# Slow-rotation weak-field limit of Kerr

Saeide
Hi all,

I want to know if slow-rotation weak-field limit of Kerr metric is also acceptable for the Earth situation or not. It has been shown that in this limit, Kerr metric is reduced to Schwarzschild metric plus a cross term that indicates rotation.

Saeide

Yes, if you don't care about the quadrupole moment (oblateness). Kerr constrains the quadrupole moment to be ma2.

Saeide
So it means that we could not consider the quadruple for earth?
Maybe it's better to ask it in this way; While we obtain a linearized GR for a weak-field limit, for a general metric g ab, and we use it for Earth case too, could we deduce that weak-field Kerr, which is just one specific metric, should be also acceptable for earth?

Mentor
"Acceptable" always depends on your required precision. If you care about effects from GR, you probably want to include higher moments of Earth's mass distribution, too. Even in geostationary orbit, they can be relevant, and for low Earth orbit they are much more pronounced (some satellites use their own path to measure the mass distribution of earth).

Saeide
Yes you're right; when I talk about GR, the higher moments would obviously be involved. But the question that I have in my mind is that, while we describe a general metric -without any specific characteristic considered for- with linearized GR, therefore we can conclude that the slow-rotation weak-field limit of Kerr -that is one of the many options for general metric- could also describes the Earth's spacetime. So why do we have limitation in caring about quadrupole moment or not?!
I would be so grateful to know if you have any idea.

If it's weak-field you're interested in, forget Kerr, its easy enough to write down the solutions for linearized gravity for any multipole, and superpose them with any coefficients you like. This looks just like the similar problem in electromagnetism, except instead of vector spherical harmonics you use tensor spherical harmonics.

Saeide
Using weak-field Kerr, I want to check the result I've got by linearized GR. That's why I'm interested in Kerr. So I could describe Earth's spacetime by weak-field Kerr, while quadrupole moment could also be obtained by superposition in linearized GR, don't I?