By adding the vector (magnitude and direction) of their relative velocity to each and every measurement or coordinate.
This works fine until we approach the speed of light which Einstein added as a limiting postulate in his Special Relativity.
Using the Lorentz Factor to adjust or transform those measurements resolved the difficulty, but how?
By rotating the system of coordinates into another dimension.
Yes, another dimension.
Sounds extreme maybe, but consider:
in one dimension we rotate into a second dimension to form a triangle;
in two dimensions we rotate into a third dimension to form a cone;
in three dimensions we rotate into a fourth dimension to form what? A four dimensional figure that we cannot picture!
But in each case the rotation accounts for the addition of a vector in another dimension and so doing it exactly fulfils the formulation of the Lorentz Transformation Equations.
So we add the relative velocity using the Lorentz Transformations and it resolves the problem of the limiting speed of light.
We can take any 3 dimensional set of coordinates – or 4 dimensional when we include Time as a temporal dimension – and, by adding the relative velocity of a moving observer by means of the Lorentz Transformation Equations, by means of an extra dimension, we can recalculate that system relative to the moving observer.
This does not effect the Newtonian Rotation that determines directions but adds the additional, Minkowski Rotation, to that set of coordinates and thereby we remain within the confines of the speed of light
This does not effect the Newtonian Rotation that determines directions but adds the additional, Minkowski Rotation, to that set of coordinates and thereby we remain within the confines of the speed of light.