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

Also,

, 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:

Some 4-vectors

4-force

Define 4-force in that

If the particle is momentarily at rest, then , then the 4-force is

Then apply Lorentz boost

4-velocity

4-momentum

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:

where
To prove this is indeed inverse Lorentz transformation:

Derivative with respect to a contravariant vector is covariant.