If photons have no mass, how can they have momentum?

As an explanation of why a large gravitational field (such as a black hole) can bend light, I have heard that light has momentum. This is given as a solution to the problem of only massive objects being affected by gravity. However, momentum is the product of mass and velocity, so, by this definition, massless photons cannot have momentum.

How can photons have momentum?

How is this momentum defined (equations)?

• See also this recent question - How can a red light photon be different from a blue light photon? Jun 7, 2020 at 1:20
• In my opinion we should only enquire; why a photon has zero mass, because the photon has momentum is demonstrable via numerous experiments- whereas zero mass is not.
Apr 29, 2021 at 16:05

The answer to this question is simple and requires only SR, not GR or quantum mechanics.

In units with $c=1$, we have $m^2=E^2-p^2$, where $m$ is the invariant mass, $E$ is the mass-energy, and $p$ is the momentum. In terms of logical foundations, there is a variety of ways to demonstrate this. One route starts with Einstein's 1905 paper "Does the inertia of a body depend upon its energy-content?" Another method is to start from the fact that a valid conservation law has to use a tensor, and show that the energy-momentum four-vector is the only tensor that goes over to Newtonian mechanics in the appropriate limit.

Once $m^2=E^2-p^2$ is established, it follows trivially that for a photon, with $m=0$, $E=|p|$, i.e., $p=E/c$ in units with $c \ne 1$.

A lot of the confusion on this topic seems to arise from people assuming that $p=m\gamma v$ should be the definition of momentum. It really isn't an appropriate definition of momentum, because in the case of $m=0$ and $v=c$, it gives an indeterminate form. The indeterminate form can, however, be evaluated as a limit in which $m$ approaches 0 and $E=m\gamma c^2$ is held fixed. The result is again $p=E/c$.

• This is the best answer, other answers that try to insist that photons have mass, (of any form, relativistic or otherwise) should be voted down in my opinion, because it obscures the fact that energy bends space-time and thus changes the direction of the lightwave. Sep 25, 2016 at 23:12
• "p=E/c" , How do you calculate the energy in this case ? Mar 6, 2017 at 20:24
• @Hammar The Planck Relation will give you the energy: $E=h\nu$, where $h$ is Planck's constant, and $\nu$ is the frequency of the light (so you'll sometimes see this written as $E=hf$). Mar 7, 2017 at 22:13
• The second part of the question asks about light being bent by gravity. Does it follow from the fact that light has momentum that it is affected by gravity, as the question suggests? Oct 19, 2018 at 11:56
• Quantum mechanics cannot be ignored when talking about elementary particles like photons; that is a ludicrous statement. From a conceptual standpoint it’s necessary, even if “quantum mechanical” formulae aren’t explicitly used. Oct 19, 2018 at 18:26

There are two important concepts here that explain the influence of gravity on light (photons).

(In the equations below $$p$$ is momentum and $$c$$ is the speed of light, $$299,792,458 \frac{m}{s}$$.)

1. The theory of Special Relativity, proved in 1905 (or rather the 2nd paper of that year on the subject) gives an equation for the relativistic energy of a particle;

$$E^2 = (m_0 c^2)^2 + p^2 c^2$$

where $$m_0$$ is the rest mass of the particle (0 in the case of a photon). Hence this reduces to $$E = pc$$. Einstein also introduced the concept of relativistic mass (and the related mass-energy equivalence) in the same paper; we can then write

$$m c^2 = pc$$

where $$m$$ is the relativistic mass here, hence

$$m = p/c$$

In other words, a photon does have relativistic mass proportional to its momentum.

2. De Broglie's relation, an early result of quantum theory (specifically wave-particle duality), states that

$$\lambda = h / p$$

where $$h$$ is simply Planck's constant. This gives

$$p = h / \lambda$$

Hence combining the two results, we get

$$E / c^2 = m = \frac{p}{c} = \frac {h} {\lambda c}$$

again, paying attention to the fact that $$m$$ is relativistic mass.

And here we have it: photons have 'mass' inversely proportional to their wavelength! Then simply by Newton's theory of gravity, they have gravitational influence. (To dispel a potential source of confusion, Einstein specifically proved that relativistic mass is an extension/generalisation of Newtonian mass, so we should conceptually be able to treat the two the same.)

There are a few different ways of thinking about this phenomenon in any case, but I hope I've provided a fairly straightforward and apparent one. (One could go into general relativity for a full explanation, but I find this the best overview.)

• Since you aren't defining all your terms, p is momentum and c is the speed of light. Apr 5, 2016 at 22:13
• I believe relativistic mass is a very confusing term to people, so I would just like to provide some comments: 1) Energy and mass are the same thing, as you wrote. If gravity affects mass, you may as well say it affects energy. 2) Light is moving, therefore it has kinetic energy, therefore it is affected by gravity. 3) Kinetic energy of an object is merely a side effect of changing the inertial frame. Any object "standing still" on Earth at any given moment, is actually drifting away from some distant star at speed c, but this doesn't mean its mass is infinitely large.
– Groo
Jun 2, 2017 at 23:02
• Which is why I find it really funny when people say that "objects moving near the speed of light would have a near infinite mass", because I don't think they understand what relativistic mass actually is.
– Groo
Jun 2, 2017 at 23:05
• @Groo: Yes, indeed. This is probably why many teachers/authors tend to avoid the concept... I can't say I blame them much, these days. Jun 2, 2017 at 23:20
• @TheodoreNorvell & Evariste: Yep, unfortunately I had written it correctly before, but when someone tried to improve the appearance of the LaTeX, they accidentally transferred the factor of $c$ from the denominator to the numerator! :-) Jul 11, 2019 at 20:19

"momentum is the product of mass and velocity, so, by this definition, massless photons cannot have momentum"

This reasoning does not hold. Momentum is the product of energy and velocity.

"How is this momentum defined (equations)?"

Inserting factors of $c$, the relativistically correct relation between momentum $p$ and velocity $v$ is $$c^2 p = E v$$ This holds for non-relativistic massive particles (total energy dominated by rest-energy: $E = m c^2$, and therefore $p=mv$) as well as for massless particles like photons ($v = c$ and hence $p=E/c$).

• How did you get c^2p=Ev in the first place? p=sqrt(1-(v^2/c^2)) (m*v), right ?
– Rian
Feb 7 at 14:25

The reason why the path of photons is bent is that the space in which they travel is distorted. The photons follow the shortest possible path (called a geodesic) in bent space. When the space is not bent, or flat, then the shortest possible path is a straight line. When the space is bent with some spherical curvature, the shortest possible path lies actually on an equatorial circumference.

Note, this is in General Relativity. In Newtonian gravitation, photons travel in straight lines.

We can associate a momentum of a photon with the De Broglie's relation

$$p=\frac{h}{\lambda}$$

where $h$ is Planck's constant and $\lambda$ is the wavelength of the photon.

This also allows us to associate a mass:

$$m=p/c=h/(\lambda c)$$

If we plug in this mass into the Newtonian gravitational formula, however, the result is not compatible with what is actually measured by experimentation.

• Please note that Newton himself calculated the deflection of light and it is his results that Einstein used to compare with. See also my answer where a reference is given that the Newoton and Einstein results with a factor of 2 are both correct if interpretted correctly. regards.
Apr 29, 2021 at 16:00

If Newton's gravitation could define the bending of light by gravity, then the general relativity wouldn't have come up. Photons don't have mass and it's clear from the fact that it travels at the speed of light. Gravity is an illusion that seems to attract things but in fact it bends spacetime; which is why a straight path seems curved. Newton's law of gravitation is still used because it's simple and we seldom encounter such massive objects like black holes in practical life, for which it does not hold.

In my opinion it is not necessary to evoke the theory of relativity or quantum physics to explain how light can have momentum but not mass. In the 19th century, it was already known that light can collide with matter; a beam of light can set a small wheel (in vacuum) rotating.

The key parameter for the study of collisions under classical mechanics is the momentum :

$$q= mv$$

(Momentum always being conserved in an isolated system)

The natural question is: Can the principle of conservation of the momentum be extended to electromagnetic radiations also?

From experience you know that the answer is positive, provided you define the momentum of light as

$$q = \frac{L}{c}$$

Where $L$ is the energy of light and $c$ the light speed.

Can you extend the analogy assuming that light has mass too?

The assumption is reasonable. In case of positive answer, you get the Einstein equation

$$m = \frac{L}{c^2}$$

However you are not allowed to make such extensions since in Physics you must stick to the experimental evidences. There is no evidence that light has also mass.

If so, how do you solve this paradox?

The light momentum and the momentum of a material particle are not the same thing.

• Let's you keep things simple when relativity doesn't come into play. Aug 21, 2018 at 3:11

Of course they have mass. When saying "photons have no mass" in LHC rap, they were referring to the rest mass, it just didn't rhyme.

(If you pack a bunch of photons into your mirror-coated box, it will be heavier, by $E/c^2$ as usual)

• The point is that the mirror coated box will be heavier, not the photons Oct 22, 2017 at 14:03

This is a fundamental question requiring fundamental thinking. I shall keep away from theories and concentrate on simple facts. From the day we knew of the Brownian motion and realizing that particles of matter are on a continuous motion and not at rest, we should have realized that motion and not rest is the true influential variable of nature. Velocity should therefore be adopted as the prime variable we use to study nature. But velocity has the units of space and time locked in an inseparable format, we should then conclude that space-time is a the variable that need to be considered in our scientific endeavor. But velocity of particles have to involve mass too. This then says that the most fundamental variable of nature is momentum with the units of mass, space and time locked together. As particles possess electrical charges too, we should also add electrical charge unit to get to the fundamental variable of nature.

When we look round we see that E.M radiation has all the above attributes. It has mechanical attributes in the form of energy and momentum flowing along the direction of propagation. This is given by the pointing vector P=E^H. Radiation also have electrical and magnetic attributes in its electric and magnetic fields that are normal to each other and normal to the direction of propagation. These attributes are all verifiable experimentally in the lab by simply directing a beam of radiation onto neutral and charged objects to see them move according to the laws of mechanics and electrodynamics.