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Relativistic mechanics -- Scalars -- 4-vectors -- 4-D velocity -- 4-momentum, rest mass -- conservation laws -- Collisions -- Photons and Compton scattering -- Velocity addition (revisited) and the Doppler shift -- 4-force

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1. Scalars A scalar is a quantity that is the same in all reference frames, or for all observers. It is an invariant number. E.g., But the time interval t, or the distance x between two events, or the length l separating two worldlines are not scalars: they do not have frame-independent values vectors This 4-vector defined above is actually a frame-independent object, although the components of it are not frame-independent, because they transform by the Lorentz transformation. E.g., in 3-space, the Different observers set up different coordinate systems and assign different coordinates to two points C and L, say Canterbury and London. --They may assign different coordinates to the point of the two cities --They agree on the 3-displacement r separating C and L., the distance between the two points, etc.

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With each 4-displacement we can associate a scalar the interval (s) 2 along the vector. The interval associated with the above defined 4-vector is Because of the similarity of this expression to that of the dot product between 3-vectors in three dimensions, we also denote this interval by a dot product and also by and we will sometimes refer to this as the magnitude or length of the 4-vector. --We can generalize this dot product to a dot product between any two 4-vectors --When frames are changed, 4-displacement transform according to the Lorentz transformation, and obeys associativity over addition and commutativity : ii) A 4-vector multiplied or divided by a scalar is another 4-vector

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3. 4-velocity In 3-dimensional space, 3-velocity is defined by where t is the time it takes the object in question to go the 3-displacement r. However, this in itself won't do, because we are dividing a 4-vector by a non-scalar (time intervals are not scalars); the quotient will not transform according to the Lorentz transformation. Can we put the 4-displacement in place of the 3-displacement r so that we have The fix is to replace t by the proper time corresponding to the interval of the 4-displacement; the 4-velocity is then whereare the components of the 3-velocity

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Although it is unpleasant to do so, we often write 4-vectors as two-component objects with the rest component a single number and the second a 3-vector. In this notation --What is the magnitude of The magnitude must be the same in all frames because is a 4-vector. Let us change into the frame in which the object in question is at rest. In this frame It is a scalar so it must have this value in all frames. You can also show this by calculating the dot product of You may find this a little strange. Some particles move quickly, some slowly, but for all particles, the magnitude of the 4-velocity is c. But this is not strange, because we need the magnitude to be a scalar, the same in all frames. If you change frames, some of the particles that were moving quickly before now move slowly, and some of them are stopped altogether. Speeds (magnitudes of 3-velocities) are relative; the magnitude of the 4-velocity has to be invariant.

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4. 4-momentum, rest mass and conservation laws In spacetime 4-momentum is mass m times 4-velocity --Under this definition, the mass must be a scalar if the 4-momentum is going to be a 4-vector. --The mass m of an object as far as we are concerned is its rest mass, or the mass we would measure if we were at rest with respect to the object. --Again, by switching into the rest frame of the particle, or by calculationg the magnitude we find that 4-momentum, we can show: As with 4-velocity, it is strange but true that the magnitude of the 4-momentum does not depend on speed. Why introduce all these 4-vectors, and in particular the 4-momentum? --all the laws of physics must be same in all uniformly moving reference frames --only scalars and 4-vectors are truly frame-independent, relativistically invariant conservation of momentum must take a slightly different form. --In all interactions, collisions and decays of objects, the total 4-momentum is conserved (of course we dont consider any external force here).

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--Furthermore, We are actually re-defining E andto be: You better forget any other expressions you learned for E or p in non-relativistic mechanics. A very useful equation suggested by the new, correct expressions for E and Taking the magnitude-squared of We get a relation between m, E and which, after multiplication by c 2 and rearrangement becomes This is the famous equation of Einstein's, which becomes when the particle is at rest

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In the low-speed limit i.e., the momentum has the classical form, and the energy is just Einstein's famous mc 2 plus the classical kinetic energy mv 2 /2. But remember, these formulae only apply when v << c. 5. Conservation laws Summed over All the 4-momenta of all the components of the whole system before interaction Summed over all the 4-momenta of all the components of the whole system after interaction For a single particle: 4-momenum before an action = that after For a multi-particle system:

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5. Collisions In non-relativistic mechanics collisions divide into two classes: elastic inelastic energy and 3-momentum are conserved. only 3-momentum is conserved In relativistic mechanics 4-momentum, and in particular the time component or energy, is conserved in all collisions; No distinction is made between elastic and inelastic collisions. mm Before the collisionAfter the collision M By conservation of 4-momentum before and after collision, which means that the two 4-vectors are equal, component by component, Non-relativistic theory gives: M=2m, v =v/2

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The ratio of these two components should provide v/c; The magnitude ofshould be M c; we use --So the non-relativistic answers are incorrect, --the mass M of the final product is greater than the sum of the masses of its progenitors, 2m. Q: Where does the extra rest mass come from?A: The answer is energy. In this classically inelastic collision, some of the kinetic energy is lost. But total energy is conserved. Even in classical mechanics the energy is not actually lost, it is just converted into other forms, like heat in the ball, or rotational energy of the final product, or in vibrational waves or sound travelling through the material of the ball.

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Strange as it may sound, this internal energy actually increases the mass of the product of the collision in relativistic mechanics. The consequences of this are strange. For example, a brick becomes more massive when one heats it up. Or, a tourist becomes less massive as he or she burns calories climbing the steps of the Effiel Tower. All these statements are true, but it is important to remember that the effect is very very small unless the internal energy of the object in question is on the same order as mc 2. For a brick of 1 kg, mc 2 is Joules, or 3 *10 13 kWh, a household energy consumption over about ten billion years (roughly the age of the Universe!) For this reason, macroscopic objects (like bricks or balls of putty) cannot possibly be put into states of relativistic motion in Earth-bound experiments. Only subatomic and atomic particles can be accelerated to relativistic speeds, and even these require huge machines (accelerators) with huge power supplies.

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6. Photons and Compton scattering i) Can something have zero rest mass? Substitute E=pc into v = p c 2 /E = c 6.1 properties of photon From E = p c(p is the magnitude of the 3-momentum) So massless particles would always have to travel at v = c, the speed of light. Strange?? Photons, or particles of light, have zero rest mass, and this is why they always travel at the speed of light. ii) The magnitude of a photon's 4-momentum but this does not mean that the components are all zero. --The time component squared, E 2 / c 2, is exactly cancelled out by the sum of the space components squared, --Thus the photon may be massless, but it carries momentum and energy, and it should obey the law of conservation of 4-momentum.

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6.2 Compton scattering. The idea of the experiment is to beam photons of known momentum Q at a target of stationary electrons,and measure the momenta Qof the scattered photons as a function of scattering angle. We therefore want to derive an expression for Q as a function of. Before the collision the 4-momenta of the photon and electron are: after they are: The conservation law is For all photonsand for all electrons Also, in this case And: Equation (a) becomes:

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But by conservation of energy, ( 1)mc is just QQ, and (a b)/ab is just 1/b1/a, so we have what we are looking for: This prediction of special relativity was confirmed in a beautiful experiment by Compton (1923) and has been reconfirmed many times since by undergraduates in physics lab courses. In addition to providing quantitative confirmation of relativistic mechanics, this experimental result is a demonstration of the fact that photons, though massless, carry momentum and energy. Quantum mechanics tells The energy E of a photon is related to its wave frequency by E = h so we can rewrite the Compton scattering equation in its traditional form: Then

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An elementary particle of rest mass M decays from rest into a photon and a new particle of rest mass M/2. Find its velocity. M M/ 2 For 3-momentum conservation, the particle moves in x direction, and the photon moves in –x direction. (2) Into (1): 7. Particle decay and pair production Solve for u: 7.1 Particle decay: By momentum conservation:

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7.2 Pair production - gamma photon can not be converted to e - and e+ Show that the following pair production cannot occur without involvement of other particles. e-e- e+ Let m be the rest mass of electrons and u, v the 3-velocities of electron and positron. Sub. (3) into (2): Sub. (3) into (1): Compare (4) and (5):For u x and v x < c (6) can not be satisfied Pair production needs an additional particle to carry off some momentum.

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8 Velocity addition (revisited) and the Doppler shift In S, a particle of mass m moves in the x-direction at speed v x, so its 4-momentum is In S moving at speed v, the 4-momentum of the particle: 8.1 Velocity addition revisited This is a much simpler derivation than that found before.

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8.2 Photon makes an angle from x axis Q(f obs ) S y x z vQ(f rest ) S y x z Equate each component on both side:

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Doppler effect from:

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Aberration of light from: Light rays emitted by source in SLight rays observed in S When v is very large so that =0.9, and cos obs =0.9, obs =26

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if we want to define a 4-vector form of acceleration, or a 4-vector force, we will need to use 9. 4-force We recall the 4-velocity and 4-momentum are defined in terms of derivatives with respect to proper time rather than coordinate time t. The definitions are Where is spacetime position and m is rest mass Also, if the rest mass m of the object in question is a constant (not true if the object in question is doing work, because then it must be using up some of its rest energy!), i.e., if the rest mass is not changing then and are orthogonal. In 3+1-dimensional spacetime, orthogonality is something quite different from orthogonality in 3-space: it has nothing to do with 90 angles.

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