Problem Set 10

Problem Set 10

Due Tuesday, April 23, by 4pm in the homework locker

Problem 1 [10 points]
Consider two 4-vectors u_μ and v_μ. Prove that the inner product u_μv_μ is a Lorentz invariant scalar.
Problem 2 [20 points]
Consider making a Lorentz transformation L₁ from inertial reference frame K to the inertial reference frame K', where K' moves with velocity v₁ in the x direction with respect to K. Then make a second Lorentz transformation L₂ from frame K' to frame K", where K" moves with velocity v₂ in the x direction with respect to K'. Find the velocity v₃ with which K" is moving with respect to the initial frame K. This gives the relativisitic law of addition of velocities!
To find the answer, use the result that the matrix of the combined transformation L₃ = L₂L₁ from K to K" is given by the product of the matrices for the transformations L₂ and L₁, i.e.
a_μν(L₃) = Σ_σ a_μσ(L₂)a_σν(L₁)
Compute the above matrix and identify the velocity v₃ by comparing it to the known form for a Lorentz transformation with velocity v₃ in the x direction.
Problem 3 [15 points]
Consider a point charge q at rest at the origin in an inertial frame K'. We know the electric and magnetic fields E'(r', t') and B'(r', t') of the charge in this frame K'. The frame K' is moving with velocity v in the x direction, as seen by an observor in the inertial frame K. Using the Lorentz transformation for electric and magnetic fields, find the electric and magnetic fields of the charge as seen by the observor in frame K.
Show that the fields you get above are the same as we found from the Lienard-Wiechert potentials as applied to the case of a charge moving with constant velocity v. Remember, you have to transform both the fields and the coordinates, i.e. your result should give E(r,t) and B(r,t).