# 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)?

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

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

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

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. – b1nary.atr0phy Apr 5 '16 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 '17 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 '17 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. – Noldorin Jun 2 '17 at 23:20
• @Groo I agree that the more distant a galaxy is from us, the light coming from it is proportionately more doppler red shifted since it is moving away from us at a faster speed than a closer galaxy. However, I beg to differ with you that this is anywhere near c. – 0tyranny 0poverty Nov 8 '17 at 23:11

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. – Haru Fujimura Sep 25 '16 at 23:12
• "p=E/c" , How do you calculate the energy in this case ? – Hammar Mar 6 '17 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 $E=hf$). – owjburnham Mar 7 '17 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? – Theodore Norvell Oct 19 '18 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. – Noldorin Oct 19 '18 at 18:26

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.

"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 $p=E/c$).

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. – Alex Jasmin Aug 21 '18 at 3:11

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.

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 – Alchimista Oct 22 '17 at 14:03

Light doesn't have momentum in the normal sense that matter has. Frequency and Wavelength soak up momentum. The more energetic the higher the frequency. Wavelength can change even though light stays at C .. Light doesn't bend, but space can be deformed. A better question is how can space be deformed when space has even less energy, mass, or wavelength than light does?? Space has no mass, momentum, yet changes (grows) between galaxies in the voids between them. Space takes a tremendous amount of energy to deform it, but what is causing space to be deformed? What is gravity exactly? There is something about space and time that is linked together. Time slows down as you encounter a gravitational field. How can time have energy to distort space? Where does the energy that time has come from and where does it go? What is time exactly? Time and gravity are unknowns yet have a constant predictable nature in physics, except quantum physics. A singular particle can phase into different multiverses popping back into existence by probability. How can a particle exist in two different locations at the same time? Where are multi-verses in relation to this universe located? Is the Weak Nuclear Force weak because it exists in all multi-verses simultaneously in comparison to the other forces in physics that can only exist in one universe at time? I have about a half dozen more questions but I will just stop at this point. The more we know about universe the bigger the unknown grows.

• Even pure classical (i.e. Maxwell's equations) light has well defined momentum, a fact which has been known since the end of the 19th century. – dmckee Jul 6 '12 at 13:17

## protected by dmckee♦Mar 20 '13 at 19:15

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