1. Unruh’s Effect
Savan Kharel

2. Historical Introduction
Unruh Effect shows us that accelerating observer in Minkowski vacuum state will observe a thermal spectrum of particles.
Historically, this was discovered to understand the physics of Hawking effect.
Hawking effect is similar but slightly different because it’s harder to solve wave equation in the curved spacetime.

3. Inertial Observers
A Minkowski vacuum is something that all inertial (not accelerating) observers agree.
If the stationary observer sees nothing, then the observer moving with constant v w.rt to the observer sees nothing. (as one is related to the other by the boost)

4. Non inertial Observer
If the velocity is not constant, then moving observer is no longer inertial. So what happens when the observers are not inertial is the premise of Unruh’s effect.
For simplicity, lets consider the Minkowski metric with just one spatial coordinate for simplicity:

5. Constant accelerating path asymptotic to the null path
Consider an observer moving at a uniform acceleration in the x direction. We can parametrize:
Its easy to see that by taking the magnitude .
The trajectory is a hyperboloid which are asymptotic to the null path x=+t and x=-t.

6. Light Cone Coordinates
The trajectory is a hyperbolae which is asymptotic to x= +t, x=-t. Also define a special convenient coordinate( L.C Coordinate) The Light Cone gives 4 wedges.
Line element,

7. Rindler’s Coordinates
Also, we will introduce new coordinates on two dimensional Minkowski space.Lets define:
The metric would look like this:

8. Some definitions Lets simplify this further and write down:
Now the metric in these coordinates take the following form. The metric in region I defines what is known as R.S.
The metric in terms of the Rindler light cone coordinate:

9. Rindler Space So the region I with this metric is known as
Rindler observer moves
along a constant
acceleration path.
Here, one sees how
equal-time lines evolve

10. A little pause
We defined the inertial versus the accelerating observer.
We introduced light cone coordinates and then define the Rindler Space.
The goal here is to relate the inertial vacuum with the Rindler (accelerating) vacuum.

11. Klein Gordon Equation
Consider a Klein Gordon Equation for some massless field
Equations of motions are given by:
As we well know this admits a positive plane wave solutions with the standard normalization factor
and the negative energy solution is given by

12. Right Moving and Left Moving
When we have the right moving positive energy wave
When we have the left moving negative energy wave.

13. General Solution to the Wave Equation
The general solution to the wave equation is given by:
Now we can define the vacuum:

14. (Almost ) same Story for Rindler Observer
However the crucial difference is that
only covers the quarter of the Minkowski space. The wave equation now is:
This leads to the positive energy solutions:

15. (Almost) Same Story When we have the right moving positive energy wave
When we have the left moving negative energy wave.

16. Analytic continuation beyond Region I
We can analytically continue these plane waves beyond region I by expressing Rindler’s light cone coordinates in terms of Minkowski light cone coordinate:

17. Rindler Wave Solution
After analytic continuation we obtain two sets of modes using analytic continuation: Also we can define the Rindler Vacuum

18. Lets think a little about the inertial observer again
Inertial Observers may be expanded in more general solution:
where,
and the vacuum of the inertial observer is defined as,

19. Relation of Modes
Using the above definitions, we can relate the inertial modes to the Rindler modes:
And we can find the commutation relation of the Minkowski mode w.rt Rindler’s mode:

20. Now lets relate these modes to Rindler Modes
Here is the Bogoliubov Transformation that relates the Rindler modes with the creation and annihilation operators of the inertial observer:

21. Number Operator Assume that system is in Minkowski vacuum
The number operator for the Rindler observer:
So using Bogoliubov transformation and definition of inertial vacuum, the expected number of such particles of some frequency will be given by (the expectation value):

22. Emergence of Black Body Spectrum
The above is the black body spectrum:
The delta function is an artifact of plane wave basis modes. If we had constructed normalized wave packet then we would have obtained a finite result with the identical spectrum.
This gives the blackbody spectrum with

23. Summary of what has happened so far
Inertial versus accelerating observer
We discussed light Cone Coordinates and Rindler Space
Related the modes of Minkowski and the Rindler modes
Showed the expectation value which lead us remarkably to the Planck Factor and the black body spectrum.

24. Conclusion An observer moving with uniform acceleration
through the Minkowski vacuum observes a thermal spectrum of particles. This is called the Unruh Effect.
This shows how different set of observers will describe the same state in very different ways.
In some sense this reveals thermal nature of vacuum in quantum field theory.