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

 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 The next slide shows a simple example to illustrate that p ≠ mv as v ➛ c. Relativistic Momentum

 To illustrate why p ≠ mv as v ➛ c, consider the following. Let us say that you double the velocity of an object (traveling in the x-direction), so that its momentum doubles, in frame S. In frame S′ moving at a velocity u = 0.05c in the x-direction with respect to frame S, how much would the momentum of this object change by? Say the initial velocity in frame S is v xi = 0.2c. In frame S′, v xi ′ = c. Say the final velocity in frame S is v xf = 0.4c. In frame S′, v xf ′ = c. So, v xf ′ / v xi ′ = c / c = 2.36; i.e., in frame S′, the momentum of this object more than doubles. This would violate the conservation of momentum, and hence p ≠ mv. Relativistic Momentum

 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. As the particle travels from its initial position x i to its final position x f, its kinetic energy changes by an amount Relativistic Energy

 Integrating the last expression by parts, assuming an initial momentum p i = 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)? 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  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? 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  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)

 Let us take a closer look at the expression for the relativistic kinetic energy  What does Eq. (4.46) 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 (there is not an infinite amount of energy in the entire Universe), 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? Energy and Momentum Assignment question

 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.  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