Newtonian limit of Gravitational Klein-Gordon Equation

[FunkyDwarf]

Hey guys,

Was wondering if anyone has seen this done? Essentially I've tried plugging in the Schwarzschild exterior metric and getting a radial wave equation then taking a series expansion for small M (gravitating mass) and comparing that to the KG radial wave equation in a radial potential V(r) but i cannot seem to recover the linear energy term (grav KG equation seems to only provide E^2)

Any ideas/help would be...helpful!

Cheers
-G

 

[FunkyDwarf]

Nobody? :(

 

[Altabeh]

Hey guys,

Was wondering if anyone has seen this done? Essentially I've tried plugging in the Schwarzschild exterior metric and getting a radial wave equation then taking a series expansion for small M (gravitating mass) and comparing that to the KG radial wave equation in a radial potential V(r) but i cannot seem to recover the linear energy term (grav KG equation seems to only provide E^2)

Any ideas/help would be...helpful!

Cheers
-G

Can you do all this here so we can get to know what actually the problem is? I know somehow what you're trying to say is that if we put all metric tensor appearing in the GKG equation equal to the Minkowski metric and modify the Christoffel symbols as in the linear approximation, we would get

$$-\square\phi+\frac{\phi}{\hbar^2}+\eta^{\mu\nu}\Gamma^{\alpha}_{\mu\nu}\partial_{\alpha}\phi=0,$$

where the convention $$c=m=1$$ is used and in the linear approximation of a static field (e.g. Schwarzschild metric) we have

$$\eta^{\mu\nu}\Gamma^{\alpha}_{\mu\nu}=-\frac{1}{2}\eta^{i\alpha}(\partial_{i}g_{ii}+\partial_{i}g_{00}).$$

(The i-indices are all summed over 1,2,3.)

Now introduce this in the main equation to get the desired form of a GKG equation in the linear approxi. Having substituted in these the components of the Schwarzschild metric, you can easily get the scalar potential $$\phi$$ by solving the reduced equation from which the linear energy term could be recovered.

AB

 

[FunkyDwarf]

What i want to be able to do is take the radial wave equation generated by considering the gravitational KG equation in the exterior schwarzschild metric, take some limits, and recover the Schrödinger equation in potential V(r) = -GM/r.

I know that making the substitution
$$\varepsilon = E +m$$
and forcing the condition
$$E^2=0$$

you can recover the Schrödinger equation from the standard special relativistic KG equation, but I'm finding it hard to find a similar set of conditions to reduce the general relativistic case for rs=2GM to reduce it to the Schrödinger case described above.