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

• MsBXL
In summary, question 11.6 in Chapter 11 of the Wald GR book asks for the calculation of angular momentum for a given sphere, with particular focus on showing its independence of the sphere it is integrated over and calculating its value for a sphere of radius R.
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

Last edited:
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.

## 1. What is the specific question from Wald's General Relativity that you need help with?

The specific question is question 11.6, which deals with the geodesic deviation equation in curved spacetime.

## 2. Can you provide any background information on the topic before attempting to answer the question?

Yes, the geodesic deviation equation is an important equation in general relativity that describes the behavior of nearby test particles in a curved spacetime. It is derived from the geodesic equation and is used to study the effects of gravity on particles.

## 3. What are some common pitfalls or misconceptions when trying to solve this type of problem?

One common misconception is assuming that the geodesic deviation equation is only applicable in the presence of a strong gravitational field. However, it can also be used in flat spacetime to study the effects of acceleration on nearby particles.

## 4. Can you explain the steps involved in solving this problem?

To solve this problem, you will first need to understand the geodesic deviation equation and its components, including the Riemann curvature tensor and the affine connection. Then, you will need to apply the equation to the given scenario and manipulate the equations to solve for the desired quantities.

## 5. Are there any helpful resources or tips for better understanding this topic and problem?

Yes, there are many resources available for better understanding general relativity and the geodesic deviation equation. Some helpful tips include practicing with similar problems, seeking clarification from a tutor or instructor, and utilizing online resources and textbooks for further explanation and examples.

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