An inertial fame is a reference frame in which all relative accelerations due to external forces are eliminated.
I. In vacuum, light propagates with respect to any inertial
frame and in all directions with the universal speed c. This
speed is a constant of nature.
II. The laws of nature are the same in all inertial reference
frames.
Consider two reference frames K and K’. Assume that the coordinate axes in the two frames are parallel
and that the origins of the coordinates coincide at t =
t’ = 0. Assume that K’ is moving with
velocity 
with respect to K.
The Lorentz transformation
gives the coordinates of a space-time
point (x0,x1,x2,x3)=(ct,x,y,z) in K in terms of
its coordinates (x'0,x'1,x'2,x'3)=(ct',x',y', z')
in K’ and vice versa.
.
.
Since 0 £ b £ 1, we may write b=tanhB, where B is the boost parameter or the rapidity.
.
Then
and
,
reminiscent of a rotation. We define as a 4-vector any set of 4 quantities which transform like (x0,x1,x2,x3) under a Lorentz transformation; (a0,a1,a2,a3) is a 4-vector if
,
or
.
The "dot product"
is invariant under a Lorentz transformation.
The 4-vector (x0,r) defines an event. The space-time interval between two events is ds.
.
In a reference frame in which two events have the same space
coordinates dr=0 and 
where
is the proper time interval. It is a Lorentz invariant quantity.
.
A particle moves in K with velocity 
.
Its
velocity in K’, 
, is given by
.
It is impossible to obtain speeds greater than c.
4-velocity:
,
a Lorentz invariant scalar.
4-vector momentum:
We define
.
Then
and
.
In every reference frame energy and momentum are conserved.
For each component pm of the 4-vector (p0,p1,p2,p3) we have
,
where i denotes the particles going into the collision and j denotes the particles emerging from the collision.
For transformations between reference frames we have
.
This is a consequence of the invariance of the dot product under a Lorentz transformation.