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Einstein field equation has many solutions. Out of them, is there any solution that is incompatible with quantum field theory?
Also, what solutions of Einstein field equation would be incompatible with some variants of string theory? (let us restrict the scope of string theory into superstring theory.)
Thanks.
I'll mention just one example of the complexity that curved space introduces into the quantization process.
Consider Minkowski space quantization of the free Klein Gordon Field, which satisfies
$$(\Box+m^2)\phi=0$$
A fundamental step in the procedure is performing the mode expansion
$$\phi(x)=\int{\frac{1}{\sqrt{2\omega_\bf{k}}}}(a_{\bf{k}}e^{-ikx}+a^*_{\bf{k}}e^{ikx})d^{3}\bf{k}$$
Here we have a splitting into a negative frequency part
$$\phi^-(x)=\int{\frac{1}{\sqrt{2\omega_\bf{k}}}a_{\bf{k}}e^{-ikx}}d^{3}\bf{k}$$ and a positive frequency part
$$\phi^+(x)=\int{\frac{1}{\sqrt{2\omega_\bf{k}}}a^*_{\bf{k}}e^{ikx}}d^{3}\bf{k}$$
Upon quantization, the $a^*_{\bf{k}}$ in the positive frequency parts become creation operators and the $a_{\bf{k}}$ in the negative frequency parts annihilation operators. The splitting is covariant - the exponentials contain Lorentz scalars.
Now if we try to do the same thing in curved space to the (covariant form of) the Klein Gordon equation, we can find spacetimes for which there's no clear way to perform this splitting because in general, there's no "natural" time coordinate. In the Minkowski case, we had the action of the Poincare group to allow us to deal with the different possible time coordinates - we even had a Poincare invariant vacuum state, but here there's no equivalent of the Poincare group action.
The particle content of the theory depends upon this splitting, and the ambiguity we have in the curved case (or even in the flat case if we allow non inertial frames) is the origin of the Hawking and Unruh effects.
That was just a single example of a problem that crops up right from the word "go" in curved space quantization. There has been a lot of effort expended over the last few decades studying quantization on curved spacetimes. For a review, see here.
