Quantcast
Loading [MathJax]/jax/output/HTML-CSS/jax.js
  • Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.
W3Counter Web Stats

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

News

PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public β tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback

Attributions

(propose a free ad)

Site Statistics

208 submissions , 166 unreviewed
5,138 questions , 2,258 unanswered
5,414 answers , 23,090 comments
1,470 users with positive rep
823 active unimported users
More ...

  Derivation of uncertainty in mode number in curved spacetime

+ 0 like - 0 dislike
594 views

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 aiai 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 . In Hawking's language =B1/2 where B is a least upper bound on |Rabcd|, so is a radius of curvature and the flat space limit is given by . 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 ω1/, there is an indeterminacy between choosing fi and its corresponding negative frequency solution fi which is of the order exp(cω). 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 ΔEΔt1. In units where =1 one has uncertainty Δω=ΔE1/Δt. Since Δt is bounded by B1/2 in the normal coordinates, we have a minimal uncertainty in frequency of order ΔωB1/2.

So we can imagine two normal distributions, one for fi and one for fi, centered at ±ω, each having standard deviation B1/2

There are two extreme cases:

 1. When ωB1/2, 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 ωB1/2, 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 12παeα2 where α=12ωB1/2. 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.

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

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol in the following word:
pysicsOverfow
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
Please complete the anti-spam verification




user contributions licensed under cc by-sa 3.0 with attribution required

Your rights
...