ADVANCED DYNAMICS — PHY 4241/5227 HOME AND CLASS WORK – SET 13 (April 6, 2009)

(56) Let T^{αβ} be a rank two tensor. 1. Write down T_{S}^{αβ}, the symmetric part of T^{αβ}. Due April 8 in class (1 point).

2. Write down T_{A}^{αβ}, the antisymmetric part of T^{αβ}. Due April 8 in class (1 point).

3. Calculate T_{A}^{αβ} T_{S} _{αβ}. Due April 8 in class (1 point).

4. Derive the continuity equation from the antisymmetry of F^{αβ}. Due April 8 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 8 in class (2 points).

(57)

1. Write down (F_{αβ}) in E⃗ and B⃗ fields starting from the corresponding results for (F^{αβ}). Due April 10 in class (3 points).

2. What is the value of ϵ_{0 1 2 3}? Due April 10 in class (1 point).

3. Write down (^{∗}F^{αβ}) in (F_{αβ}) fields. Due April 10 in class (3 points).

4. Write down (^{∗}F^{αβ}) in (F^{αβ}) fields. Due April 10 in class (3 points).

5. Write down (^{∗}F^{αβ}) in E⃗ and B⃗ fields. Due April 10 in class (3 points).

6. Write down 0 = ∂_{α} ^{∗}F^{α 0} in E⃗ and B⃗ fields. Due April 10 in class (1 point).

7. Write down 0 = ∂_{α} ^{∗}F^{α 1} in E⃗ and B⃗ fields. Due April 10 in class (1 point).

8. 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 10 in class (3 points).