Relativistic mechanics -- Scalars -- 4-vectors -- 4-D velocity -- 4-momentum, rest mass -- conservation laws -- Collisions -- Photons and Compton scattering -- Velocity addition (revisited) -- Doppler shift -- 4-force 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. For each 4-displacement we can associate a scalar: the interval 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 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 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 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. 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 calculating the magnitude, 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). --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 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