The following is an addendum to a lecture on conservation of relativistic momentum presented on UNIZOR.COM - Relativity 4 All - Conservation - Momentum.
For details on experiment considered, symbol meaning and approach watch this lecture and read notes that accompany it on the above Web site.
Calculation of γ=1/√1 − l²/c², where l is the length of the object's velocity vector.
In the β-frame l²=lβ²=v²+w² for both green and blue objects.
The α-frame is the one where we would like to check the conservation of the relativistic momentum and where we need γ-factor that depends on transformed velocity vectors.
It's assumed that β-frame is uniformly moving along X-axis of α-frame.
Green Object
l² = lα² = [(s−v)²+w²(1-s²/c²)] /(1−s·v/c²)²
| 1 − lα²/c² = 1 − |
| [(s−v)²+w²(1-s²/c²)] |
| c²·(1−s·v/c²)² |
|
| = |
| c²·(1−s·v/c²)² − [(s−v)²+w²(1-s²/c²)] |
| c²·(1−s·v/c²)² |
|
| = |
| c²·(1−s·v/c²)² − [(s−v)²+w²(1-s²/c²)] |
| c²·(1−s·v/c²)² |
|
| = |
| c²−2s·v+s²·v²/c² − s²+2s·v−v² − w²+w²·s²/c² |
| c²·(1−s·v/c²)² |
|
| = |
| c²+s²·v²/c² − s²−v² − w²+w²·s²/c² |
| c²·(1−s·v/c²)² |
|
| = |
| c²−s²−(v²+w²)+s²·(v²+w²)/c² |
| c²·(1−s·v/c²)² |
|
| = |
| (c²−s²)−(v²+w²)·(1−s²/c²) |
| c²·(1−s·v/c²)² |
|
| = |
| (1−s²/c²)−(v²+w²)·(1−s²/c²)/c² |
| (1−s·v/c²)² |
|
| = |
| (1−s²/c²)·[1−(v²+w²)/c²] |
| (1−s·v/c²)² |
|
Therefore, γ=1/√1 − l²/c² factor for green object is
| γ = |
| 1−s·v/c² |
| √1−s²/c²·√1−(v²+w²)/c² |
|
Blue Object
l² = lα² = [(s+v)²+w²(1-s²/c²)] /(1+s·v/c²)²
| 1 − lα²/c² = 1 − |
| [(s+v)²+w²(1-s²/c²)] |
| c²·(1+s·v/c²)² |
|
| = |
| c²·(1+s·v/c²)² − [(s+v)²+w²(1-s²/c²)] |
| c²·(1+s·v/c²)² |
|
| = |
| c²·(1+s·v/c²)² − [(s+v)²+w²(1-s²/c²)] |
| c²·(1+s·v/c²)² |
|
| = |
| c²+2s·v+s²·v²/c² − s²−2s·v−v² − w²+w²·s²/c² |
| c²·(1+s·v/c²)² |
|
| = |
| c²+s²·v²/c² − s²−v² − w²+w²·s²/c² |
| c²·(1+s·v/c²)² |
|
| = |
| c²−s²−(v²+w²)+s²·(v²+w²)/c² |
| c²·(1+s·v/c²)² |
|
| = |
| (c²−s²)−(v²+w²)·(1−s²/c²) |
| c²·(1+s·v/c²)² |
|
| = |
| (1−s²/c²)−(v²+w²)·(1−s²/c²)/c² |
| (1+s·v/c²)² |
|
| = |
| (1−s²/c²)·[1−(v²+w²)/c²] |
| (1+s·v/c²)² |
|
Therefore, γ=1/√1 − l²/c² factor for blue object is
| γ = |
| 1+s·v/c² |
| √1−s²/c²·√1−(v²+w²)/c² |
|
Usage of γ-factor to prove the conservation of relativistic momentum
Newtonian momentum of an object of mass m moving in two-dimensional Euclidean inertial reference frame with some velocity vector u={v,w} is a vector m·u={m·v,m·w}.
The rules of addition of velocities when observing the movement from another inertial reference frame are simple vector addition derived from Galilean transformation of coordinates.
Relativistic momentum of this object is its Newtonian-like momentum (product of mass and velocity vector) multiplied by factor γ=1/√1 − l²/c², where l is the magnitude (length) of a velocity vector.
The rules of addition of velocities when observing the movement from another inertial reference frame are more complex and derived from Lorentz transformation of coordinates.
We will check the conservation of X- and Y-components of this momentum separately.
1. The X-component of relativistic momentum of green object before a collision
Dividing by γ factor, the X-component of the green object's Newtonian-like momentum before collision
|
|
= |
| m·(1−s·v/c²)·(s−v) |
| √1−s²/c²·√1−(v²+w²)/c²·(1−s·v/c²) |
|
= |
| m·(s−v) |
| √1−s²/c²·√1−(v²+w²)/c² |
|
where m is a mass of this object.
2. The X-component of relativistic momentum of blue object before a collision
Dividing by γ factor, the X-component of the blue object's Newtonian-like momentum before collision
|
|
= |
| m·(1+s·v/c²)·(s+v) |
| √1−s²/c²·√1−(v²+w²)/c²·(1+s·v/c²) |
|
= |
| m·(s+v) |
| √1−s²/c²·√1−(v²+w²)/c² |
|
where m is a mass of this object.
3. The total X-component of relativistic momentum of both objects before a collision
The total X-component of the relativistic momentum of both objects before a collision is a sum the above calculated X-components of green and blue objects and equals to
| m·(s−v) |
| √1−s²/c²·√1−(v²+w²)/c² |
|
+ |
| m·(s+v) |
| √1−s²/c²·√1−(v²+w²)/c² |
|
= |
| 2m·s |
| √1−s²/c²·√1−(v²+w²)/c² |
|
IMPORTANT: The X-components of velocities of both objects after collision are exactly the same as before collision.
Therefore, their relativistic moments will be the same, which confirms the conservation of X-component of the relativistic momentum.
4. The Y-component of relativistic momentum of green object before a collision
Dividing by γ factor, the Y-component of the green object's Newtonian momentum before collision is
| −γ·m·w·√1−s²/c² |
| (1−s·v/c²) |
|
= |
| −m·(1−s·v/c²)·w·√1−s²/c² |
| √1−s²/c²·√1−(v²+w²)/c²·(1−s·v/c²) |
|
= |
|
where m is a mass of this object.
5. The Y-component of relativistic momentum of blue object before a collision
Dividing by γ factor, the Y-component of the blue object's Newtonian momentum before collision is
| γ·m·w·√1−s²/c² |
| (1+s·v/c²) |
|
= |
| m·(1+s·v/c²)·w·√1−s²/c² |
| √1−s²/c²·√1−(v²+w²)/c²·(1+s·v/c²) |
|
= |
|
where m is a mass of this object.
6. The total Y-component of relativistic momentum of both objects before a collision
The total Y-component of the relativistic momentum of both objects before a collision is a sum the above calculated Y-components of green and blue objects and, obviously, equals to zero.
7. The Y-component of relativistic momentum of green object after a collision
| γ·m·w·√1−s²/c² |
| (1−s·v/c²) |
|
= |
| m·(1−s·v/c²)·w·√1−s²/c² |
| √1−s²/c²·√1−(v²+w²)/c²·(1−s·v/c²) |
|
= |
|
8. The Y-component of relativistic momentum of blue object after a collision
| −γ·m·w·√1−s²/c² |
| (1−s·v/c²) |
|
= |
| −m·(1−s·v/c²)·w·√1−s²/c² |
| √1−s²/c²·√1−(v²+w²)/c²·(1−s·v/c²) |
|
= |
|
9. The total Y-component of relativistic momentum of both objects after a collision
The total Y-component of the relativistic momentum of both objects after a collision is a sum the above calculated Y-components of green and blue objects and, obviously, equals to zero.
CONCLUSION
The total X-component of the relativistic momentum of both green and blue objects before a collision equal to its value after a collision and equals to
| 2m·s |
| √1−s²/c²·√1−(v²+w²)/c² |
|
The total Y-component of the relativistic momentum of both green and blue objects before a collision equal to its value after a collision and equals to ZERO
These two results prove the Law of Conservation of Relativistic Momentum.