1. Transformation of velocities
Section 5
Transformation of velocities

2. How does x-component of velocity of material particle transform when changing to a new inertial reference system? V
vx’ vx K’ K
In the limit c ®¥, we must get the classical result: vx = vx’ + V

4. V and vx enter symmetrically since vx || V
In the limit c ®¥, we do get the classical result: vx = vx’ + V

5. Do vy and vz also change?
Yes No
Sometimes 1 2 3

6. vy and vz do also change! V
vy’ vy K’ K

7. The change in vy and vz does not happen classically
v’ and V enter unsymmetrically when they are not parallel.
This is due to the non-commutativity of the Lorentz transform.
In the limit c ®¥, we get the classical result:

8. Special case
Then
Usual undergraduate formula
V and v’ enter symmetrically in this case

9. Here are some homework problems
The sum of two velocities never exceeds c
Approximate velocity transformation formulas for V<<c.

10. Does the apparent direction of motion for material particles depend on the reference frame?
Classically?
Relativistic particles? V K q q’ K’

11. Transformation of the direction of motion (homework)V K q q’ K’

12. Light has the same speed in all frames of reference.
Does it have the same direction?

13. Aberration of light
Light always travels at speed c in every inertial reference system, but not in the same direction.
In equation for material particles, put

14. Approximate aberration formula for V << c. (HW)
q < q’ if q’ is positive
Aberration angle