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 momentum is given by P=mv, where m is mass and v is 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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177 Responses to Q: How can photons have energy and momentum, but no mass?

  1. Gojko says:

    Hello. Why sunlight cannot escape the black hole, when it doesn’t have mass? If black holes gravity wants to influence on sunlight, sunlight should have some mass.
    I think mass of light is just extremely small, because if you turn equation around, you get m=E/c^2, which is something very small.
    Thanks for all replies 🙂

  2. David Smith/TintWeezl says:

    Elaine-yes on the Faraday cage, interesting thought. Careful with the caulking though…

    And yes, photons are “bundles of energy” just as particles with mass are bundles of energy (locked, somewhat) provided by the Higgs boson within the Higgs Field. Light is bestowed its energy vis a vis motion through the electromagnetic field. The motion gives it kinetic energy which is pure and massless when not at rest (at rest it disappears since it is produced kinetically). Particles with mass are massless too when not interacting with the mysterious Higgs field that apparently permeates space and it is this field and its, until recently, hypothesized but not confirmed existence that was inferred by the Standard Formula’s suggestion that all particles should inherently be massless without a catalyst to impart it. Why photons are unaffected by Higgs remains unanswered.

    This question of light and gravity is very poorly addressed in school, unless you go for a PhD in Physics. They really ought to explain it better. This board and many like it are testament to the failure of our educators to present such a cardinal aspect of our universe intelligibly, but I’ll bet most teachers couldn’t answer it themselves.

    To wit:

    1) Light in all its forms has no rest mass.
    2) Light does not exist at rest. (This is why we can’t weigh it to be absolutely sure 😉
    3) Light behaves as a wave unless interfered with (examined. This can be by camera film or ccd, photon detector, measurement of any kind that would reveal any effect of light or it’s position/velocity)
    4) Light behaves as a particle when examined (the examiner can be a metal plate)
    5) Light is affected by gravity only in the sense that it’s straight path through curved 4-D spacetime appears to 3-D observers as bending towards objects with large mass. This would occur regardless of whether kinetic energy was present, everything that travels through space must traverse the mass-warped portions as the substrate dictates.
    6) E=mc2 is not the whole equation, it is what the media presents for simplicity. A closer approximation is E=mc2+hf with hf determining the kinetic energy.
    7) Particles with rest mass (higgs mass) can’t travel at light speed because any amount of mass would consume the entire universe at that speed due to the relation between “c” (300m k/s or “light speed”) and infinite mass. This is made impossible by physical laws which require infinite energy to attain light speed for even a single proton. Energies required to move a proton in the LHC beams require exponentially higher energies for incrementally smaller increases in speed.
    8) Light gets from point A to point B by using every possible path between the two points including circling behind your head and then Pluto and then Alpha Centauri and then back to the tree in front of you. You only see the result that probability suggests is the most likely path it should take. (E.g., when you shine a light at a tree, occasionally one photon hits Pluto). If this makes sense, you need to reread that. And youtube some Richard Feynman lectures.
    9) Time does not exist for light and if you were going along at 300m k/s you wouldn’t be notice time’s passage either. (You might notice that your body filled the entire universe all of a sudden and since time is a physical property within our universe, it wouldn’t matter since there is now no other reference frame to express time relative to you).
    10) Light only moves at “light” speed (c) in a vacuum, otherwise it is slowed down by bouncing off electrons (light is negatively charged too). This bouncing can be predicted and quantified by using Richard Feynman’s “arrows of light” tool discussed in his lectures which you can find referenced on his wikipedia page. Fascinating stuff for the curious.

  3. David Smith/TintWeezl says:

    In Newtonian gravity there is attraction to mass, but not so in General Relativity. Recall that Einstein described gravity not as any kind of force but a warping of spacetime. My favorite analogy is of a trampoline. That’s space, nice and flat. Put a bowling ball on the edge and see it roll to the center. A bowling ball rolling in space. When it comes to rest in the center, it has warped the trampoline surface somewhat. Enough so that when we place marbles or ball bearings on the edge, they roll down and “stick” to the bowling ball. That’s us on Earth. Ball bearings stuck to a bowling ball. And yet no force caused us to stick, it is what happens to the trampoline surface when a weight (mass) is introduced. Now while light cannot warp space, it still must follow the curvature like anything else. And where, like in the case of black holes, that curvature is so extreme it actually wraps back around onto itself, there is nothing to do but follow the circle of no release.

    That Einstein was pretty clever, we ought to give him some kind of recognition. An award, perhaps.

  4. DDayanov says:

    Unbelievable that hypothetical Bose-Einstein photon condensate has energy signature and doesn’t have mass. Does it mean even in gravitational field of black hole those condensate will stay where it forms? I do remember read about experiment that prove Einstein’s E=m*C^2. They heat massive object and measure dm=dE/C^2. I heard about modern materials with big dielectric conductivity that allows light travel with speed = 60km/h. Does anybody did similar experiment with light trapped inside big coil made of those materials?

  5. David Smith says:

    Nikola Tesla. But he ain’t talking.

    And I agree that the energy component of a photon sounds like a joke when you first hear it. But let it settle a while. It does make sense.

  6. Tommy Scott says:

    The explanation doesn’t seem to have answered the question. It asserts that they have no mass, but doesn’t prove it as far as I can see. And the statement that things like light that travel at the speed of light don’t experience the passage of time is ludicrous in my opinion. First, only light travels at the speed of light as far as we know. Second, if light travels at a particular speed in order to get from point a to point b then there is a passage of time. Third, since no one has actually traveled at the speed of light, what one might “perceive” is purely a hypothetical concept and not based on observational science. I honestly think all that we know about light, the speed of light, and time/space is more than likely very different from the reality.

  7. Arun says:

    Actually protons should have some mass orelse, it is impossible to have momentum as momentum doesn’t work with zero mass(p=mv)…….simply momentum(p)=force(f) ;can u guys believe that a force would be exerted with zero mass (I.e; f=ma) since m=0…..f=0×a ….f=0, hence without mass, protons possess no momentum(p/f)……A/Q Einstein derived a body of mass 1kg can converted into 90,000,000,000,000,000 joules that is a lot of energy only for converting 1kg of mass……thus ,how much light energy can it be possess in one proton?so it seems too negligent mass for a beam of light and moreover protons are more much tiny than an atom! [impossible n unbeleiveable with no mass and possess momentum(p/f) cuz…..zero is zero(nothing) result of p/f will be always zero]

  8. Alex says:

    @Tommy Scott: First, there are also gluons with zero rest mass. Second, there will be no passage of time for the light to get from A to B since point A and point B will be the same place. Third, those explanations based on what we can observe, not just made up.

  9. Tint Weezl says:

    @Tommy Scott

    It gets proven and reproven every day (that light has no rest mass). The LHC proves that not even the tiniest particle of mass (e.g., a proton) can go the speed of light regardless of how many trillions of electrin volts we muster behind it. The more energy we add, the smaller the payoff as we were able to achieve 99.9% light speed a long time ago and have since doubled the power to minimal effect (the useful effect is that by adding more energy, the collider produces higher-mass particles in collisions.

    So that demonstrates the fact that any mass seems to have a speed limit just shy of light speed. The fact that light is always measured at light speed gives you the (rest) mass right there: zero. Necessarily.

    And as for the time it takes light to get from point A to point B, it is relative to the observer. I know that from our perspective light takes time to travel distances we perceive. About 300,000 kilometres each second. But if we use light itself as the observational frame of reference (impossible to actually test, of course, as light has no way to tell us and we have nothing that can tag along at that speed and report back) … But theoretically, from the information we have about the nature of light, time and space: no time has passed from the reference frame of light.

    Nor could it! Because light travels purely in the “space” dimension of spacetime. Space and time are equivalent just as energy and mass are equivalent. Take some energy and create mass with it (LHC) and you lose the correlating amount of energy in the mass formation. Conversely, turn mass into energy (burn a log) and you lose maas in direct correlation to the energy produced.

    Time and space have similar trade-offs. In spacetime, both space and time perception are determined by how fast you are moving relative to an observer. If I am boy moving in space, I am therefore hurtling through spacetime solely in the time dimension. I am going the equivalent of light speed through time. We all do this. Every molecule in iur bodies are moving at all times (so we don’t violate Brownian Motion and cease to exist) but for purposes of spacetime: we are going as fast through time as possible and virtually none through space. If we hop in a spaceship that can go 90% the speed of light, we are now going through spacetime with only 10% dedicated to the time dimension which means time slows down for us (relative to slower moving observers)
    by an equivalent amount. So, for light, using 100% of the speed of light, there is nothing being used in the time dimension so time is not a part of light’s eternal journey through spacetime. A journey that, FOR LIGHT, never actually begins or ends.

  10. Tommy says:

    @Alex, why do you say point A and point B are the same place? They are different points, hence the reason they are labeled A and B. And no you cannot observe travel at the speed of light from the reference frame of one traveling at that speed. You are observing it as an outside observer. Back to square one.

  11. V.L.NARASIMHA says:


  12. no photons has weight because of einstein statement energy mass equilense we can say that photons have mass and energy

  13. photons have mass beacuse of einstien statement and by influence of gravity on photons we can say that photons have mass

  14. V.L.NARASIMHA says:

    It is wrong beacuse light carry photons which have energy and momentum with this we can say light carry energy with it.

  15. Elaine Puricelli says:

    I am thinking that, because I have always believed that on some level, photons have mass – we may need to be in a black hole to see this for ourselves. Mathematically can we prove the mass bearing photon? That is, it’s much more believable once it is proven in math – we can understand the mass bearing photon. When one gets an X-ray, mammogram, ultrasound, etc., one cannot feel any intrusion of photons. That’s why I think we have to perhaps be in a black hole to see this for ourselves – the black hole seems like a great proving ground for this concept.

  16. TintWeezl says:

    All black holes are filled with confetti made from shredded pistachio shells and left over from the black hole raves that take place forever unless you get there late which would be any time after they start (black hole mass > maximum gravity to experience time dimension). The rave begins precisely at critical mass for singularity status which means you need to get there right when time stops. Too soon, and you join the confetti as a non-sentient, non-observer. But you aren’t missing much since nothing can happen since without time no events can occur. Unless… you factor in THE COMPLETE SUSPENSION OF ALL PHYSICAL LAWS OF THE UNIVERSE!

    You heard right, singularity fans! All physics breaks down at the point of the singularity. All math results in zeroes or infinities from our frame of reference, which can only be used on our side of the event cone. Our physics does not apply outside the event cone! Anything can be going on in there. Sexual deviance, no doubt, is being practiced with abandon. Armani suits being sold for ridiculous prices out of the back of Roman chariots. Free chameleon tongues for all.

    And light? Well you can pick it up, a photon at a time. pet it. ask it questions. tickle it. name it elmo. but there’s only one way out. and it’s often a long wait. you can only escape the event horizon by becoming hawking radiation. and the odds of a black hole releasing even one intact molecule let alone all your molecules lined up correctly is about one chance in zero in a billion trillion years. normally they just burp an electron here or there. so wear something warm (i guess. who knows?) and bring your ungulflazer (you will understand later when it’s too late to build one).

    Happy trails!

  17. Hybycozo says:

    Well done tintweezl

    One more question, why does light, when it’s slowed down take on a mass, as in the Einstein-Bose condensate?

  18. Thomas B says:

    E = mc^2 only applies to non-moving objects.
    For moving objects, we have E = γmc^2 where γ depends on speed. And for objects with the speed of light, γ = ∞. So light cannot have a mass.

  19. Shivam says:

    the answer I got was not satisfying. It is said that everything has mass. Nothing is massless. Then how come photon does not have mass,even when a photon is moving at the speed of light.

  20. Tintweezl says:

    “It is said” where?

  21. Thomas B says:

    “It is said that everything has mass.”
    Whoever said that clearly doesn’t understand physics.

  22. markc says:

    @Thomas B the speed f light is not infinite

  23. Thomas B says:

    γ = sqrt(1/(1-β^2))
    β = v/c
    If an object is moving at the speed of light, then β = 1 and γ = ∞.

  24. Elaine Puricelli says:

    Please explain Thomas B’s equation- posting of 5/6/16. Does this favor the photon being mass bearing?

  25. Mridul rai says:

    Light has the very high velocity but the mass of photon is very very less that why the force due to light get tends to “0” but the photons has the mass….

  26. Who cares says:

    For everyone arguing E=mc^2 => Photon has mass, since it got energy.
    The complete equation is E= Mc^2 + hf. In everything above atoms hf —-> 0.

    This explains Zero mass and non-zero energy of Photon.

    Hence the saying, partial information is dangerous.

  27. Who cares says:

    E= hf. (Einstein)
    For light, E= hf. (Plank’s constant times it’s frequency.)
    What about Momentum?
    Momentum is a quantity of motion in an object. Light, I’m sure you all would agree, has a lot of motion.
    E^2= P^2C^2 + M^2C^4
    For photon, M=0,
    E^2 = P^2 x C^2
    => E=PC
    also E=hf
    => P=hf/C
    P is momentum.
    Energy is capacity to do work, and momentum is quantity of motion.
    Light by virtue of its velocity can do work and has motion.

    Philosophy: (understanding the equations)

    Any mass (no matter how insignificant) moving at the speed C will occupy the entire universe.
    To understand how something can occupy the entire space with a given velocity, consider this:
    Everyone has seen a fan I guess. When you see a moving fan, you don’t see the blades, you see a disk. Because at the speed of rotation, the blades occupy all the space at all the time, hence we see a disk.
    In a similar way, anything with a mass and moving with velocity C will occupy all the space and at ALL THE TIME.
    At “ALL THE TIME” is a tricky statement, because light doesn’t know time.
    Let me explain why light must not experience time. And what it means to say “at all the time.”
    First, let’s define Time:
    1 hour is the time it takes for the hour hand to move one unit; or the time it take a car travelling at an average speed of 60km/hr. for one hour to travel a distance of 60 km.
    But the definition has a problem: I cannot define time by using time in my definition.
    Let me try another way:
    Time is the relative difference in observable velocity. What it means is, if you can see/observe any motion, then there is time that exists for you. Time is relative to the observer and cannot exist independent of motion (relative motion) and observer. T=d/v. d is distance, v is velocity. Anything that is moving must change its position and have a non-zero d. So, if you can observe velocity, d exists for you, and (using the equation) time exists for you.
    Now what does it really mean when we say time doesn’t exist for the light?
    As I said, time is the relative difference in observable/observed velocity. (Observation is relative to the observer.) If we can prove that relative velocity doesn’t exist for light i.e., it cannot see anything moving (not even itself) then we will prove that time don’t exist for light.
    Let’s go to the reference of a light, call it L, and see what it might observe.
    Relative to L (assuming L sees itself as at rest) everything else (assuming L can differentiate b/w itself and other things) will seem to be moving at the speed C (C+x=C, also C-x=C, x is the speed of thing L observes in their frame of reference). This is simple relativity. L would observe me moving at the speed of light, C. Given that I have a mass, an undeniable truth, moving at speed of light I will become energy (e=mc^2), no matter the amount of mass I carry. Everything else that has mass will be energy too in the reference of light. (Assuming nothing is mass-less) Hence, everything will become the same thing. L won’t be able to distinguish between anything. To L everything will look the same.
    Now, the question is will this other thing (everything else) will be something different than light or light itself, to L.
    Light is defines is something that moves with a velocity C, because all it has is velocity (no mass) and at rest it ceases to exist. Therefore, anything moving with the speed of light must be light itself.
    Hence, L will not just see everything to be same, it will see everything to be same as itself. This implies, L won’t see anything at all, because everything is L and L is everything.
    If you can’t differentiate between two things, you can observe their relative velocity.
    Hence, there will be no relative velocity for L. For L everything (which is itself) will be at rest, at ZERO and at all the “time”. In it’s frame of reference, Light, L, is at ZERO velocity and at all the “time”.
    Just to recall, by the analogy of the Fan, if light had any mass, then it would ideally occupy all the space. ——— 1
    Also we argued that time doesn’t exist for light. —— 2
    From 1 & 2,
    Light (L) if it had mass would occupy all the space at all the time.
    Let’s argue the same thing in a way that can be challenged experimentally:
    If light has some mass, then that would mean that photons has mass. If this was true, then it would be possible to create a laser beam with extremely high density of light (we can talk about density if there is a mass) and then measure it’s weight. (Infinitesimally small quantity when added infinitely would generate something that is not infinitesimally small.)

    Another way would be to slow the light in a bose-einstein condensate and then observe the weight of the entire thing. If light has mass, then Gravity will give it weight.

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