Relativistic Momentum

We have seen that a photon of light carries an energy that depends on its frequency (or wavelength) according to the Einstein-Planck relation

It turns out that a photon also carries linear momentum with it, so that when an electron (for example) absorbs a photon, it not only absorbs its energy but also its momentum.  (A photon also carries angular momentum, but we won’t be exploring this interesting phenomenon in this course.) To get an expression for the photon’s momentum, it will first be necessary for us to learn just a bit of special relativity.

A Tad Bit of Relativity

In 1905, Albert Einstein published his Special Theory of Relativity which explains the physics  dealing with objects traveling very close to the speed of light (that is, objects traveling with speeds in excess of about 0.1 c = 3 x 10 7 m/s). In this theory the mass of an object is found not to be a constant, but rather to increase as the object’s speed increases.  If an object is at rest next to you, then you would measure its mass to be the so-called rest-mass, mo ; if it’s moving past you at some speed, then you would measure its mass to be the relativistic mass m.  (The rest mass mo is simply the mass that we’re used to thinking ofit’s the relativistic mass m that’s new....)

Einstein found that, if an object of rest mass mo travels with a speed v past an observer, then that observer would measure the object’s (relativistic) mass m to be

where c is the speed of light in a vacuum.  This equation shows that, for any non-zero speed of the object, the relativistic mass is always greater than the rest mass.  Indeed, it shows that, as the speed of the object approaches the speed of light in a vacuum, c, then the denominator approaches zero, and the relativistic mass becomes infinite.  The fact that it takes an infinite  amount of work to increase the speed of an object that has an infinite mass explains why we cannot get objects with a non-zero rest mass to go as fast as the speed of light in a vacuum.

The relativistic mass of an object, m, is always greater than or equal to its rest mass, mo.

Einstein further found that the total relativistic energy of a particle, E, is given by the famous equation

where m is the relativistic mass given in Eq. (8.2) above, or by the equivalent equation

where p is the particle’s relativistic momentum, defined, as in classical physics, by the equation

Note from Eq. (8.3a) that a particle has a relativistic energy even if it is not moving. In the case v = 0, we get from Eq. (8.2) that m = mo (the relativistic mass is just equal to the rest mass if  the particle is not moving), so that its relativistic energy is, from Eq. (8.3a), Eo = moc2.  This special energy is called the particle’s relativistic rest energy.

This definition then allows us to write Eq. (8.3b) in the alternate form

The total relativistic energy E of a particle is just its rest energy Eo(the energy it has when it is not moving) plus its kinetic energy KE (the energy it has by virtue of its motion): E = Eo + KE.  Solving this equation for the kinetic energy and using Eqs. (8.3a) and (8.5) then allows us to write that the relativistic kinetic energy is given by

Now, substituting in for the relativistic mass m from Eq. (8.2), we get from the equation above that the relativistic kinetic energy is

This equation gives the relativistic expression for the kinetic energy of a particle in terms of its rest mass mo and its speed v.  (Believe it or not, the results from this equation actually agree with the classical equation for kinetic energy from your first-semester physics class, KE = ½mov2, in the case that the particle’s speed v is much less than the speed of light, c = 3.0 x 108 m/s!  This is a general property of equations in relativity—they must give results that agree with the classical results when the speeds involved slow down so that the particles are not moving relativistically.)

It is important to keep in mind that the mass m is the relativistic mass as given in Eq. (8.2).  However, if the speed of the object under consideration is not too large we are justified in making the (easier) approximation that m = mothat is, that the relativistic mass is just about  equal to the rest mass (the mass that we measure with a scale).  When do we know when we should use the (correct) relativistic equations and when can we get by using the classical equations (the ones that we’re used to)?

    If the speed of an object is less than about one-tenth the speed of light in a vacuumthat is, if v < 0.1c = 3 x 10 7 m/s—then it will be a very good approximation to use the classical equations that we are used to.  Otherwise, for speeds greater than about 0.1c, we must be wary of errors introduced by using the classical instead of the correct relativistic equations.

All of the speeds that you encountered in your first physics course were well below the speed of light in a vacuum (c = 3 x 108 m/s = 186,000 mi/s), so you do not have to go and “unlearn” the  physics that you learned there.  It is not until you start considering the motion of particles at the atomic level that you really have to start worrying about such effects.

Back to Photons

A photon is a particle of pure energy—its rest mass is zero: mo,ph = 0.  (As usual, we are using the subscript “ph” to remind us that we are talking about the special case of a photon.) It then follows from Eq. (8.3b) that the total relativistic energy of a photon (we must use relativistic equations for light!) is given by

It then follows that the (relativistic) momentum carried by a photon is given by

But from the Einstein-Planck relation, Eq. (8.1), we have that Eph = hc/l, so that Eq. (8.9)  becomes

This relationship for the photon’s momentum was known in 1905 with the publication of Einstein’s Special Theory of Relativity.  This, along with the success of the Einstein-Planck relation in explaining the photoelectric effect, solidified the idea of light “waves” having particle-like characteristics (discrete energy values and momentum, E and p).