| dbp:proof
|
- Reverse formula found by using standard procedure of swapping for and for . (en)
- Since a relativistic transformation rotates space and time into each other much as geometric rotations in the plane rotate the - and -axes, it is convenient to use the same units for space and time, otherwise a unit conversion factor appears throughout relativistic formulae, being the speed of light. In a system where lengths and times are measured in the same units, the speed of light is dimensionless and equal to . A velocity is then expressed as fraction of the speed of light.
To find the relativistic transformation law, it is useful to introduce the four-velocities , which is the motion of the ship away from the shore, as measured from the shore, and which is the motion of the fly away from the ship, as measured from the ship. The four-velocity is defined to be a four-vector with relativistic length equal to , future-directed and tangent to the world line of the object in spacetime. Here, corresponds to the time component and to the component of the ship's velocity as seen from the shore. It is convenient to take the -axis to be the direction of motion of the ship away from the shore, and the -axis so that the plane is the plane spanned by the motion of the ship and the fly. This results in several components of the velocities being zero:
The ordinary velocity is the ratio of the rate at which the space coordinates are increasing to the rate at which the time coordinate is increasing:
Since the relativistic length of is ,
so
The Lorentz transformation matrix that converts velocities measured in the ship frame to the shore frame is the inverse of the transformation described on the Lorentz transformation page, so the minus signs that appear there must be inverted here:
This matrix rotates the pure time-axis vector to , and all its columns are relativistically orthogonal to one another, so it defines a Lorentz transformation.
If a fly is moving with four-velocity in the ship frame, and it is boosted by multiplying by the matrix above, the new four-velocity in the shore frame is ,
Dividing by the time component and substituting for the components of the four-vectors and in terms of the components of the three-vectors and gives the relativistic composition law as
The form of the relativistic composition law can be understood as an effect of the failure of simultaneity at a distance. For the parallel component, the time dilation decreases the speed, the length contraction increases it, and the two effects cancel out. The failure of simultaneity means that the fly is changing slices of simultaneity as the projection of onto . Since this effect is entirely due to the time slicing, the same factor multiplies the perpendicular component, but for the perpendicular component there is no length contraction, so the time dilation multiplies by a factor of . (en)
|
