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  Derivation of uncertainty in mode number in curved spacetime

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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 $a_i a_i^\dagger$ 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 $\ell$. In Hawking's language $\ell=B^{-1/2}$ where $B$ is a least upper bound on $|R_{abcd}|$, so $\ell$ is a radius of curvature and the flat space limit is given by $\ell\rightarrow \infty$. Next, since this is locally flat space, he is allowed to choose a basis of (approximately) positive frequency solutions to the wave equation, $\{f_i\}$. Finally, he writes that, when $\omega \gg 1/\ell$, there is an indeterminacy between choosing $f_i$ and its corresponding negative frequency solution $f_i^*$ which is of the order $\exp(-c \omega \ell)$. Here I have let $c$ be some constant, and $\omega$ 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 $\Delta E \Delta t \sim 1$. In units where $\hbar =1$ one has uncertainty $\Delta \omega = \Delta E \sim 1/\Delta t$. Since $\Delta t$ is bounded by $B^{-1/2}$ in the normal coordinates, we have a minimal uncertainty in frequency of order $\Delta \omega \sim B^{1/2}$.

So we can imagine two normal distributions, one for $f_i$ and one for $f_i^*$, centered at $\pm\omega$, each having standard deviation $B^{1/2}$. 

There are two extreme cases:

 1. When $\omega \gg B^{1/2}$, 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 $\omega \lesssim B^{1/2}$, 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 $\sim \frac{1}{2\sqrt{\pi} \alpha}e^{-\alpha^2}$ where $\alpha=\frac{1}{\sqrt{2}}\omega B^{-1/2}$. Whilst qualitatively this is the same as Hawking's result, it differs quantitatively - I have an $\alpha^2$ in the exponent, whilst Hawking only has $\alpha$. 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.

asked Jul 22, 2020 in Theoretical Physics by anonymous [ revision history ]
edited Jul 22, 2020

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