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Let $\phi$ be a scalar field satisfying the KG equation in the maximal extension of Schwarzschild spacetime

$$(\Box+m^2)\phi=0.$$

We introduce Kruskal-Szekeres coordinates $(T,X)$ and for simplicity discard here the angular part, so that the problem is 2-dimensional. Now, the metric tensor becomes, setting $2GM = 1$, which is equivalent to working in units of the Schwarzschild radius

$$g=\dfrac{4e^{-r}}{r}(dT^2-dX^2),$$

with $r = 1 + W((X^2-T^2)/e)$ where $W$ is the Lambert W-function.

Now the metric is digonal and since

$$\Box= g^{\mu\nu}\nabla_\mu \nabla_\nu$$

we can compute it to be

$$\Box = \dfrac{re^r}{4}(\nabla_T^2-\nabla_X^2).$$

Now, we also know that $$\nabla_\mu f=\partial_\mu f$$

and thus

$$\nabla_\mu\nabla_\nu f=\nabla_\mu (\partial_\mu f)=\partial_\mu \partial_\nu f-\Gamma_{\mu\nu}^\lambda \partial_\lambda f.$$

All of this combined lead to the KG equation written as

$$\left(\dfrac{\partial^2\phi}{\partial T^2}-\dfrac{\partial^2\phi}{\partial X^2}\right)=-\dfrac{4e^{-r}}{r}m^2\phi.$$

I want to find a complet set of modes in the KG inner product

$$(\phi,\psi)_{KG}=i\int_{\Sigma}(\phi^\ast\nabla_\mu \psi-\psi\nabla_\mu \phi^\ast)d\Sigma^\mu,$$

using these coordinates. It is interesting that in the massless case the equation is exactly like the one in Minkowski spacetime, the only difference being that $T$ is restricted by $-\sqrt{1+X^2}< T< \sqrt{1+X^2}$.

Now, when the field is massive, the term on the RHS screws everything. The radius $r$ is a quite complicated function of $T,X$. The equation is not separable, so separation of variables doesn't help. Fourier transform also doesn't seem to help.

Still, the equation simplified so much that I have the impression that: (1) or I did something quite wrong, (2) or else there is a way to actually proceed in the massive case and find the set of modes.

So, is there a workaround to find a set of modes in the massive case, and deal with the $r$ function which is in terms of the $W$ function? How could one proceed?

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