Newtonian physics

In pre-Einstein relativity (known as Galilean relativity), time is considered an absolute in all reference frames.

From this assumption, the following consequences can be derived about the perspective of an event in two reference frames, S and S', where S' is traveling at a relative speed of u to S.

v' = v - u (the velocity of a particle from the perspective of S' is slower by u than from the perspective of S)
a' = a (the acceleration of a particle remains the same regardless of reference frame)
F' = F (since F = ma) (the force on a particle remains the same regardless of reference frame; see Newton's law)
the speed of light is not a constant
the form of Maxwell's equations is not preserved in different reference frames

Example

Consider two reference frames, one of which is traveling at a relative speed of u to the other. For example, for a car passing another car at a relative speed of 10 km/h, u is 10 km/h.

Two reference frames S and S', with S' traveling at a relative speed of u to S; an event has space-time coordinates of (x,y,z,t) in S and (x',y',z',t') in S'.

The space-time coordinates of an event in Galilean-Newtonian relativity are governed by the set of formulas which defines a group transformation known as the Galilean transformation:

Assuming time is considered an absolute in all reference frames, the relationship between space-time coordinates in reference frames differing by a relative speed of u in the x direction (let x = ut when x' = 0) is:

x' = x - ut

y' = y

z' = z

t' = t

The set of formulas defines a group transformation known as the Galilean transformation (informally, the Galilean transform).

Velocity

In pre-Einstein relativity velocities are directly additive and subtractive. For example, if one car traveling at 60 km/h passes another car traveling at 50 km/h, from the perspective of the car it passes it is traveling at 60-50 = 10 km/h.

Mathematically, if we define the velocity of the second reference frame in our previous discussion above as the vector u = ux (x being the x-dimensional unit vector), following the above formulas gives us:

v' = v - u

as we would expect.