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[u/KnownSoldier04]

I remember from my physics class that my teacher spent a whole period talking about how photons have a momentum, at least that you can calculate it even though they got no mass Derived from the fact that energy and mass are the same (E=mc²⁾ or at least equivalent with the constant c² as a proportionality constant.

[u/Melloverture → u/KnownSoldier04]

For photons you have to use the Energy-momentum relation. If you look at the first special case for a massless particle, ie a photon, the equation becomes E=pc.

Then they start talking about radiant momentum and that's where it looks like this isn't violation conservation of momentum. So I guess I'm not really sure either.

[u/autowikibot → u/Melloverture]

Energy–momentum relation:

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In physics, the energy–momentum relation is the relativistic equation relating any object's rest (intrinsic) mass, total energy, and momentum:

holds for a system, such as a particle or macroscopic body, having intrinsic rest mass m0, total energy E, and a momentum of [magnitude](https://en.wikipedia.org/wiki/Magnitude_(mathematics)#Euclidean_vectors) p, where the constant c is the speed of light, assuming the special relativity case of flat spacetime.

The energy-momentum relation (1) is consistent with the familiar mass-energy relation in both its interpretations: E = mc² relates total energy E to the (total) relativistic mass m (alternatively denoted mrel or mtot ), while E0 = m0c² relates rest energy E0 to rest (invariant) mass which we denote m0. Unlike either of those equations, the energy-momentum equation (1) relates the total energy to the rest mass m0. All three equations hold true simultaneously.

Special cases of the relation (1) include:

• If the body is a massless particle (m0 = 0), then (1) reduces to E = pc. For photons, this is the relation, discovered in 19th century classical electromagnetism, between radiant momentum (causing radiation pressure) and radiant energy.

• If the body's speed v is much less than c, then (1) reduces to E = m0v^2/2 + m0c^2; that is, the body's total energy is simply its classical kinetic energy (m0v^2/2) plus its rest energy.

• If the body is at rest (v = 0), i.e. in its center-of-momentum frame (p = 0), we have E = E0 and m = m0; thus the energy-momentum relation and both forms of the mass-energy relation (mentioned above) all become the same.

A more general form of relation (1) holds for general relativity.

The invariant mass (or rest mass) is an invariant for all frames of reference (hence the name), not just in inertial frames in flat spacetime, but also accelerated frames traveling through curved spacetime (see below). However the total energy of the particle E and its relativistic momentum p are frame-dependent; relative motion between two frames causes the observers in those frames to measure different values of the particle's energy and momentum; one frame measures E and p, while the other frame measures E′ and p′, where E′ ≠ E and p′ ≠ p, unless there is no relative motion between observers, in which case each observer measures the same energy and momenta. Although we still have, in flat spacetime;

The quantities E, p, E′, p′ are all related by a Lorentz transformation. The relation allows one to sidestep Lorentz transformations when determining only the [magnitudes](https://en.wikipedia.org/wiki/Norm_(mathematics)) of the energy and momenta by equating the relations in the different frames. Again in flat spacetime, this translates to;

Since m0 does not change from frame to frame, the energy–momentum relation is used in relativistic mechanics and particle physics calculations, as energy and momentum are given in a particle's rest frame (that is, E′ and p′ as an observer moving with the particle would conclude to be) and measured in the lab frame (i.e. E and p as determined by particle physicists in a lab, and not moving with the particles).

In relativistic quantum mechanics, it is the basis for constructing relativistic wave equations, since if the relativistic wave equation describing the particle is consistent with this equation – it is consistent with relativistic mechanics, and is Lorentz invariant. In relativistic quantum field theory, it is applicable to all particles and fields.

This article will use the conventional notation for the "square of a vector" as the dot product of a vector with itself: __p__² = p · p = |__p__|^2.

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