# Why field expansion into positive and negative frequency Fourier components becomes ambiguous in curved spacetime?

Why is the expansion of a relativistic field into positive and negative frequency components $$\phi({\vec x},t)=\int\frac{d^3{k}}{~~(2\pi)^{3/2}}\left[A\left({\vec k}\right)e^{-i\omega_{\vec k}t}+A^*\left(-{\vec k}\right)e^{+i\omega_{\vec k}t}\right]e^{+i{\vec k}\cdot{\vec x}}$$ unambiguous in flat spacetime but not so in curved spacetime? I am very interested in knowing this.

In curved space-time "$$t$$" itself is ambiguous. You need some "time" translation symmetry to be able to use $$e^{i\omega t}$$ as a set for expanding something out. Technically this involves finding a timelike Killing vector field. There may be more than one of these fields and what is positive frequency with respect to the $$t$$ of one of them, may not be positive frequency with respect to the other. This problem even occurs in flat space. The metrics $$d\tau^2 = dt^2-dx^2 \quad \hbox{Minkowski}$$ and $$d\tau^2 = X^2 dT^2-dX^2 \quad \hbox{Rindler}$$ both describe flat space, and both are "time" ($$t$$ or $$T$$) independent --- but what is positive frequency with respect to $$T$$ is not necessarily positive with respect to $$t$$.
The mapping between $$T$$ and $$t$$ is implicit in the change of coordinates formulae $$x= X\cosh T\\ t= X\sinh T$$
• Why do you say that we need time-translation symmetry to expand in terms of $e^{i\omega t}$? – mithusengupta123 Nov 18 '20 at 14:45
• I mean that the field equations for whatever we are expanding need to be $t$ independent for their normal modes to have an $e^{i\omega t}$ $t$ dependence. – mike stone Nov 18 '20 at 14:57
• If the spacetime is flat and Minkowskian, why are the mode functions unique? Why two different observers will agree on the mode functions? Under Lorentz transformations, $t,{\vec x}$ and $\omega, {\vec k}$ of one observer is different from $t',{\vec x}'$ and $\omega', {\vec k}'$ of another observer. Why will they agree on mode functions? – mithusengupta123 Nov 18 '20 at 16:31