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The Theory of Special Relativity

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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 γ v mv

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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 γ v mv

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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 Frame S:

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

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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 Frame S: M = 2m

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Suppose that frame S′ has a speed u = v in the x-direction. Using the velocity transformation equation Relativistic Momentum Frame S′: mv M = 2m

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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 x i and reaches a final velocity v by the time it travels to position x f, its kinetic energy changes by Relativistic Energy v x xixi xfxf

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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

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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)

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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)

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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

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Experimental proof that light has momentum: the Compton effect. Momentum of Light

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Experimental proof that light has momentum: the Compton effect. Momentum of Light

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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 ~10 7 K. Light from the CMB gain energy and therefore is blueshifted when passing through a galaxy cluster.

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

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

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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

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The Sunyaev-Zel’dovich effect towards the cluster Abell Momentum of Light 44 GHz70 GHz 100 GHz143 GHz 217 GHz353 GHz545 GHz

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