Suppose that a function $f(t)$ can be written $$ \newcommand{\reals}{\mathbb{R}} f(t)=\int_0^\infty d\omega\ F(\omega)e^{-i\omega t} $$ for some absolutely integrable function $F(\omega)$. This is what the title means by the positive frequency property. Now let $\sigma:\reals^+\to\reals$ be a smooth and monotonically increasing function from the positive reals to the reals. This is what the title means by time distortion.
Conjecture: If the composite function $f(\sigma(t))$ has positive frequency for $t>0$ whenever $f(t)$ has positive frequency, then $\sigma(t)$ must be an affine function: $\sigma(t)=at+b$.
Is this conjecture true? A study of generic piecewise-affine $\sigma(t)$s convinced me that typical $\sigma(t)$s don't preserve the positive-frequency property, but I haven't yet distilled a proof that affine $\sigma(t)$s are the only ones that do.
Motivation from physics: Reference 1 sketches a proof that the special case $\sigma(t)=\log(t)$ does not preserve the positive-frequency property. This fact plays a famous role in the derivation of Hawking radiation. Hawking radiation is special because particles continue to be produced indefinitely (thanks to the event horizon) and because it has characteristics of blackbody radiation, but reference 1 also mentions that some production of particles is expected for quantum fields in generic non-stationary spacetime backgrounds, even without event horizons, as a result of the generic mixing of positive and negative frequencies. My conjecture came from trying to quantify just how generic this phenomenon is.
- Townsend, Black Holes (https://arxiv.org/abs/gr-qc/9707012), section 7.3