Need some help/advice/guidance for question 11.6 from Wald, General Relativity

[MsBXL]

Hey,
I was reading through chapter 11 in the Wald GR book and I came across question 11.6, which i was trying to solve, but I am having difficulties getting started on the questions a) and b), as I have only just learned about tangent vectors (and diff geometry in general) etc. and I would like to ask for some guidance at attempting to answer the question.

In particular, I am not sure how to show that the angular momentum J is independent of the sphere it is integrated over. so if you knew how to attempt this question, i would appreciate a "push into the right direction".

Thanks

 

[Nate810]

in advance!For question 11.6 a), it is useful to recall the definition of angular momentum, which is J = ∫r × (mv)dVwhere r is the position vector, m is the mass of the particle, and v is the velocity of the particle. In order to show that J is independent of the sphere it is integrated over, we must show that the integral does not depend on the surface of integration. To do this, we need to show that the integrand, r × (mv), is an exact differential.To show this, we can use the fact that the curl of a gradient is always zero. In particular, we can calculate the curl of r × (mv) by taking the cross product of the gradient of r with the vector mv. We then note that the gradient of r is the tangent vector T, so the curl of r × (mv) is the cross product of T and mv, which is zero since the curl of a gradient is always zero. This shows that r × (mv) is an exact differential and hence, the integral is independent of the surface of integration. For question 11.6 b), we must calculate the value of J for a given sphere of radius R. To do this, we can use the fact that mv is the same throughout the sphere since the velocity of the particle is constant. Furthermore, we can use spherical coordinates to simplify the integral. In particular, the integral becomesJ = ∫R^2 sinθ dθ dϕ (r × (mv)) where r is the position vector in spherical coordinates. Now, we can note that the only part of the integrand that depends on θ and ϕ is the position vector r. Since the position vector is always perpendicular to the sphere, we can see that the integrand is always zero, so J = 0. This shows that the angular momentum J is zero for a sphere with radius R.