Moving between inertial frames

In the universe, there are countless inertial frames of reference. We can move between these frames in distinct ways:

Translation - Moving from one frame to another by a constant distance.

Boost - Shifting from one frame to another that has a constant relative velocity.

When an object moves with constant velocity in one frame, it will also have a constant (though possibly different) velocity in the other frame under any of these conditions.

The principle of relativity

The principle of relativity reveals fundamental aspects of our universe:

• The translation rule implies that there is no special position; position is relative.
• The boost rule suggests that there is no special velocity; being at rest is not an absolute state.
• The rotation rule indicates that there is no special direction; direction is relative.

Transformations between inertial frames

Translated frame

For a translation, to convert a position from frame (x, y, z) to frame (x', y', z'), subtract the offset distance X from x:

x' = x - X

x = x' + X

Boosted Frames

In a boost, when one frame moves relative to another with constant velocity v along the x-axis, the distance between their origins changes by v every second. This distance is $vt$, where t is the time since the frames coincided.

Suppose the origins of frames S and S' coincided at time $t = 0$, with clocks synchronised so that $x = x'$ at this moment. Later, at time t, the origins will be separated by vt. Thus, a position x in frame (x, y) relates to position x' in (x', y') as:

$x' = x - vt$

$x = x' + vt$

The velocities u and u' of an object in the two frames are related by:

$u' = u - v$

When u and v are not in the same direction, they must be subtracted as vectors. Alternatively, treat them as scalar components in the direction of the frames' relative motion.