Israel Khan Derivation Step: From Eq. 2.9 to 2.10

[keithdow]

I am looking at the classic paper Israel and Khan "Collinear Particles and Bondi Dipoles in General Relativity"

Nuovo Cimento 33 (1964) 331.

The basic question is how to go from equation 2.9 to 2.10?

a_i is the center of rod i and b_i is the mass of rod i.

The attached file has the details.

Thanks!

 

[Ambitwistor]

Hello,

Thank you for bringing up this interesting paper. I am familiar with it and I can help explain the process of going from equation 2.9 to 2.10.

In equation 2.9, we have the expression for the Bondi dipole moment, which is defined as the integral over the entire rod system of the product of the center of the rod (a_i) and the mass of the rod (b_i). This integral is taken over all the rods in the system, hence the summation over i.

In equation 2.10, we are essentially simplifying this integral by using the definition of the center of mass of a system. The center of mass is defined as the weighted average of the positions of all the particles in the system, where the weights are given by the masses of the particles. Mathematically, it can be expressed as the sum of the product of the positions and the masses, divided by the total mass of the system.

So, in equation 2.10, we are essentially replacing the integral over all the rods with the center of mass of the system, which is given by the term in parentheses. This simplification is possible because the Bondi dipole moment is a property of the entire rod system, not just individual rods.

I hope this helps to clarify the process of going from equation 2.9 to 2.10. Let me know if you have any further questions. Thank you.