Linear FE for static, spherical body

[WannabeNewton]

How exactly would I go about finding the components of $$h_{ab}$$ of the linear vacuum field equations for the external gravitational field of a static, spherical body situated at x = y = z = 0 for all t? I assumed since x = y = z = 0 for all t all $$h_{ab}$$,x and ,y and ,z terms vanish from the Riemann tensor. Do I go about solving $$R_{ab}$$ = 0 for $$h_{ab}$$ because I can't really see where the spherical part comes in.

 

[Mentz114]

How exactly would I go about finding the components of $$h_{ab}$$ of the linear vacuum field equations for the external gravitational field of a static, spherical body situated at x = y = z = 0 for all t? I assumed since x = y = z = 0 for all t all $$h_{ab}$$,x and ,y and ,z terms vanish from the Riemann tensor. Do I go about solving $$R_{ab}$$ = 0 for $$h_{ab}$$ because I can't really see where the spherical part comes in.

Use spherical polar coordinates and let h_ab=h_ab(r) be a function of r only. Then you'll have spherical symmetry.

 

[WannabeNewton]

I just don't get what to do with the linear field equations in this case: $$\nabla$$$$_{\alpha }$$$$\nabla$$$$_{\alpha }$$$$h_{\mu \nu }$$= 0
($$\alpha$$s should be lower indexes sorry)
with $$g_{\mu \nu }$$ = diag(-1, 1, r$$^{2}$$, r$$^{2}$$sin$$^{2}\theta$$)

Do I set up a generalized metric like one would for the full field equations?

 

[Mentz114]

The linear theory is a sort of local theory and assumes we can use approximately Cartesian coords, so my first post is wrong.

It's a tricky business deriving the linearized field equations. I recommend you look it up in a textbook,
or have a look at this

http://www.lehigh.edu/~kdw5/project/howto1.pdf

 

[Bill_K]

The linearized field obeys the linear wave equation, or in this time-independent case Laplace's equation. With spherical symmetry it's just like the electrostatic potential for the Coulomb field of a charge, h ~ m/r. Put together solutions of this form that satisfy your gauge condition.

 

[WannabeNewton]

I'm actually fine on the derivation. The subsequent solution for the plane gravitational wave was pretty straight forward too. I'm just confused with what to do with the field equations and the Lorentz gauge in the case of curvilinear coordinates like in this situation.