The Theory of Special Relativity

Learning Objectives Relativistic momentum:

Why p ≠ mv as in Newtonian physics. Instead,

Energy of an object:Total Relativistic energy = Relativistic kinetic energy + Rest energy

Why objects with mass can only have velocities < c

Relation between momentum and energy

γv mv

The linear momentum of an objective with a mass m and velocity v in Newtonian physics is p = mv.

Conservation of linear momentum: in the absence of external forces, Δp = 0.

Recall the Principle of Relativity: The laws of physics are the same for all observers in uniform motion relative to one another (i.e., in all inertial reference frames). Thus, if linear momentum is conserved in one inertial reference frame, it is conserved in all inertial reference frames.

By applying the conservation of linear momentum, one can show (see proof in textbook) that the linear momentum is given by

Note that the v in the denominator is the velocity of the object relative to the observer, not the velocity of one reference frame relative to another. The next slide shows a simple example to illustrate that p ≠ mv as v ➛ c.

Relativistic Momentum

γv mv

To illustrate why p ≠ mv as v ➛ c, consider the following. In frame S, two bodies having equal masses m and speeds v but traveling in opposite x-directions collide. After the inelastic collision, both bodies have v = 0.

Relativistic Momentum

mv mv

Frame S:

Suppose that frame S′ has a speed u = v in the x-direction. Using the velocity transformation equation

Relativistic Momentum

Frame S′:

mv mv

To illustrate why p = γmv as v ➛ c, consider the following. In frame S, two bodies having equal masses m and speeds v but traveling in opposite x-directions collide. After the inelastic collision, both bodies have v = 0.

Relativistic Momentum

mv mv

Frame S:

M = 2m

Suppose that frame S′ has a speed u = v in the x-direction. Using the velocity transformation equation

Relativistic Momentum

Frame S′:

mv mvM = 2m

Consider a force of magnitude F that acts on a particle in the x-direction. This particle will experience a change in momentum given by

The particle accelerates, thereby changing its kinetic energy K. Suppose that the particle is initially at rest at position xi and reaches a final velocity v by the time it travels to position xf, its kinetic energy changes by

Relativistic Energy

vx

xi xf

Integrating the last expression by parts, assuming an initial momentum pi = 0,

Dropping the subscript f, the relativistic kinetic energy is given by

Relativistic Energy

Let us take a closer look at the expression for the relativistic kinetic energy

How much energy would I have that is related to my mass if I was not moving (v = 0)?

We call this the rest energy of an object. This equation expresses the equivalence between mass and energy.

How much energy would I have if I was moving at a velocity v?

We call this the total relativistic energy of an object (rest + kinetic energy).

Relativistic kinetic energy is therefore total relativistic energy – rest energy.

Relativistic Energy

total relativistic energy (velocity dependent)

rest energy (no velocity dependence)

Let us take a closer look at the expression for the relativistic kinetic energy

What does Eq. (4.45) tell us about the maximum speed an object can have? As v ➙ c, K ➙∞. Since we cannot put an infinite amount of energy into an object (we can never quite get to infinity), the speed of an object can never quite get to the speed of light.

Relativistic Energy

total relativistic energy (velocity dependent)

rest energy (no velocity dependence)

Relation between total energy, momentum, and rest energy

Do waves, including electromagnetic waves and therefore light, have mass? Waves – including electromagnetic waves, and therefore light – have no mass.

Do waves impart a net momentum? Waves do not impart a net momentum.

As we shall see in the next chapter, light can also behave like a particle: i.e., although not having any mass, light can impart momentum. The momentum of light is given by p = E/c.

Energy and Momentum

Assignment question

(Relativistic kinetic energy)2

(Rest energy)2

Experimental proof that light has momentum: the Compton effect.

Momentum of Light

Experimental proof that light has momentum: the Compton effect.

Momentum of Light

In the Compton effect, light transfers a part of its momentum to particles. In the inverse-Compton effect, light gains momentum from energetic particles. An example of the inverse-Compton effect in astronomy is the Sunyaev-Zel’dovich effect.

Momentum of Light

Galaxy clusters are permeated by plasma with temperatures of ~107 K. Light from the CMB gain energy and therefore is blueshifted when passing through a galaxy cluster.

The Cosmic Microwave Background (CMB) is a nearly uniform glow of light from every direction in space with an observed blackbody temperature of 2.726 K.

Momentum of Light

Image of the CMB

The Cosmic Microwave Background (CMB) is a nearly uniform glow of light from every direction in space with an observed blackbody temperature of 2.726 K.

Momentum of Light

In the Sunyaev-Zel’dovich effect, the spectrum of the CMB is blueshifted after passing through a galaxy cluster. Depending on the wavelength observed, the CMB will therefore appear dimmer or brighter towards a galaxy cluster.

Momentum of Light

CMB

The Sunyaev-Zel’dovich effect towards the cluster Abell 2319.

Momentum of Light

44 GHz 70 GHz

100 GHz 143 GHz

217 GHz 353 GHz 545 GHz