ADVANCED DYNAMICS — PHY 4241/5227

HOME AND CLASS WORK – SET 14

(April 8, 2010)

(51) An electron and a positron, each with mass equal to 0.511 MeV, annihilate at rest into two photons. Chose the rest frame of the positronium, such that one of the photons moves in positive x = x^{1} direction.

1. For each photon find the momentum 4-vector.

2. Find the 4-momentum in a frame that moves at a velocity cβx̂ with respect to the rest frame of the positronium.

3. Suppose that the annihilation took place 10^{9} years ago in a galaxy that is receding from us at β = 4/5. What is the energy of the photon that we observe?

Due April 21 before class (10 points).

(52) Let T^{αβ} be a rank two tensor.

1. Write down T_{S}^{αβ}, the symmetric part of T^{αβ}. Due April 19 in class (1 point).

2. Write down T_{A}^{αβ}, the antisymmetric part of T^{αβ}. Due April 19 in class (1 point).

3. Calculate T_{A}^{αβ} T_{S} _{αβ}. Due April 19 in class (1 point).

4. Derive the continuity equation from the antisymmetry of F^{αβ}. Due April 19 in class (1 point).

5. Show ∂_{α} ^{∗}F^{αβ} = 0, where

∗F^{αβ} = \frac{1}{2}ϵ^{αβγδ}F_{γδ}

is the dual tensor. Here ϵ^{αβγδ} is the completely antisymmetric Levi-Cevita tensors: +1 for (α, β, γ, δ) and even permutation of (0, 1, 2, 3), −1 for (α, β, γ, δ) and even permutation of (0, 1, 2, 3). Due April 19 in class (2 points).

(53) Under a gauge transformation the vector potential transform according to

A^{α} → A^{′ α} = A^{α} + ∂^{α}Λ .

Calculate the corresponding transformation of the electromagnetic field tensor

F^{αβ} → F^{′ αβ} .

Due April 21 before class (4 points).