sparknotes
Special Relativity: Dynamics
Problems on Four-vectors
Problem :
Find the inner product of the 4-momenta for the following two particles: a particle of mass
m
moving with
speed
in the lab frame and a particle of mass
M
moving with velocity
also
in the lab frame.
Problem : Calculate the same inner product as in the previous question, but now in a frame moving with one of the particles (or, if you already did it in such a frame, calculate it in the lab frame). Check that the result is the same.
Problem : Prove that if A and B are 4-vectors, A.B their inner product is independent of the frame in which it is calculated.
Problem : Derive the velocity addition formula using the invariance of the 4-velocity inner product. In other words, if in frame A, B moves to the right with speed v , and C moves to the left with speed u , find w , the speed of B with respect to C.
Problem : Again using the invariance of the inner product, determine the speed of one particle as observed by the other as two particles approach each other with speed v along trajectories separated by an angle 2θ , as shown in the figure below.


(c
2 - u
y
v
y)
γ(A
0' + vA
1')
γ(B
0' + vB
1')
-
γ(A
1' + vA
0')
γ(B
1' + vB
0')
- A
2'B
2' - A
3'B
3'
A
0'B
0' + v(A
0'B
0' + A
1'B
0') + v
2
A
1'B
1'
- γ
2
A
1'B
1' + v(A
1'B
0' + A
0'B
1') + v
2
A
0'B
0'
- A
2'B
2' - A
3'B
3'
=


