I began reading the original paper by Hawking, Particle Creation by Black Holes (1975, Commun. math. Phys 43, 199—220), but am a little confused by what he writes at the bottom of the second page. The idea is that there is some indeterminacy or uncertainty in the mode number operator aia†i in curved spacetime.
What Hawking does: He first chooses a point p and locally goes into Riemann normal coordinates. These are valid in some region around p up to some length scale, say ℓ. In Hawking's language ℓ=B−1/2 where B is a least upper bound on |Rabcd|, so ℓ is a radius of curvature and the flat space limit is given by ℓ→∞. Next, since this is locally flat space, he is allowed to choose a basis of (approximately) positive frequency solutions to the wave equation, {fi}. Finally, he writes that, when ω≫1/ℓ, there is an indeterminacy between choosing fi and its corresponding negative frequency solution f∗i which is of the order exp(−cωℓ). Here I have let c be some constant, and ω is the (modulus) frequency of the mode in question.
My Question: I have a hard time understanding what he means by this final part. What does he mean precisely by 'indeterminacy'? Why is there an exponential involved?
My Intuition: I have the following picture: it follows from the Heisenberg uncertainty principle that ΔEΔt∼1. In units where ℏ=1 one has uncertainty Δω=ΔE∼1/Δt. Since Δt is bounded by B−1/2 in the normal coordinates, we have a minimal uncertainty in frequency of order Δω∼B1/2.
So we can imagine two normal distributions, one for fi and one for f∗i, centered at ±ω, each having standard deviation B1/2.
There are two extreme cases:
1. When ω≫B1/2, the two normal distributions are far apart and one is exponentially sure that a mode which is measured to have positive frequency really is a positive frequency mode.
2. When the two distributions are close, i.e. when ω≲B1/2, one might expect increasingly equal probabilities (close to 1/2).
In the former case one can use an asymptotic of the normal distribution to show that the probability of a negative frequency mode to be measured as positive is of order ∼12√παe−α2 where α=1√2ωB−1/2. Whilst qualitatively this is the same as Hawking's result, it differs quantitatively - I have an α2 in the exponent, whilst Hawking only has α. What am I doing wrong, and what is Hawking doing?!
A bonus question: Does anyone know / can anyone give a more rigorous derivation of the uncertainty in the mode number?
Many thanks.