# Light has inertia

### 1.4 Mass and momentum of the photons

For a correct description of photons, some aids from the special theory of relativity (SRT) are necessary. For a particle of rest mass *m*_{0} applies with the gamma factor :

From this it can be read:

Squaring the relationships for *W.* and *p* and forms the difference , it results:

so:

** 1.** According to the SRT, mass and energy are equivalent. Photons have energy - do they also have mass and momentum?

An electromagnetic wave has the energy density

.

If the above equation (3) is applied to electromagnetic waves that deal with *v* = *c* spread out, so follows

It is *W.* the energy in a certain volume, which results from the integral of the energy density:

The impulse *p*the wave after the relationship **p**** = ****W.****/*** c* is correspondingly as the integral over a spatial momentum density to understand:

Comparing the relationship for an electromagnetic wave

with the general energy-momentum relation (4)

then there is only agreement for :

*The rest mass of photons is zero.*

This is also to be expected: Photons always move at the speed of light. Particles with rest mass different from zero cannot do this because of For their indolence tends towards infinity.

** 2.** Mass can be proven by its inertia and its gravity. Since mass and energy are equivalent, one can ask: Are photons also inert and heavy?

To investigate the behavior of photons in the gravitational field, the following process is considered:

A photon at A with becomes a particle with rest mass The particle overcomes height D in the gravitational field . At B the particle turns into a photon with transformed. |

Because of energy conservation applies

Difference:

Elimination of *m*_{0}:

This results in

So the frequency of the light decreases and it becomes longer-wave:

This is a shift towards the red end of the spectrum. Hence this effect is called *Gravitational redshift* designated. This shift in spectral lines can be demonstrated in light emanating from the sun. The first evidence on earth was carried out in 1962/65 by Pound, Rebka and Snider. With a height difference of D*H* = 22.6 m, the theoretical value for the relative frequency change resulted:

** 3.** The momentum of a photon results in

Particles hit by light experience an impact. This can be clearly seen from the comet's tails made up of dust particles: these always point away from the sun. Further, atoms that emit a photon experience a recoil according to the law of momentum.

** 4.** If one observes the momentum of the photon, the knowledge about the photoelectric effect is deepened.

Light of wavelength falls on a Cs photocell (*W.*_{A.} = 1.94 eV). The energy of the fastest photoelectrons is then

The momentum of the electrons results from

The photons have a momentum of

So is . The photon cannot push the electron out of the atom as a particle in the Newtonian sense would. The photon does not behave like a "little billiard ball". In the photo effect, the photon only serves as an "energy supplier". The atom must act as a "supplier of impulses".

Note on the mass

In older representations of the special theory of relativity, the one in energy *W.* and the impulse *p* occurring term referred to as "dynamic mass":

This is where the relationship comes from

The application of this relation to photons then leads to the fact that one speaks of a "mass of the photon":

This must not be taken literally, since photons are the rest mass *m*_{0} = 0 and the speed *v* = *c* exhibit. The above term for the "dynamic mass" is therefore not defined for photons.

A meaningful interpretation of the term is to understand this as the mass that a particle would have to have as rest mass so that its rest energy is equal to the photon energy.

Exercises

Do some research on the keywords:

positron

Positronium

Pair creation

Annihilation

Try to answer the following questions:

Why does a photon have a minimum frequency

*f*_{min}so that it can create an electron-positron pair? What is this minimum frequency?

Why can pair creation only take place with the assistance of an atomic nucleus?

Why does the annihilation have to produce 2 photons?

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