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.

Because the inner product is invariant it does not matter which frame we work in, therefore let us choose the simplest. This would be the frame of lab. The m particle has 4-momentum (γvymc, 0γvymvy, 0). The particle in motion has 4-momentum (γuyMc, 0, γuyMuy, 0). Thus the inner product is:

Mmγvyγuy(c2 - vyuy)    

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.

In this frame the 4-momenta of the particle with speed vy is (mc, 0, 0, 0). The particle at rest in this frame sees the other particle moving in the y-direction with speed wy = . Thus the 4-momentum is (γwyMc, 0, γwyMwy, 0). Thus the inner product is:


But c2γwy is given by:

 =(c2 - uyvy)  

So the overall result is: Mmγvyγuy(c2 - uyvy), which is the same as above.

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.

The inner product in an arbitrary frame is given by:

A.BâÉáA0B0 - A1B1 - A2B2 - A3B3    

Then using the definition of the 4-vector we have:

= γ(A0' + vA1')γ(B0' + vB1') - γ(A1' + vA0')γ(B1' + vB0') - A2'B2' - A3'B3'    
= γ2A0'B0' + v(A0'B0' + A1'B0') + v2A1'B1' - γ2A1'B1' + v(A1'B0' + A0'B1') + v2A0'B0' - A2'B2' - A3'B3'    
= A0'B0'(γ2 - γ2v2) - A1'B1'(γ2 - γ2v2) - A2'B2' - A3'B3'    
= A0'B0' - A1'B1' - A2'B2' - A3'B3'    

Which is the same as the inner product in some other inertial frame moving past with speed v.

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.

First we must compute the 4-velocities of B and C in the frame of A. They are (γvc, γvv, 0, 0) for B and (γuc, - γuu) for C. Next, what are the 4-velocities of B and C in C's frame: for B (γwc, γww, 0, 0) and for C (c, 0, 0, 0). We can take the inner product of each of these pairs and set them equal since the inner product is frame invariant:

(γvc, γvv, 0, 0).(γuc, - γuu, 0, 0) = (γwc, γw, 0, 0).(c, 0, 0, 0)    
âá’γuγv(c2 + uv) = γwc2    
âá’ =     

Solving for w yields:

w =    

Which is the velocity addition formula.

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.

Particle approaching each other at an angle.

In the frame of an outside observer the 4-velocities of the two particles are (γvc, γvv cosθ, - γvv sinθ, 0) and (γvc, γvv cosθ, γvv sinθ, 0). If we let w be the speed of one particle as viewed by the other and rotate so that the relative motion is along the x-axis, then the 4-velocities in the frame of one of the particles are (γwc, γww, 0, 0) and (c, 0, 0, 0). The invariance of the inner product says:

(γvc, γvv cosθ, - γvv sinθ, 0).(γvc, γvv cosθ, γvv sinθ, 0) = (γwc, γww, 0, 0).(c, 0, 0, 0)âá’γv2(1 - v2cos 2θ) = γw    

where we used cos2θ - sin2θ = cos 2θ in the last equality. Expanding out for the γs, squaring, and solving for w yields:

w =     
âá’w =