ADVANCED DYNAMICS — PHY 4241/5227

HOME AND CLASS WORK – SET 13

(April 8, 2009)

(46) For light from some galaxy the Balmer spectrum of the hydrogen atom is found at wavelengths λ = (729.2 nm) m^{2}/(m^{2}−4), m = 3, 4, 5, …. Neglect expansion of space and find the speed at which the galaxy moves away or towards us. Due April 14 before class (4 points).

(47) Derive for the momentum in the rest frame p^{0} = c m_{0}, where m_{0} is the rest mass, from the relativistic transformation laws and the known definition of momentum in the non-relativistic limit. Due April 12 in class (4 points).

(48) (1) Taylor expand \sqrt{x + ϵ} to leading order in ϵ. (2) Substitute x = c^{4}m_{0}^{2} and ϵ = c^{2}p^{2}. (3) Compare with mv^{2}/2. Due April 12 in class (4 points).

(49) End of the spaceship journey: Assume that the spaceship moves by transforming mass into light, which is exhausted.

1. Derive an expression for m(τ), the (remaining) mass of the spaceship at proper time τ. (Hint: Use momentum conservation in the instanteneous rest frame.)

2. Which fraction of the original mass is left, after the spacetrip has been performed?

Due April 19 before class (8 points).

(50) (1) Write down the definition of ∂_{α}. Due April 14 in class (1 point).

(2) Let A^{α} be a vector field. Write down the scalar field from this and ∂_{α} (not the scalar operator). Due April 16 in class (1 point).

(3) Use the quantities of assignments (1) and (2) to write down a rank two tensor T^{αβ} field (not a tensor operator). Due April 16 in class (1 point).

(4) Write the equation

∂_{α} F^{αβ} = \frac{4π}{c} J^{β}

explicitly out for β = 0, 1, 2, 3. Due April 16 in class (4 points).

(5) Compare the β = 0 result with ∇E⃗ = 4πρ and the β = i = 1, 2, 3 results with ∇ × B⃗ − c^{−1} (∂E⃗/∂t) = 4πJ⃗/c. Due April 16 in class (4 points).