Consider now a frame [ i.e. a set of spatial coordinates
(x,y,z) and a time coordinate t ], and another frame
with coordinates
which moves in
the x direction with uniform speed v relative to the frame
.
Common sense suggests that the two sets of coordinates are related by
These are the Galelian transformations .
If the particle has a velocity with components
in
, its velocity in
is:
or
where
More generally if the coordinate axes and the origins of
and
differ then:
where has components (x,y,z). Here
is a
rotation matrix aligning
and
,
is
the relative velocity of
with respect to
and
is the displacement of the origin from
.
Since the transformation is linear [ constant velocity
in
constant velocity in
],
is inertial if
is.
Thus there are an infinite set of inertial frames, all moving uniformly with respect to each other.
All of Newtons laws apply in any inertial frame since
and is invariant. Thus we have Newtonian
[ Galelian ] Relativity.
The laws of mechanics do not allow measurement of absolute velocity, however one can measure absolute acceleration.
Newton explained inertial frames in terms of absolute space identified with the center of mass of the solar system or a frame of ``fixed stars''. However this is unsatisfactory because: