Special Relativity is a thoery of translating coordinates in planar spacetime, which has a constant measure .
Any two inertial reference frames can be related by a Lorentz Transformation (or Poincaré transformation). Consider a photon is shot from a point , then we can move the origin of the two frame to that point. Then we can rotate the frames so that their relative speed lays on the axis. The location of the photon in these two frames can be defined as and , then the photon's motion forms a straight line in each frames of reference. Since the speed of light is constant, we have these relations:
In Newtonian transformion, frame of references are related by Galilean transformation:
Thus we have the notion of "absolute" time flow. This does not satisfy . Lorentz transformation multiplies the time coordinate by a factor , which depends on :
Pluggin it into :
In general, the Lorentz transformation relates two coordinates in different reference frames by
has a important property:
where . It is similar to rotation in Euclidean space. It is usually seen as rotation in Minkovsky space.
From now on, we set .
Define proper time that
, so the proper time in another frame is
General form of boost-only Lorentz transformations:
If we define , we then have a unified form:
We can check if this obeys the identity:
Define 4-force in that
If the particle is momentarily at rest, then , then the 4-force is
Then apply Lorentz boost
It can be defined as
then we get:
Another way to generalize momentum:
the momentum at rest is , applying Lorentz boost:
When , :
This should become the clasicall limit where , therefore
Scalars, Vectors, and Tensors
Contravariant vectors transform by Lorentz transformation; covariant vectors transform by inverse Lorentz transformation:
To prove this is indeed inverse Lorentz transformation:
Derivative with respect to a contravariant vector is covariant.