**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

t=γτ

where

as always.

**Length contraction: **the proper length (L) between two events (x_{1}, t_{1}) and (x_{2}, t_{2}) is the distance between them in the frame where t_{1}=t_{2}; 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

=

where

**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:

Position:

Momentum:

Wavevector:

Velocity:

Force:

**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.

final 4-momentum

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).