A Unruh & Minkowski Modes: Analytic Extension Explained

KDPhysics
Messages
73
Reaction score
24
TL;DR Summary
Why do the so-called unruh modes which are extensions of Rindler modes share the same vacuum as the Minkowski modes, as explained in Carroll's Spacetime and geometry?
In Carroll "Spacetime and Geometry" I found the following explanation for why the analytically extended rindler modes share the same vacuum state as the Minkowski vacuum state:
Screenshot 2021-10-31 at 23.22.40.png

I can't quite understand why the fact that the extended modes [\tex]h_k^{(1),(2)}[\tex] are analytic and bounded on the same region as the Minkowski modes proves that [\tex]h_k^{(1),(2)}[\tex] can be expressed in terms of positive-frequency Minkowski modes only. Why are negative frequency modes out of the picture?
 
  • Like
Likes Demystifier
Physics news on Phys.org
For simplicity, consider only right-moving modes. (The argument for left-moving ones will be analogous.) Positive frequency Minkowski modes are analytic in one half of the complex plane, while negative frequency Minkowski modes are analytic in the other half. Hence, a function that is analytic in one (and not in the other!) half of the plane must have expansion only in terms of one set of Minkowski modes. Modes which are not analytic in the needed half cannot contribute to a function which is analytic there.
 
Last edited:
  • Like
Likes vanhees71 and KDPhysics
But couldn't the left moving negative frequency modes be analytic in that half of the complex plane?
 
KDPhysics said:
But couldn't the left moving negative frequency modes be analytic in that half of the complex plane?
Right moving wave has an expansion only in terms of right moving modes. You cannot superimpose both left and right moving modes to get a purely right moving wave.
 
Last edited:
  • Like
Likes vanhees71 and KDPhysics
I see, thanks!
 
  • Like
Likes Demystifier
One last question, the Unruh modes as defined in Sean Carroll's "Spacetime and Geometry" are:
$$h_k^{(1)} = \frac{1}{\sqrt{2\sinh(\pi \omega/a)}}\big(e^{\pi \omega/2a} g_k^{(1)} + e^{-\pi \omega/2a} g_{-k}^{(2)}{}^*\big)$$
On the other hand this paper gives a different definition:
$$h_k^{(1)} = \frac{1}{\sqrt{2\sinh(\pi \omega/a)}}\big(e^{\pi \omega/2a} g_k^{(1)} + e^{-\pi \omega/2a} g_{k}^{(2)}{}^*\big)$$
I can't quite understand how these could be the same.
 
The other paper either has a typo or uses a different convention for definition of second modes.
 
Back
Top