic particles zig-zag randomly and scientists can keep track of them only by assigning probabilities of their being in given places at given points in time (i.e. by applying quantum mechanics). Photons have energy because they spin round.

33. exploringuniverse says:

If E=p*c for photons (m=0)
and P=m*v =0
E=0*c =0
Seems legit ! ^^

34. Nickson Joseph says:

Gravitational force can bent light ! can nuclear force ?

My doubt is electron is a specific charge which has specific mass but photon is a particle but why it has no mass. but actually particle should have mass rather than charge but in reality why photon has no mass

36. Shawn Heath says:

If light were composed of waves it would be as though every star in the galaxy was a radio station playing at the same frequency. You would see nothing but static.

Time is simply a measure of distance traveled.

The reason that time appears to slow down near a massive object is NOT because time itself has slowed down as time is not a thing but a measure of distance traveled. So if “time” seems slower for an observer at a distance what he is really seeing is a change in distance travelled.

An observer outside of the gravitational field will find her space uncompressed by a massive object displacing space. It is the compression of space on the plank scale that is causing the confusion.
X
a b c d
()()()()()()()()()()()()()()
( e )( f )( )( )( )( )
( g )( )( )( )

This is a very rude picture of a gravitational field. X being the massive object, ()= plank unit of space, a being closer to the massive object compressing space and g representing an observer outside of the field. Obviously on the plank scale there would be many trillions of discreet bits of space per cubic centimeter.

Now we assume that light has to take a plank time to cross one unit of space no matter how compressed or decompressed it is. So for an observer looking at the ground he sees units a,b,c,d. Even though a,b,c,d are 4 units of space compressed as they are by the massive object; the observer will measure them in uncompressed space to be the distance of 1 unit of space. He will then observe that it takes a beam of light on the ground 4 times longer to travel the same distance as a beam of light in his region of uncompressed space takes.

He will incorrectly conclude that time has slowed down near the massive object when in reality the distance that he measures is incorrect.

In the case of a black hole it will emit light but the space around a black hole is so compressed that by the time the light reaches the event horizon it is shifted far to the red because the individual particles when translating from more compressed to less compressed space spread out. The amount of shifts indicates the strength of the gravitational field at the point of emission.

37. Shawn Heath says:

Sorry my diagram had the spaces taken out. The brackets should be spaced further apart as they get farther from X

38. Corey says:

Let me get this straight. We start out with E = mc^2, where light has m = 0, so E = 0c^2 = 0. Einsteain then waves his magic wand and math magically changes to accomidate Einstein so that 0c^2 is really > 0. The supposed reason is because hf = E = mc^2, since hf > 0, but Einstein forgot that lights m = 0 is still in that last equation, so hf = 0c^2 = 0 again, but Einstein waves the magic wand and gets that hf = 0c^2 > 0 anyway??? I’m beginning to think physicists failed basic Algebra here. But, here is the kicker, if E = hf for light and hf > 0, and hf = mc^2, then m = hf/c^2 > 0, implying that light is hiding mass somewhere.

39. The Physicist says:

@Corey
E=mc^2 is a special case of a more general equation: the “energy momentum relation“.