1. Hawking-Unruh Temperature
Charles Hughes
05/04/14
PHYS 612 Final Project Presentation

2. Presentation Outline Reference Frames: Inertial vs. Accelerated
Rindler Space-Time Coordinates
Massless Klein-Gordon fields in Rindler Space
Vacuum Expectation Value of Number Operator: Rindler Vacuum vs. Minkowski Vacuum
Unruh Radiation

3. Reference Frames Ordinary Minkowski Space
"Henceforth, space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality." - Hermann Minkowski, Sept
Reference Frames
Ordinary Minkowski Space
1) outside the light cone: space-like world lines
2) on the light cone: light-like world lines
3) inside the light cone: time-like world lines

4. Reference Frames: Inertial vs Accelerated Observers
Curved Solid Lines = Accelerated World Lines
Straight Solid Lines = Inertial World Lines
Equation of Motion: 
Equation of Motion:  x = vt
Trajectory:
Acceleration: 

5. Reference Frames: Accelerated Observer Proper Time
At any given instant, the inertial observers time (dt) and the time in the comoving frame of the accelerated observer (    ) are related via a Lorentz transformation:
v(t) = 𝑥 = 𝑐 2 𝑡 𝑥 = 𝑐𝑡 𝑥 0 𝑐 𝑡 2
τ = 0 𝜏 𝑑𝜏 ′ = 0 𝑡 𝑑𝑡 ′ 𝑎𝑐 𝑡 ′ = 1 𝑎𝑐 sinh-1(act)
Hendrik Lorentz
cosh2(y) - sinh2(y) = 1
t = 1 𝑎𝑐 sinh (𝑎𝑐𝜏)
x = 1 𝑎 cosh (𝑎𝑐𝜏)

6. Reference Frames: Light Cone Coordinates
Trajectories are HYPERBOLAS
with asymptotes at x = ±𝑡 (natural units)
Asymptotes divide Minkowski Space into 4 distinct regions, with coordinates, 𝑥 ± =𝑥 ±𝑡
t = 1 𝑎 sinh (𝑎𝜏)
x = 1 𝑎 cosh (𝑎𝜏) x+ x- I + II -
III IV

7. Rindler Space Time Coordinates
Cartesian Polar Coordinates Minkowski Polar Coordinates
x = 1 𝑎 𝜌 cosh (𝑎𝜂)
x = r cos (𝜃)
"The marvel is that nature seems to go along with some of the ‘simplest’ models that can be constructed “
–Wolfgang Rindler, American Journal of Physics 1994
t = 1 𝑎 𝜌 sinh (𝑎𝜂)
y = r sin (𝜃)
metric
metric
𝑑𝜏 2 = − 𝑑𝜌 2 𝑎 𝜌 2 𝑑𝜂 2
𝑑𝑠 2 = 𝑑𝑟 2 + 𝑟 2 𝑑𝜃 2
Lines of constant r
Lines of constant ρ
Rindler Metric
𝑑𝜏 2 = − 𝑑𝜌 2 𝑎 𝜌 2 𝑑𝜂 2
Lines of constant θ
Lines of constant η
ρ → space-like
η → time-like

8. Rindler Space Time Coordinates
Rindler Metric:
Introduce Parametrization: ξ= 1 𝑎 ln 𝜌
Metric Becomes:
Light Cone coordinates become: ξ ± = ξ ± 𝜂= 1 𝑎 ln 𝑎 𝑥 ±
𝑑𝜏 2 = − 𝑑𝜌 2 𝑎 𝜌 2 𝑑𝜂 2
𝑑𝜏 2 = 𝑒 2𝑎ξ ( 𝑑𝜂 2 − 𝑑ξ 2 )
Not defined for x+ , x- < 0
ξ + analytic for region I , II
ξ − analytic for region I , III
𝑑𝜏 2 = 𝑒 −𝑎 ξ + + ξ − 𝑑ξ + 𝑑ξ −

9. Massless Klein Gordon Fields: Light Cone Coordinates
Massless Scalar Field in Minkowski Space:
Equation of Motion (1 –D):
Solutions -> Plane Waves:
Right Moving Waves (k > 0):
Left Moving Waves (k < 0):
𝓛 = 𝜕 𝜇 𝜑 𝜕 ν 𝜑
𝜕 𝑡 2 𝜑 − 𝜕 𝑥 2 𝜑=0
Oskar Klein

10. Massless Klein Gordon Fields: Light Cone Coordinates
Right Moving Waves (k > 0):
Left Moving Waves (k < 0):
General Solution:

11. Massless Klein Gordon Fields: Rindler Coordinates
Massless Scalar Field in Rindler Space:
Equation of Motion (1 –D):
Solutions -> Plane Waves:
Right Moving Waves (k > 0):
Left Moving Waves (k < 0):
𝓛 = 𝑒 2𝑎ξ ( 𝜕𝜂 2 𝜑 − 𝜕ξ 2 𝜑)
𝜕𝜂 2 𝜑 − 𝜕ξ 2 𝜑 = 0

12. Massless Klein Gordon Fields: Rindler Coordinates
Right Moving Waves (k > 0):
Left Moving Waves (k < 0):
Not defined for x+ , x- < 0
ξ − analytic for region I , III
ξ + analytic for region I , II

13. Massless Klein Gordon Fields: Rindler Coordinates
To get a complete coverage extended to region IV, reverse sign of coordinates; (t , x) -> (-t , -x)
Interchanges I <-> IV and II <-> III
Now two sets of modes:
General Solution (Rindler Coordinates):

14. Vacuum Expectation Value of Number Operator: Minkowski Coordinates
Number Operator (in Minkowski space) is:
Vacuum Expectation Value is:
Is this the case for the Rindler Space ?
Must relate a(k) modes to b(r)(k) (Rindler)modes
𝑁(𝑘) 𝑀𝑖𝑛𝑘𝑜𝑤𝑠𝑖 = 0 𝑎 ᵻ 𝑘 𝑎 𝑘 0 =0

15. Vacuum Expectation Value of Number Operator
Use Unruh’s Argument (Not Bogulibov Transformations)
Using:
We extend the support for 𝑢 (k) modes:
Defining factor in
Place branch cut in lower half of x- plane, ω -> π ω
W.G. Unruh
− 𝒊 𝝎 𝒂

16. Vacuum Expectation Value of Number Operator
Now combining 𝑢 𝑘 (1) + 𝑢 𝑘 2 ∗ , a full set (I , II , III ,IV) of positive energy modes is obtained
Mode is bounded in upper half plane (just like in Minkowski space !!!)
Similarly for (1)* and (2)
And the Minkowski observer can then expand in 𝑢 (k) modes.

17. Vacuum Expectation Value of Number Operator
Since the Minkowski observer can now express their modes in terms of the modes of the Rindler observer, we can find the vacuum expectation value of the Rindler Number Operator acting on the Minkowski vacuum
Minkowski KG Field
Match Terms
Rindler KG Field

18. Vacuum Expectation Value of Number Operator
After term matching:
Need to normalize C-modes:

19. Vacuum Expectation Value of Number Operator: Rindler Coordinates
Rindler modes are then expressed in terms of c-modes (Minkowski):
Can now calculate vacuum expectation value in Rindler Space 𝑁(𝑘) 𝑅𝑖𝑛𝑑𝑙𝑒𝑟 = 0 𝑏 1 ᵻ 𝑘 𝑏 (1) 𝑘 0 (accelerated observer in region I, where b(2) modes vanish !
NOT ZERO !!!!

20. Unruh Radiation 𝟏 𝒆 ℏ𝝎 𝒌 𝑩 𝑻 −𝟏
Looks Exactly Like Planck Distribution (Black-Body):
𝟏 𝒆 ℏ𝝎 𝒌 𝑩 𝑻 −𝟏
“Experimenters are the schocktroops of science… An experiment is a question which science poses to Nature, and a measurement is the recording of Nature’s answer” – Max Planck

21. Unruh Radiation: How Large an Acceleration ?
Impossibly Large …

22. Reconciliation ?
Inertial Observer sees Rindler Observer EMITTING radiation (e.g. bremmstrahlung)
Whatever is accelerating the Rindler obsever is supplying energy for the emission of radiation that the Inertial observer sees
Rindler observer sees only net effect which is the thermal bath

23. Sources http://aesop.phys.utk.edu/QFT/notes.pdf