Einstein’s postulates are where it all began.
- The laws of physics are the same in all inertial frames.
- Light propagates in a vacuum rectilinearly with the same speed at all times, in all directions and in all inertial frames.
Time dilation: proper time (τ) is the time measured in the rest frame of an object. The time taken in any other frame is given by
Length contraction: the proper length (L) between two events (x1, t1) and (x2, t2) is the distance between them in the frame where t1=t2; i.e., the frame where the two events occur simultaneously. The length in any other frame is given by
Lorentz transformations for going from one frame (S) to another frame (S’) travelling at speed v relative to S.
or, in matrix form
Lorentz transformations of energy and momentum
or, in matrix form
Natural units operate on the principle that . The reason for doing this is basically because it involves less writing and makes equations look tidier- right up to the point where you need to get a meaningful number out. Then you need to look at the dimensions of the quantities you are working with and multiply by the appropriate powers of c and to get an answer that makes physical sense.
Don’t worry if natural units sound pointless and difficult to use at first- the more you work with this field of physics, the more you get used to them.
4-vectors are just vectors with four components. Below are some common 4-vectors in natural units:
Magnitude of 4-vectors: when taking the dot product of a 4-vector, the fourth component has the opposite sign to the other three, e.g. . We could represent this with a factor of i in the 4-vector, but that’s just a convention that is not universally used.
Compton Scattering with 4-vectors: scattering of high energy photons off electrons.
Photon: initial 4-momentum where is the unit vector in the direction of motion.
Electron: initial 4-momentum as it starts at rest.
final 4-momentum – we don’t know or care about this, so it will be eliminated later.
Conservation of 4-momentum
Now let’s work out what these dot products actually are
where θ is the scattering angle.
Substitute back into conservation of momentum equation
But of course,
Where is the Compton wavelength.
Invariance: the key to solving most problems in this subject is remembering that for a system of particles, is invariant.
Remember, for a single particle , the rest mass (invariant).
For a system of particles, it just means that there is always some frame we can find where the sum of their momenta is zero (i.e. there is always a centre-of-mass frame).